Finite Group Databasebeta

← #86752

Subgroups — #86752

Order 384.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Q8C211yesmaximal, characteristic, complemented
2Q8×#67C211yesmaximal, characteristic, complemented
3Q8×#67C211yesmaximal, characteristic, complemented
4#5812C211yesmaximal, characteristic, complemented
5C2×Q8×Dic3C211yesmaximal, characteristic, complemented
6#5731C213yesmaximal, complemented
7#5721C213yesmaximal, complemented
8#5642C213yesmaximal, complemented
9#5641C213yesmaximal, complemented
10#5625C213yesmaximal, complemented
11#5622C213yesmaximal, complemented
12#5572C211yesmaximal, characteristic, complemented
13#5569C211yesmaximal, characteristic, complemented
14#5541C213yesmaximal, complemented
15#5538C213yesmaximal
16#3523—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C3×C4⋊Q8C2213yes—
21C3×C22⋊Q8C2213yes—
22C3×C22⋊Q8C2213yes—
23C3×C4×Q8C2211yescharacteristic
24C3×C4×D4C2213yes—
25#946C2213yes—
26#946C2213yes—
27Q8×Dic3C2213yes—
28Q8×Dic3C2213yes—
29Q8×Dic3C2213yes—
30Q8×Dic3C2213yes—
31Q8×Dic3C2211yescharacteristic
32Q8×Dic3C2211yescharacteristic
33Q8×Dic3C2211yescharacteristic
34#944C2213yes—
35#944C2213yes—
36#939C2211yescharacteristic
37#935C2213yes—
38C4×#67C2213yes—
39C4×#67C2213yes—
40#926C2213yes—
41#926C2213yes—
42C2×C4⋊Dic3C2213yes—
43#924C2213yes—
44#924C2213yes—
45C2×C4×Dic3C2213yes—
46#898C2216yes—
47#898C2213yes—
48#892C2213yes—
49#890C2216yes—
50#890C2213yes—
51#889C2213yes—
52#889C2216yes—
53#889C2216yes—
54#888C2213yes—
55#887C2216yes—
56#887C2213yes—
57#886C2213yes—
58#885C2213yes—
59#879C2216yes—
60#879C2213yes—
61#878C2213yes—
62#877C2216yes—
63#872C2213yes—
64#870C2213yes—
65C4×Dic6C2213yes—
66C4×Dic6C2213yes—
67#557—33no—
68#557—31no—
69#556—33no—
70#552—33no—
71#552—33no—
72D4×Q8—31no—
73D4×Q8—31no—
74D4×Q8D311yescharacteristic, complemented
75#541—33no—
76#541—33no—
77#533—33no—
78#529—33no—
79#519—31no—
80#519—31no—
81C2×C4×Q8—31no—
82C2×C3×Q8—24no—
83C2×C3×Q8—24no—
84C2×C3×Q8C2313yes—
85C2×C3×Q8C2313yes—
86C2×C3×Q8C2311yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4C2313yes—
89C22×C3×C4C2313yes—
90C2×#67C2311yescharacteristic
91C2×#67C2311yescharacteristic
92C22×Dic3C2313yes—
93C22×Dic3C2311yescharacteristic
94D3×Q8—24no—
95C2×C4×D3C2313yes—
96C2×Dic6C2313yes—
97C2×Dic6C2313yes—
98C3×C4⋊C4C2313yes—
99C3×C4⋊C4C2316yes—
100C3×C4⋊C4C2313yes—
101C3×C22⋊C4C2313yes—
102C3×C22⋊C4C2313yes—
103C3×C42C2313yes—
104#217C2313yes—
105#217C2313yes—
106#217C2311yescharacteristic
