Finite Group Databasebeta

← #29836935

Subgroups — #29836935

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458246C211yesmaximal, characteristic, complemented
2#21457538C211yesmaximal, characteristic
3#21448045C211yesmaximal, characteristic, complemented
4#21448029C211yesmaximal, characteristic, complemented
5#21448024C211yesmaximal, characteristic, complemented
6#21447955C211yesmaximal, characteristic, complemented
7#21447909C211yesmaximal, characteristic, complemented
8#21447867C211yesmaximal, characteristic, complemented
9#21447758C211yesmaximal, characteristic, complemented
10#21447757C211yesmaximal, characteristic, complemented
11#21447668C211yesmaximal, characteristic, complemented
12C16×C22⋊D61C211yesmaximal, characteristic, complemented
13#21447527C211yesmaximal, characteristic, complemented
14#21447525C211yesmaximal, characteristic, complemented
15C61×#2216C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519799C2211yescharacteristic
18#16840483—21no—
19#16840414—21no—
20C2×#14519723C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic
23#16840166C2211yescharacteristic
24#16840140C2211yescharacteristic
25#16840101—21no—
26#16840100—21no—
27#16840045—21no—
28#16840004C2211yescharacteristic
29#16840000C2211yescharacteristic
30#16839966—21no—
31#16839927—21no—
32#16839922—21no—
33Q16×Dic61C2211yescharacteristic
34#16839844C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36#16839800C2211yescharacteristic
37#16839799C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839587C2211yescharacteristic
40#16839580C2211yescharacteristic
41#16839567C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839541C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×Q32×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#366C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C2×#11758226—21no—
60C2×#11758222—21no—
61#14519800—22no—
62#14519800—41no—
63#14519799—22no—
64#14519799—41no—
65C2×Dic244C2311yescharacteristic
66C2×#11758202—21no—
67C2×#11758190—21no—
68C2×#11758201—21no—
69C2×D488C2311yescharacteristic
70C2×C8×D61C2311yescharacteristic
71#14519776—21no—
72#14519775—21no—
73#14519772—41no—
74#14519770—41no—
75#14519763—41no—
76#14519762—41no—
77#14519759—41no—
78#14519757—21no—
79#14519753—21no—
80#14519750—41no—
81Q8×Dic61—21no—
82#14519742—41no—
83#14519739—21no—
84C4×C22⋊D61D411yescharacteristic
85#14519736—21no—
86#14519735—21no—
87C4⋊C4×D61—41no—
88#14519723—41no—
89#14519720—22no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519710—21no—
96#14519709—21no—
97#14519708—21no—
98#14519706D411yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519697—21no—
103C22⋊#92418D411yescharacteristic
104#14519689D411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×Q16×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61—21no—
111Q32×C61—41no—
112C2×C16×C61—41no—
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#103×C61—21no—
118#103×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—41no—
123C2×C22⋊D61D811yescharacteristic
124#11758226—42no—
125#11758226—81no—
126#11758222—42no—
127#11758222—81no—
128C2×D244—21no—
129C2×D244—41no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61—41no—
133C2×C4×D61C2×D411yescharacteristic
134C2×Dic122—21no—
135C2×Dic122—41no—
136Q16×C61—21no—
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61D811yescharacteristic
152#11758202—81no—
153#11758201—42no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198D811yescharacteristic
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—41no—
162C4⋊Dic61—41no—
163#11758196—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196D811yescharacteristic
168C4×Dic61—41no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244—21no—
176D488—21no—
177D488#4112yes—
178#11758190—42no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61C2×D811yescharacteristic
186C22×D61—41no—
187C22×D61C2×D811yescharacteristic
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—41no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61C2×D811yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—22no—
201C22⋊D61—82no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208D244—22no—
209D244—41no—
210D244—82no—
211C4×D61—21no—
212C4×D61—21no—
213C4×D61—82no—
214C4×D61—82no—
215C4×D61—82no—
216Dic122—41no—
217Dic122—41no—
218Dic122—82no—
219C8×C61—21no—
220C8×C61C2×#4111yescharacteristic
221C8×C61H(5,𝔽2)11yescharacteristic
222#92418—21no—
223#92418—21no—
224#33869—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61C22×D811yescharacteristic
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—42no—
233C2×D61—81no—
234C2×D61—81no—
235C4×C61—21no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61—81no—
240C4×C61#54611yescharacteristic
241C4×C61#57711yescharacteristic
242Dic61—21no—
243Dic61—21no—
244Dic61—81no—
245Dic61—81no—
246Dic61—81no—
247Dic61#50812yes—
248#3342—611no—
249#3283—611no—
250#2241—611no—
251#2237—611no—
252#2237—611no—
253#2224—611no—
254#2216—611no—
255#2216D6111yescharacteristic, complemented
256#2209—611no—
257#2189—611no—
258#2189—611no—
259C4×SD32—611no—
260D4×C16—611no—
261#2138—611no—
262#2138—611no—
263C2×C61—41no—
