Finite Group Databasebeta

← #29662487

Subgroups — #29662487

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456398C211yesmaximal, characteristic, complemented
2#21455879C212yesmaximal, complemented
3#21445235C212yesmaximal, complemented
4#21445228C212yesmaximal, complemented
5#21445223C211yesmaximal, characteristic, complemented
6#21445218C211yesmaximal, characteristic, complemented
7#21445179C211yesmaximal, characteristic, complemented
8#21445174C211yesmaximal, characteristic, complemented
9#21444567C211yesmaximal, characteristic, complemented
10#21444479C212yesmaximal, complemented
11C8×#41×C61C211yesmaximal, characteristic, complemented
12#16840312C2212yes—
13#16840167C2211yescharacteristic
14#16840139C2211yescharacteristic
15#16840109C2212yes—
16#16840086—22no—
17#16840057C2212yes—
18#16840052C2212yes—
19#16839998—21no—
20#16839998—21no—
21#16839962C2212yes—
22#16839957C2212yes—
23#16839907—22no—
24C4×#11758201—22no—
25C4×D488—21no—
26C4×D488—21no—
27C4×#11758199—22no—
28#16839815C2212yes—
29#16839515C2211yescharacteristic
30#16839515C2211yescharacteristic
31#16839513C2212yes—
32#16839510C2212yes—
33#16839510C2212yes—
34#16839505C2212yes—
35#16839505C2212yes—
36C4×#41×C61C2211yescharacteristic
37C8×Q8×C61C2211yescharacteristic
38C8×Q8×C61C2211yescharacteristic
39C8×D4×C61—21no—
40C8×D4×C61—21no—
41C8×D4×C61C2211yescharacteristic
42C8×D4×C61C2211yescharacteristic
43C61×#431—21no—
44C61×#431C2211yescharacteristic
45C2×C4×C8×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758201—22no—
49C2×D488—21no—
50C2×D488—21no—
51C2×#11758199—22no—
52#14519777C2312yes—
53#14519774C2312yes—
54#14519765—22no—
55#14519751—22no—
56#14519751—22no—
57C4×D244—22no—
58C4×D244—22no—
59#14519711—21no—
60#14519711—21no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65#14519701—22no—
66#14519699—22no—
67#14519699—22no—
68#14519698—22no—
69#14519697—22no—
70#14519697—22no—
71C22⋊#92418—22no—
72C4×#92418—22no—
73C2×#41×C61D411yescharacteristic
74C22×C8×C61—21no—
75C22×C8×C61—21no—
76C22×C8×C61D411yescharacteristic
77C22×C8×C61D411yescharacteristic
78C4×Q8×C61D411yescharacteristic
79C4×Q8×C61C2311yescharacteristic
80C4×D4×C61—21no—
81C4×D4×C61—21no—
82C4×D4×C61D411yescharacteristic
83C4×D4×C61C2311yescharacteristic
84#117×C61—21no—
85#117×C61D411yescharacteristic
86C2×C42×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2311yescharacteristic
91C4⋊C8×C61C2311yescharacteristic
92C22⋊C8×C61—21no—
93C22⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95C4×C8×C61—21no—
96C4×C8×C61—21no—
97C4×C8×C61—21no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102#41×C61—22no—
103#41×C61—22no—
104C2×Q8×C61C2×D411yescharacteristic
105C2×D4×C61—21no—
106C2×D4×C61C2×D411yescharacteristic
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C2×C22⋊D61—42no—
114#11758222—84no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×C4×D61—42no—
119C2×Dic122—22no—
120C2×C8×C61—22no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—41no—
124C2×C8×C61—42no—
125C2×C8×C61—42no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61C2×D411yescharacteristic
133C4⋊C4×C61C2×D411yescharacteristic
134C22⋊C4×C61—21no—
135C22⋊C4×C61—21no—
136C22⋊C4×C61—21no—
137C42×C61—21no—
138C42×C61—21no—
139C42×C61—21no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144#11758201—44no—
145#11758199—44no—
146#11758198—42no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C2×#92418—22no—
151C2×#92418—22no—
152D488—42no—
153D488—42no—
154C23×C61—21no—
155C23×C61#12111yescharacteristic
156C22×D61—42no—
157Q8×C61—21no—
158Q8×C61#13512yes—
159D4×C61—24no—
160D4×C61—21no—
161D4×C61#13512yes—
162C2×C4×C61—24no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C22⋊D61—84no—
181C2×Dic61—42no—
182D244—42no—
183D244—42no—
184D244—42no—
185D244—42no—
186D244—82no—
187C4×D61—84no—
188Dic122—82no—
189C8×C61—21no—
190C8×C61—21no—
191C8×C61—21no—
192C8×C61—41no—
193C8×C61—41no—
194C8×C61#13512yes—
195#92418—42no—
196#92418—42no—
197#19128—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—22no—
201C22×C61—21no—
202C22×C61—41no—
203C22×C61—41no—
204C22×C61C2×#12111yescharacteristic
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—41no—
213C4×C61—41no—
214C4×C61#47212yes—
215C4×C61C2×#13511yescharacteristic
216C4×C61C2×#13511yescharacteristic
217C4×C61C2×#13511yescharacteristic
218C4×C61#57511yescharacteristic
219Dic61—82no—
220#3169—611no—
221#3106—612no—
222C8×#41D6111yescharacteristic, complemented
223#1678—612no—
224#1677—612no—
225#1676—611no—
226#1675—611no—
227#1668—611no—
228#1667—611no—
229#1564—611no—
230#1543—612no—
231C2×C61—41no—
232C2×C61—42no—
233C2×C61#303511yescharacteristic
234C2×C61#305011yescharacteristic
235C2×C61C2×#47211yescharacteristic
236D61—162no—
237#532—612no—
