Finite Group Databasebeta

← #80196

Subgroups — #80196

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
1Q8×#67C211yesmaximal, characteristic
2#5639C211yesmaximal, characteristic
3C3×#451C211yesmaximal, characteristic, complemented
4#5225C211yesmaximal, characteristic, complemented
5#5188C211yesmaximal, characteristic, complemented
6#5186C211yesmaximal, characteristic, complemented
7#5184C211yesmaximal, characteristic, complemented
8#5182C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#4805C211yesmaximal, characteristic, complemented
11D3×#103C211yesmaximal, characteristic, complemented
12#4797C211yesmaximal, characteristic, complemented
13#4796C211yesmaximal, characteristic, complemented
14#4728C211yesmaximal, characteristic
15#4723C211yesmaximal, characteristic, complemented
16#3312—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#946—21no—
23#946C2211yescharacteristic
24#946C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic
27#944—21no—
28#944C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C4×#67—21no—
31C4×#67C2211yescharacteristic
32#926C2211yescharacteristic
33#924—21no—
34#924C2211yescharacteristic
35C2×#208C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#892—21no—
38D3×C4⋊C4C2211yescharacteristic
39#890C2211yescharacteristic
40#889—21no—
41#883—21no—
42#879—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#835C2211yescharacteristic
50#835C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#818C2211yescharacteristic
53#817C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#808C2211yescharacteristic
60#552—31no—
61D4×Q8—31no—
62#491—31no—
63#485—31no—
64#483—31no—
65#479—31no—
66#470—31no—
67#467—31no—
68#462—31no—
69#451—31no—
70#451D311yescharacteristic, complemented
71#441—31no—
72#435—31no—
73#419—31no—
74C2×#103—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67D411yescharacteristic
84D3×Q8—22no—
85D3×Q8—41no—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C3×Q16—22no—
92C3×Q16—41no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98#217—21no—
99#217—21no—
100#217D411yescharacteristic
101#216—41no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4D411yescharacteristic
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4D411yescharacteristic
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123#208—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C22×Q8—31no—
128C22×Q8C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×Q16C2×D311yescharacteristic
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—31no—
151#117—61no—
152C2×C4⋊C4—31no—
153#107—31no—
154#106—31no—
155#106—31no—
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—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—42no—
178C3×Q8#4112yes—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—22no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—42no—
199C4×D3—42no—
200Dic6—21no—
201Dic6—21no—
202Dic6—42no—
203C3×C8—21no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—31no—
210C2×Q8—41no—
211C2×Q8—61no—
212C2×Q8—62no—
213C2×Q8—121no—
214C2×Q8C22×D311yescharacteristic
215C2×Q8C22×D311yescharacteristic
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C22×C4—21no—
219C22×C4—31no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C22×D311yescharacteristic
223Q16—22no—
224Q16—41no—
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—31no—
236C4⋊C4—31no—
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⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—31no—
250C22⋊C4—31no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—31no—
256C42—61no—
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—22no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4C2×#4111yescharacteristic
272C3×C4H(5,𝔽2)11yescharacteristic
273Dic3—21no—
274Dic3—21no—
275Dic3—21no—
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279Dic3—41no—
280C23—31no—
281C23D3×D411yescharacteristic
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—42no—
288Q8—61no—
289Q8—61no—
290Q8—122no—
291Q8#23712yes—
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—42no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
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—22no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—61no—
335C4—61no—
336C4—61no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4C2×#23711yescharacteristic
342C4#100411yescharacteristic
343C3#331211yescharacteristic, complemented, Sylow
344C2—41no—
345C2—62no—
346C2#559111yescharacteristic
347C2#576611yescharacteristic
348C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#451D3
hypocenterC3#3312
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. C3×#451C2semidirect
4. #5225C2semidirect
5. #5188C2semidirect
6. #5186C2semidirect
7. #5184C2semidirect
8. #5182C2semidirect
9. #5052C2semidirect
10. #4805C2semidirect
11. D3×#103C2semidirect
12. #4797C2semidirect
13. #4796C2semidirect
15. #4723C2semidirect
70. #451D3semidirect
343. C3#3312semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#56392
C3×#4512
#52252
#51882
#51862
#51842
#51822
#50522
#48052
D3×#1032
#47972
#47962
#47282
#47232
#33123