Finite Group Databasebeta

← #78905

Subgroups — #78905

Order 384.

380 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#5831C211yesmaximal, characteristic
2#5513C211yesmaximal, characteristic, complemented
3C3×#417C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5137C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#4970C211yesmaximal, characteristic, complemented
8#4809C211yesmaximal, characteristic, complemented
9#4802C211yesmaximal, characteristic, complemented
10#4799C211yesmaximal, characteristic, complemented
11#4790C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4779C211yesmaximal, characteristic, complemented
14#4768C211yesmaximal, characteristic, complemented
15#4767C211yesmaximal, characteristic, complemented
16#3327—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic
22C2×#216C2211yescharacteristic
23#938—21no—
24#938C2211yescharacteristic
25#936—21no—
26#936C2211yescharacteristic
27C2×#214C2211yescharacteristic
28#929—21no—
29C4×#67—21no—
30C4×#67C2211yescharacteristic
31#926—21no—
32#924—21no—
33#924C2211yescharacteristic
34#924C2211yescharacteristic
35C2×Dic12C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#898—21no—
38#897C2211yescharacteristic
39#892C2211yescharacteristic
40#890C2211yescharacteristic
41#889—21no—
42#887C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#552—31no—
61#548—31no—
62#488—31no—
63#484—31no—
64#482—31no—
65#479—31no—
66#470—31no—
67#468—31no—
68#464—31no—
69#452—31no—
70#451—31no—
71C4×Q16—31no—
72#436—31no—
73#419—31no—
74#417D311yescharacteristic, complemented
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67D411yescharacteristic
85D3×Q8—22no—
86D3×Q8—41no—
87C2×C4×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C3×M16—41no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217D411yescharacteristic
104#216—22no—
105#214—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4D411yescharacteristic
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41C2×D311yescharacteristic
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C2×M16C2×D311yescharacteristic
134#126—61no—
135#125—31no—
136#125—61no—
137#124—31no—
138#124—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151#117—61no—
152C2×C4⋊C4C2×D311yescharacteristic
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#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8#4112yes—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—42no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—41no—
188#67—42no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—42no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—22no—
209#41—41no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—61no—
213C2×Q8—62no—
214C2×Q8—121no—
215C2×Q8C22×D311yescharacteristic
216C2×D4—21no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4C22×D311yescharacteristic
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—62no—
226Q16—121no—
227SD16—121no—
228M16—41no—
229M16—121no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—31no—
248C22⋊C4—31no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C42—31no—
257C42—61no—
258C42—61no—
259C42—61no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C3×C4—22no—
271C3×C4—21no—
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—61no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—122no—
292Q8#23512yes—
293D4—21no—
294D4—21no—
295D4—42no—
296D4—121no—
297D4—122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3—81no—
323C2×C3#54611yescharacteristic
324C2×C3#57511yescharacteristic
325C2×C3#57811yescharacteristic
326D3—41no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—61no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4C2×#23511yescharacteristic
349C4#100411yescharacteristic
350C3#332711yescharacteristic, complemented, Sylow
351C2—41no—
352C2—81no—
353C2—121no—
354C2#549011yescharacteristic
355C2#577311yescharacteristic
356C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
prime radical#5831C2
hypercenter#417D3
hypocenterC3#3327
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #5513C2semidirect
3. C3×#417C2semidirect
4. #5239C2semidirect
5. #5137C2semidirect
6. #5128C2semidirect
7. #4970C2semidirect
8. #4809C2semidirect
9. #4802C2semidirect
10. #4799C2semidirect
11. #4790C2semidirect
12. #4780C2semidirect
13. #4779C2semidirect
14. #4768C2semidirect
15. #4767C2semidirect
74. #417D3semidirect
350. C3#3327semidirect

Maximal subgroups

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

subgroupindex
#58312
#55132
C3×#4172
#52392
#51372
#51282
#49702
#48092
#48022
#47992
#47902
#47802
#47792
#47682
#47672
#33273