238C4×#41C2×D6111yescharacteristic
239#500—611no—
240#493—611no—
241#486—612no—
242#482—1222no—
243#477—612no—
244#476—612no—
245#466—1221no—
246#466—1221no—
247#460—612no—
248#459—612no—
249#449—1222no—
250C8×Q8C2×D6111yescharacteristic
251C8×Q8C2×D6111yescharacteristic
252C4×SD16—1222no—
253C4×D8—1222no—
254C4×D8—1221no—
255C4×D8—1221no—
256C8×D4—21no—
257C8×D4—21no—
258C8×D4C2×D6111yescharacteristic
259C8×D4C2×D6111yescharacteristic
260#431—21no—
261#431—612no—
262#431C2×D6111yescharacteristic
263C2×C4×C8—21no—
264C2×C4×C8C2×D6111yescharacteristic
265#335—611no—
266#335—611no—
267#334—612no—
268#333—612no—
269#333—612no—
270#331—612no—
271#331—612no—
272C61#1912811yescharacteristic, complemented, Sylow
273C2×#41—1222no—
274C2×#41D24411yescharacteristic
275C2×SD16—1222no—
276C2×D8—1222no—
277C2×D8—1221no—
278C2×D8—1221no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8C22⋊D6111yescharacteristic
282C22×C8C22⋊D6111yescharacteristic
283C4⋊Q8—612no—
284#127—612no—
285#124—1222no—
286#121—1222no—
287#121—1222no—
288C4×Q8D24411yescharacteristic
289C4×Q8C22×D6111yescharacteristic
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—1222no—
293C4×D4—1222no—
294C4×D4D24411yescharacteristic
295C4×D4C22×D6111yescharacteristic
296#117—21no—
297#117D24411yescharacteristic
298C2×C42—21no—
299C2×C42C22×D6111yescharacteristic
300#107—1221no—
301#107—1221no—
302#107—1222no—
303#106—1222no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—612no—
307C4⋊C8—612no—
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6111yescharacteristic
310#103—1222no—
311#102—1222no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—1222no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—1222no—
324C4×C8C22⋊D6111yescharacteristic
325C4×C8C22⋊D6111yescharacteristic
326C4×C8C22×D6111yescharacteristic
327C4×C8C22×D6111yescharacteristic
328#41—22no—
329#41—22no—
330#41—4884no—
331C2×Q8—1222no—
332C2×Q8C2×D24411yescharacteristic
333C2×D4—21no—
334C2×D4—1222no—
335C2×D4—1222no—
336C2×D4—1222no—
337C2×D4—2442no—
338C2×D4C2×D24411yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—2442no—
343C22×C4C2×D24411yescharacteristic
344C22×C4#1175822211yescharacteristic
345C22×C4C2×C22⋊D6111yescharacteristic
346SD16—2444no—
347D8—2442no—
348D8—2444no—
349D8—2442no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—1222no—
357C2×C8—1222no—
358C2×C8C2×C22⋊D6111yescharacteristic
359C2×C8C2×C22⋊D6111yescharacteristic
360C2×C8C2×C22⋊D6111yescharacteristic
361C2×C8C2×C22⋊D6111yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—1222no—
365C4⋊C4—1222no—
366C4⋊C4C2×D24411yescharacteristic
367C4⋊C4C2×D24411yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—2442no—
372C22⋊C4—2442no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42C2×D24411yescharacteristic
377C42#1175822211yescharacteristic
378C42C2×C22⋊D6111yescharacteristic
379C42C23×D6111yescharacteristic
380C23—21no—
381C23—2442no—
382C23#1451975111yescharacteristic
383Q8—21no—
384Q8—4882no—
385Q8#1451979912yes—
386D4—24no—
387D4—21no—
388D4—4882no—
389D4—4884no—
390D4—2442no—
391D4—2442no—
392D4—2442no—
393D4—2442no—
394D4#1451979912yes—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—4884no—
408C2×C4—2442no—
409C2×C4#1451975111yescharacteristic
410C2×C4#1451975111yescharacteristic
411C2×C4#1451975111yescharacteristic
412C2×C4C22×C22⋊D6111yescharacteristic
413C2×C4C22×D24411yescharacteristic
414C2×C4C2×#1175822211yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8—41no—
419C8—41no—
420C8—2442no—
421C8—2442no—
422C8#1451979712yes—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—4882no—
430C22—4882no—
431C22—4882no—
432C22C2×#1451975111yescharacteristic
433C4—21no—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—4882no—
440C4#1684003212yes—
441C4C2×#1451979711yescharacteristic
442C4C2×#1451979911yescharacteristic
443C4C2×#1451979911yescharacteristic
444C4#1684056011yescharacteristic
445C2—41no—
446C2—42no—
447C2—9762no—
448C2#2145527411yescharacteristic
449C2#2145540011yescharacteristic
450C2C2×#1684003211yescharacteristic
subgroupG/H
centerC2×C4#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×#41×C61C2
socleC22×C61C2×#121
hypercenterC8×#41D61
hypocenterC61#19128
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. #21455879C2semidirect
3. #21445235C2semidirect
4. #21445228C2semidirect
5. #21445223C2semidirect
6. #21445218C2semidirect
7. #21445179C2semidirect
8. #21445174C2semidirect
9. #21444567C2semidirect
10. #21444479C2semidirect
11. C8×#41×C61C2semidirect
222. C8×#41D61semidirect
272. C61#19128semidirect

Maximal subgroups

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

subgroupindex
#214563982
#214558792
#214452352
#214452282
#214452232
#214452182
#214451792
#214451742
#214445672
#214444792
C8×#41×C612
#1912861