Finite Group Databasebeta

← #29662453

Subgroups — #29662453

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456399C211yesmaximal, characteristic, complemented
2#21455277C212yesmaximal, complemented
3C8×#11758226C211yesmaximal, characteristic, complemented
4#21445265C212yesmaximal, complemented
5#21445259C212yesmaximal, complemented
6#21445205C211yesmaximal, characteristic, complemented
7#21445179C211yesmaximal, characteristic, complemented
8#21445175C211yesmaximal, characteristic, complemented
9#335×D61C211yesmaximal, characteristic, complemented
10#21444467C212yesmaximal, complemented
11C61×#1674C211yesmaximal, characteristic, complemented
12C2×#14519800—22no—
13#16840313C2212yes—
14C4×#11758226C2211yescharacteristic
15#16840139C2211yescharacteristic
16#16840110C2212yes—
17#16839999—21no—
18#16839999—21no—
19#16839986—22no—
20#16839980—22no—
21#16839964C2212yes—
22#16839962C2212yes—
23#16839958C2212yes—
24Q8×#92418C2211yescharacteristic
25#16839844—21no—
26#16839844—21no—
27C8×D244—21no—
28C8×D244C2211yescharacteristic
29#16839827—21no—
30#16839827C2211yescharacteristic
31#16839814—21no—
32#16839814C2211yescharacteristic
33C4⋊C8×D61C2212yes—
34#16839757—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×#11758226D411yescharacteristic
49#14519800—48no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×D488—22no—
53C2×D488—21no—
54C2×D488—21no—
55C2×C8×D61—22no—
56C2×C8×D61—21no—
57C2×C8×D61—21no—
58C2×C8×D61D411yescharacteristic
59C2×C8×D61D411yescharacteristic
60#14519774C2312yes—
61#14519766—22no—
62#14519752—22no—
63#14519752—22no—
64Q8×Dic61D411yescharacteristic
65C4×D244—21no—
66C4×D244C2311yescharacteristic
67#14519739—21no—
68#14519739—22no—
69#14519739—22no—
70#14519739D411yescharacteristic
71#14519735—21no—
72#14519735D411yescharacteristic
73C42×D61—21no—
74C42×D61C2311yescharacteristic
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77C4⋊#92418C2311yescharacteristic
78#14519706—22no—
79#14519701—22no—
80#14519699—22no—
81#14519697—22no—
82#14519697—22no—
83#14519697—22no—
84#14519689—21no—
85#14519689—21no—
86#14519689—21no—
87C4×#92418—21no—
88C4×#92418C2311yescharacteristic
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—22no—
106#11758226—44no—
107#11758226—44no—
108#11758226D814yescomplemented
109C2×D244—21no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—42no—
114C2×D244C2×D411yescharacteristic
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—42no—
119C2×C4×D61C2×D411yescharacteristic
120C2×C4×D61C2×D411yescharacteristic
121C2×C4×D61#4111yescharacteristic
122Q16×C61—44no—
123C2×C8×C61—21no—
124C2×C8×C61—22no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61C2×D411yescharacteristic
131C42×C61—21no—
132C42×C61C2411yescharacteristic
133#11758201—44no—
134#11758201—44no—
135#11758198—21no—
136#11758198—21no—
137#11758198—21no—
138#11758198—42no—
139#11758198—42no—
140C4⋊Dic61—21no—
141C4⋊Dic61C2×D411yescharacteristic
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—44no—
153D488—42no—
154C8×D61—22no—
155C8×D61—41no—
156C8×D61—42no—
157C8×D61—44no—
158C8×D61—42no—
159C22×D61—21no—
160C22×D61—42no—
