Finite Group Databasebeta

← #87026

Subgroups — #87026

Order 384.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#557C211yesmaximal, characteristic, complemented
2#5734C213yesmaximal, complemented
3#5733C213yesmaximal, complemented
4#5697C213yesmaximal, complemented
5#5695C213yesmaximal, complemented
6#5694C213yesmaximal, complemented
7#5691C213yesmaximal, complemented
8#5585C213yesmaximal, complemented
9#5580C213yesmaximal, complemented
10#5577C211yesmaximal, characteristic, complemented
11#5576C211yesmaximal, characteristic, complemented
12#5576C211yesmaximal, characteristic, complemented
13#5574C211yesmaximal, characteristic, complemented
14C4×#239C211yesmaximal, characteristic, complemented
15#5572C211yesmaximal, characteristic, complemented
16#3524—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C3×C4⋊Q8C2213yes—
21C3×#125C2213yes—
22C3×#125C2213yes—
23C3×C4×Q8C2211yescharacteristic
24C3×C4×Q8C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×Q8C2213yes—
27#947C2216yes—
28#947C2213yes—
29#946C2213yes—
30Q8×Dic3C2211yescharacteristic
31#898C2213yes—
32#898C2216yes—
33#898C2211yescharacteristic
34#897C2213yes—
35#897C2213yes—
36#895C2213yes—
37#895C2216yes—
38#895C2213yes—
39#895C2213yes—
40#895C2213yes—
41#895C2213yes—
42#894C2216yes—
43#894C2216yes—
44#894C2213yes—
45#894C2213yes—
46#893C2216yes—
47#893C2213yes—
48#893C2213yes—
49#892C2216yes—
50#892C2213yes—
51#892C2213yes—
52D3×C4⋊C4C2213yes—
53#890C2213yes—
54#887C2213yes—
55#875C2213yes—
56#875C2213yes—
57#874C2213yes—
58C4×D12C2213yes—
59C4×D12C2213yes—
60C4×D12C2213yes—
61C4×D12C2213yes—
62C4×D12C2213yes—
63#872C2213yes—
64#872C2213yes—
65C42×D3C2213yes—
66C4×Dic6C2213yes—
67#559—33no—
68#557D311yescharacteristic, complemented
69#555—33no—
70#553—33no—
71#552—33no—
72#550—31no—
73#548—31no—
74#548—31no—
75#542—33no—
76#540—33no—
77#529—33no—
78#529—33no—
79#520—31no—
80#519—31no—
81C4×#41—31no—
82C2×C3×Q8C2313yes—
83C2×C3×Q8C2311yescharacteristic
84#239—24no—
85#239—24no—
86D3×Q8—24no—
87C2×D12C2313yes—
88C2×D12C2313yes—
89C2×D12C2313yes—
90C2×D12C2313yes—
91C2×D12C2313yes—
92C2×D12C2316yes—
93C2×C4×D3C2313yes—
94C2×C4×D3C2313yes—
95C2×C4×D3C2313yes—
96C2×C4×D3C2313yes—
97C2×C4×D3C2313yes—
98C2×C4×D3C2316yes—
99C2×Dic6C2313yes—
100C3×C4⋊C4C2313yes—
101C3×C4⋊C4C2313yes—
102C3×C4⋊C4C2313yes—
103C3×C4⋊C4C2313yes—
104C3×C4⋊C4C2316yes—
105C3×C4⋊C4C2313yes—
106C3×C4⋊C4C2311yescharacteristic
107C3×C42C2313yes—
108C3×C42C2313yes—
109C3×C42C2313yes—
110(C2×D3)⋊C4C2311yescharacteristic
111(C2×D3)⋊C4C2313yes—
