Finite Group Databasebeta

← #81302

Subgroups — #81302

Order 384.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5609C211yesmaximal, characteristic, complemented
2#5507C211yesmaximal, characteristic, complemented
3C3×#480C211yesmaximal, characteristic, complemented
4#5038C211yesmaximal, characteristic, complemented
5C2×#809C211yesmaximal, characteristic, complemented
6#4890C211yesmaximal, characteristic, complemented
7#4883C211yesmaximal, characteristic, complemented
8#4882C211yesmaximal, characteristic, complemented
9#4872C211yesmaximal, characteristic, complemented
10#4786C211yesmaximal, characteristic, complemented
11#4773C211yesmaximal, characteristic, complemented
12D3×#102C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16#3294—31nomaximal, Sylow
17C2×D3×D4—21no—
18C2×D3×D4C2211yescharacteristic
19C2×#235—21no—
20C22×D12C2211yescharacteristic
21C3×#121C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic, complemented
23#940C2211yescharacteristic
24#937—21no—
25C2×#213C2211yescharacteristic, complemented
26#930C2211yescharacteristic, complemented
27#929—21no—
28C4×#67C2211yescharacteristic
29C22×C3⋊C8C2211yescharacteristic, complemented
30C2×D24C2211yescharacteristic
31C2×D24C2211yescharacteristic, complemented
32C2×D3×C8C2211yescharacteristic, complemented
33#896—21no—
34#895C2211yescharacteristic
35#895C2211yescharacteristic, complemented
36#894—21no—
37#893C2211yescharacteristic
38#893C2211yescharacteristic, complemented
39D3×C4⋊C4C2211yescharacteristic, complemented
40#888C2211yescharacteristic, complemented
41#884—21no—
42#882—21no—
43C3×#107C2211yescharacteristic, complemented
44C3×#107C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47C3×C22⋊C8C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52Dic3⋊C8C2211yescharacteristic, complemented
53C8×Dic3C2211yescharacteristic, complemented
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#547—31no—
61D42—31no—
62#486—31no—
63#480—31no—
64#480D311yescharacteristic, complemented
65#474—31no—
66#466—31no—
67#466—31no—
68#459—31no—
69#447—31no—
70#447—31no—
71C4×D8—31no—
72C8×D4—31no—
73C2×#102—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67D411yescharacteristic
86D3×D4—22no—
87D3×D4—41no—
88D3×D4—44no—
89#235—44no—
90C2×D12—22no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—41no—
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4—21no—
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110#217D411yescharacteristic
111#213—41no—
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)⋊C4D411yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D24—22no—
127D24—41no—
128D3×C8—41no—
129C2×#41—61no—
130C22×D4—31no—
131C22×D4—31no—
132C22×D4—61no—
133C2×D8—31no—
134C2×D8—31no—
135C2×D8—31no—
136C22×C8—31no—
137C22×C8—31no—
138C4⋊Q8—31no—
139#127—31no—
140#123—61no—
141#123—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—31no—
145#121—31no—
146#121—61no—
147#121C2×D311yescharacteristic
148#120—61no—
149#120—61no—
150C4×D4—31no—
151C4×D4—31no—
152C4×D4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4C2×D311yescharacteristic, complemented
155#107—31no—
156#107C2×D311yescharacteristic, complemented
157#107C2×D311yescharacteristic, complemented
158C4⋊C8—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic, complemented
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic, complemented
171C4×C8—31no—
172C23×C3—21no—
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—42no—
181C22×D3—81no—
182C22×D3—81no—
183C22×D3C2×D411yescharacteristic
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—42no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—42no—
196#67—42no—
197#67—42no—
198#67—81no—
199#67D814yescomplemented
200C2×Dic3—21no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—21no—
209D12—42no—
210D12—42no—
211D12—42no—
212D12#4112yescomplemented
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—42no—
216C4×D3—42no—
217C4×D3—42no—
218Dic6—42no—
219C3×C8—21no—
220C3×C8—21no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C24—61no—
224C24—61no—
225#41—124no—
226C2×Q8—61no—
227C2×D4—21no—
228C2×D4—31no—
