Finite Group Databasebeta

← #29835224

Subgroups — #29835224

Order 15616.

544 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456462C211yesmaximal, characteristic
2#21454686C211yesmaximal, characteristic, complemented
3#21448010C211yesmaximal, characteristic, complemented
4#21447986C211yesmaximal, characteristic, complemented
5#21447982C211yesmaximal, characteristic, complemented
6#21447858C211yesmaximal, characteristic, complemented
7#21447858C211yesmaximal, characteristic, complemented
8#21447738C211yesmaximal, characteristic, complemented
9#21447735C211yesmaximal, characteristic, complemented
10#21447682C211yesmaximal, characteristic, complemented
11#21447680C211yesmaximal, characteristic, complemented
12#21447674C211yesmaximal, characteristic, complemented
13#21447670C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2234C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17#16840365—21no—
18#16840239—21no—
19C2×#14519722C2211yescharacteristic
20C2×#14519722C2211yescharacteristic
21C2×#14519721C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23#16840163C2211yescharacteristic
24#16840137C2211yescharacteristic
25#16840125—21no—
26#16840013—21no—
27#16839998C2211yescharacteristic
28#16839871—21no—
29#16839868—21no—
30#16839868—21no—
31#16839856C2211yescharacteristic
32#16839856C2211yescharacteristic
33#16839856C2211yescharacteristic
34#16839856C2211yescharacteristic
35D8×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic
37#16839770—21no—
38#16839717C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C61×#466C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C22×Dic122—21no—
60C2×Dic244—21no—
61C2×Dic244—22no—
62C2×Dic244—22no—
63C2×Dic244C2311yescharacteristic
64C2×Dic244C2311yescharacteristic
65C2×D488C2311yescharacteristic
66C2×#11758199—21no—
67C2×#11758194D411yescharacteristic
68#14519773—21no—
69#14519768—21no—
70#14519761—41no—
71#14519759—41no—
72#14519751—21no—
73#14519749—41no—
74#14519748—41no—
75C4×Dic122—41no—
76C4×Dic122—41no—
77#14519742—21no—
78#14519742—21no—
79#14519742—21no—
80#14519742—21no—
81D4×Dic61—21no—
82#14519739—21no—
83#14519736D411yescharacteristic
84#14519733—41no—
85#14519733—41no—
86#14519722—22no—
87#14519722—22no—
88#14519721—22no—
89#14519721—22no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519708—21no—
94#14519700—21no—
95#14519700—21no—
96#14519700—21no—
97#14519700—21no—
98#14519699—21no—
99#14519698—21no—
100#14519697—21no—
101#14519687D411yescharacteristic
102#14519687D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107#121×C61—21no—
108#117×C61—21no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115#102×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C22×C4×C61C2×D411yescharacteristic
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—41no—
123C22×Dic61—41no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—24no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—42no—
130C2×Dic122—41no—
131D8×C61—21no—
132D8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2411yescharacteristic
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C22⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C42×C61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61D811yescharacteristic
145C22⋊Dic61D811yescharacteristic
146#11758199—42no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61D811yescharacteristic
152C4⋊Dic61D811yescharacteristic
153#11758196—41no—
154#11758196—41no—
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158C4×Dic61—41no—
159C4×Dic61—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164#11758194—41no—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167Dic244—24no—
168Dic244—21no—
169Dic244—21no—
170Dic244#4112yes—
171Dic244#4112yes—
172D488—21no—
173D488—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61C2×D811yescharacteristic
180C22×D61—41no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—161no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61C2×D811yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C22⋊D61—161no—
192C22⋊D61—161no—
193C22⋊D61—161no—
194C2×Dic61—44no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—161no—
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203D244—41no—
204D244—41no—
205C4×D61—161no—
206Dic122—22no—
207Dic122—22no—
208Dic122—44no—
209Dic122—41no—
210Dic122—41no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33686—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C22×D811yescharacteristic
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—21no—
228C4×C61—81no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61#52511yescharacteristic
232C4×C61#57511yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—21no—
236Dic61—21no—
237Dic61—42no—
238Dic61—42no—
239Dic61—81no—
240#3176—611no—
241#2950—611no—
242#2234—611no—
243#2234D6111yescharacteristic, complemented
244#2229—611no—
245#2229—611no—
246#2207—611no—