112(C2×D3)⋊C4C2313yes—
113(C2×D3)⋊C4C2313yes—
114(C2×D3)⋊C4C2311yescharacteristic
115(C2×D3)⋊C4C2311yescharacteristic
116(C2×D3)⋊C4C2313yes—
117(C2×D3)⋊C4C2313yes—
118(C2×D3)⋊C4C2313yes—
119(C2×D3)⋊C4C2316yes—
120(C2×D3)⋊C4C2313yes—
121(C2×D3)⋊C4C2316yes—
122(C2×D3)⋊C4C2316yes—
123C4⋊Dic3C2313yes—
124C4⋊Dic3C2311yescharacteristic
125C4⋊Dic3C2313yes—
126C4⋊Dic3C2313yes—
127Dic3⋊C4C2313yes—
128Dic3⋊C4C2311yescharacteristic
129Dic3⋊C4C2313yes—
130Dic3⋊C4C2313yes—
131C4×Dic3C2316yes—
132C4×Dic3C2311yescharacteristic
133C4×Dic3C2313yes—
134C4×Dic3C2313yes—
135C2×#41—31no—
136C2×#41—31no—
137C22×Q8—31no—
138C4⋊Q8C2×D313yes—
139#127—33no—
140#126—31no—
141#126—33no—
142#126—36no—
143#125—33no—
144#125C2×D313yes—
145#125C2×D313yes—
146#124—33no—
147#124—36no—
148#124—33no—
149#124—33no—
150#123—36no—
151#123—33no—
152#123—33no—
153#123—36no—
154C22⋊Q8—33no—
155C22⋊Q8—33no—
156C22⋊Q8—33no—
157#121—33no—
158#121—33no—
159#121—33no—
160#121—33no—
161#121—36no—
162#121—33no—
163C4×Q8—31no—
164C4×Q8—33no—
165C4×Q8—33no—
166C4×Q8C2×D311yescharacteristic
167C4×Q8C2×D311yescharacteristic
168C4×Q8C2×D311yescharacteristic
169C4×Q8C2×D313yes—
170C4×D4—33no—
171C4×D4—33no—
172C4×D4—36no—
173C4×D4—33no—
174C4×D4—33no—
175C4×D4—33no—
176C4×D4—33no—
177C4×D4—33no—
178#117—33no—
179#117—33no—
180#117—36no—
181#117—33no—
182#117—33no—
183C2×C4⋊C4—33no—
184C2×C42—33no—
185C22×D3C2411yescharacteristic
186C22×D3C2413yes—
187C22×D3C2413yes—
188C3×Q8—43no—
189C3×Q8#4114yes—
190C2×C3×C4C2413yes—
191C2×C3×C4C2413yes—
192C2×C3×C4C2411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194C2×C3×C4C2413yes—
195C2×C3×C4C2411yescharacteristic
196C2×C3×C4C2413yes—
197C2×Dic3C2411yescharacteristic
198C2×Dic3C2413yes—
199C2×Dic3C2411yescharacteristic
200C2×Dic3C2413yes—
201C2×Dic3C2411yescharacteristic
202D12—26no—
203D12—26no—
204D12—43no—
205D12—43no—
206D12—43no—
207D12—46no—
208C4×D3—26no—
209C4×D3—26no—
210C4×D3—26no—
211C4×D3—43no—
212C4×D3—46no—
213C4×D3—43no—
214Dic6—26no—
215#41—64no—
216#41—64no—
217C2×Q8—33no—
218C2×Q8—64no—
219C2×Q8C22×D313yes—
220C2×Q8C22×D311yescharacteristic
221C2×D4—33no—
222C2×D4—33no—
223C2×D4—33no—
224C2×D4—33no—
225C2×D4—36no—
226C2×D4—33no—
227C22×C4—33no—
228C22×C4—33no—
229C22×C4—33no—
230C22×C4—33no—
231C22×C4—33no—
232C22×C4—36no—
233C4⋊C4—33no—
234C4⋊C4—33no—
235C4⋊C4—31no—
236C4⋊C4—33no—
237C4⋊C4—33no—
238C4⋊C4—33no—
239C4⋊C4—31no—
240C4⋊C4—33no—
241C4⋊C4C22×D313yes—
242C4⋊C4C22×D313yes—
243C4⋊C4C22×D313yes—
244C4⋊C4C22×D316yes—
245C4⋊C4C22×D313yes—