161C22×D61#12111yescharacteristic
162Q8×C61—21no—
163Q8×C61—42no—
164Q8×C61—42no—
165Q8×C61C2×D812yes—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—42no—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×Dic61—21no—
173C2×Dic61#12111yescharacteristic
174C2×Dic61#12111yescharacteristic
175C2×Dic61#12111yescharacteristic
176D244—24no—
177D244—21no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—82no—
183D244—84no—
184D244C2×D812yes—
185C4×D61—21no—
186C4×D61—21no—
187C4×D61—24no—
188C4×D61—41no—
189C4×D61—42no—
190C4×D61—41no—
191C4×D61—41no—
192C4×D61—84no—
193C4×D61C2×D812yes—
194C4×D61#13512yes—
195C8×C61—21no—
196C8×C61—42no—
197C8×C61—41no—
198C8×C61#13512yescomplemented
199#92418—21no—
200#92418—21no—
201#92418—42no—
202#92418—41no—
203#19125—611nomaximal, Sylow
204C22×C61C2×#12111yescharacteristic
205C2×D61—21no—
206C2×D61—21no—
207C2×D61—22no—
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—82no—
211C2×D61—82no—
212C2×D61—82no—
213C2×D61#46612yes—
214C4×C61—22no—
215C4×C61—21no—
216C4×C61—41no—
217C4×C61—82no—
218C4×C61C22×D811yescharacteristic
219C4×C61C2×#13511yescharacteristic
220C4×C61C2×#13511yescharacteristic
221C4×C61#57511yescharacteristic
222Dic61—21no—
223Dic61—21no—
224Dic61—41no—
225Dic61#46612yes—
226#3169—611no—
227#3035—612no—
228C8×#41—611no—
229#1682—612no—
230#1681—612no—
231#1674D6111yescharacteristic, complemented
232#1673—611no—
233#1668—611no—
234#1667—611no—
235C2×#335—611no—
236#1540—612no—
237C2×C61#303511yescharacteristic
238C2×C61C2×#46611yescharacteristic
239C2×C61#306711yescharacteristic
240D61—22no—
241D61—42no—
242D61—162no—
243C2×#135—1222no—
244#532—612no—
245C4×#41—611no—
246#500C2×D6111yescharacteristic
247#493—611no—
248#486—612no—
249#466—1221no—
250#466—1221no—
251#464—1222no—
252#463—1222no—
253#462C2×D6112yes—
254#461—612no—
255#460—612no—
256#459—612no—
257C8×Q8—611no—
258C8×Q8C2×D6111yescharacteristic
259C4×D8—1221no—
260C4×D8—1221no—
261C8×D4—611no—
262C8×D4—611no—
263C8×D4—1221no—
264C8×D4—1221no—
265#431—611no—
266#431—1221no—
267C2×C4⋊C8—612no—
268#416—1222no—
269C2×C4×C8—611no—
270C2×C4×C8—1221no—
271#335—612no—
272#335—611no—
273#335C2×D6111yescharacteristic
274#332—612no—
275#332C2×D6112yes—
276#331—612no—
277#331—612no—
278C61#1912511yescharacteristic, complemented, Sylow
279C2×#41—611no—
280C2×#41—1222no—
281#135—2448no—
282C2×Q16—22no—
283C2×SD16—1222no—
284C2×SD16—1222no—
285C2×D8—1222no—
286C2×D8—1221no—
287C2×D8—1221no—
288C22×C8—611no—
289C22×C8—611no—
290C22×C8—1221no—
291C22×C8—1222no—
292C22×C8—1221no—
293C4⋊Q8C22×D6112yes—
294#127—612no—
295#124—1222no—
296#121—1222no—
297#121—1222no—
298C4×Q8—611no—
299C4×Q8C22×D6111yescharacteristic
300C4×D4—611no—
301C4×D4—611no—
302C4×D4—1221no—
303C4×D4—1222no—
304C4×D4—1222no—
305C4×D4—1221no—
306#117—611no—
307#117—1221no—
308C2×C42—611no—
309C2×C42—1221no—
310#107—21no—
311#107—21no—
312C4⋊C8—21no—
313C4⋊C8—611no—
314C4⋊C8—612no—
315C4⋊C8—1221no—
316C4⋊C8—1222no—
317C4⋊C8C22×D6111yescharacteristic
318C4⋊C8C22×D6112yes—
319#103—22no—
320#103—1222no—
321#102—1222no—
322#102—1222no—
323#102—1222no—
324#102—1222no—
325C22⋊C8—1221no—
326C22⋊C8—1221no—
