Finite Group Databasebeta

← #79847

Subgroups — #79847

Order 384.

392 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#5580C211yesmaximal, characteristic, complemented
2C4×#239C211yesmaximal, characteristic, complemented
3C3×#441C211yesmaximal, characteristic, complemented
4#5187C211yesmaximal, characteristic, complemented
5#5185C211yesmaximal, characteristic, complemented
6#5026C211yesmaximal, characteristic, complemented
7#5026C211yesmaximal, characteristic, complemented
8#4871C211yesmaximal, characteristic, complemented
9#4866C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4815C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic
13#4799C211yesmaximal, characteristic, complemented
14#4711C211yesmaximal, characteristic, complemented
15#4705C211yesmaximal, characteristic, complemented
16#3111—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×C3×Q16C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21#947C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24C2×#215C2211yescharacteristic
25C2×#215C2211yescharacteristic
26C2×D24C2211yescharacteristic
27#898—21no—
28#895C2211yescharacteristic
29#894—21no—
30#893—21no—
31#893C2211yescharacteristic
32#892—21no—
33#892C2211yescharacteristic
34#890C2211yescharacteristic
35C4×D12—21no—
36C4×D12—21no—
37C4×D12C2211yescharacteristic
38C4×D12C2211yescharacteristic
39#872—21no—
40C42×D3—21no—
41C42×D3C2211yescharacteristic
42C3×#106C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#97C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#817C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#529—31no—
61C4×#41—31no—
62#485—31no—
63#482—31no—
64#479—31no—
65#478—31no—
66#464—31no—
67#463—31no—
68#452—31no—
69#449—31no—
70#442—31no—
71#441D311yescharacteristic, complemented
72#440—31no—
73#440—31no—
74#432—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77#239—22no—
78#239—22no—
79C2×D12—21no—
80C2×D12—21no—
81C2×D12C2311yescharacteristic
82C2×D12C2311yescharacteristic
83C2×C4×D3—21no—
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3D411yescharacteristic
88C2×C4×D3D411yescharacteristic
89C2×C4×D3C2311yescharacteristic
90C3×Q16—22no—
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C42—21no—
98C3×C42—21no—
99C3×C42C2311yescharacteristic
100#215—22no—
101#215—22no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3D411yescharacteristic
119C4×Dic3D411yescharacteristic
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123D24—22no—
124C2×#41—31no—
125C2×Q16C2×D311yescharacteristic
126C2×SD16—31no—
127C2×SD16—31no—
128C2×D8—31no—
129#126—61no—
130#125—31no—
131#124—31no—
132#123—61no—
133C22⋊Q8—31no—
134#121—31no—
135C4×Q8—31no—
136C4×Q8C2×D311yescharacteristic
137C4×Q8C2×D311yescharacteristic
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—61no—
142C4×D4—61no—
143C4×D4—61no—
144#117—31no—
145#117—61no—
146#117—61no—
147C2×C42—31no—
148C2×C42—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106C2×D311yescharacteristic
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103C2×D311yescharacteristic
158#103C2×D311yescharacteristic
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165#97—31no—
166#97C2×D311yescharacteristic
167C22×D3—21no—
168C22×D3—21no—
169C22×D3C2×D411yescharacteristic
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8—21no—
173C3×Q8#4112yes—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186D12—24no—
187D12—21no—
188D12—21no—
189D12—21no—
190D12—42no—
191D12#4112yes—
192C4×D3—24no—
193C4×D3—21no—
194C4×D3—21no—
195C4×D3—41no—
196C4×D3—41no—
197C4×D3—42no—
198C4×D3—42no—
199C4×D3—42no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204#41—62no—
205#41—62no—
206C2×Q8C22×D311yescharacteristic
207C2×Q8C22×D311yescharacteristic
208C2×D4—31no—
209C2×D4—31no—
210C2×D4—61no—
211C2×D4—61no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219Q16—22no—
220SD16—62no—
221SD16—62no—
222D8—62no—
223C2×C8—31no—
224C2×C8—31no—
225C2×C8C22×D311yescharacteristic
226C2×C8C22×D311yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4C22×D311yescharacteristic
236C4⋊C4C22×D311yescharacteristic
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C42—21no—
246C42—21no—
247C42—31no—
248C42—31no—
249C42—31no—
250C42—61no—
251C42—61no—
252C42C22×D311yescharacteristic
253C22×C3C22×D411yescharacteristic
254C2×D3—21no—
255C2×D3—22no—
256C2×D3—21no—
257C2×D3—21no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4C2×#4111yescharacteristic
270C3×C4C2×#4111yescharacteristic
271Dic3—21no—
272Dic3—21no—
273Dic3—21no—
274Dic3—41no—
275Dic3—41no—
276C23—31no—
277C23—61no—
278C23—61no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8#23512yes—
283D4—32no—
284D4—64no—
285D4—61no—
286D4—61no—
287D4—61no—
288D4—122no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—64no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4D3×D411yescharacteristic
307C2×C4D3×D411yescharacteristic
308C2×C4C23×D311yescharacteristic
309C8—21no—
310C8—21no—
311C8—61no—
312C8—61no—
313C2×C3#52511yescharacteristic
314C2×C3#57511yescharacteristic
315C2×C3#57511yescharacteristic
316D3—41no—
317D3—42no—
318D3—81no—
319C22—61no—
320C22—61no—
321C22—62no—
322C22—61no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22C2×D3×D411yescharacteristic
329C4—21no—
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—61no—
336C4—61no—
337C4—61no—
338C4—121no—
339C4—121no—
340C4C2×#23511yescharacteristic
341C4D3×#4111yescharacteristic
342C3#311111yescharacteristic, complemented, Sylow
343C2—121no—
344C2—122no—
345C2—241no—
346C2#554711yescharacteristic
347C2#577711yescharacteristic
348C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#441C2
socleC22×C3C22×D4
prime radical#4810C2
hypercenter#441D3
hypocenterC3#3111
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #5580C2semidirect
2. C4×#239C2semidirect
3. C3×#441C2semidirect
4. #5187C2semidirect
5. #5185C2semidirect
6. #5026C2semidirect
7. #5026C2semidirect
8. #4871C2semidirect
9. #4866C2semidirect
10. #4819C2semidirect
11. #4815C2semidirect
13. #4799C2semidirect
14. #4711C2semidirect
15. #4705C2semidirect
71. #441D3semidirect
342. C3#3111semidirect

Maximal subgroups

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

subgroupindex
#55802
C4×#2392
C3×#4412
#51872
#51852
#50262
#48712
#48662
#48192
#48152
#48102
#47992
#47112
#47052
#31113