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D313yes—
248C22⋊C4—31no—
249C22⋊C4—33no—
250C22⋊C4—33no—
251C22⋊C4—33no—
252C22⋊C4—36no—
253C22⋊C4—33no—
254C22⋊C4—36no—
255C22⋊C4—31no—
256C22⋊C4—33no—
257C22⋊C4—31no—
258C22⋊C4—33no—
259C22⋊C4—36no—
260C22⋊C4—33no—
261C42—31no—
262C42—33no—
263C42—33no—
264C42—36no—
265C42C22×D313yes—
266C42C22×D313yes—
267C42C22×D313yes—
268C22×C3C2511yescharacteristic
269C2×D3—21no—
270C2×D3—23no—
271C2×D3—23no—
272C2×D3—23no—
273C2×D3—23no—
274C2×D3—23no—
275C2×D3—21no—
276C2×D3—23no—
277C2×D3—21no—
278C3×C4—21no—
279C3×C4—23no—
280C3×C4—23no—
281C3×C4—21no—
282C3×C4—23no—
283C3×C4—21no—
284C3×C4C2×#4116yes—
285Dic3—23no—
286Dic3—21no—
287Dic3—21no—
288Dic3—21no—
289Dic3—23no—
290C23—31no—
291C23—33no—
292C23—33no—
293Q8—43no—
294Q8—66no—
295Q8#23914yes—
296D4—66no—
297D4—66no—
298D4—123no—
299D4—126no—
300D4—123no—
301D4—123no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—33no—
305C2×C4—33no—
306C2×C4—31no—
307C2×C4—66no—
308C2×C4—66no—
309C2×C4—66no—
310C2×C4—123no—
311C2×C4—123no—
312C2×C4—126no—
313C2×C4C23×D313yes—
314C2×C4C23×D311yescharacteristic
315C2×C4C23×D313yes—
316C2×C4C23×D311yescharacteristic
317C2×C4C23×D311yescharacteristic
318C2×C4C23×D313yes—
319C2×C4C23×D313yes—
320C2×C3C22×#4111yescharacteristic
321C2×C3#58511yescharacteristic
322C2×C3#58511yescharacteristic
323D3—41no—
324D3—43no—
325D3—43no—
326C22—61no—
327C22—63no—
328C22—63no—
329C22—63no—
330C22—61no—
331C22—63no—
332C22—63no—
333C22—63no—
334C22—61no—
335C22C24×D311yescharacteristic
336C4—23no—
337C4—23no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—21no—
342C4—61no—
343C4—61no—
344C4—63no—
345C4—61no—
346C4—63no—
347C4C2×#23916yes—
348C3#352411yescharacteristic, complemented, Sylow
349C2—121no—
350C2—123no—
351C2—123no—
352C2C22×#23911yescharacteristic
353C2#596411yescharacteristic
354C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#557C2
hypercenter#557D3
hypocenterC3#3524
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. #5734C2semidirect
3. #5733C2semidirect
4. #5697C2semidirect
5. #5695C2semidirect
6. #5694C2semidirect
7. #5691C2semidirect
8. #5585C2semidirect
9. #5580C2semidirect
10. #5577C2semidirect
11. #5576C2semidirect
12. #5576C2semidirect
13. #5574C2semidirect
14. C4×#239C2semidirect
15. #5572C2semidirect
68. #557D3semidirect
348. C3#3524semidirect

Maximal subgroups

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

subgroupindex
C3×#5572
#57342
#57332
#56972
#56952
#56942
#56912
#55852
#55802
#55772
#55762
#55742
C4×#2392
#55722
#35243