229C2×D4—31no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—62no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—62no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—124no—
244C2×D4C22×D311yescharacteristic
245C22×C4—21no—
246C22×C4—31no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C22×D311yescharacteristic
251D8—62no—
252D8—121no—
253D8—121no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—121no—
257C2×C8—121no—
258C2×C8C22×D311yescharacteristic
259C2×C8C22×D311yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—31no—
262C4⋊C4—31no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4C22×D311yescharacteristic
266C4⋊C4C22×D311yescharacteristic
267C4⋊C4C22×D311yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—31no—
270C22⋊C4—31no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C42—31no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3C2×D812yes—
283C22×C3C22×D411yescharacteristic
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—22no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—84no—
296C2×D3—84no—
297C2×D3C2×D812yes—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4C2×#4111yescharacteristic
303C3×C4H(5,𝔽2)11yescharacteristic
304Dic3—21no—
305Dic3—21no—
306Dic3—41no—
307Dic3C2×D812yes—
308C23—21no—
309C23—31no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—122no—
314C23—122no—
315C23—122no—
316C23—122no—
317C23—241no—
318C23—241no—
319C23D3×D411yescharacteristic
320Q8—122no—
321D4—21no—
322D4—21no—
323D4—34no—
324D4—32no—
325D4—42no—
326D4—61no—
327D4—61no—
328D4—61no—
329D4—61no—
330D4—61no—
331D4—122no—
332D4—122no—
333D4—122no—
334D4—122no—
335D4—122no—
336D4—122no—
337D4—241no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—42no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4D3×D411yescharacteristic
354C2×C4C23×D311yescharacteristic
355C8—21no—
356C8—21no—
357C8—61no—
358C8—61no—
359C2×C3—22no—
360C2×C3—81no—
361C2×C3#54611yescharacteristic
362C2×C3C22×D811yescharacteristic
363C2×C3#57711yescharacteristic
364D3—22no—
365D3—42no—
366D3—81no—
367D3—81no—
368C22—21no—
369C22—21no—
370C22—32no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—61no—
375C22—61no—
376C22—62no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—244no—
386C22—244no—
387C22D3×D812yes—
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—32no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—61no—
395C4—61no—
396C4—121no—
397C4D3×#4111yescharacteristic
398C4#100911yescharacteristic
399C3#329411yescharacteristic, complemented, Sylow
400C2—22no—
401C2—62no—
402C2—81no—
403C2—122no—
404C2—241no—
405C2—241no—
406C2#565411yescharacteristic
407C2C2×D3×D811yescharacteristic
408C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#480C2
socleC22×C3C22×D4
hypercenter#480D3
hypocenterC3#3294
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #5609C2semidirect
2. #5507C2semidirect
3. C3×#480C2semidirect
4. #5038C2semidirect
5. C2×#809C2semidirect
6. #4890C2semidirect
7. #4883C2semidirect
8. #4882C2semidirect
9. #4872C2semidirect
10. #4786C2semidirect
11. #4773C2semidirect
12. D3×#102C2semidirect
13. #4760C2semidirect
14. #4732C2semidirect
15. #67⋊C8C2semidirect
22. C2×C3×C4⋊C4C22semidirect
25. C2×#213C22semidirect
26. #930C22semidirect
29. C22×C3⋊C8C22semidirect
31. C2×D24C22semidirect
32. C2×D3×C8C22semidirect
35. #895C22semidirect
38. #893C22semidirect
39. D3×C4⋊C4C22semidirect
40. #888C22semidirect
43. C3×#107C22semidirect
44. C3×#107C22semidirect
45. C3×#102C22semidirect
46. C3×#102C22semidirect
47. C3×C22⋊C8C22semidirect
48. #832C22semidirect
50. #821C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. Dic3⋊C8C22semidirect
53. C8×Dic3C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #809C22semidirect
59. #807C22semidirect
64. #480D3semidirect
154. C2×C4⋊C4C2×D3semidirect
156. #107C2×D3semidirect
157. #107C2×D3semidirect
167. #102C2×D3semidirect
168. #102C2×D3semidirect
170. C22⋊C8C2×D3semidirect
199. #67D8semidirect
212. D12#41semidirect
399. C3#3294semidirect

Maximal subgroups

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

subgroupindex
#56092
#55072
C3×#4802
#50382
C2×#8092
#48902
#48832
#48822
#48722
#47862
#47732
D3×#1022
#47602
#47322
#67⋊C82
#32943