Finite Group Databasebeta

← #79811

Subgroups — #79811

Order 384.

430 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
1C4×#239C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#440C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5163C211yesmaximal, characteristic, complemented
6#5026C211yesmaximal, characteristic, complemented
7#5014C211yesmaximal, characteristic, complemented
8#4890C211yesmaximal, characteristic, complemented
9#4885C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4791C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4779C211yesmaximal, characteristic, complemented
14#4711C211yesmaximal, characteristic, complemented
15#4705C211yesmaximal, characteristic, complemented
16#3115—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#235—21no—
19C2×C3×SD16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23C2×#215C2211yescharacteristic
24#940C2211yescharacteristic
25#938C2211yescharacteristic
26C2×#213C2211yescharacteristic
27C4×#67—21no—
28C2×D24C2211yescharacteristic
29#895C2211yescharacteristic
30#893—21no—
31#893C2211yescharacteristic
32#892—21no—
33#892C2211yescharacteristic
34#888C2211yescharacteristic
35#885—21no—
36#884—21no—
37C4×D12—21no—
38C4×D12C2211yescharacteristic
39C4×D12C2211yescharacteristic
40C42×D3—21no—
41C42×D3C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#97C2211yescharacteristic
46#835C2211yescharacteristic
47#832C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#817C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#532—31no—
61C4×#41—31no—
62#482—31no—
63#482—31no—
64#475—31no—
65#474—31no—
66#460—31no—
67#459—31no—
68#449—31no—
69#449—31no—
70#442—31no—
71#442—31no—
72#440—31no—
73#440D311yescharacteristic, complemented
74#432—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81#239—22no—
82#239—22no—
83#235—44no—
84C2×D12—21no—
85C2×D12—21no—
86C2×D12C2311yescharacteristic
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×C4×D3D411yescharacteristic
93C2×C4×D3D411yescharacteristic
94C2×C4×D3C2311yescharacteristic
95C2×Dic6—21no—
96C3×SD16—22no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C42—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#215—22no—
107#213—22no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3D411yescharacteristic
120C4×Dic3D411yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124D24—22no—
125C2×#41—31no—
126C2×#41—61no—
127C2×SD16—31no—
128C2×SD16C2×D311yescharacteristic
129C2×D8—31no—
130C2×D8—31no—
131C4⋊Q8—31no—
132#127—31no—
133#124—61no—
134#121—31no—
135#121—31no—
136#121—61no—
137C4×Q8—31no—
138C4×Q8C2×D311yescharacteristic
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—61no—
143C4×D4—61no—
144C4×D4—61no—
145C4×D4C2×D311yescharacteristic
146#117—31no—
147#117—61no—
148C2×C42—31no—
149C2×C42—61no—
150#107—31no—
151#107C2×D311yescharacteristic
152#106—31no—
153#106—31no—
154C4⋊C8—31no—
155C4⋊C8—31no—
156#103—31no—
157#103C2×D311yescharacteristic
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102C2×D311yescharacteristic
164C22⋊C8—31no—
165C22⋊C8—31no—
166#97—31no—
167#97C2×D311yescharacteristic
168C23×C3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8#4112yescomplemented
174C3×D4—21no—
175C3×D4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—42no—
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183#67—42no—
184#67—42no—
185#67—42no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191D12—24no—
192D12—21no—
193D12—21no—
194D12—21no—
195D12—42no—
196D12#4112yescomplemented
197C4×D3—24no—
198C4×D3—21no—
199C4×D3—21no—
200C4×D3—22no—
201C4×D3—41no—
202C4×D3—42no—
203C4×D3—42no—
204C4×D3—42no—
205Dic6—42no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—62no—
211#41—62no—
212#41—124no—
213C2×Q8—61no—
214C2×Q8C22×D311yescharacteristic
215C2×D4—31no—
216C2×D4—31no—
217C2×D4—61no—
218C2×D4—61no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4C22×D311yescharacteristic
223C22×C4—21no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231SD16—22no—
232SD16—62no—
233D8—62no—
234D8—62no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8C22×D311yescharacteristic
238C2×C8C22×D311yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C4⋊C4C22×D311yescharacteristic
246C22⋊C4—21no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—21no—
254C42—31no—
255C42—31no—
256C42—31no—
257C42—61no—
258C42—61no—
259C42C22×D311yescharacteristic
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3C22×D411yescharacteristic
264C2×D3—21no—
265C2×D3—22no—
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4C2×#4111yescharacteristic
279C3×C4C2×#4111yescharacteristic
280Dic3—21no—
281Dic3—21no—
282Dic3—41no—
283Dic3—41no—
284Dic3AGL(1,ℤ/8)12yes—
285C23—21no—
286C23—31no—
287C23—61no—
288C23—61no—
289Q8—21no—
290Q8—122no—
291Q8#23512yescomplemented
292D4—21no—
293D4—21no—
294D4—32no—
295D4—64no—
296D4—61no—
297D4—61no—
298D4—61no—
299D4—122no—
300D4—122no—
301D4—122no—
302D4—122no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—42no—
310C2×C4—62no—
311C2×C4—64no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—121no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4D3×D411yescharacteristic
321C2×C4D3×D411yescharacteristic
322C2×C4C23×D311yescharacteristic
323C8—21no—
324C8—21no—
325C8—61no—
326C8—61no—
327C2×C3—81no—
328C2×C3#52511yescharacteristic
329C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
330C2×C3#57511yescharacteristic
331D3—41no—
332D3—42no—
333D3—81no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—61no—
338C22—61no—
339C22—62no—
340C22—61no—
341C22—121no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—21no—
349C4—22no—
350C4—32no—
351C4—41no—
352C4—41no—
353C4—61no—
354C4—61no—
355C4—121no—
356C4—121no—
357C4C2×#23511yescharacteristic
358C4D3×#4111yescharacteristic
359C3#311511yescharacteristic, complemented, Sylow
360C2—81no—
361C2—121no—
362C2—122no—
363C2—241no—
364C2#554711yescharacteristic
365C2D3×AGL(1,ℤ/8)11yescharacteristic
366C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3115
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C4×#239C2semidirect
2. #5548C2semidirect
3. C3×#440C2semidirect
4. #5172C2semidirect
5. #5163C2semidirect
6. #5026C2semidirect
7. #5014C2semidirect
8. #4890C2semidirect
9. #4885C2semidirect
10. #4810C2semidirect
11. #4791C2semidirect
12. #4786C2semidirect
13. #4779C2semidirect
14. #4711C2semidirect
15. #4705C2semidirect
73. #440D3semidirect
173. C3×Q8#41semidirect
196. D12#41semidirect
291. Q8#235semidirect
359. C3#3115semidirect

Maximal subgroups

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

subgroupindex
C4×#2392
#55482
C3×#4402
#51722
#51632
#50262
#50142
#48902
#48852
#48102
#47912
#47862
#47792
#47112
#47052
#31153