Finite Group Databasebeta

← #29740155

Subgroups — #29740155

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839984C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3#21449617C211yesmaximal, characteristic
4#21447348C211yesmaximal, characteristic
5#21447310C211yesmaximal, characteristic, complemented
6#21447228C211yesmaximal, characteristic, complemented
7#21447060C211yesmaximal, characteristic
8#21446501C211yesmaximal, characteristic
9#21446378C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446174C211yesmaximal, characteristic
12#21446160C211yesmaximal, characteristic, complemented
13#21446073C211yesmaximal, characteristic, complemented
14#21446072C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839984—24no—
21C4×#11758202—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
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×Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#389—21no—
54C61×#389—21no—
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#384—21no—
58C61×#384C2211yescharacteristic
59C4×C4⋊C4×C61—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202D414yes—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×#11758201—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#14519701—22no—
92#14519701—22no—
93#14519700—22no—
94#14519700—22no—
95#14519700—22no—
96#14519697—22no—
97#14519697—22no—
98#14519697—41no—
99C4×#92418—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61C2311yescharacteristic
104C4×Q8×C61—22no—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—22no—
108C4×Q8×C61—41no—
109C4×Q8×C61—41no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61—21no—
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#95×C61—21no—
129#95×C61—21no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×D412yes—
137C2×Q8×C61#4112yes—
138C2×Q8×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C2×D412yes—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—42no—
176C4⋊C4×C61—81no—
177C4⋊C4×C61—81no—
178C4⋊C4×C61—81no—
179C4⋊C4×C61#4112yes—
180C42×C61—24no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—42no—
188C42×C61—42no—
189#11758202—28no—
190#11758201—44no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195#11758198—81no—
196#11758198—81no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201#11758196—42no—
202#11758196—81no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61#12314yes—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
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×C61—41no—
256C2×C4×C61#13512yes—
257C2×C4×C61#13712yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273D244—41no—
274D244—41no—
275D244—41no—
276D244—41no—
277Dic122—22no—
278Dic122—22no—
279Dic122—41no—
280Dic122—41no—
281#92418—22no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#26169—611nomaximal, Sylow
286C22×C61#52511yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61#54811yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61C2×#13711yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57811yescharacteristic
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—22no—
301C4×C61—24no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61C2×#12311yescharacteristic
311C4×C61#53911yescharacteristic
312C4×C61#54711yescharacteristic
313C4×C61#55311yescharacteristic
314Dic61—42no—
315Dic61—81no—
316Dic61—81no—
317C2×#464—611no—
318C2×C4×Q16—611no—
319#2448—611no—
320Q8×C4⋊C4D6111yescharacteristic, complemented
321#2093—611no—
322#2086—611no—
323#2072—611no—
324#2042—611no—
325#1934—611no—
326#1916—611no—
327#1880—611no—
328#1880—611no—
329#1878—611no—
330#1863—611no—
331#1863—611no—
332C2×C61#244311yescharacteristic
333C2×C61#310811yescharacteristic
334C2×C61#312411yescharacteristic
335C2×C61#327011yescharacteristic
336C2×C61#328711yescharacteristic
337C2×C61#336411yescharacteristic
338C2×C61#337511yescharacteristic
339D61—161no—
340C22×Q16—611no—
341C22×SD16—611no—
342C2×#124—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8C2×D6111yescharacteristic
345C2×C4×Q8C2×D6111yescharacteristic
346C2×C4×Q8C2×D6111yescharacteristic
347#464—1224no—
348C4×Q16—1224no—
349C2×#107—611no—
350C2×#107—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×C4×C8—611no—
361#399—1221no—
362#397—1221no—
363#391—611no—
364#390—611no—
365#390—611no—
366#389—21no—
367#389—21no—
368#389—611no—
369#389C2×D6111yescharacteristic
370#389C2×D6111yescharacteristic
371#388—1221no—
372#388—1221no—
373#384—21no—
374#384C2×D6111yescharacteristic
375C4×C4⋊C4—21no—
376C4×C4⋊C4C2×D6111yescharacteristic
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384#336—611no—
385C61#2616911yescharacteristic, complemented, Sylow
386C22×Q8—611no—
387C22×Q8C22×D6111yescharacteristic
388C22×D4—611no—
389C2×Q16—614no—
390C2×Q16—1222no—
391C2×Q16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399#124—1222no—
400#124—2441no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—41no—
406C4×Q8—41no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22⋊D6111yescharacteristic
421C2×C4⋊C4C22⋊D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42—1221no—
435C2×C42C22×D6111yescharacteristic
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438#107—1222no—
439#107—2441no—
440C4⋊C8—1222no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—1222no—
445#103—1222no—
446#102—1222no—
447#102—1222no—
448#102—2441no—
449C4×C8—1222no—
450#95—21no—
451#95—21no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95C22⋊D6111yescharacteristic
456#95C22⋊D6111yescharacteristic
457C24—1221no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—612no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8#1175822212yes—
468C2×Q8D4×D6112yes—
469C2×Q8#1175822612yes—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490Q16—1228no—
491SD16—2444no—
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—2442no—
524C4⋊C4#1175822212yes—
525C22⋊C4—4881no—
526C22⋊C4—4881no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—24no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—42no—
540C42—2442no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23C22×C22⋊D6111yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8—1222no—
553Q8—1222no—
554Q8—2441no—
555Q8—2441no—
556Q8#1451976214yes—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560D4—2441no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—1222no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4#1451979712yes—
599C2×C4#1451980812yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×#1175822211yescharacteristic
602C2×C4C2×#1175822211yescharacteristic
603C2×C4C2×#1175822611yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982511yescharacteristic
606C8—1222no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22#1684027311yescharacteristic
618C22D4×C22⋊D6111yescharacteristic
619C22#1684042311yescharacteristic
620C22C2×#1451979711yescharacteristic
621C22C2×#1451980811yescharacteristic
622C22#1684056511yescharacteristic
623C22#1684057711yescharacteristic
624C4—22no—
625C4—24no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—4881no—
635C4—4881no—
636C4—2442no—
637C4C2×#1451976211yescharacteristic
638C4#1684035911yescharacteristic
639C4#1684041311yescharacteristic
640C4#1684046011yescharacteristic
641C2—9761no—
642C2#2144955411yescharacteristic
643C2#2145589911yescharacteristic
644C2#2145608111yescharacteristic
645C2#2145739011yescharacteristic
646C2#2145757511yescharacteristic
647C2#2145850411yescharacteristic
648C2#2145863311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterQ8×C4⋊C4D61
hypocenterC61#26169
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
5. #21447310C2semidirect
6. #21447228C2semidirect
9. #21446378C2semidirect
12. #21446160C2semidirect
13. #21446073C2semidirect
14. #21446072C2semidirect
15. Q8×C4⋊C4×C61C2semidirect
320. Q8×C4⋊C4D61semidirect
385. C61#26169semidirect

Maximal subgroups

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

subgroupindex
C2×#168399842
C2×C4×#117582022
#214496172
#214473482
#214473102
#214472282
#214470602
#214465012
#214463782
#214461742
#214461602
#214460732
#214460722
Q8×C4⋊C4×C612
#2616961