Finite Group Databasebeta

← #86772

Subgroups — #86772

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×#550C211yesmaximal, characteristic, complemented
2#5834C211yesmaximal, characteristic, complemented
3#5834C211yesmaximal, characteristic, complemented
4#5827C211yesmaximal, characteristic, complemented
5Dic3×#41C211yesmaximal, characteristic, complemented
6#5719C213yesmaximal, complemented
7#5718C213yesmaximal, complemented
8#5619C213yesmaximal, complemented
9#5614C213yesmaximal, complemented
10#5611C213yesmaximal, complemented
11#5596C213yesmaximal, complemented
12#5574C211yesmaximal, characteristic, complemented
13#5569C211yesmaximal, characteristic, complemented
14#5565C213yesmaximal, complemented
15#5538C213yesmaximal, complemented
16#3517—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C2×#239C2211yescharacteristic
20C2×#237C2216yes—
21C3×#127C2213yes—
22C3×#121C2213yes—
23C3×#121C2213yes—
24C3×C4×Q8C2211yescharacteristic
25C3×C4×D4C2213yes—
26Q8×Dic3C2211yescharacteristic
27Q8×Dic3C2211yescharacteristic
28Q8×Dic3C2211yescharacteristic
29#940C2213yes—
30#940C2213yes—
31#939C2213yes—
32#939C2216yes—
33#939C2216yes—
34#938C2216yes—
35#938C2213yes—
36#938C2213yes—
37#935C2213yes—
38#935C2216yes—
39D4×Dic3C2213yes—
40D4×Dic3C2213yes—
41D4×Dic3C2213yes—
42D4×Dic3C2213yes—
43#930C2213yes—
44#930C2213yes—
45C4×#67C2213yes—
46C4×#67C2213yes—
47#926C2213yes—
48#926C2213yes—
49C2×C4⋊Dic3C2213yes—
50C2×C4×Dic3C2213yes—
51#898C2213yes—
52#893C2213yes—
53D3×C4⋊C4C2213yes—
54#890C2213yes—
55#889C2211yescharacteristic
56#888C2213yes—
57#886C2213yes—
58#885C2213yes—
59#883C2216yes—
60#67⋊C4C2216yes—
61#879C2213yes—
62#878C2213yes—
63C4×D12C2213yes—
64#872C2213yes—
65#870C2213yes—
66C4×Dic6C2213yes—
67#557—31no—
68#553—33no—
69#553—33no—
70#550—31no—
71#550—31no—
72#550—33no—
73#550D311yescharacteristic, complemented
74#547—33no—
75#540—33no—
76#540—33no—
77#532—33no—
78#529—33no—
79#520—31no—
80#520—31no—
81C4×#41—31no—
82C3×#41—24no—
83C3×#41—24no—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4C2313yes—
86C2×C3×D4C2313yes—
87C2×C3×D4C2316yes—
88C2×C3×D4C2313yes—
89C22×C3×C4C2313yes—
90C22×C3×C4C2313yes—
91C2×#67C2313yes—
92C2×#67C2316yes—
93C2×#67C2316yes—
94C2×#67C2313yes—
95C22×Dic3C2313yes—
96C22×Dic3C2316yes—
97C22×Dic3C2313yes—
98#239—24no—
99#237—412no—
100C2×D12C2313yes—
101C2×C4×D3C2313yes—
102C2×C4×D3C2316yes—
103C2×Dic6C2313yes—
104C3×C4⋊C4C2313yes—
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4C2313yes—
107C3×C22⋊C4C2313yes—
108C3×C42C2313yes—
109#217C2313yes—
110#217C2313yes—
111#217C2313yes—
112#217C2316yes—
113#217C2313yes—
114(C2×D3)⋊C4C2313yes—
115(C2×D3)⋊C4C2313yes—
116(C2×D3)⋊C4C2313yes—
117(C2×D3)⋊C4C2313yes—
118C4⋊Dic3C2313yes—
119C4⋊Dic3C2313yes—
120C4⋊Dic3C2313yes—
121C4⋊Dic3C2313yes—
122C4⋊Dic3C2313yes—
123C4⋊Dic3C2311yescharacteristic
124Dic3⋊C4C2311yescharacteristic
125Dic3⋊C4C2313yes—
126Dic3⋊C4C2316yes—
127Dic3⋊C4C2311yescharacteristic
128Dic3⋊C4C2313yes—
129Dic3⋊C4C2311yescharacteristic
130Dic3⋊C4C2311yescharacteristic
131C4×Dic3C2313yes—
132C4×Dic3C2313yes—
133C4×Dic3C2313yes—
134C4×Dic3C2313yes—
135C4×Dic3C2311yescharacteristic
136C2×#41—31no—
137C2×#41—36no—
138C2×#41C2×D311yescharacteristic
139C2×#41C2×D311yescharacteristic
140C4⋊Q8—33no—
141#127—33no—
142#127—33no—
143#127C2×D313yes—
144#126—33no—
145#126—33no—
146#125—31no—
147#125—33no—
148#125—33no—
149#124—33no—
150#123—33no—
151#123—36no—
152#123—33no—
153#123—36no—
154C22⋊Q8—33no—
155#121—33no—
156#121—36no—
157#121—33no—
158#121—36no—
159#121—33no—
160#121—36no—
161#121—33no—
162#121—33no—
163#121C2×D313yes—
164#121C2×D313yes—
165C4×Q8—31no—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×Q8—33no—
169C4×Q8C2×D311yescharacteristic
170C4×D4—33no—
171C4×D4—33no—
172C4×D4—33no—