247#2207—611no—
248#2185—611no—
249#2185—611no—
250#2175—611no—
251#2175—611no—
252C4×SD32—611no—
253C4×SD32—611no—
254#2164—611no—
255C2×C61—41no—
256C2×C61—161no—
257C2×C61#310011yescharacteristic
258C2×C61#341611yescharacteristic
259C2×C61#341611yescharacteristic
260D61—161no—
261C22×Q16—611no—
262#540—1221no—
263#519—1221no—
264C2×SD32—611no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×SD32—611no—
268#499—611no—
269#493—611no—
270#490—1221no—
271#469—1221no—
272#466—611no—
273#466C2×D6111yescharacteristic
274#442—1221no—
275#441—1221no—
276#441—1221no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×Q16—611no—
281C4×D8—611no—
282C4×D8—611no—
283#427C2×D6111yescharacteristic
284#419—1221no—
285#405—611no—
286#367—611no—
287#367C2×D6111yescharacteristic
288#366—611no—
289#366C2×D6111yescharacteristic
290C4⋊C16—611no—
291C4⋊C16—611no—
292#358—611no—
293#358—611no—
294#358—611no—
295#358—611no—
296#357—611no—
297#357—611no—
298#357C2×D6111yescharacteristic
299#357C2×D6111yescharacteristic
300C22⋊C16—611no—
301C22⋊C16C2×D6111yescharacteristic
302#346—611no—
303C4×C16—611no—
304C61#3368611yescharacteristic, complemented, Sylow
305C22×Q8—1221no—
306C2×Q16—611no—
307C2×Q16—611no—
308C2×Q16—1222no—
309C2×Q16—1222no—
310C2×Q16—1221no—
311C2×D8—611no—
312C2×D8—1221no—
313C2×D8C22×D6111yescharacteristic
314C2×M16—611no—
315C22×C8C22×D6111yescharacteristic
316#127—1221no—
317#125—1221no—
318#123—2441no—
319#123—2441no—
320#121—21no—
321#121—1221no—
322#121—2441no—
323#121—2441no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×Q8—1221no—
328C4×Q8—2441no—
329C4×Q8—2441no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—21no—
333#117—611no—
334#117—2441no—
335#117—2441no—
336SD32—1222no—
337SD32—1222no—
338SD32—1222no—
339SD32—1222no—
340C2×C16—611no—
341C2×C16—611no—
342C2×C16C22×D6111yescharacteristic
343C2×C16C22×D6111yescharacteristic
344#107—611no—
345#107C22×D6111yescharacteristic
346#107C22×D6111yescharacteristic
347#107C22×D6111yescharacteristic
348#106—21no—
349#106—1221no—
350#103—1221no—
351#103—1221no—
352#103—1221no—
353#103—1221no—
354#102—21no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#97—611no—
359#97—611no—
360C4×C8—611no—
361C4×C8—611no—
362C2×Q8—1224no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—2442no—
366C2×Q8—2441no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—1221no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4D4×D6111yescharacteristic
377Q16—612no—
378Q16—612no—
379Q16—1224no—
380Q16—1221no—
381Q16—1221no—
382D8—21no—
383D8—21no—
384D8—1221no—
385D8—1221no—
386D8—2442no—
387M16—2441no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8D4×D6111yescharacteristic
393C2×C8C23×D6111yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—611no—
398C4⋊C4—611no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—611no—
409C22⋊C4—611no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C42—41no—
414C42—611no—
415C42—611no—
416C42—2441no—
417C42—2441no—
418C42—2441no—
419C42—2441no—
420C16—21no—
421C16—21no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—2441no—
426C23D8×D6111yescharacteristic
427Q8—1222no—
428Q8—1222no—
429Q8—2444no—
430Q8—2441no—
431Q8—2441no—
432D4—41no—
433D4—41no—
434D4—161no—
435D4—9761no—
436D4—9761no—
437D4—9761no—
438D4—2441no—
439D4—2441no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—9761no—
450C2×C4—9761no—
451C2×C4—2441no—
452C2×C4—2444no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4D8×D6111yescharacteristic
456C2×C4C2×D4×D6111yescharacteristic
457C8—21no—
458C8—1221no—
459C8—1221no—
460C8#41×D6111yescharacteristic
461C8#41×D6111yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22C2×D8×D6111yescharacteristic
472C4—21no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—1221no—
477C4—1221no—
478C4—1221no—
479C4—1221no—
480C4—4881no—
481C4—2442no—
482C4—2442no—
483C4#1684027011yescharacteristic
484C4#1684056111yescharacteristic
485C2—41no—
486C2—161no—
487C2—9761no—
488C2#2145581611yescharacteristic
489C2#2145896211yescharacteristic
490C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2234C2
socleC22×C61C22×D8
prime radical#21456462C2
hypercenter#2234D61
hypocenterC61#33686
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #21454686C2semidirect
3. #21448010C2semidirect
4. #21447986C2semidirect
5. #21447982C2semidirect
6. #21447858C2semidirect
7. #21447858C2semidirect
8. #21447738C2semidirect
9. #21447735C2semidirect
10. #21447682C2semidirect
11. #21447680C2semidirect
12. #21447674C2semidirect
13. #21447670C2semidirect
14. #21447623C2semidirect
15. C61×#2234C2semidirect
243. #2234D61semidirect
304. C61#33686semidirect

Maximal subgroups

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

subgroupindex
#214564622
#214546862
#214480102
#214479862
#214479822
#214478582
#214477382
#214477352
#214476822
#214476802
#214476742
#214476702
#214476232
C61×#22342
#3368661