Finite Group Databasebeta

← #87022

Subgroups — #87022

Order 384.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#557C211yesmaximal, characteristic, complemented
2#5733C213yesmaximal, complemented
3#5730C213yesmaximal, complemented
4#5698C213yesmaximal, complemented
5#5695C213yesmaximal, complemented
6#5694C213yesmaximal, complemented
7#5691C213yesmaximal, complemented
8#5584C213yesmaximal, complemented
9#5580C213yesmaximal, complemented
10#5576C211yesmaximal, characteristic, complemented
11#5576C211yesmaximal, characteristic, complemented
12Q8×D12C211yesmaximal, characteristic, complemented
13#5574C211yesmaximal, characteristic, complemented
14#5572C211yesmaximal, characteristic, complemented
15C4×D3×Q8C211yesmaximal, characteristic, complemented
16#3516—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C3×C4⋊Q8C2213yes—
21C3×#125C2213yes—
22C3×#125C2213yes—
23C3×C4×Q8C2211yescharacteristic
24C3×C4×Q8C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×Q8C2213yes—
27#947C2213yes—
28#946C2213yes—
29#946C2216yes—
30Q8×Dic3C2211yescharacteristic
31#898C2216yes—
32#898C2213yes—
33#898C2216yes—
34#898C2213yes—
35#897C2213yes—
36#897C2213yes—
37#897C2213yes—
38#897C2213yes—
39#895C2213yes—
40#895C2213yes—
41#895C2213yes—
42#894C2213yes—
43#894C2216yes—
44#894C2211yescharacteristic
45#893C2216yes—
46#893C2213yes—
47#892C2216yes—
48#892C2213yes—
49D3×C4⋊C4C2213yes—
50D3×C4⋊C4C2213yes—
51#890C2216yes—
52#890C2213yes—
53#887C2213yes—
54#887C2213yes—
55#875C2213yes—
56#875C2213yes—
57C4×D12C2213yes—
58C4×D12C2213yes—
59C4×D12C2213yes—
60C4×D12C2213yes—
61#872C2213yes—
62#872C2213yes—
63C42×D3C2213yes—
64#869C2213yes—
65C4×Dic6C2213yes—
66C4×Dic6C2213yes—
67#557D311yescharacteristic, complemented
68#556—33no—
69#554—33no—
70#553—33no—
71#552—33no—
72D4×Q8—31no—
73#548—31no—
74#548—31no—
75#542—33no—
76#541—33no—
77#529—33no—
78#529—33no—
79#520—31no—
80#519—31no—
81C2×C4×Q8—31no—
82C2×C3×Q8C2313yes—
83C2×C3×Q8C2311yescharacteristic
84#239—24no—
85D3×Q8—24no—
86D3×Q8—24no—
87C2×D12C2311yescharacteristic
88C2×D12C2313yes—
89C2×D12C2313yes—
90C2×D12C2313yes—
91C2×C4×D3C2311yescharacteristic
92C2×C4×D3C2313yes—
93C2×C4×D3C2313yes—
94C2×C4×D3C2313yes—
95C2×C4×D3C2313yes—
96C2×C4×D3C2316yes—
97C2×Dic6C2313yes—
98C2×Dic6C2313yes—
99C3×C4⋊C4C2313yes—
100C3×C4⋊C4C2313yes—
101C3×C4⋊C4C2316yes—
102C3×C4⋊C4C2313yes—
103C3×C4⋊C4C2313yes—
104C3×C4⋊C4C2313yes—
105C3×C4⋊C4C2311yescharacteristic
106C3×C42C2313yes—
107C3×C42C2313yes—
108C3×C42C2313yes—
109(C2×D3)⋊C4C2311yescharacteristic
