Finite Group Databasebeta

← #28402252

Subgroups — #28402252

Order 14464.

526 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813643C211yesmaximal, characteristic, complemented
2C22×#14268632C211yesmaximal, characteristic, complemented
3#20299242C211yesmaximal, characteristic, complemented
4#20299239C211yesmaximal, characteristic, complemented
5#20298979C211yesmaximal, characteristic, complemented
6#20298837C211yesmaximal, characteristic, complemented
7#20298833C211yesmaximal, characteristic, complemented
8#20298815C211yesmaximal, characteristic, complemented
9#20298812C211yesmaximal, characteristic, complemented
10#20298751C211yesmaximal, characteristic, complemented
11#20298744C211yesmaximal, characteristic, complemented
12#20298742C211yesmaximal, characteristic, complemented
13#20298678C211yesmaximal, characteristic, complemented
14#20298661C211yesmaximal, characteristic, complemented
15#450×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic, complemented
17C2×#14268646C2211yescharacteristic, complemented
18C22×D452C2211yescharacteristic
19#16813643—24no—
20C2×#14268637C2211yescharacteristic, complemented
21C2×#14268636C2211yescharacteristic, complemented
22C2×#14268636C2211yescharacteristic, complemented
23C2×#14268632—24no—
24C2×#14268632C2211yescharacteristic
25C2×#14268632C2211yescharacteristic
26C2×#14268632C2211yescharacteristic
27C2×#14268632C2211yescharacteristic
28C22×#11744751C2211yescharacteristic
29C2×C8×D113C2211yescharacteristic, complemented
30#16813611C2211yescharacteristic, complemented
31#16813609C2211yescharacteristic, complemented
32#16813607—21no—
33#16813603—21no—
34#16813601C2211yescharacteristic
35#16813597—21no—
36#16813597C2211yescharacteristic, complemented
37#16813594—21no—
38#16813593C2211yescharacteristic, complemented
39#16813592—21no—
40#16813591—21no—
41D4×Dic113C2211yescharacteristic, complemented
42#16813584C2211yescharacteristic, complemented
43C4×C22⋊D113C2211yescharacteristic
44C2×C22⋊Dic113—21no—
45C22⋊C4×D113—21no—
46#16813555C2211yescharacteristic, complemented
47#16813551C2211yescharacteristic, complemented
48#16813546C2211yescharacteristic, complemented
49#16813545C2211yescharacteristic, complemented
50#16813544C2211yescharacteristic, complemented
51#16813544C2211yescharacteristic, complemented
52#16813534C2211yescharacteristic, complemented
53C8×Dic113C2211yescharacteristic, complemented
54C22×D4×C113C2211yescharacteristic
55C2×SD16×C113C2211yescharacteristic, complemented
56C2×SD16×C113C2211yescharacteristic, complemented
57C22⋊Q8×C113C2211yescharacteristic
58#102×C113C2211yescharacteristic, complemented
59#102×C113C2211yescharacteristic, complemented
60C22⋊C8×C113C2211yescharacteristic
61C23×D113—21no—
62#14268648—22no—
63#14268648—41no—
64#14268646—22no—
65#14268646—41no—
66C2×Dic226C2311yescharacteristic
67C2×D452—22no—
68C2×D452—21no—
69C2×D452—21no—
70C2×D452—21no—
71C2×D452—41no—
