Finite Group Databasebeta

← #81305

Subgroups — #81305

Order 384.

386 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#5598C211yesmaximal, characteristic
2#5513C211yesmaximal, characteristic, complemented
3C3×#480C211yesmaximal, characteristic, complemented
4#5041C211yesmaximal, characteristic, complemented
5C2×#810C211yesmaximal, characteristic, complemented
6#4888C211yesmaximal, characteristic, complemented
7#4887C211yesmaximal, characteristic, complemented
8#4877C211yesmaximal, characteristic, complemented
9#4873C211yesmaximal, characteristic, complemented
10#4783C211yesmaximal, characteristic, complemented
11#4772C211yesmaximal, characteristic, complemented
12#4768C211yesmaximal, characteristic, complemented
13#4767C211yesmaximal, characteristic, complemented
14#4739C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16#3300—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#939—21no—
22#936C2211yescharacteristic
23D4×Dic3—21no—
24C2×#214C2211yescharacteristic
25#929—21no—
26C4×#67C2211yescharacteristic
27#926—21no—
28#924C2211yescharacteristic
29C22×C3⋊C8C2211yescharacteristic
30C2×Dic12C2211yescharacteristic
31C2×Dic12C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic
33#898—21no—
34#897C2211yescharacteristic
35#897C2211yescharacteristic
36#892C2211yescharacteristic
37#890C2211yescharacteristic
38#889—21no—
39#887C2211yescharacteristic
40#887C2211yescharacteristic
41#885—21no—
42#878—21no—
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47C3×C22⋊C8C2211yescharacteristic
48#832C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#816C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#810C2211yescharacteristic
59#808C2211yescharacteristic
60#552—31no—
61#548—31no—
62#488—31no—
63#485—31no—
64#480D311yescharacteristic, complemented
65#479—31no—
66#467—31no—
67#465—31no—
68#464—31no—
69#452—31no—
70#451—31no—
71C4×Q16—31no—
72C8×D4—31no—
73#416—31no—
74C2×#103—31no—
75C2×C3×D4—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67D411yescharacteristic
81C22×Dic3—21no—
82C22×Dic3—21no—
83#237—22no—
84#237—41no—
85C2×C4×D3C2311yescharacteristic
86C2×Dic6—22no—
87C2×Dic6—21no—
88C2×Dic6—41no—
89C2×Dic6C2311yescharacteristic
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217D411yescharacteristic
103#214—41no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4D411yescharacteristic
108C4⋊Dic3—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124Dic12—22no—
125Dic12—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C22×C8—31no—
133C22×C8—31no—
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—61no—
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151#117—61no—
152C2×C4⋊C4C2×D311yescharacteristic
153#107C2×D311yescharacteristic
154#107C2×D311yescharacteristic
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#103—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—42no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—42no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184#67—22no—
185#67—42no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C4×D3—21no—
196C4×D3—21no—
197Dic6—21no—
198Dic6—21no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—42no—
203Dic6#4112yes—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—62no—
209#41—121no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—31no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—121no—
216C2×D4—21no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4C22×D311yescharacteristic
220C22×C4—21no—
221C22×C4—31no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—62no—
226Q16—121no—
227SD16—121no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8—121no—
231C2×C8—121no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—31no—
249C22⋊C4—31no—
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—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3#13512yes—
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—22no—
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—21no—
284C23—31no—
285C23D3×D411yescharacteristic
286Q8—32no—
287Q8—61no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292Q8—122no—
293D4—21no—
294D4—21no—
295D4—42no—
296D4—62no—
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—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—22no—
322C2×C3—81no—
323C2×C3#54611yescharacteristic
324C2×C3C2×#13511yescharacteristic
325C2×C3#57811yescharacteristic
326D3—41no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—61no—
333C22—61no—
334C22—61no—
335C22#91212yes—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—61no—
342C4—62no—
343C4—61no—
344C4—61no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4D3×#4111yescharacteristic
349C4#100911yescharacteristic
350C3#330011yescharacteristic, complemented, Sylow
351C2—22no—
352C2—81no—
353C2—121no—
354C2#565411yescharacteristic
355C2C2×#91211yescharacteristic
356C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#480C2
socleC22×C3C22×D4
prime radical#5598C2
hypercenter#480D3
hypocenterC3#3300
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #5513C2semidirect
3. C3×#480C2semidirect
4. #5041C2semidirect
5. C2×#810C2semidirect
6. #4888C2semidirect
7. #4887C2semidirect
8. #4877C2semidirect
9. #4873C2semidirect
10. #4783C2semidirect
11. #4772C2semidirect
12. #4768C2semidirect
13. #4767C2semidirect
14. #4739C2semidirect
15. #67⋊C8C2semidirect
64. #480D3semidirect
350. C3#3300semidirect

Maximal subgroups

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

subgroupindex
#55982
#55132
C3×#4802
#50412
C2×#8102
#48882
#48872
#48772
#48732
#47832
#47722
#47682
#47672
#47392
#67⋊C82
#33003