107#217C2313yes—
108(C2×D3)⋊C4C2311yescharacteristic
109(C2×D3)⋊C4C2313yes—
110(C2×D3)⋊C4C2313yes—
111(C2×D3)⋊C4C2313yes—
112C4⋊Dic3C2313yes—
113C4⋊Dic3C2313yes—
114C4⋊Dic3C2313yes—
115C4⋊Dic3C2313yes—
116C4⋊Dic3C2313yes—
117C4⋊Dic3C2316yes—
118C4⋊Dic3C2313yes—
119Dic3⋊C4C2311yescharacteristic
120Dic3⋊C4C2313yes—
121Dic3⋊C4C2313yes—
122Dic3⋊C4C2316yes—
123Dic3⋊C4C2313yes—
124Dic3⋊C4C2316yes—
125Dic3⋊C4C2313yes—
126Dic3⋊C4C2316yes—
127Dic3⋊C4C2313yes—
128C4×Dic3C2316yes—
129C4×Dic3C2313yes—
130C4×Dic3C2313yes—
131C4×Dic3C2313yes—
132C4×Dic3C2316yes—
133C4×Dic3C2313yes—
134C4×Dic3C2313yes—
135C22×Q8—31no—
136C22×Q8C2×D311yescharacteristic
137C22×Q8C2×D311yescharacteristic
138C4⋊Q8—33no—
139C4⋊Q8—33no—
140C4⋊Q8—33no—
141C4⋊Q8C2×D313yes—
142#126—33no—
143#126—36no—
144#126—33no—
145#126—36no—
146#125—33no—
147#125—36no—
148#125—36no—
149#125—36no—
150#125—33no—
151#125—33no—
152#124—33no—
153#123—33no—
154#123—33no—
155C22⋊Q8—33no—
156C22⋊Q8—33no—
157C22⋊Q8—33no—
158C22⋊Q8—33no—
159C22⋊Q8—33no—
160C22⋊Q8C2×D313yes—
161C22⋊Q8C2×D313yes—
162#121—31no—
163C4×Q8—31no—
164C4×Q8—33no—
165C4×Q8—31no—
166C4×Q8—33no—
167C4×Q8—33no—
168C4×Q8—31no—
169C4×Q8—33no—
170C4×Q8—33no—
171C4×Q8—36no—
172C4×Q8—33no—
173C4×Q8—33no—
174C4×Q8C2×D311yescharacteristic
175C4×D4—33no—
176C4×D4—33no—
177C4×D4C2×D313yes—
178#117—33no—
179#117—33no—
180#117—33no—
181#117—36no—
182#117—33no—
183C2×C4⋊C4—33no—
184C2×C42—33no—
185C23×C3C2411yescharacteristic
186C23×C3C2411yescharacteristic
187C22×D3C2411yescharacteristic
188C3×Q8—26no—
189C3×Q8—26no—
190C3×Q8#4114yes—
191C3×D4—22no—
192C2×C3×C4—26no—
193C2×C3×C4—26no—
194C2×C3×C4C2413yes—
195C2×C3×C4C2413yes—
196C2×C3×C4C2413yes—
197C2×C3×C4C2411yescharacteristic
198C2×C3×C4C2413yes—
199#67—22no—
200#67—22no—
201C2×Dic3—26no—
202C2×Dic3—22no—
203C2×Dic3C2411yescharacteristic
204C2×Dic3C2413yes—
205C2×Dic3C2413yes—
206C2×Dic3C2411yescharacteristic
207C2×Dic3C2413yes—
208C2×Dic3C2413yes—
209C2×Dic3C2411yescharacteristic
210C4×D3—26no—
211Dic6—26no—
212Dic6—26no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—33no—
216C2×Q8—33no—
217C2×Q8—64no—
218C2×Q8C22×D313yes—
219C2×Q8C22×D313yes—
220C2×Q8C22×D311yescharacteristic
221C2×D4—31no—
222C2×D4—31no—
223C2×D4C22×D311yescharacteristic
224C22×C4—31no—
225C22×C4—33no—
226C22×C4—33no—
227C22×C4C22×D313yes—
228C22×C4C22×D313yes—
229C4⋊C4—33no—
230C4⋊C4—33no—
231C4⋊C4—36no—
232C4⋊C4—33no—
233C4⋊C4—33no—
234C4⋊C4—33no—
235C4⋊C4—33no—
236C4⋊C4—33no—
237C4⋊C4—36no—
238C4⋊C4—36no—