72C2×D452C2311yescharacteristic
73C2×D452C2311yescharacteristic
74C2×C22⋊D113—21no—
75C2×C22⋊D113—21no—
76C2×C22⋊D113D411yescharacteristic
77C22×Dic113—21no—
78C22×Dic113—21no—
79C2×C4×D113—21no—
80C2×C4×D113—21no—
81C2×C4×D113C2311yescharacteristic
82#14268637—22no—
83#14268636—22no—
84#14268636—41no—
85#14268632—22no—
86#14268632—22no—
87#14268632—22no—
88#14268632D414yescomplemented
89C2×#11744751—22no—
90C2×#11744751C2311yescharacteristic
91C2×#11744751C2311yescharacteristic
92C8×D113—22no—
93C4⋊Dic113C2311yescharacteristic
94#14268626—21no—
95#14268626D411yescharacteristic
96C22⋊Dic113—21no—
97C22⋊Dic113—21no—
98C22⋊Dic113—21no—
99C22⋊Dic113—21no—
100C22⋊Dic113—21no—
101C22⋊Dic113D411yescharacteristic
102#14268624—21no—
103#14268624—21no—
104#14268624—21no—
105#14268624—21no—
106#14268624D411yescharacteristic
107C4×Dic113C2311yescharacteristic
108C24×C113—21no—
109C2×Q8×C113C2311yescharacteristic
110C2×D4×C113—22no—
111C2×D4×C113—21no—
112C2×D4×C113—41no—
113C2×D4×C113C2311yescharacteristic
114C2×D4×C113C2311yescharacteristic
115C22×C4×C113C2311yescharacteristic
116SD16×C113—22no—
117SD16×C113—41no—
118C2×C8×C113C2311yescharacteristic
119C2×C8×C113C2311yescharacteristic
120C4⋊C4×C113—21no—
121C4⋊C4×C113C2311yescharacteristic
122C22⋊C4×C113—21no—
123C23×C113—21no—
124C23×C113—21no—
125C23×C113—42no—
126C23×C113—42no—
127C23×C113—81no—
128C23×C113C2×D411yescharacteristic
129C22×D113—21no—
130C22×D113—21no—
131C22×D113—42no—
132C22×D113—42no—
133C22×D113—81no—
134C22×D113C2×D411yescharacteristic
135Q8×C113—21no—
136Q8×C113—21no—
137D4×C113—21no—
138D4×C113—21no—
139D4×C113—21no—
140D4×C113—42no—
141D4×C113C2×D412yes—
142C2×C4×C113—21no—
143C2×C4×C113—21no—
144C2×C4×C113—21no—
145C2×C4×C113—21no—
146C2×C4×C113C2×D411yescharacteristic
147C2×C4×C113C2411yescharacteristic
148C22⋊D113—22no—
149C22⋊D113—42no—
150C22⋊D113—81no—
151C2×Dic113—21no—
152C2×Dic113—21no—
153C2×Dic113—42no—
154C2×Dic113—81no—
155C2×Dic113C2×D411yescharacteristic
156C2×Dic113C2×D411yescharacteristic
157C2×Dic113C2×D411yescharacteristic
158D452—21no—
159D452—21no—
160D452—21no—
161D452—42no—
162D452—42no—
163D452—81no—
164D452C2×D412yes—
165C4×D113—21no—
166C4×D113—21no—
167C4×D113—42no—
168C4×D113—81no—
169Dic226—21no—
170Dic226—21no—
171C8×C113—21no—
172C8×C113—21no—
173#11744751—21no—
174#11744751C2×D412yes—
175C22×C113—21no—
176C22×C113—21no—
177C22×C113—22no—
178C22×C113—41no—
179C22×C113—41no—
180C22×C113—41no—
181C22×C113—41no—
182C22×C113—41no—
183C22×C113—84no—
184C22×C113#13512yescomplemented
185C22×C113C22×D411yescharacteristic
186C2×D113—21no—
187C2×D113—22no—
188C2×D113—21no—
189C2×D113—21no—
190C2×D113—41no—
191C2×D113—41no—
192C2×D113—41no—
193C2×D113—41no—
194C2×D113—41no—
195C2×D113—84no—
196C4×C113—21no—
197C4×C113—41no—
198C4×C113—41no—
199C4×C113C22×D411yescharacteristic
200C4×C113H(5,𝔽2)11yescharacteristic
201Dic113—21no—
202Dic113—21no—
203Dic113—21no—
204Dic113—41no—
