Finite Group Databasebeta

← #80793

Subgroups — #80793

Order 384.

348 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5622C211yesmaximal, characteristic, complemented
2#5600C211yesmaximal, characteristic
3C3×#468C211yesmaximal, characteristic
4#5166C211yesmaximal, characteristic, complemented
5#5162C211yesmaximal, characteristic, complemented
6#5137C211yesmaximal, characteristic, complemented
7#5118C211yesmaximal, characteristic, complemented
8#5052C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic
10#4877C211yesmaximal, characteristic, complemented
11#4875C211yesmaximal, characteristic, complemented
12#4799C211yesmaximal, characteristic, complemented
13#4795C211yesmaximal, characteristic, complemented
14#4767C211yesmaximal, characteristic, complemented
15#4761C211yesmaximal, characteristic, complemented
16#3252—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#216C2211yescharacteristic
24#936C2211yescharacteristic
25#935—21no—
26D4×Dic3—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29#924C2211yescharacteristic
30#924C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32C2×#208C2211yescharacteristic
33C2×Dic12C2211yescharacteristic
34#890C2211yescharacteristic
35#889—21no—
36#888C2211yescharacteristic
37#887—21no—
38#887C2211yescharacteristic
39#887C2211yescharacteristic
40#879—21no—
41#878—21no—
42#877—21no—
43#877—21no—
44C3×#107C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#817C2211yescharacteristic
50#816C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#533—31no—
61#529—31no—
62#479—31no—
63#479—31no—
64#477—31no—
65#476—31no—
66#470—31no—
67#470—31no—
68#468—31no—
69#468D311yescharacteristic
70#464—31no—
71#464—31no—
72#462—31no—
73#461—31no—
74#423—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×Dic3—21no—
80C22×Dic3D411yescharacteristic
81C22×Dic3D411yescharacteristic
82C2×Dic6—21no—
83C2×Dic6C2311yescharacteristic
84C2×Dic6C2311yescharacteristic
85C3×SD16—41no—
86C3×M16—41no—
87C2×C3×C8C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C3×C4⋊C4—21no—
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93C3×C22⋊C4—21no—
94#217—21no—
95#217—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#216—41no—
100#214—41no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104C4⋊Dic3C2311yescharacteristic
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3D411yescharacteristic
117C4×Dic3D411yescharacteristic
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120#208—41no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123Dic12—41no—
124C2×Q16—31no—
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×M16—31no—
129C2×M16C2×D311yescharacteristic
130C4⋊Q8—31no—
131C4⋊Q8—31no—
132#126—61no—
133#125—31no—
134#125—61no—
135#124—31no—
136#123—61no—
137C22⋊Q8—31no—
138C22⋊Q8—31no—
139C22⋊Q8—61no—
140C22⋊Q8C2×D311yescharacteristic
141#121C2×D311yescharacteristic
142C4×Q8—31no—
143C4×Q8—31no—
144C4×Q8—31no—
145C4×Q8—61no—
146C4×D4—31no—
147C4×D4—61no—
148#117—61no—
149#117—61no—
150C2×C42—31no—
151#107—31no—
152#107C2×D311yescharacteristic
153#106—31no—
154#106—31no—
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#102—31no—
166#102C2×D311yescharacteristic
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C3×Q8—21no—
170C3×Q8—21no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—42no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181C2×Dic3—22no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—41no—
187C2×Dic3—42no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192Dic6—21no—
193Dic6—21no—
194Dic6—21no—
195Dic6—21no—
196Dic6—42no—
197C3×C8—21no—
198C3×C8—21no—
199C3⋊C8—21no—
200C3⋊C8—21no—
201C2×Q8—31no—
202C2×Q8—31no—
203C2×Q8—61no—
204C2×Q8C22×D311yescharacteristic
205C2×D4—21no—
206C2×D4C22×D311yescharacteristic
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—61no—
210C22×C4C22×D311yescharacteristic
211Q16—121no—
212Q16—121no—
213SD16—41no—
214SD16—121no—
215M16—41no—
216M16—121no—
217C2×C8—31no—
218C2×C8—31no—
219C2×C8C22×D311yescharacteristic
220C2×C8C22×D311yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4C22×D311yescharacteristic
235C4⋊C4C22×D311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C42—31no—
244C42—31no—
245C42—31no—
246C42—31no—
247C42—61no—
248C42—61no—
249C42—61no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C22×D411yescharacteristic
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4H(5,𝔽2)11yescharacteristic
262C3×C42−1+411yescharacteristic
263Dic3—21no—
264Dic3—21no—
265Dic3—21no—
266Dic3—41no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270Dic3#13712yes—
271C23—21no—
272C23D3×D411yescharacteristic
273Q8—21no—
274Q8—21no—
275Q8—61no—
276Q8—61no—
277Q8—61no—
278Q8—61no—
279Q8—122no—
280D4—21no—
281D4—21no—
282D4—42no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—62no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—122no—
298C2×C4—121no—
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—61no—
304C8—61no—
305C2×C3—41no—
306C2×C3—81no—
307C2×C3#53711yescharacteristic
308C2×C3C2×#13711yescharacteristic
309C2×C3#57511yescharacteristic
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22C2×D3×D411yescharacteristic
317C4—21no—
318C4—32no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—61no—
323C4—61no—
324C4—61no—
325C4—121no—
326C4—121no—
327C4—121no—
328C4—121no—
329C4#100411yescharacteristic
330C4#101011yescharacteristic
331C3#325211yescharacteristic, complemented, Sylow
332C2—41no—
333C2—81no—
334C2#559711yescharacteristic
335C2#577311yescharacteristic
336C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radicalC3×#121C22
hypercenter#468D3
hypocenterC3#3252
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #5622C2semidirect
4. #5166C2semidirect
5. #5162C2semidirect
6. #5137C2semidirect
7. #5118C2semidirect
8. #5052C2semidirect
10. #4877C2semidirect
11. #4875C2semidirect
12. #4799C2semidirect
13. #4795C2semidirect
14. #4767C2semidirect
15. #4761C2semidirect
331. C3#3252semidirect

Maximal subgroups

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

subgroupindex
#56222
#56002
C3×#4682
#51662
#51622
#51372
#51182
#50522
#50422
#48772
#48752
#47992
#47952
#47672
#47612
#32523