Finite Group Databasebeta

← #86724

Subgroups — #86724

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×#548C211yesmaximal, characteristic, complemented
2#5834C211yesmaximal, characteristic, complemented
3#5827C211yesmaximal, characteristic, complemented
4#5817C211yesmaximal, characteristic, complemented
5#5812C211yesmaximal, characteristic, complemented
6#5681C213yesmaximal, complemented
7#5680C213yesmaximal, complemented
8#5643C213yesmaximal, complemented
9#5642C213yesmaximal, complemented
10#5619C213yesmaximal, complemented
11#5616C213yesmaximal, complemented
12C4×#239C211yesmaximal, characteristic, complemented
13#5569C211yesmaximal, characteristic, complemented
14#5565C213yesmaximal, complemented
15#5557C213yesmaximal, complemented
16#3524—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×#239C2211yescharacteristic
20C3×#124C2213yes—
21C3×C22⋊Q8C2213yes—
22C3×#121C2213yes—
23C3×C4×Q8C2211yescharacteristic
24C3×C4×D4C2213yes—
25#947C2213yes—
26#946C2213yes—
27Q8×Dic3C2213yes—
28Q8×Dic3C2213yes—
29Q8×Dic3C2211yescharacteristic
30Q8×Dic3C2211yescharacteristic
31Q8×Dic3C2211yescharacteristic
32#940C2213yes—
33#939C2216yes—
34#938C2213yes—
35#938C2213yes—
36#936C2213yes—
37D4×Dic3C2213yes—
38D4×Dic3C2213yes—
39#930C2213yes—
40#930C2213yes—
41C4×#67C2213yes—
42C4×#67C2213yes—
43#926C2213yes—
44#926C2213yes—
45#926C2213yes—
46C2×C4⋊Dic3C2213yes—
47#898C2216yes—
48#897C2213yes—
49#893C2213yes—
50#892C2213yes—
51#890C2213yes—
52#890C2213yes—
53#888C2213yes—
54#886C2216yes—
55#885C2216yes—
56#883C2216yes—
57#883C2213yes—
58#883C2213yes—
59#67⋊C4C2216yes—
60#879C2213yes—
61#879C2211yescharacteristic
62#877C2216yes—
63C4×D12C2213yes—
64C42×D3C2213yes—
65#870C2213yes—
66C4×Dic6C2213yes—
67#559—33no—
68#557—31no—
69#555—33no—
70#553—33no—
71#552—33no—
72#550—31no—
73#548—31no—
74#548D311yescharacteristic, complemented
75#542—33no—
76#540—33no—
77#529—33no—
78#529—33no—
79#520—31no—
80#519—31no—
81C4×#41—31no—
82C3×#41—24no—
83C2×C3×Q8—24no—
84C2×C3×Q8C2313yes—
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4C2313yes—
87C2×C3×D4C2313yes—
88C22×C3×C4C2313yes—
89C22×C3×C4C2313yes—
90C2×#67C2316yes—
91C2×#67C2313yes—
92C2×#67C2313yes—
93C22×Dic3C2316yes—
94#239—24no—
95C2×D12C2313yes—
96C2×C4×D3C2313yes—
97C2×C4×D3C2313yes—
98C2×C4×D3C2313yes—
99C2×Dic6C2313yes—
100C3×C4⋊C4C2313yes—
101C3×C4⋊C4C2313yes—
102C3×C22⋊C4C2316yes—
103C3×C22⋊C4C2313yes—
104C3×C22⋊C4C2311yescharacteristic
105C3×C42C2313yes—
106#217C2313yes—
107#217C2313yes—
108#217C2311yescharacteristic
109#217C2311yescharacteristic
110#217C2313yes—
111#217C2313yes—
112(C2×D3)⋊C4C2316yes—
113(C2×D3)⋊C4C2316yes—
114(C2×D3)⋊C4C2313yes—
115(C2×D3)⋊C4C2313yes—
116C4⋊Dic3C2313yes—
117C4⋊Dic3C2313yes—
118C4⋊Dic3C2313yes—
119C4⋊Dic3C2313yes—
120C4⋊Dic3C2313yes—
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2313yes—
123C4⋊Dic3C2313yes—
124Dic3⋊C4C2311yescharacteristic
125Dic3⋊C4C2313yes—
126Dic3⋊C4C2316yes—
127Dic3⋊C4C2313yes—
128Dic3⋊C4C2311yescharacteristic
129C4×Dic3C2313yes—
130C4×Dic3C2313yes—
131C4×Dic3C2311yescharacteristic
132C4×Dic3C2313yes—
133C4×Dic3C2313yes—
134C4×Dic3C2316yes—
135C2×#41—31no—
136C2×#41C2×D311yescharacteristic
137C22×Q8C2×D311yescharacteristic
138C4⋊Q8—33no—
139#127—33no—
140#126—36no—
141#126—31no—
142#126—33no—
143#125—33no—
144#125—33no—
145#125—33no—
146#124—33no—
147#124—33no—
148#124—36no—
149#124C2×D313yes—
150#123—36no—
151#123—33no—
152#123—33no—
153#123—36no—
154C22⋊Q8—33no—
155C22⋊Q8—33no—
156C22⋊Q8C2×D313yes—
157#121—33no—
158#121—33no—
159#121—36no—
160#121—33no—
161#121—33no—
162#121C2×D313yes—
163C4×Q8—31no—
164C4×Q8—33no—
165C4×Q8—31no—
166C4×Q8—33no—
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—36no—
176C4×D4—33no—
177C4×D4C2×D313yes—
178#117—33no—
