Finite Group Databasebeta

← #79503

Subgroups — #79503

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5550C211yesmaximal, characteristic
2#5543C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5134C211yesmaximal, characteristic, complemented
5#5133C211yesmaximal, characteristic, complemented
6#5116C211yesmaximal, characteristic, complemented
7C2×#821C211yesmaximal, characteristic, complemented
8#4846C211yesmaximal, characteristic, complemented
9#4845C211yesmaximal, characteristic, complemented
10#4736C211yesmaximal, characteristic, complemented
11#4733C211yesmaximal, characteristic, complemented
12#4731C211yesmaximal, characteristic, complemented
13#4722C211yesmaximal, characteristic, complemented
14#4714C211yesmaximal, characteristic, complemented
15#4710C211yesmaximal, characteristic, complemented
16#3314—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic, complemented
20C22×C3×C8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#937—21no—
23D4×Dic3—21no—
24#930C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic, complemented
27#924C2211yescharacteristic
28#924C2211yescharacteristic
29#924C2211yescharacteristic, complemented
30C2×Dic12C2211yescharacteristic, complemented
31C2×#204C2211yescharacteristic
32C2×#204C2211yescharacteristic
33#894—21no—
34#890—21no—
35#884—21no—
36D3×C22⋊C4—21no—
37#878—21no—
38#878—21no—
39#875C2211yescharacteristic
40C4×D12C2211yescharacteristic
41#869C2211yescharacteristic, complemented
42C4×Dic6C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic, complemented
46C3×#97C2211yescharacteristic, complemented
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#818C2211yescharacteristic, complemented
54#818C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#554—31no—
61#546—31no—
62#489—31no—
63#484—31no—
64#480—31no—
65#474—31no—
66#468—31no—
67#467—31no—
68#464—31no—
69#452—31no—
70#450—31no—
71#440—31no—
72#435D311yescharacteristic, complemented
73#417—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×D4D411yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C22×Dic3—21no—
82C22×Dic3—21no—
83#235—22no—
84#235—41no—
85C2×D12—22no—
86C2×D12—21no—
87C2×D12—41no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C2×Dic6C2311yescharacteristic
95C3×M16—41no—
96C2×C3×C8—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4D411yescharacteristic
102C3×C22⋊C4D411yescharacteristic
103C3×C22⋊C4D411yescharacteristic
104C3×C42C2311yescharacteristic
105#217—21no—
106#217—21no—
107#217—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124Dic12—41no—
125#204—22no—
126#204—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16C2×D311yescharacteristic, complemented
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135#125—61no—
136#124—31no—
137#123—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—61no—
145#120—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—61no—
153#107—31no—
154#107—31no—
155#107—31no—
156C4⋊C8C2×D311yescharacteristic, complemented
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8C2×D311yescharacteristic
168C22⋊C8C2×D311yescharacteristic, complemented
169#97C2×D311yescharacteristic, complemented
170C23×C3C2×D411yescharacteristic
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—81no—
177C3×D4—22no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—42no—
188#67—81no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197D12—21no—
198D12—21no—
199D12—21no—
200D12—42no—
201D12#4112yes—
202C4×D3—42no—
203C4×D3—81no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—21no—
208Dic6—81no—
209C3×C8—21no—
210C3×C8—21no—
211C3×C8—21no—
212C3×C8—21no—
213C24—61no—
214#41—62no—
215#41—121no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—61no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—62no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—121no—
226C2×D4D1211yescharacteristic
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233Q16—121no—
234SD16—62no—
235SD16—121no—
236M16—41no—
237C2×C8—41no—
238C2×C8C22×D311yescharacteristic
239C2×C8C22×D311yescharacteristic
240C2×C8C22×D311yescharacteristic
241C2×C8C22×D311yescharacteristic
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4D1211yescharacteristic
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4D1211yescharacteristic
262C22⋊C4D1211yescharacteristic
263C42—61no—
264C42C22×D311yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#13712yes—
271C22×C3C22×D411yescharacteristic
272C2×D3—22no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—84no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4C2×#4111yescharacteristic
284C3×C4H(5,𝔽2)11yescharacteristic
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—61no—
292C23—61no—
293C23—122no—
294C23—122no—
295C23—241no—
296C23C2×D1211yescharacteristic
297C23C2×D1211yescharacteristic
298Q8—61no—
299Q8—61no—
300Q8—61no—
301Q8—61no—
302Q8—241no—
303D4—22no—
304D4—32no—
305D4—61no—
306D4—61no—
307D4—61no—
308D4—122no—
309D4—122no—
310D4—241no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—241no—
323C2×C4—241no—
324C2×C4—241no—
325C2×C4C2×D1211yescharacteristic
326C2×C4C2×D1211yescharacteristic
327C2×C4C2×D1211yescharacteristic
328C2×C4C2×D1211yescharacteristic
329C2×C4C23×D311yescharacteristic
330C8—21no—
331C8—21no—
332C8—21no—
333C8—21no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3#54611yescharacteristic
337C2×C3C2×#13711yescharacteristic
338C2×C3#57611yescharacteristic
339D3—42no—
340D3—81no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—62no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22—244no—
353C22#90912yes—
354C22C22×D1211yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—121no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4#100411yescharacteristic
366C4D3×#4111yescharacteristic
367C3#331411yescharacteristic, complemented, Sylow
368C2—22no—
369C2—41no—
370C2—122no—
371C2—241no—
372C2#555411yescharacteristic
373C2C2×#90911yescharacteristic
374C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
prime radical#5550C2
hypercenter#435D3
hypocenterC3#3314
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #5543C2semidirect
3. C3×#435C2semidirect
4. #5134C2semidirect
5. #5133C2semidirect
6. #5116C2semidirect
7. C2×#821C2semidirect
8. #4846C2semidirect
9. #4845C2semidirect
10. #4736C2semidirect
11. #4733C2semidirect
12. #4731C2semidirect
13. #4722C2semidirect
14. #4714C2semidirect
15. #4710C2semidirect
19. C2×C3×M16C22semidirect
26. C2×C4⋊Dic3C22semidirect
29. #924C22semidirect
30. C2×Dic12C22semidirect
41. #869C22semidirect
42. C4×Dic6C22semidirect
43. C3×C4⋊C8C22semidirect
45. C3×C22⋊C8C22semidirect
46. C3×#97C22semidirect
53. #818C22semidirect
54. #818C22semidirect
55. #818C22semidirect
56. #816C22semidirect
57. #816C22semidirect
58. #816C22semidirect
59. #816C22semidirect
72. #435D3semidirect
132. C2×M16C2×D3semidirect
156. C4⋊C8C2×D3semidirect
168. C22⋊C8C2×D3semidirect
169. #97C2×D3semidirect
367. C3#3314semidirect

Maximal subgroups

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

subgroupindex
#55502
#55432
C3×#4352
#51342
#51332
#51162
C2×#8212
#48462
#48452
#47362
#47332
#47312
#47222
#47142
#47102
#33143