Finite Group Databasebeta

← #29663857

Subgroups — #29663857

Order 15616.

710 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#21456399C211yesmaximal, characteristic
2#21455292C212yesmaximal, complemented
3C8×Q8×D61C211yesmaximal, characteristic, complemented
4#21445266C212yesmaximal, complemented
5#21445265C212yesmaximal, complemented
6#21445210C211yesmaximal, characteristic, complemented
7#21445180C211yesmaximal, characteristic, complemented
8#21445179C211yesmaximal, characteristic, complemented
9#21444576C211yesmaximal, characteristic, complemented
10#21444495C212yesmaximal, complemented
11C61×#1674C211yesmaximal, characteristic, complemented
12C2×#14519809—22no—
13#16840313C2212yes—
14C4×Q8×D61C2211yescharacteristic
15#16840139C2211yescharacteristic
16#16840110C2212yes—
17#16839999—21no—
18#16839999—21no—
19#16839987—22no—
20#16839986—22no—
21#16839969C2212yes—
22#16839963C2212yes—
23#16839962C2212yes—
24Q8×#92418C2211yescharacteristic
25C8×Dic122—21no—
26C8×Dic122C2211yescharacteristic
27#16839886—21no—
28#16839886C2211yescharacteristic
29#16839844—21no—
30#16839844—21no—
31#16839784C2212yes—
32C4⋊C8×D61—21no—
33C4⋊C8×D61C2211yescharacteristic
34#16839762—22no—
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839508C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C61×#500C2211yescharacteristic
43C61×#462C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758226—22no—
48C2×Q8×D61D411yescharacteristic
49#14519809—48no—
50C2×#11758202—22no—
51C2×#11758190—22no—
52C2×#11758201—22no—
53C2×D488—21no—
54C2×D488—21no—
55C2×#11758189—22no—
56C2×C8×D61—21no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61D411yescharacteristic
59#14519774C2312yes—
60#14519766—22no—
61#14519752—22no—
62#14519752—22no—
63Q8×Dic61D411yescharacteristic
64C4×Dic122—21no—
65C4×Dic122C2311yescharacteristic
66#14519742—21no—
67#14519742D411yescharacteristic
68#14519739—22no—
69#14519739—22no—
70C4⋊C4×D61—21no—
71C4⋊C4×D61D411yescharacteristic
72C42×D61—21no—
73C42×D61C2311yescharacteristic
74C4⋊#92418—21no—
75C4⋊#92418C2312yes—
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—22no—
79#14519706—21no—
80#14519706—21no—
81#14519701—22no—
82#14519699—22no—
83#14519697—22no—
84#14519697—22no—
85#14519697—22no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61—21no—
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×Q16×C61—22no—
93C4⋊Q8×C61C2312yes—
94C4×Q8×C61C2311yescharacteristic
95#107×C61—21no—
96#107×C61—21no—
97C4⋊C8×C61—21no—
98C4⋊C8×C61C2312yes—
99C4⋊C8×C61C2311yescharacteristic
100#103×C61—22no—
101C4×C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×Q8×C61C2×D411yescharacteristic
105#11758226—44no—
106#11758226—44no—
107Q8×D61—22no—
108Q8×D61D814yescomplemented
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—42no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—42no—
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61#4111yescharacteristic
119C2×Dic122—21no—
120C2×Dic122C2×D411yescharacteristic
121Q16×C61—44no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D411yescharacteristic
130C42×C61—21no—
131C42×C61C2411yescharacteristic
132#11758202—44no—
133#11758201—44no—
134#11758198—42no—
135#11758198—42no—
136C4⋊Dic61—21no—
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—21no—
