Finite Group Databasebeta

← #85896

Subgroups — #85896

Order 384.

1122 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#520C211yesmaximal, characteristic, complemented
2#5834C211yesmaximal, characteristic, complemented
3#5834C211yesmaximal, characteristic, complemented
4#5577C211yesmaximal, characteristic, complemented
5#5574C211yesmaximal, characteristic, complemented
6C4×#239C211yesmaximal, characteristic, complemented
7#5569C211yesmaximal, characteristic, complemented
8#5562C213yesmaximal, complemented
9#5558C213yesmaximal, complemented
10#5553C213yesmaximal, complemented
11#5548C213yesmaximal, complemented
12#5535C213yesmaximal, complemented
13#5528C213yesmaximal, complemented
14#5514C213yesmaximal, complemented
15#5507C213yesmaximal, complemented
16#3517—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#239C2211yescharacteristic
20C2×#235C2216yes—
21C3×C4×Q8C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic
23C3×C4×D4C2213yes—
24C3×C4×D4C2213yes—
25C3×#117C2213yes—
26C2×C3×C4⋊C4C2213yes—
27Q8×Dic3C2211yescharacteristic
28Q8×Dic3C2211yescharacteristic
29#940C2213yes—
30#940C2213yes—
31#938C2213yes—
32#938C2213yes—
33#930C2213yes—
34#930C2213yes—
35#930C2216yes—
36#929C2216yes—
37C4×#67C2213yes—
38C4×#67C2213yes—
39C4×#67C2216yes—
40#898C2213yes—
41#896C2213yes—
42#895C2213yes—
43#895C2213yes—
44#894C2216yes—
45#894C2213yes—
46#893C2213yes—
47#893C2213yes—
48#892C2213yes—
49D3×C4⋊C4C2213yes—
50#890C2213yes—
51#889C2211yescharacteristic
52#888C2213yes—
53#885C2213yes—
54#884C2213yes—
55#884C2216yes—
56#884C2216yes—
57#883C2213yes—
58#879C2213yes—
59#874C2213yes—
60C4×D12C2213yes—
61C4×D12C2213yes—
62C4×D12C2213yes—
63#872C2213yes—
64C42×D3C2213yes—
65#870C2213yes—
66C4×Dic6C2213yes—
67#557—31no—
68#553—33no—
69#553—33no—
70#550—31no—
71#550—31no—
72#550—31no—
73#550—33no—
74#547—33no—
75#540—33no—
76#540—33no—
77#532—33no—
78#529—33no—
79#520—31no—
80#520D311yescharacteristic, complemented
81C4×#41—31no—
82C3×#41—24no—
83C2×C3×Q8C2311yescharacteristic
84C2×C3×D4C2313yes—
85C22×C3×C4C2313yes—
86C22×C3×C4C2316yes—
87C2×#67C2313yes—
88C2×#67C2313yes—
89C2×#67C2316yes—
90C2×#67C2316yes—
91#239—24no—
92#239—24no—
93#235—412no—
94C2×D12C2313yes—
95C2×D12C2313yes—
96C2×D12C2313yes—
97C2×D12C2316yes—
98C2×C4×D3C2313yes—
99C2×C4×D3C2313yes—
100C2×C4×D3C2313yes—
101C2×C4×D3C2313yes—
102C2×C4×D3C2316yes—
103C2×Dic6C2313yes—
104C3×C4⋊C4C2313yes—
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2313yes—
107C3×C4⋊C4C2313yes—
108C3×C22⋊C4C2313yes—
109C3×C22⋊C4C2313yes—
110C3×C42C2313yes—
111C3×C42C2313yes—
112#217C2313yes—
113#217C2313yes—
