Finite Group Databasebeta

← #29740440

Subgroups — #29740440

Order 15616.

924 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839977C211yesmaximal, characteristic
2C2×#16839862C211yesmaximal, characteristic, complemented
3#21449617C211yesmaximal, characteristic, complemented
4#21447348C211yesmaximal, characteristic, complemented
5#21447315C211yesmaximal, characteristic, complemented
6#21447228C211yesmaximal, characteristic, complemented
7#21447060C211yesmaximal, characteristic, complemented
8#21446515C211yesmaximal, characteristic, complemented
9#21446385C211yesmaximal, characteristic, complemented
10#21446167C211yesmaximal, characteristic
11#21446167C211yesmaximal, characteristic
12#21446153C211yesmaximal, characteristic, complemented
13#21446072C211yesmaximal, characteristic, complemented
14#21446066C211yesmaximal, characteristic, complemented
15D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839977—24no—
21#16839862—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34#16839697—21no—
35#16839691—21no—
36#16839668C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839652—21no—
41#16839651—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C22×C4⋊C4×C61—21no—
54C61×#389C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#385—21no—
57C61×#385—21no—
58C61×#384C2211yescharacteristic
59C61×#380—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200D414yes—
66C2×#11758199—22no—
67C2×#11758199—22no—
68C2×#11758199—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519765—22no—
74#14519765—41no—
75C4×Dic122—22no—
76C4×Dic122—41no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88#14519710—22no—
89#14519710—41no—
90C4⋊#92418—22no—
91#14519700—22no—
92#14519700—22no—
93#14519700—22no—
94#14519698—22no—
95#14519698—22no—
96#14519697—22no—
97#14519697—22no—
98#14519697—41no—
99#14519688—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61—21no—
106C23×C4×C61—21no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—41no—
112C4×D4×C61—41no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—44no—
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C4⋊C4×C61D411yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#95×C61—21no—
130#95×C61—21no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C24×C61—21no—
134C23×D61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C2×D412yes—
139C2×D4×C61#4112yes—
140C2×D4×C61#4112yes—
141C22×C4×C61—24no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—81no—
152C22×C4×C61—81no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122C2×D412yes—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—81no—
186C4⋊C4×C61#4112yes—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—42no—
189C22⋊C4×C61—42no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—21no—
196#11758200—28no—
197#11758199—44no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201#11758198—42no—
202#11758198—81no—
203#11758198—81no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758196—42no—
209#11758196—81no—
210C4×Dic61—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D412yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C22×D411yescharacteristic
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61#12314yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—48no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—84no—
270C2×C4×C61—84no—
271C2×C4×C61AGL(1,ℤ/8)12yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C2×C4×C612−1+411yescharacteristic
278C2×Dic61—22no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287D244—41no—
288D244—41no—
289D244—41no—
290D244—41no—
291Dic122—22no—
292Dic122—22no—
293Dic122—41no—
294Dic122—41no—
295#92418—22no—
296#92418—41no—
297#92418—41no—
298#92418—41no—
299#26188—611nomaximal, Sylow
300C22×C61—24no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61#52511yescharacteristic
308C22×C61D4211yescharacteristic
309C22×C61#54811yescharacteristic
310C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C61#57511yescharacteristic
312C22×C61#57611yescharacteristic
313C22×C61#57811yescharacteristic
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C4×C61—22no—
322C4×C61—42no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61C2×#12311yescharacteristic
331C4×C61#53911yescharacteristic
332C4×C61#54711yescharacteristic
333C4×C61#55311yescharacteristic
334Dic61—42no—
335Dic61—81no—
336Dic61—81no—
337C2×#463—611no—
338C2×#440—611no—
339#2448—611no—
340D4×C4⋊C4D6111yescharacteristic, complemented
341#2093—611no—
342#2087—611no—
343#2072—611no—
344#2042—611no—
345#1936—611no—
346#1917—611no—
347#1879—611no—
348#1879—611no—
349#1877—611no—
350#1863—611no—
351#1862—611no—
352C2×C61—44no—
353C2×C61#244311yescharacteristic
354C2×C61#311411yescharacteristic
355C2×C61#312411yescharacteristic
356C2×C61#326911yescharacteristic