327C22⋊C8—1221no—
328C4×C8—21no—
329C4×C8—611no—
330C4×C8—611no—
331C4×C8—611no—
332C4×C8—1221no—
333C4×C8—1221no—
334C4×C8C22×D6111yescharacteristic
335#41—614no—
336#41—1222no—
337#41—2444no—
338#41—2444no—
339C2×Q8—22no—
340C2×Q8D4×D6111yescharacteristic
341C2×D4—611no—
342C2×D4—1221no—
343C2×D4—1222no—
344C2×D4—1222no—
345C2×D4—1222no—
346C2×D4—2442no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—2442no—
354Q16—44no—
355SD16—2444no—
356SD16—2444no—
357D8—2442no—
358D8—2444no—
359D8—2442no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—611no—
363C2×C8—611no—
364C2×C8—1222no—
365C2×C8—1221no—
366C2×C8—1222no—
367C2×C8—2441no—
368C2×C8—2442no—
369C2×C8—2444no—
370C2×C8—2442no—
371C2×C8D4×D6111yescharacteristic
372C2×C8D4×D6111yescharacteristic
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—611no—
377C4⋊C4—1221no—
378C4⋊C4D4×D6111yescharacteristic
379C22⋊C4—1221no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—2442no—
383C22⋊C4—2442no—
384C42—21no—
385C42—611no—
386C42—611no—
387C42—611no—
388C42—1221no—
389C42—1221no—
390C42C23×D6111yescharacteristic
391C23—611no—
392C23—1221no—
393C23—2442no—
394Q8—21no—
395Q8—42no—
396Q8—42no—
397Q8D8×D6112yes—
398D4—612no—
399D4—1224no—
400D4—1221no—
401D4—4882no—
402D4—4884no—
403D4—2442no—
404D4—2442no—
405D4—2442no—
406D4—2442no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—611no—
411C2×C4—612no—
412C2×C4—612no—
413C2×C4—611no—
414C2×C4—611no—
415C2×C4—1224no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—4884no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4C2×D4×D6111yescharacteristic
425C2×C4C2×D4×D6111yescharacteristic
426C2×C4#41×D6111yescharacteristic
427C8—21no—
428C8—42no—
429C8—41no—
430C8—1221no—
431C8—1221no—
432C8—2442no—
433C8—2441no—
434C8#1451980012yescomplemented
435C22—612no—
436C22—1221no—
437C22—1221no—
438C22—1222no—
439C22—4882no—
440C22—4882no—
441C22—4882no—
442C22—2441no—
443C22—2441no—
444C22#121×D6111yescharacteristic
445C4—22no—
446C4—21no—
447C4—41no—
448C4—82no—
449C4—612no—
450C4—1221no—
451C4—1221no—
452C4—2441no—
453C4C2×D8×D6111yescharacteristic
454C4#135×D6111yescharacteristic
455C4C2×#1451980011yescharacteristic
456C4#1684056111yescharacteristic
457C2—1222no—
458C2—9762no—
459C2—2442no—
460C2#2145527711yescharacteristic
461C2#466×D6111yescharacteristic
462C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1674C2
socleC22×C61C2×#121
hypercenter#1674D61
hypocenterC61#19125
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
1. #21456399C2semidirect
2. #21455277C2semidirect
3. C8×#11758226C2semidirect
4. #21445265C2semidirect
5. #21445259C2semidirect
6. #21445205C2semidirect
7. #21445179C2semidirect
8. #21445175C2semidirect
9. #335×D61C2semidirect
10. #21444467C2semidirect
11. C61×#1674C2semidirect
108. #11758226D8semidirect
198. C8×C61#135semidirect
231. #1674D61semidirect
278. C61#19125semidirect
434. C8#14519800semidirect

Maximal subgroups

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

subgroupindex
#214563992
#214552772
C8×#117582262
#214452652
#214452592
#214452052
#214451792
#214451752
#335×D612
#214444672
C61×#16742
#1912561