Finite Group Databasebeta

← #81349

Subgroups — #81349

Order 384.

434 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5596C211yesmaximal, characteristic, complemented
2#5522C211yesmaximal, characteristic, complemented
3C3×#482C211yesmaximal, characteristic, complemented
4#5037C211yesmaximal, characteristic, complemented
5#5003C211yesmaximal, characteristic, complemented
6#4875C211yesmaximal, characteristic, complemented
7#4872C211yesmaximal, characteristic, complemented
8#4870C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4786C211yesmaximal, characteristic, complemented
11#4762C211yesmaximal, characteristic, complemented
12#4758C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4718C211yesmaximal, characteristic, complemented
16#3106—31nomaximal, Sylow
17C2×#237—21no—
18C2×#235C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#940C2211yescharacteristic
22#939—21no—
23D4×Dic3C2211yescharacteristic
24C2×#214C2211yescharacteristic
25C2×#213C2211yescharacteristic
26#930C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28C2×D24C2211yescharacteristic
29C2×#204C2211yescharacteristic
30#893C2211yescharacteristic
31#893C2211yescharacteristic
32#893C2211yescharacteristic
33#888C2211yescharacteristic
34#887C2211yescharacteristic
35#887C2211yescharacteristic
36#885—21no—
37#67⋊C4—21no—
38#67⋊C4—21no—
39#67⋊C4—21no—
40#872—21no—
41C42×D3—21no—
42C3×#107C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic
47#830C2211yescharacteristic
48#821C2211yescharacteristic
49#816C2211yescharacteristic
50Dic3⋊C8C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52C8×Dic3C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#810C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#532—31no—
61C4×#41—31no—
62#482—31no—
63#482D311yescharacteristic, complemented
64#477—31no—
65#476—31no—
66#460—31no—
67#459—31no—
68#449—31no—
69#449—31no—
70C4×SD16—31no—
71C4×SD16—31no—
72C4×D8—31no—
73C4×D8—31no—
74#431—31no—
75C2×C3×D4—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83C22×Dic3D411yescharacteristic
84C22×Dic3D411yescharacteristic
85#237—44no—
86#235—22no—
87#235—22no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×Dic6—21no—
95C2×Dic6C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C42—21no—
104#217—21no—
105#214—22no—
106#213—22no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3D411yescharacteristic
118C4×Dic3D411yescharacteristic
119C4×Dic3C2311yescharacteristic
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123D24—22no—
124#204—22no—
125C2×#41—31no—
126C2×#41—61no—
127C2×SD16—31no—
128C2×SD16—31no—
129C2×D8—31no—
130C2×D8—31no—
131C4⋊Q8—31no—
132#127—31no—
133#124—61no—
134#121—31no—
135#121—61no—
136#121C2×D311yescharacteristic
137C4×Q8—31no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145C4×D4—61no—
146#117—61no—
147#117C2×D311yescharacteristic
148C2×C42—31no—
149C2×C42—61no—
150#107—31no—
151#107C2×D311yescharacteristic
152#106—31no—
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155C4⋊C8—31no—
156#103—31no—
157#103—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102C2×D311yescharacteristic
163#102C2×D311yescharacteristic
164C22⋊C8—31no—
165C22⋊C8C2×D311yescharacteristic
166C4×C8—31no—
167C4×C8—31no—
168C23×C3—21no—
169C23×C3C2×D411yescharacteristic
170C22×D3—21no—
171C22×D3—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—42no—
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—24no—
183#67—42no—
184#67—42no—
185#67—42no—
186C2×Dic3—21no—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—21no—
190C2×Dic3—42no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195D12—21no—
196D12—21no—
197D12—21no—
198D12#4112yescomplemented
199C4×D3—24no—
200C4×D3—42no—
201C4×D3—42no—
202C4×D3—42no—
203Dic6—21no—
204Dic6—42no—
205Dic6#4112yescomplemented
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—62no—
211#41—62no—
212#41—124no—
213C2×Q8—31no—
214C2×Q8—61no—
215C2×D4—21no—
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4C22×D311yescharacteristic
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231SD16—62no—
232SD16—62no—
233D8—62no—
234D8—62no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8C22×D311yescharacteristic
238C2×C8C22×D311yescharacteristic
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4C22×D311yescharacteristic
244C4⋊C4C22×D311yescharacteristic
245C4⋊C4C22×D311yescharacteristic
246C22⋊C4—21no—
247C22⋊C4—21no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—21no—
254C42—31no—
255C42—31no—
256C42—31no—
257C42—31no—
258C42—61no—
259C42—61no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3C22×D411yescharacteristic
267C2×D3—22no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4C2×#4111yescharacteristic
279Dic3—22no—
280Dic3—21no—
281Dic3—21no—
282Dic3—41no—
283Dic3#13512yes—
284Dic3AGL(1,ℤ/8)12yes—
285C23—21no—
286C23—61no—
287C23—61no—
288C23D3×D411yescharacteristic
289Q8—32no—
290Q8—61no—
291Q8—122no—
292D4—21no—
293D4—21no—
294D4—32no—
295D4—42no—
296D4—64no—
297D4—61no—
298D4—61no—
299D4—61no—
300D4—122no—
301D4—122no—
302D4—122no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—64no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4D3×D411yescharacteristic
322C2×C4C23×D311yescharacteristic
323C8—21no—
324C8—21no—
325C8—61no—
326C8—61no—
327C2×C3—41no—
328C2×C3—81no—
329C2×C3#52511yescharacteristic
330C2×C3C2×#13511yescharacteristic
331C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
332D3—42no—
333D3—81no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—62no—
341C22—121no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—32no—
349C4—32no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—62no—
354C4—61no—
355C4—61no—
356C4—121no—
357C4D3×#4111yescharacteristic
358C4D3×#4111yescharacteristic
359C3#310611yescharacteristic, complemented, Sylow
360C2—41no—
361C2—81no—
362C2—122no—
363C2—241no—
364C2#565611yescharacteristic
365C2D3×#13511yescharacteristic
366C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#482C2
socleC22×C3C22×D4
hypercenter#482D3
hypocenterC3#3106
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #5596C2semidirect
2. #5522C2semidirect
3. C3×#482C2semidirect
4. #5037C2semidirect
5. #5003C2semidirect
6. #4875C2semidirect
7. #4872C2semidirect
8. #4870C2semidirect
9. #4852C2semidirect
10. #4786C2semidirect
11. #4762C2semidirect
12. #4758C2semidirect
13. #4756C2semidirect
14. #4734C2semidirect
15. #4718C2semidirect
63. #482D3semidirect
198. D12#41semidirect
205. Dic6#41semidirect
359. C3#3106semidirect

Maximal subgroups

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

subgroupindex
#55962
#55222
C3×#4822
#50372
#50032
#48752
#48722
#48702
#48522
#47862
#47622
#47582
#47562
#47342
#47182
#31063