Finite Group Databasebeta

← #81313

Subgroups — #81313

Order 384.

384 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#5603C211yesmaximal, characteristic, complemented
2#5504C211yesmaximal, characteristic, complemented
3C3×#481C211yesmaximal, characteristic, complemented
4#5037C211yesmaximal, characteristic
5#4971C211yesmaximal, characteristic, complemented
6#4870C211yesmaximal, characteristic, complemented
7#4854C211yesmaximal, characteristic, complemented
8#4853C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4787C211yesmaximal, characteristic, complemented
11#4785C211yesmaximal, characteristic, complemented
12#4767C211yesmaximal, characteristic, complemented
13#4758C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3125—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C3×#121C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20#936—21no—
21#936C2211yescharacteristic
22#935—21no—
23D4×Dic3C2211yescharacteristic
24C2×#214C2211yescharacteristic
25C2×#213C2211yescharacteristic
26C4×#67—21no—
27C4×#67—21no—
28#926C2211yescharacteristic
29#924C2211yescharacteristic
30C2×Dic12C2211yescharacteristic
31C2×#204C2211yescharacteristic
32#893C2211yescharacteristic
33#893C2211yescharacteristic
34#892—21no—
35D3×C4⋊C4—21no—
36#888C2211yescharacteristic
37#887C2211yescharacteristic
38#887C2211yescharacteristic
39#887C2211yescharacteristic
40#879—21no—
41#878—21no—
42C3×#106C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic
47#830C2211yescharacteristic
48#821C2211yescharacteristic
49#816C2211yescharacteristic
50#815C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#542—31no—
61#520—31no—
62#483—31no—
63#481D311yescharacteristic, complemented
64#476—31no—
65#475—31no—
66#464—31no—
67#463—31no—
68#452—31no—
69#449—31no—
70#442—31no—
71#441—31no—
72C4×SD16—31no—
73C4×SD16—31no—
74#433—31no—
75C2×C3×D4—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C22×Dic3—21no—
81#235—22no—
82#235—22no—
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×Dic6—21no—
88C2×Dic6C2311yescharacteristic
89C2×Dic6C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217D411yescharacteristic
102#217D411yescharacteristic
103#214—22no—
104#213—22no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107C4⋊Dic3D411yescharacteristic
108C4⋊Dic3D411yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122Dic12—22no—
123#204—22no—
124C2×#41—31no—
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×D8—31no—
129C4⋊Q8—31no—
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121C2×D311yescharacteristic
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145#117—31no—
146#117—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4C2×D311yescharacteristic
149#106—31no—
150#106—31no—
151#106C2×D311yescharacteristic
152#106C2×D311yescharacteristic
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic
162#102C2×D311yescharacteristic
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C22×D3—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—81no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—81no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181#67—24no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—81no—
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190D12—21no—
191D12#4112yes—
192C4×D3—24no—
193C4×D3—81no—
194C4×D3—81no—
195Dic6—21no—
196Dic6—21no—
197Dic6—21no—
198Dic6—81no—
199Dic6#4112yes—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204#41—62no—
205#41—62no—
206C2×Q8—31no—
207C2×Q8—31no—
208C2×Q8—61no—
209C2×D4—21no—
210C2×D4—31no—
211C2×D4—61no—
212C2×D4C22×D311yescharacteristic
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219Q16—62no—
220SD16—62no—
221SD16—62no—
222D8—62no—
223C2×C8—31no—
224C2×C8—31no—
225C2×C8C22×D311yescharacteristic
226C2×C8C22×D311yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4C22×D311yescharacteristic
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—31no—
241C22⋊C4—31no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—31no—
249C42—31no—
250C42—61no—
251C42—61no—
252C42—61no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3C22×D411yescharacteristic
260C2×D3—22no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4C2×#4111yescharacteristic
268C3×C4C2×#4111yescharacteristic
269Dic3—22no—
270Dic3—21no—
271Dic3—21no—
272Dic3—21no—
273Dic3—21no—
274Dic3—41no—
275Dic3—41no—
276C23—21no—
277C23—61no—
278C23D3×D411yescharacteristic
279Q8—32no—
280Q8—61no—
281Q8—61no—
282Q8—61no—
283Q8—241no—
284D4—21no—
285D4—21no—
286D4—32no—
287D4—64no—
288D4—61no—
289D4—81no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—64no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—81no—
304C2×C4—241no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4D3×D411yescharacteristic
308C2×C4C23×D311yescharacteristic
309C8—21no—
310C8—21no—
311C8—61no—
312C8—61no—
313C2×C3—41no—
314C2×C3—81no—
315C2×C3#52511yescharacteristic
316C2×C3#57711yescharacteristic
317C2×C3#57711yescharacteristic
318D3—42no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—62no—
326C22—121no—
327C22—121no—
328C22C2×D3×D411yescharacteristic
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—62no—
334C4—61no—
335C4—61no—
336C4—61no—
337C4—61no—
338C4—121no—
339C4—121no—
340C4D3×#4111yescharacteristic
341C4D3×#4111yescharacteristic
342C3#312511yescharacteristic, complemented, Sylow
343C2—41no—
344C2—81no—
345C2—122no—
346C2#565611yescharacteristic
347C2#576211yescharacteristic
348C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#481C2
socleC22×C3C22×D4
prime radical#5037C2
hypercenter#481D3
hypocenterC3#3125
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #5603C2semidirect
2. #5504C2semidirect
3. C3×#481C2semidirect
5. #4971C2semidirect
6. #4870C2semidirect
7. #4854C2semidirect
8. #4853C2semidirect
9. #4852C2semidirect
10. #4787C2semidirect
11. #4785C2semidirect
12. #4767C2semidirect
13. #4758C2semidirect
14. #4733C2semidirect
15. #4720C2semidirect
63. #481D3semidirect
342. C3#3125semidirect

Maximal subgroups

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

subgroupindex
#56032
#55042
C3×#4812
#50372
#49712
#48702
#48542
#48532
#48522
#47872
#47852
#47672
#47582
#47332
#47202
#31253