Finite Group Databasebeta

← #80392

Subgroups — #80392

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5715C211yesmaximal, characteristic, complemented
2#5582C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic, complemented
5#5173C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5168C211yesmaximal, characteristic, complemented
8#5072C211yesmaximal, characteristic, complemented
9#5028C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4765C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C3×#127C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23#947C2211yescharacteristic, complemented
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27#940C2211yescharacteristic
28#939—21no—
29#939—21no—
30#937—21no—
31#937—21no—
32#936C2211yescharacteristic
33C2×#213C2211yescharacteristic
34C2×#213C2211yescharacteristic, complemented
35C2×D24C2211yescharacteristic
36C2×#204C2211yescharacteristic, complemented
37#896—21no—
38#895C2211yescharacteristic
39#894—21no—
40#888C2211yescharacteristic, complemented
41#876—21no—
42#875—21no—
43#875C2211yescharacteristic
44C4×D12C2211yescharacteristic, complemented
45#872C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#102C2211yescharacteristic
48C3×#102C2211yescharacteristic, complemented
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic, complemented
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#542—31no—
61#535—31no—
62#496—31no—
63#495—31no—
64#489—31no—
65#486—31no—
66#464—31no—
67#463—31no—
68#460D311yescharacteristic, complemented
69#459—31no—
70#452—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#237—41no—
87#237—41no—
88D3×D4—41no—
89D3×D4—41no—
90C2×D12—21no—
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×Dic6—21no—
96C2×Dic6C2311yescharacteristic
97C3×SD16—41no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C42—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#217—21no—
107#216—41no—
108#215—41no—
109#213—41no—
110#213—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4D411yescharacteristic
119(C2×D3)⋊C4D411yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124Dic3⋊C4D411yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128D24—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16C2×D311yescharacteristic
136C2×SD16C2×D311yescharacteristic, complemented
137C2×D8—31no—
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8—31no—
141#127—31no—
142#127C2×D311yescharacteristic
143#126—61no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#123—61no—
149C22⋊Q8—31no—
150C22⋊Q8—61no—
151#121—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8C2×D311yescharacteristic, complemented
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C3×Q8—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—81no—
188C3×D4—81no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—42no—
196#67—81no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—42no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—81no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—81no—
212Dic6—21no—
213Dic6—21no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—121no—
221#41—121no—
222C2×Q8—31no—
223C2×Q8—61no—
224C2×Q8C22×D311yescharacteristic
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—31no—
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4C22×D311yescharacteristic
237C2×D4C22×D311yescharacteristic
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241Q16—121no—
242SD16—41no—
243SD16—41no—
244SD16—121no—
245SD16—121no—
246D8—121no—
247D8—121no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C22⋊C4—31no—
261C22⋊C4—31no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—21no—
272C42—31no—
273C42C22×D311yescharacteristic
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3C22×D411yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—84no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4H(5,𝔽2)11yescharacteristic
296C3×C4H(5,𝔽2)11yescharacteristic
297Dic3—21no—
298Dic3—21no—
299Dic3—21no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—31no—
305C23—61no—
306C23—61no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—21no—
311Q8—21no—
312Q8—61no—
313Q8—61no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—81no—
324D4—81no—
325D4—122no—
326D4—122no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—122no—
340C2×C4—241no—
341C2×C4D3×D411yescharacteristic
342C2×C4D3×D411yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—21no—
345C8—21no—
346C8—61no—
347C8—61no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3#53511yescharacteristic
351C2×C3#57611yescharacteristic
352C2×C3#57711yescharacteristic
353D3—41no—
354D3—81no—
355D3—81no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—244no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—61no—
378C4—61no—
379C4—61no—
380C4—121no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100911yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—81no—
386C2—81no—
387C2—121no—
388C2—241no—
389C2—241no—
390C2#560711yescharacteristic
391C2#576211yescharacteristic
392C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
hypercenter#460D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #5715C2semidirect
2. #5582C2semidirect
3. C3×#460C2semidirect
4. #5176C2semidirect
5. #5173C2semidirect
6. #5169C2semidirect
7. #5168C2semidirect
8. #5072C2semidirect
9. #5028C2semidirect
10. #4844C2semidirect
11. #4840C2semidirect
12. #4781C2semidirect
13. #4773C2semidirect
14. #4765C2semidirect
15. #4760C2semidirect
20. C2×C3×SD16C22semidirect
22. C3×C4×Q8C22semidirect
23. #947C22semidirect
25. C2×#216C22semidirect
34. C2×#213C22semidirect
36. C2×#204C22semidirect
40. #888C22semidirect
44. C4×D12C22semidirect
46. C3×C4⋊C8C22semidirect
48. C3×#102C22semidirect
50. #832C22semidirect
54. #816C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
68. #460D3semidirect
136. C2×SD16C2×D3semidirect
156. C4×Q8C2×D3semidirect
160. C4⋊C8C2×D3semidirect
168. #102C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#57152
#55822
C3×#4602
#51762
#51732
#51692
#51682
#50722
#50282
#48442
#48402
#47812
#47732
#47652
#47602
#32243