Finite Group Databasebeta

← #29725144

Subgroups — #29725144

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840036C211yesmaximal, characteristic
2#21454683C211yesmaximal, characteristic, complemented
3#21447289C211yesmaximal, characteristic, complemented
4#21447101C211yesmaximal, characteristic
5#21447097C211yesmaximal, characteristic
6#21446733C211yesmaximal, characteristic, complemented
7#21446606C211yesmaximal, characteristic, complemented
8#21446314C211yesmaximal, characteristic
9#21446311C211yesmaximal, characteristic
10#21445941C211yesmaximal, characteristic, complemented
11#21445932C211yesmaximal, characteristic, complemented
12#21445931C211yesmaximal, characteristic, complemented
13#21445872C211yesmaximal, characteristic
14#21445618C211yesmaximal, characteristic
15C61×#2082C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519808C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20#16840240C2211yescharacteristic
21#16840239C2211yescharacteristic
22#16840036—22no—
23#16840036—22no—
24#16839866—22no—
25#16839854—22no—
26#16839810C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28C2×#14519712C2211yescharacteristic
29#16839798C2211yescharacteristic
30#16839780C2211yescharacteristic
31#16839778C2211yescharacteristic
32#16839772C2211yescharacteristic
33#16839770C2211yescharacteristic
34#16839744C2211yescharacteristic
35C2×#14519696C2211yescharacteristic
36C2×#14519695C2211yescharacteristic
37C2×#14519695C2211yescharacteristic
38#16839735C2211yescharacteristic
39#16839720C2211yescharacteristic
40#16839718C2211yescharacteristic
41#16839534C2211yescharacteristic
42#16839534C2211yescharacteristic
43#16839534C2211yescharacteristic
44#16839533C2211yescharacteristic
45#16839533C2211yescharacteristic
46#16839532C2211yescharacteristic
47#16839518C2211yescharacteristic
48C61×#430C2211yescharacteristic
49C61×#419C2211yescharacteristic
50C61×#419C2211yescharacteristic
51C2×#101×C61C2211yescharacteristic
52C61×#344C2211yescharacteristic
53C61×#344C2211yescharacteristic
54C61×H(3,ℤ/4)C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57#14519810—22no—
58#14519810—41no—
59#14519808—22no—
60#14519808—41no—
61C2×Dic244—21no—
62C2×#11758202—21no—
63C2×#11758202—21no—
64C2×#11758202—21no—
65C2×#11758202—21no—
66C2×#11758202—21no—
67C2×#11758202D412yes—
68C2×#11758202D412yes—
69C2×#11758190—21no—
70C2×#11758201—21no—
71#14519785—21no—
72#14519785—21no—
73#14519785—21no—
74#14519785—21no—
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C2×#11758189—21no—
79C22×#92418C2311yescharacteristic
80C2×C8×D61—21no—
81Q8×Dic61—22no—
82Q8×Dic61—41no—
83C4×Dic122—41no—
84#14519742—22no—
85#14519736—22no—
86#14519736C2311yescharacteristic
87#14519736C2311yescharacteristic
88#14519733—22no—
89#14519712—21no—
90#14519712—21no—
91#14519712—21no—
92#14519712—21no—
93#14519712—41no—
94#14519710—21no—
95#14519708—21no—
96#14519705—41no—
97#14519703—41no—
98#14519702—22no—
99#14519700—41no—
100#14519696—41no—
101#14519695—22no—
102#14519695—21no—
103#14519695—41no—
104#14519693—21no—
105#14519691—22no—
106#14519688—21no—
107#14519687—21no—
108C4×#92418—21no—
109C8×Dic61—21no—
110C22×Q8×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112C2×M16×C61C2311yescharacteristic
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2311yescharacteristic
115#117×C61C2311yescharacteristic
116#117×C61C2311yescharacteristic
117#108×C61—41no—
118#103×C61—22no—
119#103×C61—41no—
120#101×C61—22no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61C2×D412yes—
127C22×C4×C61—21no—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—21no—
130C22×Dic61—21no—
131#11758222—21no—
132#11758222—21no—
133#11758222—21no—
134#11758222—21no—
135C2×D244#4111yescharacteristic