239C4⋊C4—33no—
240C4⋊C4—33no—
241C4⋊C4—31no—
242C4⋊C4—36no—
243C4⋊C4—33no—
244C4⋊C4—33no—
245C4⋊C4C22×D313yes—
246C4⋊C4C22×D313yes—
247C4⋊C4C22×D316yes—
248C22⋊C4—31no—
249C22⋊C4—33no—
250C22⋊C4—33no—
251C22⋊C4—33no—
252C22⋊C4—31no—
253C22⋊C4—33no—
254C22⋊C4—33no—
255C22⋊C4—33no—
256C22⋊C4C22×D313yes—
257C22⋊C4C22×D313yes—
258C42—33no—
259C42—33no—
260C42—33no—
261C42—33no—
262C42—36no—
263C42—36no—
264C42—33no—
265C42C22×D313yes—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C32−1+412yes—
272C22×C3C2511yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—21no—
276C3×C4—23no—
277C3×C4—23no—
278C3×C4—23no—
279C3×C4—21no—
280C3×C4C2×#4116yes—
281Dic3—23no—
282Dic3—23no—
283Dic3—23no—
284Dic3—23no—
285Dic3—21no—
286Dic3—21no—
287Dic32−1+412yes—
288C23—31no—
289C23C23×D311yescharacteristic
290C23C23×D311yescharacteristic
291Q8—26no—
292Q8—26no—
293Q8—66no—
294Q8—66no—
295Q8#23714yes—
296D4—22no—
297D4—62no—
298D4—62no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—31no—
302C2×C4—33no—
303C2×C4—31no—
304C2×C4—33no—
305C2×C4—31no—
306C2×C4—33no—
307C2×C4—33no—
308C2×C4—62no—
309C2×C4—66no—
310C2×C4—66no—
311C2×C4C23×D313yes—
312C2×C4C23×D313yes—
313C2×C4C23×D313yes—
314C2×C4C23×D311yescharacteristic
315C2×C4C23×D313yes—
316C2×C3—22no—
317C2×C3—41no—
318C2×C3C22×#4111yescharacteristic
319C2×C3C2×2−1+411yescharacteristic
320C2×C3C2×2−1+411yescharacteristic
321D3—41no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22#100712yes—
331C22C24×D311yescharacteristic
332C4—23no—
333C4—23no—
334C4—21no—
335C4—23no—
336C4—32no—
337C4—63no—
338C4—63no—
339C4—63no—
340C4—61no—
341C4—61no—
342C4—63no—
343C4C2×#23716yes—
344C3#352311yescharacteristic, complemented, Sylow
345C2—22no—
346C2—41no—
347C2—121no—
348C2C22×#23711yescharacteristic
349C2C2×#100711yescharacteristic
350C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D4×Q8C2
prime radical#939C22
hypercenterD4×Q8D3
hypocenterC3#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×D4×Q8C2semidirect
2. Q8×#67C2semidirect
3. Q8×#67C2semidirect
4. #5812C2semidirect
5. C2×Q8×Dic3C2semidirect
6. #5731C2semidirect
7. #5721C2semidirect
8. #5642C2semidirect
9. #5641C2semidirect
10. #5625C2semidirect
11. #5622C2semidirect
12. #5572C2semidirect
13. #5569C2semidirect
14. #5541C2semidirect
74. D4×Q8D3semidirect
344. C3#3523semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Q82
Q8×#672
#58122
C2×Q8×Dic32
#57312
#57212
#56422
#56412
#56252
#56222
#55722
#55692
#55412
#55382
#35233