Finite Group Databasebeta

← #86712

Subgroups — #86712

Order 384.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#548C211yesmaximal, characteristic, complemented
2#5831C211yesmaximal, characteristic, complemented
3Dic3×#41C211yesmaximal, characteristic, complemented
4Q8×#67C211yesmaximal, characteristic, complemented
5C2×Q8×Dic3C211yesmaximal, characteristic, complemented
6#5680C213yesmaximal, complemented
7#5671C213yesmaximal, complemented
8#5639C213yesmaximal, complemented
9#5622C213yesmaximal, complemented
10#5611C213yesmaximal, complemented
11#5600C213yesmaximal, complemented
12C4×D3×Q8C211yesmaximal, characteristic, complemented
13#5569C211yesmaximal, characteristic, complemented
14#5541C213yesmaximal, complemented
15C4×#237C213yesmaximal, complemented
16#3482—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C2×#237C2213yes—
21C3×#124C2213yes—
22C3×C22⋊Q8C2213yes—
23C3×#121C2213yes—
24C3×C4×Q8C2211yescharacteristic
25C3×C4×D4C2213yes—
26#946C2213yes—
27Q8×Dic3C2213yes—
28Q8×Dic3C2213yes—
29Q8×Dic3C2211yescharacteristic
30Q8×Dic3C2211yescharacteristic
31Q8×Dic3C2211yescharacteristic
32#944C2213yes—
33#938C2213yes—
34#936C2213yes—
35#935C2216yes—
36D4×Dic3C2213yes—
37D4×Dic3C2213yes—
38D4×Dic3C2213yes—
39C4×#67C2213yes—
40C4×#67C2213yes—
41#926C2213yes—
42C2×C4⋊Dic3C2213yes—
43#924C2213yes—
44#924C2213yes—
45C2×C4×Dic3C2213yes—
46C2×C4×Dic3C2213yes—
47#892C2213yes—
48D3×C4⋊C4C2213yes—
49#890C2213yes—
50#889C2216yes—
51#888C2213yes—
52#887C2213yes—
53#887C2213yes—
54#883C2216yes—
55#67⋊C4C2213yes—
56#67⋊C4C2211yescharacteristic
57#879C2216yes—
58#879C2216yes—
59#878C2216yes—
60#877C2213yes—
61#877C2216yes—
62#877C2213yes—
63C42×D3C2213yes—
64#870C2213yes—
65C4×Dic6C2213yes—
66C4×Dic6C2213yes—
67#557—31no—
68#556—33no—
69#552—33no—
70#552—33no—
71D4×Q8—31no—
72#548—31no—
73#548D311yescharacteristic, complemented
74#547—33no—
75#533—33no—
76#529—33no—
77#529—33no—
78C4×#41—31no—
79C4×#41—33no—
80C2×C4×Q8—31no—
81C2×C4×Q8—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×#67C2313yes—
91C2×#67C2311yescharacteristic
92C22×Dic3C2313yes—
93C22×Dic3C2313yes—
94C22×Dic3C2313yes—
95C22×Dic3C2311yescharacteristic
96D3×Q8—24no—
97#237—212no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2313yes—
100C2×C4×D3C2313yes—
101C2×Dic6C2313yes—
102C2×Dic6C2313yes—
103C3×C4⋊C4C2313yes—
104C3×C4⋊C4C2313yes—
105C3×C22⋊C4C2311yescharacteristic
106C3×C22⋊C4C2316yes—
107C3×C22⋊C4C2313yes—
108C3×C42C2313yes—
109#217C2316yes—
110#217C2313yes—
111#217C2313yes—
112#217C2316yes—
113(C2×D3)⋊C4C2313yes—
114(C2×D3)⋊C4C2311yescharacteristic