110(C2×D3)⋊C4C2313yes—
111(C2×D3)⋊C4C2313yes—
112(C2×D3)⋊C4C2313yes—
113(C2×D3)⋊C4C2311yescharacteristic
114(C2×D3)⋊C4C2311yescharacteristic
115(C2×D3)⋊C4C2313yes—
116(C2×D3)⋊C4C2313yes—
117(C2×D3)⋊C4C2313yes—
118(C2×D3)⋊C4C2313yes—
119(C2×D3)⋊C4C2316yes—
120(C2×D3)⋊C4C2316yes—
121C4⋊Dic3C2313yes—
122C4⋊Dic3C2313yes—
123C4⋊Dic3C2316yes—
124C4⋊Dic3C2313yes—
125C4⋊Dic3C2313yes—
126Dic3⋊C4C2313yes—
127Dic3⋊C4C2316yes—
128Dic3⋊C4C2311yescharacteristic
129Dic3⋊C4C2313yes—
130Dic3⋊C4C2313yes—
131C4×Dic3C2313yes—
132C4×Dic3C2316yes—
133C4×Dic3C2313yes—
134C4×Dic3C2313yes—
135C2×#41—31no—
136C22×Q8—31no—
137C22×Q8—31no—
138C4⋊Q8—33no—
139C4⋊Q8C2×D313yes—
140#126—36no—
141#126—33no—
142#126—33no—
143#126—36no—
144#125—33no—
145#125—36no—
146#125C2×D313yes—
147#125C2×D313yes—
148#124—33no—
149#124—33no—
150#124—33no—
151#123—31no—
152#123—33no—
153#123—36no—
154C22⋊Q8—33no—
155C22⋊Q8—36no—
156C22⋊Q8—33no—
157C22⋊Q8—33no—
158C22⋊Q8—33no—
159C22⋊Q8—33no—
160#121—33no—
161#121—33no—
162#121—33no—
163C4×Q8—31no—
164C4×Q8—33no—
165C4×Q8—33no—
166C4×Q8—33no—
167C4×Q8—33no—
168C4×Q8C2×D311yescharacteristic
169C4×Q8C2×D311yescharacteristic
170C4×Q8C2×D311yescharacteristic
171C4×Q8C2×D313yes—
172C4×D4—33no—
173C4×D4—36no—
174C4×D4—33no—
175C4×D4—33no—
176C4×D4—33no—
177C4×D4—33no—
178#117—33no—
179#117—36no—
180#117—33no—
181#117—33no—
182C2×C4⋊C4—33no—
183C2×C4⋊C4—33no—
184C2×C42—33no—
185C22×D3C2411yescharacteristic
186C22×D3C2413yes—
187C22×D3C2411yescharacteristic
188C3×Q8—43no—
189C3×Q8#4114yes—
190C2×C3×C4C2411yescharacteristic
191C2×C3×C4C2413yes—
192C2×C3×C4C2413yes—
193C2×C3×C4C2411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195C2×C3×C4C2413yes—
196C2×C3×C4C2413yes—
197C2×Dic3C2411yescharacteristic
198C2×Dic3C2413yes—
199C2×Dic3C2413yes—
200C2×Dic3C2413yes—
201C2×Dic3C2411yescharacteristic
202D12—22no—
203D12—26no—
204D12—26no—
205D12—43no—
206C4×D3—22no—
207C4×D3—26no—
208C4×D3—26no—
209C4×D3—26no—
210C4×D3—46no—
211C4×D3—43no—
212Dic6—26no—
213Dic6—26no—
214#41—64no—
215C2×Q8—33no—
216C2×Q8—33no—
217C2×Q8—64no—
218C2×Q8—64no—
219C2×Q8C22×D313yes—
220C2×Q8C22×D311yescharacteristic
221C2×D4—33no—
222C2×D4—31no—
223C2×D4—33no—
224C2×D4—33no—
225C22×C4—31no—
226C22×C4—33no—
227C22×C4—33no—
228C22×C4—33no—
229C22×C4—33no—
230C22×C4—36no—
231C4⋊C4—33no—
232C4⋊C4—33no—
233C4⋊C4—36no—
234C4⋊C4—31no—
235C4⋊C4—33no—
236C4⋊C4—33no—
237C4⋊C4—36no—
238C4⋊C4—33no—
239C4⋊C4—33no—
240C4⋊C4—33no—
241C4⋊C4C22×D313yes—
242C4⋊C4C22×D313yes—
243C4⋊C4C22×D313yes—
244C4⋊C4C22×D313yes—
245C4⋊C4C22×D316yes—