114(C2×D3)⋊C4C2313yes—
115(C2×D3)⋊C4C2313yes—
116(C2×D3)⋊C4C2313yes—
117(C2×D3)⋊C4C2316yes—
118(C2×D3)⋊C4C2313yes—
119(C2×D3)⋊C4C2313yes—
120(C2×D3)⋊C4C2313yes—
121C4⋊Dic3C2313yes—
122C4⋊Dic3C2313yes—
123C4⋊Dic3C2313yes—
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4C2311yescharacteristic
126Dic3⋊C4C2313yes—
127Dic3⋊C4C2316yes—
128Dic3⋊C4C2311yescharacteristic
129Dic3⋊C4C2313yes—
130Dic3⋊C4C2311yescharacteristic
131Dic3⋊C4C2311yescharacteristic
132C4×Dic3C2313yes—
133C4×Dic3C2313yes—
134C4×Dic3C2311yescharacteristic
135C4×Dic3C2313yes—
136C2×#41—31no—
137C2×#41—31no—
138C2×#41—36no—
139C2×#41C2×D311yescharacteristic
140C4⋊Q8—33no—
141#127—33no—
142#127—33no—
143#127—33no—
144#126—33no—
145#126—33no—
146#125—31no—
147#125—33no—
148#125—33no—
149#124—33no—
150#123—36no—
151#123—33no—
152#123—36no—
153#123—33no—
154C22⋊Q8—33no—
155#121—33no—
156#121—33no—
157#121—33no—
158#121—36no—
159#121—33no—
160#121—33no—
161#121—33no—
162#121—33no—
163#121—36no—
164#121—36no—
165C4×Q8—31no—
166C4×Q8—31no—
167C4×Q8—33no—
168C4×Q8C2×D311yescharacteristic
169C4×Q8C2×D311yescharacteristic
170C4×D4—33no—
171C4×D4—33no—
172C4×D4—36no—
173C4×D4—33no—
174C4×D4—33no—
175C4×D4—33no—
176C4×D4—33no—
177C4×D4—33no—
178C4×D4C2×D313yes—
179C4×D4C2×D313yes—
180#117—33no—
181#117—33no—
182#117C2×D313yes—
183C2×C4⋊C4—33no—
184C2×C4⋊C4C2×D313yes—
185C2×C42—33no—
186C23×C3C2413yes—
187C22×D3C2413yes—
188C22×D3C2413yes—
189C3×Q8#4114yescomplemented
190C3×D4—26no—
191C2×C3×C4—26no—
192C2×C3×C4—46no—
193C2×C3×C4C2411yescharacteristic
194C2×C3×C4C2413yes—
195C2×C3×C4C2411yescharacteristic
196C2×C3×C4C2413yes—
197C2×C3×C4C2411yescharacteristic
198C2×C3×C4C2413yes—
199#67—26no—
200#67—46no—
201#67—46no—
202#67—43no—
203C2×Dic3C2411yescharacteristic
204C2×Dic3C2411yescharacteristic
205C2×Dic3C2413yes—
206C2×Dic3C2413yes—
207C2×Dic3C2411yescharacteristic
208C2×Dic3C2411yescharacteristic
209D12—26no—
210D12—26no—
211D12—43no—
212D12—46no—
213C4×D3—26no—
214C4×D3—26no—
215C4×D3—26no—
216C4×D3—43no—
217C4×D3—46no—
218Dic6—26no—
219#41—24no—
220#41—64no—
221#41—64no—
222#41—1212no—
223C2×Q8—33no—
224C2×Q8C22×D311yescharacteristic
225C2×D4—33no—
226C2×D4—36no—
227C2×D4—33no—
228C2×D4—36no—
229C2×D4—33no—
230C2×D4—33no—
231C2×D4—33no—
232C2×D4—36no—
233C2×D4C22×D313yes—
234C22×C4—33no—
235C22×C4—33no—
236C22×C4—33no—
237C22×C4—33no—
238C22×C4—36no—
239C22×C4C22×D313yes—
240C22×C4C22×D316yes—
241C4⋊C4—33no—
242C4⋊C4—33no—
243C4⋊C4—31no—
244C4⋊C4—33no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—36no—
249C4⋊C4—31no—
250C4⋊C4—33no—
251C4⋊C4—33no—
252C4⋊C4C22×D313yes—
253C4⋊C4C22×D313yes—
254C4⋊C4C22×D313yes—