115C4⋊Dic3C2313yes—
116C4⋊Dic3C2313yes—
117C4⋊Dic3C2313yes—
118C4⋊Dic3C2313yes—
119C4⋊Dic3C2316yes—
120Dic3⋊C4C2311yescharacteristic
121Dic3⋊C4C2313yes—
122Dic3⋊C4C2316yes—
123Dic3⋊C4C2316yes—
124Dic3⋊C4C2313yes—
125Dic3⋊C4C2316yes—
126Dic3⋊C4C2313yes—
127C4×Dic3C2313yes—
128C4×Dic3C2313yes—
129C4×Dic3C2313yes—
130C4×Dic3C2313yes—
131C4×Dic3C2313yes—
132C4×Dic3C2313yes—
133C4×Dic3C2313yes—
134C4×Dic3C2313yes—
135C4×Dic3C2311yescharacteristic
136C2×#41—33no—
137C2×#41C2×D311yescharacteristic
138C22×Q8—31no—
139C22×Q8C2×D311yescharacteristic
140C4⋊Q8—33no—
141C4⋊Q8—33no—
142#126—36no—
143#126—36no—
144#125—33no—
145#125—36no—
146#125—33no—
147#124—33no—
148#124C2×D313yes—
149#123—36no—
150#123—36no—
151C22⋊Q8—33no—
152C22⋊Q8—33no—
153C22⋊Q8—33no—
154C22⋊Q8—36no—
155C22⋊Q8C2×D313yes—
156#121—33no—
157#121C2×D313yes—
158C4×Q8—31no—
159C4×Q8—33no—
160C4×Q8—31no—
161C4×Q8—33no—
162C4×Q8—31no—
163C4×Q8—33no—
164C4×Q8—33no—
165C4×Q8—33no—
166C4×Q8—33no—
167C4×Q8C2×D311yescharacteristic
168C4×D4—33no—
169C4×D4—33no—
170C4×D4—33no—
171C4×D4—33no—
172C4×D4—33no—
173C4×D4—31no—
174C4×D4—33no—
175C4×D4C2×D313yes—
176#117—33no—
177#117—33no—
178#117—33no—
179#117—33no—
180#117—36no—
181C2×C4⋊C4—33no—
182C2×C4⋊C4—33no—
183C2×C42—33no—
184C2×C42—33no—
185C2×C42—33no—
186C23×C3C2413yes—
187C23×C3C2411yescharacteristic
188C22×D3C2411yescharacteristic
189C3×Q8—26no—
190C3×Q8#4114yescomplemented
191C3×D4—26no—
192C3×D4—26no—
193C2×C3×C4—26no—
194C2×C3×C4—26no—
195C2×C3×C4C2411yescharacteristic
196C2×C3×C4C2411yescharacteristic
197C2×C3×C4C2413yes—
198C2×C3×C4C2413yes—
199C2×C3×C4C2413yes—
200#67—26no—
201#67#4114yescomplemented
202C2×Dic3—26no—
203C2×Dic3—26no—
204C2×Dic3—22no—
205C2×Dic3—26no—
206C2×Dic3C2411yescharacteristic
207C2×Dic3C2413yes—
208C2×Dic3C2413yes—
209C2×Dic3C2413yes—
210C2×Dic3C2413yes—
211C2×Dic3C2411yescharacteristic
212C2×Dic3C2411yescharacteristic
213C4×D3—26no—
214C4×D3—26no—
215C4×D3—22no—
216Dic6—26no—
217Dic6—26no—
218#41—24no—
219#41—612no—
220C2×Q8—24no—
221C2×Q8—33no—
222C2×Q8—33no—
223C2×Q8—64no—
224C2×Q8C22×D313yes—
225C2×Q8C22×D311yescharacteristic
226C2×D4—31no—
227C2×D4—33no—
228C2×D4C22×D313yes—
229C2×D4C22×D313yes—
230C22×C4—31no—
231C22×C4—33no—
232C22×C4—33no—
233C22×C4—33no—
234C22×C4—31no—
235C22×C4—33no—
236C22×C4—33no—
237C22×C4C22×D313yes—
238C22×C4C22×D313yes—
239C4⋊C4—33no—
240C4⋊C4—33no—
241C4⋊C4—36no—
242C4⋊C4—36no—
243C4⋊C4—31no—
244C4⋊C4—33no—
245C4⋊C4—33no—
246C4⋊C4—33no—
247C4⋊C4—33no—
248C4⋊C4—33no—
249C4⋊C4—36no—
250C4⋊C4—36no—
251C4⋊C4C22×D313yes—
252C4⋊C4C22×D313yes—
253C22⋊C4—33no—
