Finite Group Databasebeta

← #78806

Subgroups — #78806

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic
2#5511C211yesmaximal, characteristic
3C2×C3×#103C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5116C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic
7#4970C211yesmaximal, characteristic, complemented
8#4813C211yesmaximal, characteristic, complemented
9#4811C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11D3×#103C211yesmaximal, characteristic, complemented
12#4795C211yesmaximal, characteristic, complemented
13#4793C211yesmaximal, characteristic, complemented
14#4792C211yesmaximal, characteristic, complemented
15#4790C211yesmaximal, characteristic, complemented
16#3299—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#946—21no—
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#944—21no—
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C4×#67—21no—
29C4×#67—21no—
30C4×#67C2211yescharacteristic
31#924—21no—
32#924—21no—
33#924C2211yescharacteristic
34#924C2211yescharacteristic
35C2×Dic12C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#897C2211yescharacteristic
38#896—21no—
39D3×C4⋊C4C2211yescharacteristic
40#889—21no—
41#888C2211yescharacteristic
42#887C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#818C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#554—31no—
61D4×Q8—31no—
62#487—31no—
63#484—31no—
64#484—31no—
65#477—31no—
66#467—31no—
67#467—31no—
68#462—31no—
69#451—31no—
70#451—31no—
71C4×Q16—31no—
72C8×D4—31no—
73C2×#103—31no—
74C2×#103D311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C22×C3×C4—21no—
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×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C2×C3×C8—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99#217—21no—
100#217—21no—
101#217D411yescharacteristic
102#216—22no—
103#216—41no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4D411yescharacteristic
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
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⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126D3×C8—41no—
127C22×Q8—31no—
128C22×Q8C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16—31no—
131C2×Q16—31no—
132C22×C8—31no—
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—61no—
137#125—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4—61no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4C2×D311yescharacteristic
153#107—31no—
154#107—31no—
155#107—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#103C2×D311yescharacteristic
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—42no—
176C3×Q8#4112yes—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—42no—
184C2×C3×C4—81no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67Q1614yescomplemented
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—42no—
198C4×D3—81no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204Dic6—81no—
205C3×C8—21no—
206C3×C8—21no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C2×Q8—21no—
210C2×Q8—22no—
211C2×Q8—31no—
212C2×Q8—31no—
213C2×Q8—41no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8C22×D311yescharacteristic
219C2×Q8C22×D311yescharacteristic
220C2×D4—31no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—31no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4C22×D311yescharacteristic
227Q16—62no—
228Q16—121no—
229Q16—121no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—41no—
233C2×C8—121no—
234C2×C8C22×D311yescharacteristic
235C2×C8C22×D311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4C22×D311yescharacteristic
251C4⋊C4C22×D311yescharacteristic
252C22⋊C4—31no—
253C22⋊C4—31no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C42—31no—
259C42—61no—
260C42—61no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3C2×Q1612yes—
264C22×C3C22×D411yescharacteristic
265C2×D3—21no—
266C2×D3—21no—
267C2×D3C2×Q1612yes—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4C2×#4111yescharacteristic
274C3×C4H(5,𝔽2)11yescharacteristic
275Dic3—21no—
276Dic3—21no—
277Dic3—41no—
278Dic3—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3C2×Q1612yes—
282C23—31no—
283C23D3×D411yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—42no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292Q8—122no—
293Q8—241no—
294Q8#23512yes—
295D4—34no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—42no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—81no—
313C2×C4—122no—
314C2×C4—241no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—22no—
322C2×C3#54611yescharacteristic
323C2×C3C22×Q1611yescharacteristic
324C2×C3#57711yescharacteristic
325D3—22no—
326C22—21no—
327C22—21no—
328C22—32no—
329C22—61no—
330C22—61no—
331C22D3×Q1612yes—
332C22C2×D3×D411yescharacteristic
333C4—21no—
334C4—22no—
335C4—32no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—61no—
340C4—61no—
341C4—121no—
342C4—121no—
343C4—121no—
344C4—121no—
345C4C2×#23511yescharacteristic
346C4#100411yescharacteristic
347C3#329911yescharacteristic, complemented, Sylow
348C2—22no—
349C2—62no—
350C2#549011yescharacteristic
351C2#576211yescharacteristic
352C2C2×D3×Q1611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#103C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenterC2×#103D3
hypocenterC3#3299
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×C3×#103C2semidirect
4. #5228C2semidirect
5. #5116C2semidirect
7. #4970C2semidirect
8. #4813C2semidirect
9. #4811C2semidirect
10. #4809C2semidirect
11. D3×#103C2semidirect
12. #4795C2semidirect
13. #4793C2semidirect
14. #4792C2semidirect
15. #4790C2semidirect
74. C2×#103D3semidirect
187. #67Q16semidirect
347. C3#3299semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#55112
C2×C3×#1032
#52282
#51162
C8×#672
#49702
#48132
#48112
#48092
D3×#1032
#47952
#47932
#47922
#47902
#32993