136C2×C4×D61—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122C2×D412yes—
143C2×Dic122#4111yescharacteristic
144M16×C61—21no—
145M16×C61—21no—
146M16×C61—21no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C22⋊C4×C61—21no—
159C22⋊C4×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—21no—
163C22⋊Dic61#4112yes—
164#11758202—22no—
165#11758202—22no—
166#11758202—22no—
167#11758202—22no—
168#11758202—42no—
169#11758201—42no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—42no—
173#11758196—42no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—42no—
177C4×Dic61—42no—
178#11758194—21no—
179#11758194—21no—
180#11758194—21no—
181#11758194—21no—
182#11758194—21no—
183#11758194—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418#4111yescharacteristic
190C2×#92418#4111yescharacteristic
191C2×#92418#4111yescharacteristic
192C2×#92418#4111yescharacteristic
193Dic244—42no—
194#11758190—42no—
195#11758189—41no—
196#11758189—41no—
197C8×D61—41no—
198C8×D61—41no—
199C23×C61C2×#4111yescharacteristic
200C22×D61—21no—
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—41no—
204Q8×C61—41no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×C4×C61H(5,𝔽2)11yescharacteristic
216C22⋊D61—41no—
217C22⋊D61—41no—
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224D244—21no—
225D244—21no—
226C4×D61—41no—
227C4×D61—41no—
228Dic122—22no—
229Dic122—22no—
230Dic122—21no—
231Dic122—21no—
232Dic122—41no—
233Dic122—41no—
234C8×C61—41no—
235C8×C61—41no—
236C8×C61—41no—
237C8×C61—41no—
238#92418—41no—
239#92418—41no—
240#92418—41no—
241#92418#12311yescharacteristic
242#92418#12311yescharacteristic
243#92418#12311yescharacteristic
244#92418#12311yescharacteristic
245#24989—611nomaximal, Sylow
246C22×C61—41no—
247C22×C61#52911yescharacteristic
248C22×C61#54611yescharacteristic
249C22×C61#55511yescharacteristic
250C2×D61—41no—
251C2×D61—81no—
252C4×C61—42no—
253C4×C61—81no—
254C4×C61—81no—
255C4×C61C2×#12311yescharacteristic
256C4×C61#54011yescharacteristic
257C4×C61#54611yescharacteristic
258C4×C61#55311yescharacteristic
259Dic61—42no—
260Dic61—42no—
261Dic61—41no—
262Dic61—81no—
263C2×#473—611no—
264#2950—611no—
265#2082—611no—
266#2082D6111yescharacteristic, complemented
267#2049—611no—
268#2049—611no—
269#1979—611no—
270#1953—611no—
271#1905—611no—
272#1905—611no—
273#1833—611no—
274#1830—611no—
275#1830—611no—
276#1814—611no—
277#1757—611no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61#242011yescharacteristic
282D61—81no—
283C2×#137—611no—
284C2×#137—611no—
285C22×Q16—611no—
286C2×#131—611no—
287#519—611no—
288#519—611no—
289#473—1222no—
290#473—1222no—
291#441—1222no—
292C4×Q16—1222no—
293#430—611no—
294#430C2×D6111yescharacteristic
295C2×#108—611no—
296C2×#108—611no—
297#427—611no—
298#421—611no—
299#421—611no—
300#419—611no—
301#419—611no—
302#419C2×D6111yescharacteristic
303#419C2×D6111yescharacteristic
304#413—611no—
305C2×#101—611no—
306C2×#101—611no—
307C2×#101—611no—
308C2×#101C2×D6111yescharacteristic
309#410—611no—
310#405—611no—
311#405—611no—
312#344—611no—
313#344—611no—
314#344—611no—
315#344—611no—
316#344—611no—
317#344—611no—
318#344C2×D6111yescharacteristic
319#344C2×D6111yescharacteristic
320H(3,ℤ/4)—611no—
321H(3,ℤ/4)C2×D6111yescharacteristic
322C61#2498911yescharacteristic, complemented, Sylow
323C2×#41—611no—
324C22×Q8—611no—
325C22×Q8C22×D6111yescharacteristic
326#137—1222no—
327#137—1222no—
328#137—2441no—
329#137—2441no—
330C2×Q16—612no—
331C2×Q16—612no—
332C2×Q16—1221no—
333C2×Q16—1221no—
334C2×Q16—1221no—
335C2×Q16—1221no—
336C2×Q16—1221no—
337C2×Q16—1221no—
338C2×SD16—1221no—
339C2×SD16—1221no—
340#131—1221no—
341#131—1221no—
342#131—1221no—
343#131—1221no—
344C2×M16—611no—
345C2×M16—611no—
346C2×M16—611no—
347C2×M16—1221no—
348C2×M16C22×D6111yescharacteristic