205Dic113—41no—
206C2×C113—22no—
207C2×C113—42no—
208C2×C113—81no—
209C2×C113D4211yescharacteristic
210C2×C113C2×#13511yescharacteristic
211C2×C113#57611yescharacteristic
212D113—41no—
213D113—42no—
214D113—81no—
215#3280—1131nomaximal, Sylow
216C113#328011yescharacteristic, complemented, Sylow
217C2×#135—1131no—
218C22×D8—1131no—
219#546—1131no—
220#546—1131no—
221#495—1131no—
222#467—1131no—
223#466—1131no—
224#463—1131no—
225#463—1131no—
226#450—1131no—
227#450D11311yescharacteristic, complemented
228#449—1131no—
229#449—1131no—
230C4×D8—1131no—
231C8×D4—1131no—
232C2×#41—1131no—
233C2×#41—1131no—
234C22×D4—1131no—
235C22×D4C2×D11311yescharacteristic
236#135—2264no—
237C2×Q16—1131no—
238C2×SD16—1131no—
239C2×SD16—1131no—
240C2×SD16C2×D11311yescharacteristic, complemented
241C2×SD16C2×D11311yescharacteristic, complemented
242C2×D8—1131no—
243C2×D8—1131no—
244C2×D8—1131no—
245C2×D8—1131no—
246C2×D8—2264no—
247C22×C8—1131no—
248C22×C8—1131no—
249#124—1131no—
250#124—1131no—
251#123—2261no—
252#123—2261no—
253C22⋊Q8—1131no—
254C22⋊Q8C2×D11311yescharacteristic
255#121—1131no—
256#121—1131no—
257#121—2261no—
258#121—2261no—
259#120—2261no—
260#120—2261no—
261C4×D4—1131no—
262C4×D4—1131no—
263C4×D4—1131no—
264C2×C22⋊C4—2261no—
265C2×C22⋊C4—2261no—
266#107—1131no—
267C4⋊C8—1131no—
268#103—1131no—
269#103—1131no—
270#102—1131no—
271#102—1131no—
272#102C2×D11311yescharacteristic, complemented
273#102C2×D11311yescharacteristic, complemented
274C22⋊C8—1131no—
275C22⋊C8C2×D11311yescharacteristic
276C4×C8—1131no—
277C24—21no—
278C24—2261no—
279#41—4521no—
280#41—4521no—
281#41—2262no—
282#41—2262no—
283C2×Q8—1131no—
284C2×Q8C22×D11311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—41no—
288C2×D4—1131no—
289C2×D4—1131no—
290C2×D4—1131no—
291C2×D4—4521no—
292C2×D4—2262no—
293C2×D4—2261no—
294C2×D4—2261no—
295C2×D4—2261no—
296C2×D4—2261no—
297C2×D4—2261no—
298C2×D4C22×D11311yescharacteristic
299C2×D4C22×D11311yescharacteristic
300C22×C4—1131no—
301C22×C4—2261no—
302C22×C4—2261no—
303C22×C4—2261no—
304C22×C4—2261no—
305C22×C4C22×D11311yescharacteristic
306Q16—2262no—
307SD16—22no—
308SD16—41no—
309SD16—4521no—
310SD16—2262no—
311D8—1134no—
312D8—2262no—
313D8—2262no—
314D8—2262no—
315C2×C8—1131no—
316C2×C8—1131no—
317C2×C8—2262no—
318C2×C8—2262no—
319C2×C8C22×D11311yescharacteristic
320C2×C8C22×D11311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1131no—
323C4⋊C4—1131no—
324C4⋊C4—2261no—
325C4⋊C4C22×D11311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—1131no—
328C22⋊C4—1131no—
329C22⋊C4—2261no—
330C22⋊C4—2261no—
331C22⋊C4—2261no—
332C22⋊C4—2261no—
333C22⋊C4—2261no—
334C22⋊C4—2261no—
335C22⋊C4—2261no—
336C22⋊C4—2261no—
337C22⋊C4—2261no—
338C42—1131no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—1131no—
345C23—9041no—
346C23—4522no—
347C23—4522no—
348C23—2261no—
349C23—2261no—
350C23D4×D11311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2261no—