140#11758196—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61C2×D411yescharacteristic
145C4×Dic61C2×D411yescharacteristic
146C4×Dic61#4111yescharacteristic
147C2×#92418—21no—
148C2×#92418—22no—
149C2×#92418C2×D411yescharacteristic
150C2×#92418C2×D411yescharacteristic
151D488—42no—
152D488—42no—
153#11758190—44no—
154#11758189—44no—
155C8×D61—22no—
156C8×D61—41no—
157C8×D61—42no—
158C22×D61—42no—
159C22×D61#12111yescharacteristic
160Q8×C61—21no—
161Q8×C61—42no—
162Q8×C61—42no—
163Q8×C61C2×D812yes—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—42no—
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C2×#4111yescharacteristic
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174C2×Dic61#12111yescharacteristic
175D244—42no—
176D244—42no—
177D244—42no—
178D244—42no—
179D244—82no—
180D244—84no—
181C4×D61—24no—
182C4×D61—21no—
183C4×D61—21no—
184C4×D61—41no—
185C4×D61—42no—
186C4×D61—84no—
187C4×D61C2×D812yes—
188C4×D61#13512yes—
189Dic122—21no—
190Dic122—24no—
191Dic122C2×D812yes—
192C8×C61—21no—
193C8×C61—42no—
194C8×C61—41no—
195C8×C61#13712yescomplemented
196#92418—21no—
197#92418—21no—
198#92418—42no—
199#92418—41no—
200#19261—611nomaximal, Sylow
201C22×C61C2×#12111yescharacteristic
202C2×D61—21no—
203C2×D61—21no—
204C2×D61—82no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61#46612yes—
208C4×C61—22no—
209C4×C61—21no—
210C4×C61—41no—
211C4×C61—82no—
212C4×C61C22×D811yescharacteristic
213C4×C61C2×#13511yescharacteristic
214C4×C61C2×#13711yescharacteristic
215C4×C61#57511yescharacteristic
216Dic61—21no—
217Dic61—22no—
218Dic61—21no—
219Dic61—41no—
220Dic61#46612yes—
221#3169—611no—
222#3037—612no—
223C2×C8×Q8—611no—
224#1682—612no—
225#1682—612no—
226#1674—611no—
227#1674D6111yescharacteristic, complemented
228#1668—611no—
229#1668—611no—
230#1567—611no—
231#1547—612no—
232C2×C61#303711yescharacteristic
233C2×C61C2×#46611yescharacteristic
234C2×C61#306711yescharacteristic
235D61—22no—
236D61—162no—
237C2×#137—1222no—
238#532—612no—
239C2×C4×Q8—611no—
240#500C2×D6111yescharacteristic
241#493—611no—
242#486—612no—
243#466—1221no—
244#466—1221no—
245#464—1222no—
246#464—1222no—
247#462—612no—
248#462C2×D6112yes—
249#460—612no—
250#460—612no—
251C8×Q8—611no—
252C8×Q8—611no—
253C8×Q8—611no—
254C8×Q8—1221no—
255C8×Q8—1221no—
256C8×Q8C2×D6111yescharacteristic
257C4×D8—1221no—
258C4×D8—1221no—
259#423—612no—
260C2×C4⋊C8—611no—
261C2×C4⋊C8—1221no—
262#417—1222no—
263C2×C4×C8—611no—
264C2×C4×C8—1221no—
265#335—612no—
266#335—611no—
267#335C2×D6111yescharacteristic
268#332—612no—
269#332C2×D6112yes—
270#331—612no—
271#331—612no—
272C61#1926111yescharacteristic, complemented, Sylow
273C2×#41—1222no—
274C22×Q8—611no—
275#137—2448no—
276C2×Q16—22no—
277C2×Q16—1222no—
278C2×SD16—1222no—
279C2×SD16—1222no—
280C2×D8—1221no—
281C2×D8—1221no—
282C2×M16—1222no—
283C22×C8—611no—
284C22×C8—611no—
285C22×C8—1221no—
286C4⋊Q8C22×D6112yes—
287#127—612no—
288#124—1222no—
289#121—1222no—
290#121—1222no—
291C4×Q8—611no—
292C4×Q8—611no—
293C4×Q8—611no—
294C4×Q8—1221no—
295C4×Q8—1221no—
296C4×Q8C22×D6111yescharacteristic
297C4×D4—1222no—
298C4×D4—1222no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—1221no—
301C2×C42—611no—
302C2×C42—1221no—
303#107—21no—
304#107—21no—
305C4⋊C8—21no—
306C4⋊C8—611no—
307C4⋊C8—612no—
308C4⋊C8—1221no—
309C4⋊C8—1221no—
