Finite Group Databasebeta

← #29742965

Subgroups — #29742965

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic, complemented
2C2×#16839842C211yesmaximal, characteristic
3#21449671C211yesmaximal, characteristic, complemented
4#21448720C211yesmaximal, characteristic, complemented
5#21447345C211yesmaximal, characteristic, complemented
6#21447294C211yesmaximal, characteristic, complemented
7#21447147C211yesmaximal, characteristic, complemented
8#21446553C211yesmaximal, characteristic, complemented
9#21446373C211yesmaximal, characteristic, complemented
10#21446170C211yesmaximal, characteristic
11#21446164C211yesmaximal, characteristic
12#21446068C211yesmaximal, characteristic, complemented
13#21446067C211yesmaximal, characteristic, complemented
14#21446063C211yesmaximal, characteristic, complemented
15C61×#2067C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519754—21no—
19C2×C4×D244C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24#16840043—24no—
25#16839842—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519710C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839685—21no—
36#16839674—21no—
37#16839665C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839637—21no—
42#16839636—21no—
43#16839633—21no—
44#16839631C2211yescharacteristic
45#16839621—21no—
46C4×#11758196C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C23×Dic61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758199—22no—
68C2×#11758199—22no—
69C2×#11758199Q814yescomplemented
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519777—22no—
73#14519777—41no—
74#14519754—44no—
75C4×D244—22no—
76C4×D244—41no—
77D4×Dic61—22no—
78D4×Dic61—41no—
79#14519739—22no—
80#14519739—22no—
81#14519739—22no—
82#14519739—41no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196D411yescharacteristic
93C2×#11758196D411yescharacteristic
94C4⋊C4×D61—44no—
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519711—22no—
104#14519711—22no—
105#14519710—22no—
106#14519710—41no—
107#14519699—22no—
108#14519699D414yes—
109#14519698—41no—
110#14519697—22no—
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#107×C61—22no—
128C4⋊C8×C61—22no—
129#102×C61—22no—
130#102×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C23×D61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C2×Q812yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C2×D412yes—
160C2×D244C2×Q812yes—
161C2×D244#4112yes—
162C2×C4×D61—24no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—81no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D412yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61#4112yes—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—42no—
189C42×C61—21no—
190C42×C61—21no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—81no—
194#11758199—28no—
195#11758198—24no—
196#11758198—42no—
197#11758198—42no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61—42no—
205C4⋊Dic61—42no—
206C4⋊Dic61C2×D412yes—
207#11758196—41no—
208#11758196—42no—
209#11758196—42no—
210#11758196—81no—
211#11758196D814yescomplemented
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C2×Q812yes—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—41no—
240D4×C61—41no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C2×D812yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—84no—
282C2×Dic61C2×D812yes—
283C2×Dic61C2×D812yes—
284D244—22no—
285D244—22no—
286D244—22no—
287D244C22⋊Q814yescomplemented
288C4×D61—48no—
289C4×D61—84no—
290C4×D61—84no—
291Dic122—84no—
292C8×C61—22no—
293C8×C61—41no—
294#92418—22no—
295#92418—41no—
296#26368—611nomaximal, Sylow
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61#52511yescharacteristic
305C22×C61D4211yescharacteristic
306C22×C61D4×Q811yescharacteristic
307C22×C61C22×D811yescharacteristic
308C22×C61#57511yescharacteristic
309C22×C61#57611yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—24no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61C2×C22⋊Q811yescharacteristic
324C4×C61#53911yescharacteristic
325C4×C61#54711yescharacteristic
326C4×C61#55411yescharacteristic
327Dic61—22no—
328Dic61—22no—
329Dic61—41no—
330Dic61—41no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334C2×#474—611no—
335C2×C4×D8—611no—
336#2452—611no—
337D4×C4⋊C4—611no—
338#2093—611no—
339#2083—611no—
340#2067D6111yescharacteristic, complemented
341#2057—611no—
342#1942—611no—
343#1916—611no—
344#1879—611no—
345#1879—611no—
346#1862—611no—
347#1862—611no—
348#1862—611no—
349C2×C61—82no—
350C2×C61#244911yescharacteristic
351C2×C61#310011yescharacteristic
352C2×C61#312411yescharacteristic
353C2×C61#326911yescharacteristic
354C2×C61D4×D811yescharacteristic
355C2×C61Q8×D811yescharacteristic
356C2×C61#338911yescharacteristic
357D61—44no—
358C22×D8—611no—
359C2×C4⋊Q8—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×C4×D4—611no—
365C22×C4⋊C4—1221no—
366#474—1224no—
367C4×D8—1224no—
