Finite Group Databasebeta

← #29662489

Subgroups — #29662489

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#21455880C212yesmaximal, complemented
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21445236C212yesmaximal, complemented
5#21445229C212yesmaximal, complemented
6#21445224C211yesmaximal, characteristic, complemented
7#21445219C211yesmaximal, characteristic, complemented
8#21445180C211yesmaximal, characteristic, complemented
9#21445175C211yesmaximal, characteristic, complemented
10#21444479C212yesmaximal, complemented
11C61×#1564C211yesmaximal, characteristic, complemented
12#16840312C2212yes—
13C4×#11758222C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840139C2211yescharacteristic
16#16840109C2212yes—
17#16840087—22no—
18#16840058C2212yes—
19#16840053C2212yes—
20#16839998—21no—
21#16839998—21no—
22#16839963C2212yes—
23#16839958C2212yes—
24#16839909—22no—
25C8×Dic122C2211yescharacteristic
26C8×Dic122C2211yescharacteristic
27C4×#11758190—22no—
28C4×D488—22no—
29C4×D488—21no—
30C4×D488—21no—
31C8×D244C2211yescharacteristic
32C8×D244C2211yescharacteristic
33C8×C22⋊D61—21no—
34C8×C22⋊D61—21no—
35#16839815C2211yescharacteristic
36#16839813—21no—
37C4×C8×D61—21no—
38#16839510C2212yes—
39#16839510C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C61×#431C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#334C2212yes—
47C2×#11758222—22no—
48C2×#11758222D411yescharacteristic
49C2×#11758190—22no—
50C2×D488—22no—
51C2×D488—21no—
52C2×D488—21no—
53C2×C8×D61—21no—
54C2×C8×D61—21no—
55#14519777C2312yes—
56#14519774C2312yes—
57#14519765—22no—
58#14519751—22no—
59#14519751—22no—
60C4×Dic122D411yescharacteristic
61C4×Dic122C2311yescharacteristic
62C4×D244—22no—
63C4×D244—22no—
64C4×D244D411yescharacteristic
65C4×D244C2311yescharacteristic
66C4×C22⋊D61—21no—
67C4×C22⋊D61—21no—
68#14519736D411yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71#14519711—21no—
72#14519711—21no—
73#14519711—22no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519702—22no—
80#14519699—22no—
81#14519699—22no—
82#14519699—22no—
83#14519699—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418—21no—
89C8×Dic61—21no—
90C8×Dic61—21no—
91C22×C8×C61D411yescharacteristic
92C22×C8×C61D411yescharacteristic
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61C2312yes—
95C4⋊C8×C61C2312yes—
96C22⋊C8×C61—22no—
97C4×C8×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106C2×C22⋊D61—42no—
107#11758222—22no—
108#11758222—22no—
109#11758222—84no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244C2×D411yescharacteristic
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—42no—
118C2×Dic122—22no—
119C2×Dic122C2×D411yescharacteristic
120C2×C8×C61—22no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61—41no—
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61C2411yescharacteristic
132C22⋊Dic61—21no—
133#11758198—21no—
134#11758198—21no—
135#11758198—42no—
136#11758198—42no—
137C4⋊Dic61—22no—
138C4⋊Dic61—22no—
139C4⋊Dic61C2×D411yescharacteristic
140C4⋊Dic61C2×D411yescharacteristic
141#11758196—21no—
142#11758196—21no—
143C4×Dic61—21no—
144C4×Dic61—21no—
145C4×Dic61—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148D488—42no—
149D488—44no—
150D488—42no—
151#11758190—44no—
152C8×D61—42no—
153C8×D61—42no—
154C23×C61#12111yescharacteristic
155C22×D61—21no—
156C22×D61—42no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—22no—
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C22⋊D61—24no—
169C22⋊D61—84no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—42no—
174D244—21no—
175D244—42no—
176D244—42no—
177D244—42no—
178D244—42no—
179D244—82no—
180D244#13512yes—
181C4×D61—24no—
182C4×D61—41no—
183C4×D61—41no—
184C4×D61—42no—
185C4×D61—84no—
186Dic122—21no—
187Dic122—82no—
188Dic122#13512yes—
189C8×C61—21no—
190C8×C61—21no—
191C8×C61—21no—
192C8×C61—42no—
193C8×C61—42no—
194C8×C61#13512yes—
195#92418—41no—
196#92418—41no—
197#19128—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61C2×#12111yescharacteristic
202C2×D61—22no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61#47212yes—
212C4×C61C2×#13511yescharacteristic
213C4×C61C2×#13511yescharacteristic
214C4×C61C2×#13511yescharacteristic
215C4×C61#57511yescharacteristic
216Dic61—22no—
217Dic61—41no—
218Dic61—41no—
219Dic61—82no—
220#3169—611no—
221#3106—612no—
222C8×#41—611no—
223#1678—612no—
224#1677—612no—
225#1676—611no—
226#1675—611no—
227#1668—611no—
228#1667—611no—
229#1564D6111yescharacteristic, complemented
230#1543—612no—
231C2×C61—41no—
232C2×C61#303511yescharacteristic
233C2×C61#305011yescharacteristic
234C2×C61C2×#47211yescharacteristic
