Finite Group Databasebeta

← #80253

Subgroups — #80253

Order 384.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5824C211yesmaximal, characteristic
2#5639C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5233C211yesmaximal, characteristic, complemented
5#5189C211yesmaximal, characteristic, complemented
6#5182C211yesmaximal, characteristic, complemented
7#5173C211yesmaximal, characteristic, complemented
8#5164C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#4811C211yesmaximal, characteristic, complemented
11#4797C211yesmaximal, characteristic, complemented
12#4771C211yesmaximal, characteristic, complemented
13#4763C211yesmaximal, characteristic, complemented
14#4730C211yesmaximal, characteristic, complemented
15#4723C211yesmaximal, characteristic, complemented
16#3315—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#946C2211yescharacteristic
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27#939—21no—
28#935—21no—
29D4×Dic3C2211yescharacteristic
30#929—21no—
31C4×#67C2211yescharacteristic
32#926C2211yescharacteristic
33#924C2211yescharacteristic
34C2×Dic3⋊C4—21no—
35C22×C3⋊C8C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#892—21no—
38D3×C4⋊C4C2211yescharacteristic
39#890C2211yescharacteristic
40#889—21no—
41#883—21no—
42#879—21no—
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#818C2211yescharacteristic
53#817C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#807C2211yescharacteristic
60#552—31no—
61#547—31no—
62#491—31no—
63#485—31no—
64#484—31no—
65#478—31no—
66#470—31no—
67#467—31no—
68#462—31no—
69#452—31no—
70#452D311yescharacteristic, complemented
71#440—31no—
72#435—31no—
73#417—31no—
74#416—31no—
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67D411yescharacteristic
85C22×Dic3—21no—
86C22×Dic3—21no—
87#237—22no—
88#237—41no—
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C3×Q16—41no—
93C3×SD16—22no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217D411yescharacteristic
103#216—41no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4D411yescharacteristic
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C2×#41C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×SD16C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#126—61no—
136#125—31no—
137#125—61no—
138#123—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8C2×D311yescharacteristic
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149#117—31no—
150#117—61no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4—42no—
179C3×D4#4112yes—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—41no—
189#67—42no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C4×D3—21no—
200C4×D3—21no—
201C4×D3—42no—
202Dic6—21no—
203Dic6—21no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—22no—
209#41—41no—
210#41—62no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8C22×D311yescharacteristic
214C2×Q8C22×D311yescharacteristic
215C2×D4—21no—
216C2×D4—31no—
217C2×D4—61no—
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—41no—
226Q16—121no—
227SD16—22no—
228M16—121no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—121no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4C22×D311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—31no—
251C22⋊C4—31no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C42—31no—
258C42—61no—
259C42—61no—
260C22×C3—22no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4C2×#4111yescharacteristic
275C3×C4H(5,𝔽2)11yescharacteristic
276Dic3—21no—
277Dic3—21no—
278Dic3—21no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—21no—
284C23—31no—
285C23D3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—61no—
291Q8—61no—
292D4—21no—
293D4—42no—
294D4—121no—
295D4—122no—
296D4#23712yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—42no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3—42no—
323C2×C3#54611yescharacteristic
324C2×C3#57511yescharacteristic
325C2×C3#57811yescharacteristic
326D3—41no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—61no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4C2×#23711yescharacteristic
349C4#100411yescharacteristic
350C3#331511yescharacteristic, complemented, Sylow
351C2—41no—
352C2—42no—
353C2—121no—
354C2#559111yescharacteristic
355C2#576811yescharacteristic
356C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radical#5824C2
hypercenter#452D3
hypocenterC3#3315
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #5639C2semidirect
3. C3×#452C2semidirect
4. #5233C2semidirect
5. #5189C2semidirect
6. #5182C2semidirect
7. #5173C2semidirect
8. #5164C2semidirect
9. #5051C2semidirect
10. #4811C2semidirect
11. #4797C2semidirect
12. #4771C2semidirect
13. #4763C2semidirect
14. #4730C2semidirect
15. #4723C2semidirect
70. #452D3semidirect
350. C3#3315semidirect

Maximal subgroups

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

subgroupindex
#58242
#56392
C3×#4522
#52332
#51892
#51822
#51732
#51642
#50512
#48112
#47972
#47712
#47632
#47302
#47232
#33153