310C4⋊C8—1222no—
311C4⋊C8—1221no—
312C4⋊C8—1221no—
313C4⋊C8C22×D6111yescharacteristic
314C4⋊C8C22×D6112yes—
315#103—22no—
316#103—1222no—
317#102—1222no—
318#102—1222no—
319#102—1222no—
320#102—1222no—
321C4×C8—21no—
322C4×C8—611no—
323C4×C8—611no—
324C4×C8—611no—
325C4×C8—1221no—
326C4×C8—1221no—
327C4×C8—1221no—
328C4×C8C22×D6111yescharacteristic
329#41—2444no—
330#41—2444no—
331C2×Q8—22no—
332C2×Q8—614no—
333C2×Q8—611no—
334C2×Q8—1221no—
335C2×Q8—1222no—
336C2×Q8D4×D6111yescharacteristic
337C2×D4—1222no—
338C2×D4—1222no—
339C2×D4—1222no—
340C2×D4—2442no—
341C22×C4—611no—
342C22×C4—611no—
343C22×C4—611no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—2442no—
347Q16—44no—
348Q16—2444no—
349SD16—2444no—
350SD16—2444no—
351D8—2442no—
352D8—2442no—
353M16—2444no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—611no—
357C2×C8—611no—
358C2×C8—1222no—
359C2×C8—1221no—
360C2×C8—1222no—
361C2×C8—2441no—
362C2×C8—2442no—
363C2×C8D4×D6111yescharacteristic
364C2×C8D4×D6111yescharacteristic
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—611no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4D4×D6111yescharacteristic
374C22⋊C4—2442no—
375C22⋊C4—2442no—
376C42—21no—
377C42—611no—
378C42—611no—
379C42—611no—
380C42—1221no—
381C42—1221no—
382C42—1221no—
383C42C23×D6111yescharacteristic
384C23—611no—
385C23—2442no—
386Q8—21no—
387Q8—42no—
388Q8—42no—
389Q8—612no—
390Q8—1224no—
391Q8—1221no—
392Q8D8×D6112yes—
393D4—4882no—
394D4—4884no—
395D4—2442no—
396D4—2442no—
397D4—2442no—
398D4—2442no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—42no—
402C2×C4—611no—
403C2×C4—612no—
404C2×C4—612no—
405C2×C4—611no—
406C2×C4—611no—
407C2×C4—1224no—
408C2×C4—1221no—
409C2×C4—1221no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—4884no—
413C2×C4—2441no—
414C2×C4—2442no—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4#41×D6111yescharacteristic
418C8—21no—
419C8—42no—
420C8—41no—
421C8—1221no—
422C8—1221no—
423C8—2442no—
424C8—2441no—
425C8#1451980912yescomplemented
426C22—612no—
427C22—1221no—
428C22—1221no—
429C22—4882no—
430C22—4882no—
431C22—4882no—
432C22#121×D6111yescharacteristic
433C4—22no—
434C4—21no—
435C4—41no—
436C4—82no—
437C4—612no—
438C4—1222no—
439C4—1221no—
440C4—1221no—
441C4—2441no—
442C4C2×D8×D6111yescharacteristic
443C4#135×D6111yescharacteristic
444C4C2×#1451980911yescharacteristic
445C4#1684056111yescharacteristic
446C2—1222no—
447C2—9762no—
448C2#2145529211yescharacteristic
449C2#466×D6111yescharacteristic
450C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1674C2
socleC22×C61C2×#121
prime radical#21456399C2
hypercenter#1674D61
hypocenterC61#19261
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
2. #21455292C2semidirect
3. C8×Q8×D61C2semidirect
4. #21445266C2semidirect
5. #21445265C2semidirect
6. #21445210C2semidirect
7. #21445180C2semidirect
8. #21445179C2semidirect
9. #21444576C2semidirect
10. #21444495C2semidirect
11. C61×#1674C2semidirect
108. Q8×D61D8semidirect
195. C8×C61#137semidirect
227. #1674D61semidirect
272. C61#19261semidirect
425. C8#14519809semidirect

Maximal subgroups

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

subgroupindex
#214563992
#214552922
C8×Q8×D612
#214452662
#214452652
#214452102
#214451802
#214451792
#214445762
#214444952
C61×#16742
#1926161