235D61—42no—
236D61—162no—
237#532—612no—
238C4×#41—611no—
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×Q8—611no—
251C8×Q8—611no—
252C4×SD16—1222no—
253C4×D8—1222no—
254C4×D8—1221no—
255C4×D8—1221no—
256C8×D4—611no—
257C8×D4—611no—
258C8×D4—1221no—
259C8×D4—1221no—
260#431—611no—
261#431—1221no—
262#431C2×D6112yes—
263C2×C4×C8—1221no—
264C2×C4×C8C2×D6111yescharacteristic
265#335C2×D6111yescharacteristic
266#335C2×D6111yescharacteristic
267#334C2×D6112yes—
268#333—612no—
269#333—612no—
270#331—612no—
271#331—612no—
272C61#1912811yescharacteristic, complemented, Sylow
273C2×#41—611no—
274C2×#41—1222no—
275C2×SD16—1222no—
276C2×D8—1222no—
277C2×D8—1221no—
278C2×D8—1221no—
279C22×C8—1221no—
280C22×C8—1221no—
281C22×C8D24411yescharacteristic
282C22×C8D24411yescharacteristic
283C4⋊Q8—612no—
284#127—612no—
285#124—1222no—
286#121—1222no—
287#121—1222no—
288C4×Q8—611no—
289C4×Q8—611no—
290C4×D4—611no—
291C4×D4—611no—
292C4×D4—1221no—
293C4×D4—1221no—
294C4×D4—1222no—
295C4×D4—1222no—
296#117—611no—
297#117—1221no—
298C2×C42—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301#107—1221no—
302#107—1221no—
303#106—1222no—
304C4⋊C8—611no—
305C4⋊C8—611no—
306C4⋊C8—1221no—
307C4⋊C8—1221no—
308C4⋊C8C22×D6112yes—
309C4⋊C8C22×D6112yes—
310#103—1222no—
311#102—1222no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316C22⋊C8—22no—
317C22⋊C8—1221no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C4×C8—22no—
321C4×C8—1221no—
322C4×C8—1221no—
323C4×C8—1221no—
324C4×C8D24411yescharacteristic
325C4×C8D24411yescharacteristic
326C4×C8C22×D6111yescharacteristic
327C4×C8C22×D6111yescharacteristic
328#41—1222no—
329#41—1222no—
330#41—4884no—
331C2×Q8—611no—
332C2×Q8—1222no—
333C2×D4—611no—
334C2×D4—1221no—
335C2×D4—1222no—
336C2×D4—1222no—
337C2×D4—1222no—
338C2×D4—2442no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—2442no—
343C22×C4C2×D24411yescharacteristic
344C22×C4D4×D6111yescharacteristic
345C22×C4#1175822611yescharacteristic
346SD16—2444no—
347D8—2442no—
348D8—2444no—
349D8—2442no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—1221no—
355C2×C8—1221no—
356C2×C8—2442no—
357C2×C8—2442no—
358C2×C8C2×D24411yescharacteristic
359C2×C8C2×D24411yescharacteristic
360C2×C8C2×D24411yescharacteristic
361C2×C8C2×D24411yescharacteristic
362C4⋊C4—611no—
363C4⋊C4—611no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C22⋊C4—1221no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—2442no—
372C22⋊C4—2442no—
373C42—1221no—
374C42—1221no—
375C42—1221no—
376C42C2×D24411yescharacteristic
377C42D4×D6111yescharacteristic
378C42#1175822611yescharacteristic
379C42C23×D6111yescharacteristic
380C23—1221no—
381C23—2442no—
382C23#1451975211yescharacteristic
383Q8—612no—
384Q8—1221no—
385Q8—4882no—
386D4—612no—
387D4—1224no—
388D4—1221no—
389D4—4882no—
390D4—4884no—
391D4—2442no—
392D4—2442no—
393D4—2442no—
394D4—2442no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—1224no—
401C2×C4—1221no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—4884no—
405C2×C4—2442no—
406C2×C4—2441no—
407C2×C4—2442no—
408C2×C4—2441no—
409C2×C4#1451975211yescharacteristic
410C2×C4#1451975211yescharacteristic
411C2×C4#1451975211yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C2×C4C22×D24411yescharacteristic
414C2×C4C2×#1175822611yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8—42no—
419C8—42no—
420C8—2441no—
421C8—2441no—
422C8#1451979912yes—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—1222no—
427C22—4882no—
428C22—4882no—
429C22—4882no—
430C22—2441no—
431C22—2441no—
432C22C2×#1451975211yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—1222no—
437C4—4882no—
438C4—2441no—
439C4—2441no—
440C4#1684003312yes—
441C4#135×D6111yescharacteristic
442C4#135×D6111yescharacteristic
443C4C2×#1451979911yescharacteristic
444C4#1684056211yescharacteristic
445C2—41no—
446C2—9762no—
447C2—2442no—
448C2#2145527611yescharacteristic
449C2#2145540111yescharacteristic
450C2C2×#1684003311yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
hypercenter#1564D61
hypocenterC61#19128
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. #21455880C2semidirect
3. C8×#11758222C2semidirect
4. #21445236C2semidirect
5. #21445229C2semidirect
6. #21445224C2semidirect
7. #21445219C2semidirect
8. #21445180C2semidirect
9. #21445175C2semidirect
10. #21444479C2semidirect
11. C61×#1564C2semidirect
229. #1564D61semidirect
272. C61#19128semidirect

Maximal subgroups

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

subgroupindex
#214563982
#214558802
C8×#117582222
#214452362
#214452292
#214452242
#214452192
#214451802
#214451752
#214444792
C61×#15642
#1912861