368C2×#107—611no—
369C2×#107—611no—
370C2×#107—611no—
371C2×#107—611no—
372C2×#107C2×D6111yescharacteristic
373C2×C4⋊C8C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102C2×D6111yescharacteristic
379C2×#102C2×D6111yescharacteristic
380C2×C4×C8—611no—
381#396—1221no—
382#393—1221no—
383#390—611no—
384#390C2×D6111yescharacteristic
385#389—611no—
386#389—611no—
387#389—611no—
388#389C2×D6111yescharacteristic
389#385—1221no—
390#385—1221no—
391#385—1221no—
392#384—611no—
393#380—1221no—
394C4×C4⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#336—611no—
400#336—611no—
401#336—611no—
402#336—611no—
403C61#2636811yescharacteristic, complemented, Sylow
404C22×Q8—1221no—
405C22×D4—611no—
406C22×D4C22×D6111yescharacteristic
407C23×C4—1221no—
408C23×C4—1221no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×D8—614no—
412C2×D8—1222no—
413C2×D8—1222no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—1222no—
419C4⋊Q8—2441no—
420C22⋊Q8—2444no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—2441no—
427C4×D4—2441no—
428C4×D4—2441no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—2444no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42C22×D6111yescharacteristic
456#107—22no—
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#107—2441no—
461C4⋊C8—22no—
462#102—22no—
463#102—22no—
464#102—614no—
465#102—1222no—
466#102—1222no—
467#102—2441no—
468C4×C8—1222no—
469#95—21no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475C24—21no—
476C24—1221no—
477C2×Q8—4881no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—612no—
486C2×D4—612no—
487C2×D4—612no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4Q8×D6112yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1224no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—4881no—
504C22×C4—4881no—
505C22×C4—4881no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4C23×D6111yescharacteristic
517D8—1228no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8D4×D6112yes—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—614no—
537C4⋊C4—612no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2441no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4#1175822412yes—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—1224no—
554C22⋊C4—4881no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—1221no—
562C42—1221no—
563C42—1221no—
564C42—1221no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—1222no—
573C23—1222no—
574C23—1222no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23C2×D4×D6111yescharacteristic
580Q8—4884no—
581D4—22no—
582D4—22no—
583D4—41no—
584D4—41no—
585D4—614no—
586D4—1222no—
587D4—1222no—
588D4—1222no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—4884no—
617C2×C4—4884no—
618C2×C4—4884no—
619C2×C4—2448no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4D8×D6112yes—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×Q8×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4#1451982311yescharacteristic
634C2×C4#1451982411yescharacteristic
635C8—22no—
636C8—41no—
637C8—1222no—
638C8—2441no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—1224no—
647C22—2442no—
648C22—2442no—
649C22—2442no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22#1684027211yescharacteristic
654C22#1684039611yescharacteristic
655C22#1684042511yescharacteristic
656C22C2×D8×D6111yescharacteristic
657C22#1684055811yescharacteristic
658C22#1684056411yescharacteristic
659C22#1684057811yescharacteristic
660C4—22no—
661C4—42no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—1222no—
666C4—1222no—
667C4—4881no—
668C4—4881no—
669C4—2442no—
670C4—2441no—
671C4—2441no—
672C4C22⋊Q8×D6111yescharacteristic
673C4#1684035311yescharacteristic
674C4#1684041211yescharacteristic
675C4#1684047011yescharacteristic
676C2—82no—
677C2—2444no—
678C2#2144962911yescharacteristic
679C2#2145581811yescharacteristic
680C2#2145608011yescharacteristic
681C2#2145738111yescharacteristic
682C2#2145749311yescharacteristic
683C2#2145866911yescharacteristic
684C2#2145877611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2067C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2067D61
hypocenterC61#26368
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. C2×#16840043C2semidirect
3. #21449671C2semidirect
4. #21448720C2semidirect
5. #21447345C2semidirect
6. #21447294C2semidirect
7. #21447147C2semidirect
8. #21446553C2semidirect
9. #21446373C2semidirect
12. #21446068C2semidirect
13. #21446067C2semidirect
14. #21446063C2semidirect
15. C61×#2067C2semidirect
69. C2×#11758199Q8semidirect
211. #11758196D8semidirect
287. D244C22⋊Q8semidirect
340. #2067D61semidirect
403. C61#26368semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
C2×#168398422
#214496712
#214487202
#214473452
#214472942
#214471472
#214465532
#214463732
#214461702
#214461642
#214460682
#214460672
#214460632
C61×#20672
#2636861