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D313yes—
248C22⋊C4—33no—
249C22⋊C4—31no—
250C22⋊C4—33no—
251C22⋊C4—33no—
252C22⋊C4—33no—
253C22⋊C4—36no—
254C22⋊C4—33no—
255C22⋊C4—31no—
256C22⋊C4—31no—
257C22⋊C4—36no—
258C22⋊C4—33no—
259C22⋊C4—33no—
260C42—33no—
261C42—33no—
262C42—33no—
263C42—36no—
264C42C22×D313yes—
265C42C22×D313yes—
266C42C22×D313yes—
267C22×C3C2511yescharacteristic
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—23no—
272C2×D3—21no—
273C2×D3—23no—
274C2×D3—23no—
275C2×D3—21no—
276C2×D32−1+412yes—
277C3×C4—21no—
278C3×C4—23no—
279C3×C4—21no—
280C3×C4—23no—
281C3×C4—23no—
282C3×C4—21no—
283C3×C4C2×#4116yes—
284Dic3—23no—
285Dic3—23no—
286Dic3—21no—
287Dic3—21no—
288Dic3—23no—
289C23—31no—
290C23—31no—
291C23—33no—
292Q8—43no—
293Q8—66no—
294Q8—66no—
295Q8#23914yes—
296D4—66no—
297D4—62no—
298D4—66no—
299D4—123no—
300C2×C4—31no—
301C2×C4—33no—
302C2×C4—33no—
303C2×C4—33no—
304C2×C4—31no—
305C2×C4—62no—
306C2×C4—66no—
307C2×C4—66no—
308C2×C4—66no—
309C2×C4—126no—
310C2×C4—123no—
311C2×C4C23×D311yescharacteristic
312C2×C4C23×D313yes—
313C2×C4C23×D313yes—
314C2×C4C23×D311yescharacteristic
315C2×C4C23×D311yescharacteristic
316C2×C4C23×D313yes—
317C2×C4C23×D313yes—
318C2×C3C22×#4111yescharacteristic
319C2×C3C2×2−1+411yescharacteristic
320C2×C3#58511yescharacteristic
321D3—22no—
322D3—41no—
323D3—43no—
324C22—32no—
325C22—61no—
326C22—61no—
327C22—61no—
328C22—63no—
329C22—63no—
330C22—61no—
331C22—61no—
332C22—63no—
333C22C24×D311yescharacteristic
334C4—23no—
335C4—23no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—21no—
340C4—61no—
341C4—63no—
342C4—63no—
343C4—63no—
344C4—61no—
345C4C2×#23916yes—
346C3#351611yescharacteristic, complemented, Sylow
347C2—62no—
348C2—121no—
349C2—123no—
350C2C22×#23911yescharacteristic
351C2#596411yescharacteristic
352C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#557C2
hypercenter#557D3
hypocenterC3#3516
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C3×#557C2semidirect
2. #5733C2semidirect
3. #5730C2semidirect
4. #5698C2semidirect
5. #5695C2semidirect
6. #5694C2semidirect
7. #5691C2semidirect
8. #5584C2semidirect
9. #5580C2semidirect
10. #5576C2semidirect
11. #5576C2semidirect
12. Q8×D12C2semidirect
13. #5574C2semidirect
14. #5572C2semidirect
15. C4×D3×Q8C2semidirect
67. #557D3semidirect
346. C3#3516semidirect

Maximal subgroups

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

subgroupindex
C3×#5572
#57332
#57302
#56982
#56952
#56942
#56912
#55842
#55802
#55762
Q8×D122
#55742
#55722
C4×D3×Q82
#35163