179#117—33no—
180#117—33no—
181#117—33no—
182#117—36no—
183C2×C4⋊C4—33no—
184C2×C42—33no—
185C23×C3C2411yescharacteristic
186C23×C3C2413yes—
187C22×D3C2413yes—
188C3×Q8—26no—
189C3×Q8#4114yes—
190C3×D4—26no—
191C3×D4—43no—
192C2×C3×C4—26no—
193C2×C3×C4—26no—
194C2×C3×C4C2413yes—
195C2×C3×C4C2411yescharacteristic
196C2×C3×C4C2413yes—
197C2×C3×C4C2411yescharacteristic
198C2×C3×C4C2413yes—
199#67—46no—
200#67—43no—
201#67—43no—
202C2×Dic3—46no—
203C2×Dic3C2411yescharacteristic
204C2×Dic3C2413yes—
205C2×Dic3C2411yescharacteristic
206C2×Dic3C2413yes—
207C2×Dic3C2411yescharacteristic
208C2×Dic3C2411yescharacteristic
209C2×Dic3C2413yes—
210D12—26no—
211C4×D3—26no—
212C4×D3—43no—
213C4×D3—43no—
214Dic6—43no—
215#41—24no—
216#41—64no—
217C2×Q8—24no—
218C2×Q8—33no—
219C2×Q8C22×D313yes—
220C2×Q8C22×D311yescharacteristic
221C2×D4—33no—
222C2×D4—36no—
223C2×D4—33no—
224C2×D4—33no—
225C2×D4C22×D313yes—
226C2×D4C22×D313yes—
227C22×C4—36no—
228C22×C4—33no—
229C22×C4—33no—
230C22×C4—33no—
231C22×C4C22×D313yes—
232C22×C4C22×D313yes—
233C4⋊C4—33no—
234C4⋊C4—33no—
235C4⋊C4—33no—
236C4⋊C4—31no—
237C4⋊C4—33no—
238C4⋊C4—33no—
239C4⋊C4—33no—
240C4⋊C4—33no—
241C4⋊C4—36no—
242C4⋊C4—31no—
243C4⋊C4—33no—
244C4⋊C4—31no—
245C4⋊C4—33no—
246C4⋊C4C22×D313yes—
247C4⋊C4C22×D313yes—
248C22⋊C4—31no—
249C22⋊C4—33no—
250C22⋊C4—33no—
251C22⋊C4—33no—
252C22⋊C4—33no—
253C22⋊C4—31no—
254C22⋊C4—36no—
255C22⋊C4—33no—
256C22⋊C4—36no—
257C22⋊C4—33no—
258C22⋊C4C22×D316yes—
259C22⋊C4C22×D313yes—
260C22⋊C4C22×D311yescharacteristic
261C42—31no—
262C42—33no—
263C42—33no—
264C42—36no—
265C42—33no—
266C42—33no—
267C42C22×D313yes—
268C22×C3—21no—
269C22×C3—23no—
270C22×C3—23no—
271C22×C3—21no—
272C22×C3—23no—
273C22×C3—21no—
274C22×C3C2511yescharacteristic
275C2×D3—23no—
276C2×D3—23no—
277C2×D3—23no—
278C3×C4—21no—
279C3×C4—23no—
280C3×C4—23no—
281C3×C4—21no—
282C3×C4C2×#4116yes—
283Dic3—21no—
284Dic3—21no—
285Dic3—23no—
286Dic3—21no—
287Dic3—23no—
288Dic3—23no—
289Dic3—21no—
290C23—33no—
291C23C23×D311yescharacteristic
292C23C23×D313yes—
293Q8—26no—
294Q8—123no—
295Q8#23714yes—
296D4—26no—
297D4—43no—
298D4—66no—
299D4—126no—
300D4—123no—
301D4—123no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—33no—
307C2×C4—31no—
308C2×C4—33no—
309C2×C4—31no—
310C2×C4—33no—
311C2×C4—66no—
312C2×C4—123no—
313C2×C4—123no—
314C2×C4—126no—
315C2×C4C23×D313yes—
316C2×C4C23×D311yescharacteristic
317C2×C4C23×D313yes—
318C2×C4C23×D311yescharacteristic
319C2×C4C23×D313yes—
320C2×C3—41no—
321C2×C3—43no—
322C2×C3C22×#4111yescharacteristic
323C2×C3#58511yescharacteristic
324C2×C3#58511yescharacteristic
325D3—43no—
326C22—21no—
327C22—23no—
328C22—21no—
329C22—23no—
330C22—23no—
331C22—21no—
332C22—63no—
333C22—63no—
334C22—63no—
335C22C24×D311yescharacteristic
336C4—21no—
337C4—23no—
338C4—23no—
339C4—21no—
340C4—61no—
341C4—61no—
342C4—63no—
343C4—61no—
344C4—63no—
345C4—63no—
346C4—61no—
347C4C2×#23716yes—
348C3#352411yescharacteristic, complemented, Sylow
349C2—41no—
350C2—43no—
351C2—123no—
352C2C22×#23711yescharacteristic
353C2#596411yescharacteristic
354C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3524
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C3×#548C2semidirect
2. #5834C2semidirect
3. #5827C2semidirect
4. #5817C2semidirect
5. #5812C2semidirect
6. #5681C2semidirect
7. #5680C2semidirect
8. #5643C2semidirect
9. #5642C2semidirect
10. #5619C2semidirect
11. #5616C2semidirect
12. C4×#239C2semidirect
13. #5569C2semidirect
14. #5565C2semidirect
15. #5557C2semidirect
74. #548D3semidirect
348. C3#3524semidirect

Maximal subgroups

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

subgroupindex
C3×#5482
#58342
#58272
#58172
#58122
#56812
#56802
#56432
#56422
#56192
#56162
C4×#2392
#55692
#55652
#55572
#35243