349C2×M16C22×D6111yescharacteristic
350C2×M16C22×D6111yescharacteristic
351C2×M16C22×D6111yescharacteristic
352C22×C8—611no—
353C22×C8—1221no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356C4×Q8—2441no—
357C4×Q8—2441no—
358#117—611no—
359#117—611no—
360#117—1222no—
361#117—1222no—
362#117C22×D6111yescharacteristic
363#117C22×D6111yescharacteristic
364#108—41no—
365#108—1221no—
366#108—1221no—
367#108—1221no—
368#108—1221no—
369#108—2441no—
370#107—1221no—
371#106—1221no—
372#104—2441no—
373#104—2441no—
374#103—22no—
375#103—41no—
376#103—1222no—
377#103—2441no—
378#101—22no—
379#101—1222no—
380#101—1221no—
381#101—2441no—
382#101—2441no—
383#100—1221no—
384#99—1222no—
385#97—1221no—
386#97—1221no—
387C4×C8—1221no—
388C4×C8—1221no—
389#41—1221no—
390#41—1221no—
391#41—1221no—
392#41—1221no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—612no—
399C2×Q8—611no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8D4×D6112yes—
406C2×D4—611no—
407C2×D4—1221no—
408C22×C4—21no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4C23×D6111yescharacteristic
412Q16—1222no—
413Q16—1222no—
414Q16—1222no—
415Q16—1222no—
416Q16—2442no—
417Q16—2442no—
418SD16—2442no—
419SD16—2442no—
420M16—21no—
421M16—21no—
422M16—21no—
423M16—21no—
424M16—21no—
425M16—21no—
426M16—21no—
427M16—21no—
428M16—1221no—
429M16—1221no—
430M16—1221no—
431M16—1221no—
432M16—1221no—
433M16—1221no—
434M16—2441no—
435M16—2441no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—2441no—
450C2×C8—2441no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—2442no—
456C4⋊C4—2442no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—612no—
460C22⋊C4—1221no—
461C42—21no—
462C42—21no—
463C42—1221no—
464C42—1221no—
465C42—2442no—
466C42—2442no—
467C23—1221no—
468C23#41×D6111yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473Q8—1221no—
474Q8—1222no—
475Q8—1222no—
476Q8—1221no—
477Q8—2441no—
478Q8—2441no—
479D4—1221no—
480D4—1221no—
481D4—2441no—
482D4—2441no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—1221no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4C2×D4×D6111yescharacteristic
497C2×C4#41×D6111yescharacteristic
498C2×C4#41×D6111yescharacteristic
499C2×C4#41×D6111yescharacteristic
500C2×C4#1451982211yescharacteristic
501C2×C4#1451982311yescharacteristic
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—611no—
507C8—611no—
508C8—611no—
509C8—611no—
510C8—2441no—
511C8—2441no—
512C8—2441no—
513C22—41no—
514C22—4881no—
515C22—2441no—
516C22#1684029511yescharacteristic
517C22#1684040411yescharacteristic
518C22#1684047411yescharacteristic
519C4—42no—
520C4—81no—
521C4—81no—
522C4—4881no—
523C4—2442no—
524C4—2441no—
525C4—2442no—
526C4#123×D6111yescharacteristic
527C4#1684036411yescharacteristic
528C4#1684040111yescharacteristic
529C4#1684046311yescharacteristic
530C2—21no—
531C2—21no—
532C2—21no—
533C2—4881no—
534C2#2144930611yescharacteristic
subgroupG/H
centerC2#21449306
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2082C2
socleC2×C61#2420
prime radicalC2×#11758222C23
hypercenter#2082D61
hypocenterC61#24989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2082

Complements

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

NH ≅ G/Ntype
2. #21454683C2semidirect
3. #21447289C2semidirect
6. #21446733C2semidirect
7. #21446606C2semidirect
10. #21445941C2semidirect
11. #21445932C2semidirect
12. #21445931C2semidirect
15. C61×#2082C2semidirect
266. #2082D61semidirect
322. C61#24989semidirect

Maximal subgroups

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

subgroupindex
C2×#168400362
#214546832
#214472892
#214471012
#214470972
#214467332
#214466062
#214463142
#214463112
#214459412
#214459322
#214459312
#214458722
#214456182
C61×#20822
#2498961