Finite Group Databasebeta

← #80776

Subgroups — #80776

Order 384.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5641C211yesmaximal, characteristic, complemented
2#5631C211yesmaximal, characteristic
3C3×#467C211yesmaximal, characteristic, complemented
4#5187C211yesmaximal, characteristic, complemented
5#5182C211yesmaximal, characteristic, complemented
6#5113C211yesmaximal, characteristic, complemented
7C8×#67C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5045C211yesmaximal, characteristic, complemented
10#4890C211yesmaximal, characteristic, complemented
11#4888C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4804C211yesmaximal, characteristic, complemented
14#4792C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3302—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#215C2211yescharacteristic
28C4×#67C2211yescharacteristic
29#926C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#898—21no—
32#897C2211yescharacteristic
33#894—21no—
34#893C2211yescharacteristic
35#892C2211yescharacteristic
36D3×C4⋊C4—21no—
37#888C2211yescharacteristic
38#887—21no—
39#885—21no—
40#884—21no—
41#883—21no—
42#879—21no—
43C3×#107C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#830C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#556—31no—
61#547—31no—
62#487—31no—
63#485—31no—
64#485—31no—
65#474—31no—
66#467—31no—
67#467D311yescharacteristic, complemented
68#462—31no—
69#452—31no—
70#452—31no—
71C4×SD16—31no—
72C8×D4—31no—
73#416—31no—
74#416—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67D411yescharacteristic
80#239—22no—
81#239—41no—
82#235—22no—
83#235—41no—
84C2×D12—21no—
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C3×Q16—41no—
93C2×C3×C8—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102#217—21no—
103#217D411yescharacteristic
104#216—41no—
105#215—22no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4D411yescharacteristic
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41—31no—
128C2×#41—31no—
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×SD16—31no—
132C22×C8—31no—
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136#126—61no—
137#126—61no—
138#124—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8C2×D311yescharacteristic
144C22⋊Q8C2×D311yescharacteristic
145#121—61no—
146C4×Q8—31no—
147C4×Q8—61no—
148C4×D4—31no—
149C4×D4—31no—
150#117—31no—
151#117—31no—
152C2×C4⋊C4—61no—
153#107—31no—
154#107C2×D311yescharacteristic
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—22no—
186#67—42no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193D12—21no—
194D12—42no—
195D12#4112yes—
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—42no—
199C4×D3—42no—
200C4×D3—42no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—62no—
209#41—62no—
210#41—121no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8C22×D311yescharacteristic
215C2×Q8C22×D311yescharacteristic
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4—61no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—41no—
226Q16—121no—
227SD16—62no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8—41no—
231C2×C8—121no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—31no—
250C22⋊C4—31no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C42—31no—
257C42—61no—
258C42—61no—
259C42—61no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3C22×D411yescharacteristic
264C2×D3—21no—
265C2×D3—22no—
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—41no—
269C2×D3—41no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4C2×#4111yescharacteristic
276C3×C4H(5,𝔽2)11yescharacteristic
277Dic3—21no—
278Dic3—21no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3#13512yes—
283C23—31no—
284C23—61no—
285C23D3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—61no—
291Q8—61no—
292Q8—122no—
293D4—32no—
294D4—62no—
295D4—61no—
296D4—122no—
297D4—122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3#54611yescharacteristic
323C2×C3C2×#13511yescharacteristic
324C2×C3#57811yescharacteristic
325D3—41no—
326D3—42no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—61no—
331C22—61no—
332C22—61no—
333C22—62no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—32no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—61no—
344C4—61no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4#100411yescharacteristic
349C4D3×#4111yescharacteristic
350C3#330211yescharacteristic, complemented, Sylow
351C2—41no—
352C2—121no—
353C2—122no—
354C2#561011yescharacteristic
355C2#576611yescharacteristic
356C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#467C2
socleC22×C3C22×D4
prime radical#5631C2
hypercenter#467D3
hypocenterC3#3302
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #5641C2semidirect
3. C3×#467C2semidirect
4. #5187C2semidirect
5. #5182C2semidirect
6. #5113C2semidirect
7. C8×#67C2semidirect
8. #5047C2semidirect
9. #5045C2semidirect
10. #4890C2semidirect
11. #4888C2semidirect
12. #4813C2semidirect
13. #4804C2semidirect
14. #4792C2semidirect
15. #4788C2semidirect
67. #467D3semidirect
350. C3#3302semidirect

Maximal subgroups

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

subgroupindex
#56412
#56312
C3×#4672
#51872
#51822
#51132
C8×#672
#50472
#50452
#48902
#48882
#48132
#48042
#47922
#47882
#33023