357C2×C61#327411yescharacteristic
358C2×C61#336711yescharacteristic
359C2×C61#337011yescharacteristic
360D61—161no—
361C22×SD16—611no—
362C22×D8—611no—
363C2×#124—611no—
364C2×C4×Q8—611no—
365C2×C4×D4C2×D6111yescharacteristic
366C2×C4×D4C2×D6111yescharacteristic
367C2×C4×D4C2×D6111yescharacteristic
368C22×C4⋊C4—21no—
369#463—1224no—
370#440—1224no—
371C2×#107—611no—
372C2×#107—611no—
373C2×C4⋊C8—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#97—611no—
383#399—1221no—
384#397—1221no—
385#391—611no—
386#390—611no—
387#390—611no—
388#389—611no—
389#389C2×D6111yescharacteristic
390#389C2×D6111yescharacteristic
391#388—1221no—
392#388—1221no—
393#385—21no—
394#385—21no—
395#384C2×D6111yescharacteristic
396#380—21no—
397C4×C4⋊C4C2×D6111yescharacteristic
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#336—611no—
403#336—611no—
404#336—611no—
405#336—611no—
406C61#2618811yescharacteristic, complemented, Sylow
407C22×Q8—611no—
408C22×D4—611no—
409C22×D4C22×D6111yescharacteristic
410C23×C4—21no—
411C23×C4—21no—
412C23×C4—21no—
413C2×SD16—614no—
414C2×SD16—1222no—
415C2×SD16—1222no—
416C2×D8—1222no—
417C2×D8—1222no—
418C2×D8—1222no—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8—611no—
423#124—1222no—
424#124—2441no—
425C4×Q8—1222no—
426C4×Q8—2441no—
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—41no—
432C4×D4—41no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—44no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22⋊D6111yescharacteristic
443C2×C4⋊C4C22⋊D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458C2×C42C22×D6111yescharacteristic
459C2×C42C22×D6111yescharacteristic
460#107—1222no—
461#107—2441no—
462C4⋊C8—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#102—1222no—
467#102—1222no—
468#102—1222no—
469#102—1222no—
470#102—2441no—
471#97—1222no—
472#95—21no—
473#95—21no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95C22⋊D6111yescharacteristic
478#95C22⋊D6111yescharacteristic
479C24—21no—
480C24—1221no—
481C2×Q8—612no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4#1175822212yes—
497C2×D4D4×D6112yes—
498C2×D4#1175822612yes—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521SD16—1228no—
522D8—2444no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—41no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—81no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—4881no—
551C4⋊C4—2442no—
552C4⋊C4#1175822212yes—
553C22⋊C4—24no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—4881no—
557C22⋊C4—4881no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—2442no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23C22×C22⋊D6111yescharacteristic
584Q8—1222no—
585Q8—1222no—
586Q8—2441no—
587Q8—2441no—
588D4—22no—
589D4—22no—
590D4—22no—
591D4—2441no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595D4#1451976214yes—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—1222no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4#1451980312yes—
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C2×C4C2×#1175822211yescharacteristic
638C2×C4C2×#1175822611yescharacteristic
639C2×C4#1451982211yescharacteristic
640C2×C4#1451982511yescharacteristic
641C8—1222no—
642C8—2441no—
643C8—2441no—
644C8—2441no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—4881no—
653C22—4881no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22—4881no—
658C22—4881no—
659C22#1684027311yescharacteristic
660C22D4×C22⋊D6111yescharacteristic
661C22#1684042311yescharacteristic
662C22C2×#1451980311yescharacteristic
663C22#1684056011yescharacteristic
664C22#1684056511yescharacteristic
665C22#1684057711yescharacteristic
666C4—22no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—4881no—
676C4—4881no—
677C4—2442no—
678C4C2×#1451976211yescharacteristic
679C4#1684035911yescharacteristic
680C4#1684041311yescharacteristic
681C4#1684046011yescharacteristic
682C2—44no—
683C2—9761no—
684C2#2144955411yescharacteristic
685C2#2145595711yescharacteristic
686C2#2145608111yescharacteristic
687C2#2145737011yescharacteristic
688C2#2145743511yescharacteristic
689C2#2145853711yescharacteristic
690C2#2145857211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterD4×C4⋊C4D61
hypocenterC61#26188
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839862C2semidirect
3. #21449617C2semidirect
4. #21447348C2semidirect
5. #21447315C2semidirect
6. #21447228C2semidirect
7. #21447060C2semidirect
8. #21446515C2semidirect
9. #21446385C2semidirect
12. #21446153C2semidirect
13. #21446072C2semidirect
14. #21446066C2semidirect
15. D4×C4⋊C4×C61C2semidirect
340. D4×C4⋊C4D61semidirect
406. C61#26188semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399772
C2×#168398622
#214496172
#214473482
#214473152
#214472282
#214470602
#214465152
#214463852
#214461672
#214461532
#214460722
#214460662
D4×C4⋊C4×C612
#2618861