255C4⋊C4C22×D311yescharacteristic
256C22⋊C4—33no—
257C22⋊C4—33no—
258C22⋊C4—33no—
259C22⋊C4—36no—
260C22⋊C4—33no—
261C22⋊C4—33no—
262C22⋊C4—33no—
263C22⋊C4—33no—
264C22⋊C4—33no—
265C22⋊C4C22×D313yes—
266C22⋊C4C22×D313yes—
267C42—31no—
268C42—33no—
269C42—33no—
270C42—33no—
271C42C22×D313yes—
272C42C22×D313yes—
273C22×C3—23no—
274C22×C3—23no—
275C22×C3—23no—
276C22×C3C2511yescharacteristic
277C2×D3—23no—
278C2×D3—23no—
279C2×D3—23no—
280C2×D3—23no—
281C2×D3—23no—
282C2×D3—23no—
283C3×C4—21no—
284C3×C4—23no—
285C3×C4—23no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4C2×#4116yes—
289Dic3—21no—
290Dic3—21no—
291Dic3—23no—
292Dic3—23no—
293Dic3—21no—
294Dic3H(5,𝔽2)12yes—
295C23—33no—
296C23—33no—
297C23C23×D313yes—
298Q8—66no—
299Q8#23514yescomplemented
300D4—26no—
301D4—66no—
302D4—66no—
303D4—66no—
304D4—126no—
305D4—123no—
306D4—126no—
307D4—126no—
308D4—123no—
309C2×C4—26no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—33no—
314C2×C4—33no—
315C2×C4—31no—
316C2×C4—46no—
317C2×C4—66no—
318C2×C4—66no—
319C2×C4—66no—
320C2×C4—126no—
321C2×C4—123no—
322C2×C4C23×D311yescharacteristic
323C2×C4C23×D313yes—
324C2×C4C23×D311yescharacteristic
325C2×C4C23×D313yes—
326C2×C4C23×D311yescharacteristic
327C2×C4C23×D313yes—
328C2×C3—43no—
329C2×C3C22×#4111yescharacteristic
330C2×C3C2×H(5,𝔽2)11yescharacteristic
331C2×C3#58511yescharacteristic
332D3—43no—
333D3—43no—
334C22—23no—
335C22—23no—
336C22—23no—
337C22—63no—
338C22—63no—
339C22—63no—
340C22—63no—
341C22—63no—
342C22—63no—
343C22C24×D311yescharacteristic
344C4—21no—
345C4—23no—
346C4—23no—
347C4—21no—
348C4—21no—
349C4—32no—
350C4—61no—
351C4—61no—
352C4—63no—
353C4—63no—
354C4—61no—
355C4C2×#23516yes—
356C3#351711yescharacteristic, complemented, Sylow
357C2—43no—
358C2—123no—
359C2—123no—
360C2C22×#23511yescharacteristic
361C2D3×H(5,𝔽2)11yescharacteristic
362C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#520C2
hypercenter#520D3
hypocenterC3#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C3×#520C2semidirect
2. #5834C2semidirect
3. #5834C2semidirect
4. #5577C2semidirect
5. #5574C2semidirect
6. C4×#239C2semidirect
7. #5569C2semidirect
8. #5562C2semidirect
9. #5558C2semidirect
10. #5553C2semidirect
11. #5548C2semidirect
12. #5535C2semidirect
13. #5528C2semidirect
14. #5514C2semidirect
15. #5507C2semidirect
80. #520D3semidirect
189. C3×Q8#41semidirect
299. Q8#235semidirect
356. C3#3517semidirect

Maximal subgroups

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

subgroupindex
C3×#5202
#58342
#55772
#55742
C4×#2392
#55692
#55622
#55582
#55532
#55482
#55352
#55282
#55142
#55072
#35173