Finite Group Databasebeta

← #81527

Subgroups — #81527

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5632C211yesmaximal, characteristic
2#5529C211yesmaximal, characteristic
3C3×#485C211yesmaximal, characteristic, complemented
4#5052C211yesmaximal, characteristic, complemented
5#5004C211yesmaximal, characteristic, complemented
6#4887C211yesmaximal, characteristic, complemented
7#4877C211yesmaximal, characteristic, complemented
8#4862C211yesmaximal, characteristic, complemented
9#4853C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11D3×#103C211yesmaximal, characteristic, complemented
12#4797C211yesmaximal, characteristic, complemented
13#4795C211yesmaximal, characteristic, complemented
14#4739C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic
16#3312—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#944C2211yescharacteristic
22C2×#216C2211yescharacteristic
23C4×#67C2211yescharacteristic
24#924C2211yescharacteristic
25C2×#208C2211yescharacteristic
26C2×Dic12C2211yescharacteristic
27C2×Dic12C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29#897C2211yescharacteristic
30#897C2211yescharacteristic
31#896—21no—
32D3×C4⋊C4C2211yescharacteristic
33#890C2211yescharacteristic
34#888—21no—
35#887C2211yescharacteristic
36#887C2211yescharacteristic
37#883—21no—
38#67⋊C4—21no—
39#879—21no—
40#878—21no—
41#872—21no—
42#870—21no—
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×C22⋊C8C2211yescharacteristic
48#835C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#816C2211yescharacteristic
51#816C2211yescharacteristic
52#815C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#810C2211yescharacteristic
59#808C2211yescharacteristic
60#552—31no—
61D4×Q8—31no—
62#491—31no—
63#485D311yescharacteristic, complemented
64#483—31no—
65#479—31no—
66#470—31no—
67#467—31no—
68#462—31no—
69#451—31no—
70#451—31no—
71#441—31no—
72#435—31no—
73#419—31no—
74C2×#103—31no—
75C2×C3×Q8C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C2×#67D411yescharacteristic
78C22×Dic3—21no—
79D3×Q8—22no—
80D3×Q8—41no—
81C2×C4×D3—21no—
82C2×C4×D3—21no—
83C2×C4×D3C2311yescharacteristic
84C2×Dic6—22no—
85C2×Dic6—21no—
86C2×Dic6—21no—
87C2×Dic6—41no—
88C2×Dic6C2311yescharacteristic
89C2×Dic6C2311yescharacteristic
90C2×Dic6C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C22⋊C4—21no—
99C3×C42—21no—
100#217—21no—
101#217D411yescharacteristic
102#216—41no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4D411yescharacteristic
106C4⋊Dic3—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121#208—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124Dic12—22no—
125Dic12—41no—
126D3×C8—41no—
127C22×Q8—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×Q16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135C4⋊Q8—61no—
136#126—61no—
137#125—31no—
138#125—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—61no—
151#117C2×D311yescharacteristic
152C2×C4⋊C4—31no—
153#107C2×D311yescharacteristic
154#106—31no—
155#106C2×D311yescharacteristic
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2411yescharacteristic
182#67—22no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—42no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C4×D3—21no—
192C4×D3—21no—
193C4×D3—42no—
194C4×D3—42no—
195Dic6—21no—
196Dic6—21no—
197Dic6—21no—
198Dic6—21no—
199Dic6—21no—
200Dic6—42no—
201Dic6—42no—
202Dic6#4112yes—
203C3×C8—21no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C2×Q8—31no—
208C2×Q8—31no—
209C2×Q8—31no—
210C2×Q8—61no—
211C2×Q8—62no—
212C2×Q8—61no—
213C2×Q8—62no—
214C2×Q8—121no—
215C2×Q8—121no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C22×C4—31no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C22×D311yescharacteristic
223Q16—62no—
224Q16—121no—
225Q16—121no—
226M16—121no—
227C2×C8—31no—
228C2×C8—31no—
229C2×C8—121no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—31no—
251C22⋊C4—31no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—21no—
256C42—31no—
257C42—61no—
258C42—61no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3C22×D411yescharacteristic
263C2×D3—21no—
264C2×D3—21no—
265C2×D3#13712yes—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4C2×#4111yescharacteristic
272C3×C4H(5,𝔽2)11yescharacteristic
273Dic3—22no—
274Dic3—21no—
275Dic3—21no—
276Dic3—21no—
277Dic3—41no—
278Dic3—41no—
279Dic3—41no—
280C23—31no—
281C23D3×D411yescharacteristic
282Q8—21no—
283Q8—21no—
284Q8—32no—
285Q8—61no—
286Q8—61no—
287Q8—61no—
288Q8—61no—
289Q8—61no—
290Q8—122no—
291Q8—122no—
292D4—62no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4D3×D411yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—21no—
314C8—21no—
315C8—61no—
316C8—61no—
317C2×C3—41no—
318C2×C3#54611yescharacteristic
319C2×C3C2×#13711yescharacteristic
320C2×C3#57811yescharacteristic
321D3—22no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—32no—
326C22—61no—
327C22—61no—
328C22C2×D3×D411yescharacteristic
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—61no—
335C4—62no—
336C4—61no—
337C4—61no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4D3×#4111yescharacteristic
342C4#100911yescharacteristic
343C3#331211yescharacteristic, complemented, Sylow
344C2—41no—
345C2—62no—
346C2#565411yescharacteristic
347C2#577111yescharacteristic
348C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#485D3
hypocenterC3#3312
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. C3×#485C2semidirect
4. #5052C2semidirect
5. #5004C2semidirect
6. #4887C2semidirect
7. #4877C2semidirect
8. #4862C2semidirect
9. #4853C2semidirect
10. #4809C2semidirect
11. D3×#103C2semidirect
12. #4797C2semidirect
13. #4795C2semidirect
14. #4739C2semidirect
63. #485D3semidirect
343. C3#3312semidirect

Maximal subgroups

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

subgroupindex
#56322
#55292
C3×#4852
#50522
#50042
#48872
#48772
#48622
#48532
#48092
D3×#1032
#47972
#47952
#47392
#47282
#33123