Finite Group Databasebeta

← #80610

Subgroups — #80610

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#5674C211yesmaximal, characteristic, complemented
2#5582C211yesmaximal, characteristic, complemented
3C3×#464C211yesmaximal, characteristic, complemented
4#5191C211yesmaximal, characteristic, complemented
5#5187C211yesmaximal, characteristic, complemented
6#5176C211yesmaximal, characteristic, complemented
7#5169C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5027C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4773C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic, complemented
20C2×C3×SD16C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24#947C2211yescharacteristic, complemented
25#946C2211yescharacteristic
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29#940C2211yescharacteristic
30C2×#213C2211yescharacteristic
31C2×D24C2211yescharacteristic
32C2×D24C2211yescharacteristic, complemented
33#896—21no—
34#895C2211yescharacteristic
35#894—21no—
36#888C2211yescharacteristic, complemented
37#884—21no—
38#884—21no—
39#882—21no—
40#882—21no—
41#876—21no—
42#875—21no—
43#874C2211yescharacteristic
44C4×D12C2211yescharacteristic, complemented
45#872C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic
54(C2×D3)⋊C8C2211yescharacteristic
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#809C2211yescharacteristic, complemented
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#542—31no—
61#535—31no—
62#496—31no—
63#495—31no—
64#489—31no—
65#486—31no—
66#464D311yescharacteristic, complemented
67#463—31no—
68#460—31no—
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×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82#239—41no—
83#239—41no—
84D3×D4—41no—
85D3×D4—41no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12C2311yescharacteristic
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×Dic6—21no—
97C3×Q16—41no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C42—21no—
106C3×C42C2311yescharacteristic
107#215—41no—
108#215—41no—
109#215—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—
129D24—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×Q16C2×D311yescharacteristic, complemented
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic
137C2×D8—31no—
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8—31no—
141#127—31no—
142#127—31no—
143#126—61no—
144#124—31no—
145#124—31no—
146#124—61no—
147#124C2×D311yescharacteristic
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#103C2×D311yescharacteristic, complemented
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C22×D3—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×Q8—21no—
184C3×Q8—21no—
185C3×D4—21no—
186C3×D4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—81no—
195#67—81no—
196C2×Dic3—21no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—21no—
206D12—42no—
207D12—81no—
208D12—81no—
209D12—81no—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—42no—
213C4×D3—81no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—121no—
221#41—121no—
222C2×Q8—61no—
223C2×Q8C22×D311yescharacteristic
224C2×Q8C22×D311yescharacteristic
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4C22×D311yescharacteristic
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241Q16—41no—
242SD16—41no—
243SD16—121no—
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—21no—
261C22⋊C4—21no—
262C22⋊C4—31no—
263C22⋊C4—31no—
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×C3C22×D411yescharacteristic
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
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×C4—41no—
296C3×C4H(5,𝔽2)11yescharacteristic
297C3×C4H(5,𝔽2)11yescharacteristic
298Dic3—21no—
299Dic3—21no—
300Dic3—21no—
301Dic3—41no—
302C23—21no—
303C23—31no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—61no—
318D4—61no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—122no—
324D4—122no—
325D4—241no—
326D4—241no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
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#53511yescharacteristic
350C2×C3#57611yescharacteristic
351C2×C3#57711yescharacteristic
352D3—41no—
353D3—81no—
354D3—81no—
355D3—81no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—61no—
360C22—61no—
361C22—61no—
362C22—121no—
363C22—121no—
364C22—121no—
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—41no—
378C4—61no—
379C4—61no—
380C4—61no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100911yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—81no—
386C2—121no—
387C2—241no—
388C2—241no—
389C2—241no—
390C2#560711yescharacteristic
391C2#576911yescharacteristic
392C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
hypercenter#464D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #5674C2semidirect
2. #5582C2semidirect
3. C3×#464C2semidirect
4. #5191C2semidirect
5. #5187C2semidirect
6. #5176C2semidirect
7. #5169C2semidirect
8. #5057C2semidirect
9. #5027C2semidirect
10. #4843C2semidirect
11. #4840C2semidirect
12. #4810C2semidirect
13. #4794C2semidirect
14. #4773C2semidirect
15. #4760C2semidirect
19. C2×C3×Q16C22semidirect
22. C3×C4×Q8C22semidirect
24. #947C22semidirect
27. C2×#215C22semidirect
28. C2×#215C22semidirect
32. C2×D24C22semidirect
36. #888C22semidirect
44. C4×D12C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
49. #835C22semidirect
52. #821C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
58. #809C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
66. #464D3semidirect
132. C2×Q16C2×D3semidirect
156. C4×Q8C2×D3semidirect
160. C4⋊C8C2×D3semidirect
162. #103C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#56742
#55822
C3×#4642
#51912
#51872
#51762
#51692
#50572
#50272
#48432
#48402
#48102
#47942
#47732
#47602
#32243