Finite Group Databasebeta

← #81001

Subgroups — #81001

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#5733C211yesmaximal, characteristic, complemented
2#5574C211yesmaximal, characteristic, complemented
3C3×#475C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5028C211yesmaximal, characteristic, complemented
6#4869C211yesmaximal, characteristic, complemented
7#4863C211yesmaximal, characteristic, complemented
8#4862C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic
13#4808C211yesmaximal, characteristic, complemented
14#4789C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3125—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C3×C4⋊Q8C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20#947—21no—
21#946—21no—
22Q8×Dic3C2211yescharacteristic
23C2×#216C2211yescharacteristic
24C2×#215C2211yescharacteristic
25C2×D24C2211yescharacteristic
26C2×#204C2211yescharacteristic
27#898—21no—
28#897C2211yescharacteristic
29#895C2211yescharacteristic
30#894—21no—
31#893—21no—
32#893C2211yescharacteristic
33#893C2211yescharacteristic
34D3×C4⋊C4—21no—
35D3×C4⋊C4C2211yescharacteristic
36#887C2211yescharacteristic
37#875C2211yescharacteristic
38C4×D12—21no—
39C4×D12C2211yescharacteristic
40#872—21no—
41#872C2211yescharacteristic
42C3×#106C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×C4⋊C8C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#821C2211yescharacteristic
48(C2×D3)⋊C8C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#816C2211yescharacteristic
51#815C2211yescharacteristic
52C8×Dic3C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#542—31no—
61#520—31no—
62#483—31no—
63#481—31no—
64#476—31no—
65#475D311yescharacteristic, complemented
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×Q8—21no—
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×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×Dic6C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C42—21no—
98C3×C42C2311yescharacteristic
99#216—22no—
100#215—22no—
101(C2×D3)⋊C4—21no—
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)⋊C4D411yescharacteristic
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122D24—22no—
123#204—22no—
124C2×#41—31no—
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×D8—31no—
129C4⋊Q8C2×D311yescharacteristic
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8C2×D311yescharacteristic
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145#117—31no—
146#117—61no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149#106—31no—
150#106—31no—
151#106C2×D311yescharacteristic
152#106C2×D311yescharacteristic
153C4⋊C8—31no—
154C4⋊C8C2×D311yescharacteristic
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—
166C4×C8—31no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3C2×D411yescharacteristic
170C3×Q8—21no—
171C3×Q8—81no—
172C3×Q8#4112yes—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185D12—24no—
186D12—21no—
187D12—21no—
188D12—21no—
189D12—81no—
190D12#4112yes—
191C4×D3—24no—
192C4×D3—21no—
193C4×D3—21no—
194C4×D3—81no—
195C4×D3—81no—
196C4×D3—81no—
197C4×D3—81no—
198Dic6—21no—
199Dic6—21no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204#41—62no—
205#41—62no—
206C2×Q8—21no—
207C2×Q8—31no—
208C2×Q8C22×D311yescharacteristic
209C2×D4—31no—
210C2×D4—31no—
211C2×D4—61no—
212C2×D4—61no—
213C22×C4—31no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
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—21no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4C22×D311yescharacteristic
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—31no—
240C22⋊C4—31no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—21no—
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—61no—
281Q8—61no—
282Q8—81no—
283Q8#23912yes—
284D4—32no—
285D4—64no—
286D4—61no—
287D4—61no—
288D4—61no—
289D4—241no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—64no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—241no—
303C2×C4—241no—
304C2×C4—241no—
305C2×C4—241no—
306C2×C4D3×D411yescharacteristic
307C2×C4D3×D411yescharacteristic
308C2×C4C23×D311yescharacteristic
309C8—21no—
310C8—21no—
311C8—61no—
312C8—61no—
313C2×C3#52511yescharacteristic
314C2×C3#57711yescharacteristic
315C2×C3#57711yescharacteristic
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×#23911yescharacteristic
341C4D3×#4111yescharacteristic
342C3#312511yescharacteristic, complemented, Sylow
343C2—121no—
344C2—122no—
345C2—241no—
346C2#562711yescharacteristic
347C2#576211yescharacteristic
348C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
prime radical#4810C2
hypercenter#475D3
hypocenterC3#3125
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #5733C2semidirect
2. #5574C2semidirect
3. C3×#475C2semidirect
4. #5084C2semidirect
5. #5028C2semidirect
6. #4869C2semidirect
7. #4863C2semidirect
8. #4862C2semidirect
9. #4852C2semidirect
10. #4844C2semidirect
11. #4840C2semidirect
13. #4808C2semidirect
14. #4789C2semidirect
15. #4788C2semidirect
65. #475D3semidirect
342. C3#3125semidirect

Maximal subgroups

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

subgroupindex
#57332
#55742
C3×#4752
#50842
#50282
#48692
#48632
#48622
#48522
#48442
#48402
#48102
#48082
#47892
#47882
#31253