254C22⋊C4—36no—
255C22⋊C4—33no—
256C22⋊C4—36no—
257C22⋊C4—31no—
258C22⋊C4—33no—
259C22⋊C4C22×D311yescharacteristic
260C22⋊C4C22×D313yes—
261C22⋊C4C22×D316yes—
262C42—31no—
263C42—33no—
264C42—33no—
265C42—33no—
266C42—33no—
267C42—33no—
268C42—33no—
269C42—33no—
270C42—33no—
271C42C22×D313yes—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—23no—
275C22×C3—23no—
276C22×C3—23no—
277C22×C3C2×#4112yes—
278C22×C3C2511yescharacteristic
279C2×D3—21no—
280C2×D3—21no—
281C2×D3C2×#4112yes—
282C3×C4—23no—
283C3×C4—21no—
284C3×C4—23no—
285C3×C4—21no—
286C3×C4C2×#4116yes—
287Dic3—23no—
288Dic3—23no—
289Dic3—23no—
290Dic3—21no—
291Dic3—23no—
292Dic3C2×#4112yes—
293Dic32−1+412yes—
294C23—31no—
295C23C23×D313yes—
296C23C23×D311yescharacteristic
297Q8—26no—
298Q8—66no—
299Q8—66no—
300Q8#23714yescomplemented
301D4—26no—
302D4—26no—
303D4—34no—
304D4—66no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—33no—
311C2×C4—33no—
312C2×C4—33no—
313C2×C4—33no—
314C2×C4—62no—
315C2×C4—66no—
316C2×C4—66no—
317C2×C4—62no—
318C2×C4—66no—
319C2×C4—66no—
320C2×C4—66no—
321C2×C4C23×D311yescharacteristic
322C2×C4C23×D311yescharacteristic
323C2×C4C23×D313yes—
324C2×C4C23×D313yes—
325C2×C4C23×D313yes—
326C2×C3—22no—
327C2×C3—43no—
328C2×C3C22×#4111yescharacteristic
329C2×C3C22×#4111yescharacteristic
330C2×C3C2×2−1+411yescharacteristic
331D3—22no—
332C22—21no—
333C22—21no—
334C22—23no—
335C22—23no—
336C22—23no—
337C22—32no—
338C22—61no—
339C22—61no—
340C22D3×#4112yes—
341C22C24×D311yescharacteristic
342C4—21no—
343C4—23no—
344C4—23no—
345C4—21no—
346C4—32no—
347C4—32no—
348C4—63no—
349C4—63no—
350C4—63no—
351C4—61no—
352C4—63no—
353C4C2×#23716yes—
354C3#348211yescharacteristic, complemented, Sylow
355C2—22no—
356C2—43no—
357C2—62no—
358C2C22×#23711yescharacteristic
359C2C2×D3×#4111yescharacteristic
360C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3482
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. #5831C2semidirect
3. Dic3×#41C2semidirect
4. Q8×#67C2semidirect
5. C2×Q8×Dic3C2semidirect
6. #5680C2semidirect
7. #5671C2semidirect
8. #5639C2semidirect
9. #5622C2semidirect
10. #5611C2semidirect
11. #5600C2semidirect
12. C4×D3×Q8C2semidirect
13. #5569C2semidirect
14. #5541C2semidirect
15. C4×#237C2semidirect
73. #548D3semidirect
190. C3×Q8#41semidirect
201. #67#41semidirect
300. Q8#237semidirect
354. C3#3482semidirect

Maximal subgroups

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

subgroupindex
C3×#5482
#58312
Dic3×#412
Q8×#672
C2×Q8×Dic32
#56802
#56712
#56392
#56222
#56112
#56002
C4×D3×Q82
#55692
#55412
C4×#2372
#34823