173C4×D4—33no—
174C4×D4—33no—
175C4×D4—33no—
176C4×D4—33no—
177C4×D4—36no—
178C4×D4—33no—
179C4×D4C2×D313yes—
180#117—33no—
181#117—33no—
182#117—33no—
183C2×C4⋊C4—33no—
184C2×C4⋊C4—33no—
185C2×C42—33no—
186C23×C3C2413yes—
187C23×C3C2413yes—
188C22×D3C2413yes—
189C3×Q8#4114yescomplemented
190C3×D4—26no—
191C3×D4—26no—
192C3×D4—46no—
193C3×D4—43no—
194C2×C3×C4—26no—
195C2×C3×C4—26no—
196C2×C3×C4C2411yescharacteristic
197C2×C3×C4C2411yescharacteristic
198C2×C3×C4C2413yes—
199C2×C3×C4C2411yescharacteristic
200C2×C3×C4C2413yes—
201#67—26no—
202#67—46no—
203#67—46no—
204#67—43no—
205C2×Dic3—26no—
206C2×Dic3—46no—
207C2×Dic3—43no—
208C2×Dic3C2411yescharacteristic
209C2×Dic3C2411yescharacteristic
210C2×Dic3C2413yes—
211C2×Dic3C2413yes—
212C2×Dic3C2413yes—
213C2×Dic3C2411yescharacteristic
214C2×Dic3C2411yescharacteristic
215D12—26no—
216C4×D3—26no—
217C4×D3—46no—
218Dic6—26no—
219#41—24no—
220#41—24no—
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×D4C22×D313yes—
231C2×D4C22×D313yes—
232C2×D4C22×D313yes—
233C2×D4C22×D316yes—
234C22×C4—33no—
235C22×C4—33no—
236C22×C4—36no—
237C22×C4—33no—
238C22×C4—36no—
239C22×C4C22×D313yes—
240C22×C4C22×D313yes—
241C4⋊C4—33no—
242C4⋊C4—33no—
243C4⋊C4—33no—
244C4⋊C4—33no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—33no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—36no—
251C4⋊C4—31no—
252C4⋊C4—33no—
253C4⋊C4—33no—
254C4⋊C4C22×D313yes—
255C4⋊C4C22×D311yescharacteristic
256C22⋊C4—33no—
257C22⋊C4—36no—
258C22⋊C4—33no—
259C22⋊C4—33no—
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—
271C42—33no—
272C42C22×D313yes—
273C22×C3—23no—
274C22×C3—23no—
275C22×C3—23no—
276C22×C3—23no—
277C22×C3—23no—
278C22×C3—23no—
279C22×C3C2511yescharacteristic
280C2×D3—23no—
281C2×D3—23no—
282C2×D3—23no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—23no—
286C3×C4—21no—
287C3×C4C2×#4116yes—
288Dic3—21no—
289Dic3—23no—
290Dic3—23no—
291Dic3—21no—
292Dic3—23no—
293Dic3—21no—
294Dic3H(5,𝔽2)12yes—
295C23—33no—
296C23C23×D313yes—
297C23C23×D313yes—
298Q8—66no—
299Q8#23714yescomplemented
300D4—26no—
301D4—26no—
302D4—46no—
303D4—43no—
304D4—66no—
305D4—66no—
306D4—126no—
307D4—126no—
308D4—123no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—33no—
315C2×C4—33no—
316C2×C4—33no—
317C2×C4—31no—
318C2×C4—66no—
319C2×C4—66no—
320C2×C4—126no—
321C2×C4—126no—
322C2×C4—123no—
323C2×C4C23×D311yescharacteristic
324C2×C4C23×D311yescharacteristic
325C2×C4C23×D313yes—
326C2×C4C23×D311yescharacteristic
327C2×C4C23×D313yes—
328C2×C3—43no—
329C2×C3—43no—
330C2×C3C22×#4111yescharacteristic
331C2×C3C2×H(5,𝔽2)11yescharacteristic
332C2×C3#58511yescharacteristic
333D3—43no—
334C22—23no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—63no—
341C22—63no—
342C22—63no—
343C22C24×D311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—23no—
348C4—32no—
349C4—61no—
350C4—61no—
351C4—63no—
352C4—61no—
353C4—63no—
354C4—63no—
355C4C2×#23716yes—
356C3#351711yescharacteristic, complemented, Sylow
357C2—43no—
358C2—43no—
359C2—123no—
360C2C22×#23711yescharacteristic
361C2#596411yescharacteristic
362C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#550C2
hypercenter#550D3
hypocenterC3#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5834C2semidirect
3. #5834C2semidirect
4. #5827C2semidirect
5. Dic3×#41C2semidirect
6. #5719C2semidirect
7. #5718C2semidirect
8. #5619C2semidirect
9. #5614C2semidirect
10. #5611C2semidirect
11. #5596C2semidirect
12. #5574C2semidirect
13. #5569C2semidirect
14. #5565C2semidirect
15. #5538C2semidirect
73. #550D3semidirect
189. C3×Q8#41semidirect
299. Q8#237semidirect
356. C3#3517semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#58342
#58272
Dic3×#412
#57192
#57182
#56192
#56142
#56112
#55962
#55742
#55692
#55652
#55382
#35173