354Q8—2261no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—1132no—
360D4—9041no—
361D4—9041no—
362D4—4522no—
363D4—4522no—
364D4—4522no—
365D4—2262no—
366D4—2261no—
367D4—2261no—
368D4—2261no—
369D4D4×D11312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—1131no—
375C2×C4—1131no—
376C2×C4—1131no—
377C2×C4—9041no—
378C2×C4—9041no—
379C2×C4—4522no—
380C2×C4—4522no—
381C2×C4—2261no—
382C2×C4—2261no—
383C2×C4—2261no—
384C2×C4—2261no—
385C2×C4D4×D11311yescharacteristic
386C2×C4C23×D11311yescharacteristic
387C8—21no—
388C8—21no—
389C8—1132no—
390C8—2261no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—84no—
400C22—9044no—
401C22—4521no—
402C22—4521no—
403C22—4521no—
404C22—4521no—
405C22—4521no—
406C22—2261no—
407C22—2261no—
408C22—2262no—
409C22—2261no—
410C22#1681364312yescomplemented
411C22C2×D4×D11311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—4521no—
416C4—4521no—
417C4—2261no—
418C4—2261no—
419C4—2261no—
420C4C2×D4×D11311yescharacteristic
421C4#1681366711yescharacteristic
422C2—22no—
423C2—42no—
424C2—81no—
425C2—9041no—
426C2—4522no—
427C2—4521no—
428C2#2029923011yescharacteristic
429C2C2×#1681364311yescharacteristic
430C2#2029940211yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
hypercenter#450D113
hypocenterC113#3280
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#16813643C2semidirect
2. C22×#14268632C2semidirect
3. #20299242C2semidirect
4. #20299239C2semidirect
5. #20298979C2semidirect
6. #20298837C2semidirect
7. #20298833C2semidirect
8. #20298815C2semidirect
9. #20298812C2semidirect
10. #20298751C2semidirect
11. #20298744C2semidirect
12. #20298742C2semidirect
13. #20298678C2semidirect
14. #20298661C2semidirect
15. #450×C113C2semidirect
16. C2×#14268648C22semidirect
17. C2×#14268646C22semidirect
20. C2×#14268637C22semidirect
21. C2×#14268636C22semidirect
22. C2×#14268636C22semidirect
29. C2×C8×D113C22semidirect
30. #16813611C22semidirect
31. #16813609C22semidirect
36. #16813597C22semidirect
38. #16813593C22semidirect
41. D4×Dic113C22semidirect
42. #16813584C22semidirect
46. #16813555C22semidirect
47. #16813551C22semidirect
48. #16813546C22semidirect
49. #16813545C22semidirect
50. #16813544C22semidirect
51. #16813544C22semidirect
52. #16813534C22semidirect
53. C8×Dic113C22semidirect
55. C2×SD16×C113C22semidirect
56. C2×SD16×C113C22semidirect
58. #102×C113C22semidirect
59. #102×C113C22semidirect
88. #14268632D4semidirect
184. C22×C113#135semidirect
216. C113#3280semidirect
227. #450D113semidirect
240. C2×SD16C2×D113semidirect
241. C2×SD16C2×D113semidirect
272. #102C2×D113semidirect
273. #102C2×D113semidirect
410. C22#16813643semidirect

Maximal subgroups

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

subgroupindex
C2×#168136432
C22×#142686322
#202992422
#202992392
#202989792
#202988372
#202988332
#202988152
#202988122
#202987512
#202987442
#202987422
#202986782
#202986612
#450×C1132
#3280113