264C2×C61#329411yescharacteristic
265C2×C61C2×#50811yescharacteristic
266C2×C61#341711yescharacteristic
267D61—41no—
268D61—82no—
269C2×#135—611no—
270C2×#135—611no—
271#556—1221no—
272#547—1221no—
273C2×Q32—611no—
274C2×Q32C2×D6111yescharacteristic
275C2×SD32—611no—
276C22×C16—611no—
277C22×C16C2×D6111yescharacteristic
278#500—611no—
279#494—611no—
280#485—1221no—
281#485—1221no—
282#475—1221no—
283#467—611no—
284#467—611no—
285#467C2×D6111yescharacteristic
286#467C2×D6111yescharacteristic
287#461—1221no—
288#452—1221no—
289#452—1221no—
290C4×Q16—611no—
291C4×D8—611no—
292C8×D4—611no—
293#427—611no—
294#427—611no—
295#367—611no—
296#366—611no—
297#366C2×D6111yescharacteristic
298C4⋊C16—611no—
299#358—611no—
300#358—611no—
301#358—611no—
302#358—611no—
303#358—611no—
304#358C2×D6111yescharacteristic
305#358C2×D6111yescharacteristic
306#357—611no—
307#357—611no—
308#357—611no—
309C22⋊C16—611no—
310C22⋊C16—611no—
311C4×C16—611no—
312C61#3386911yescharacteristic, complemented, Sylow
313C2×#41—1221no—
314C2×#41—1221no—
315#135—1222no—
316#135—1222no—
317#135—2441no—
318#135—2441no—
319C2×Q16—611no—
320C2×Q16—1221no—
321C2×Q16C22×D6111yescharacteristic
322C2×Q16C22×D6111yescharacteristic
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×D8—611no—
326C22×C8—611no—
327C22×C8C22×D6111yescharacteristic
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330#126—2441no—
331#126—2441no—
332#124—2441no—
333#123—2441no—
334#123—2441no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—1221no—
338C22⋊Q8—1221no—
339#121—2441no—
340C4×Q8—1221no—
341C4×Q8—2441no—
342C4×D4—611no—
343C4×D4—1221no—
344#117—1221no—
345#117—1221no—
346C2×C4⋊C4—2441no—
347Q32—41no—
348Q32—2441no—
349SD32—1222no—
350C2×C16—41no—
351C2×C16—611no—
352C2×C16—611no—
353C2×C16—2441no—
354C2×C16C22×D6111yescharacteristic
355C2×C16C22×D6111yescharacteristic
356#107—611no—
357#107—611no—
358#107—1221no—
359#107C22×D6111yescharacteristic
360#107C22×D6111yescharacteristic
361#106—1221no—
362#106—1221no—
363C4⋊C8—611no—
364#103—21no—
365#103—21no—
366#103—1221no—
367#103—1221no—
368#103—1221no—
369#102—1221no—
370C22⋊C8—611no—
371C22⋊C8—611no—
372C4×C8—611no—
373#41—4881no—
374#41—4881no—
375#41—2442no—
376#41—2442no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—1221no—
380C2×Q8—2441no—
381C2×D4—611no—
382C2×D4—1221no—
383C2×D4—2441no—
384C2×D4—2441no—
385C22×C4—611no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4D4×D6111yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—21no—
394Q16—1221no—
395Q16—1221no—
396Q16—4881no—
397SD16—2442no—
398SD16—2442no—
399D8—612no—
400D8—1221no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—611no—
404C2×C8—611no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8D4×D6111yescharacteristic
408C2×C8C23×D6111yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—611no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4—2441no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—611no—
425C22⋊C4—611no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4—2441no—
430C22⋊C4—2441no—
431C42—611no—
432C42—2441no—
433C42—2441no—
434C42—2441no—
435C16—21no—
436C16—21no—
437C16—1221no—
438C16—1221no—
439C23—611no—
440C23—2441no—
441C23D8×D6111yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—4882no—
447Q8—2441no—
448Q8—2441no—
449D4—1222no—
450D4—1222no—
451D4—4882no—
452D4—4882no—
453D4—2441no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—611no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—4882no—
466C2×C4—4882no—
467C2×C4—4882no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4D8×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C8—21no—
474C8—1221no—
475C8—1221no—
476C8#41×D6111yescharacteristic
477C8#1451982211yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—1221no—
482C22—1221no—
483C22—1221no—
484C22—4881no—
485C22—4881no—
486C22—2442no—
487C22C2×D8×D6111yescharacteristic
488C4—21no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—612no—
494C4—1221no—
495C4—1221no—
496C4—4881no—
497C4—4881no—
498C4—4881no—
499C4#1684040111yescharacteristic
500C4#1684056911yescharacteristic
501C2—41no—
502C2—4882no—
503C2—2441no—
504C2#2145762711yescharacteristic
505C2#508×D6111yescharacteristic
506C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2216C2
socleC22×C61C22×D8
prime radical#21457538C2
hypercenter#2216D61
hypocenterC61#33869
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2216

Complements

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

NH ≅ G/Ntype
1. #21458246C2semidirect
3. #21448045C2semidirect
4. #21448029C2semidirect
5. #21448024C2semidirect
6. #21447955C2semidirect
7. #21447909C2semidirect
8. #21447867C2semidirect
9. #21447758C2semidirect
10. #21447757C2semidirect
11. #21447668C2semidirect
12. C16×C22⋊D61C2semidirect
13. #21447527C2semidirect
14. #21447525C2semidirect
15. C61×#2216C2semidirect
255. #2216D61semidirect
312. C61#33869semidirect

Maximal subgroups

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

subgroupindex
#214582462
#214575382
#214480452
#214480292
#214480242
#214479552
#214479092
#214478672
#214477582
#214477572
#214476682
C16×C22⋊D612
#214475272
#214475252
C61×#22162
#3386961