Finite Group Databasebeta

← #74470

Subgroups — #74470

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#385C211yesmaximal, characteristic
2#5211C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4986C211yesmaximal, characteristic, complemented
5#4980C211yesmaximal, characteristic, complemented
6#4978C211yesmaximal, characteristic, complemented
7#4973C211yesmaximal, characteristic, complemented
8#4952C211yesmaximal, characteristic
9#4951C211yesmaximal, characteristic
10#4947C211yesmaximal, characteristic
11#4944C211yesmaximal, characteristic
12C3⋊#401C211yesmaximal, characteristic, complemented
13#4652C211yesmaximal, characteristic, complemented
14#4648C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2527—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C3×#95C2211yescharacteristic
35C3×#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C3⋊#95C2211yescharacteristic
41C3⋊#95C2211yescharacteristic
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#401—31no—
53#400—31no—
54#397—31no—
55#393—31no—
56#391—31no—
57#391—31no—
58#388—31no—
59#388—31no—
60#387—31no—
61#387—31no—
62#387—31no—
63#385D311yescharacteristic
64#381—31no—
65C4×C22⋊C4—31no—
66#376—31no—
67C24×C3C2311yescharacteristic
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4—41no—
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C2×Dic6—22no—
86C2×Dic6—22no—
87C2×Dic6—41no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—41no—
90C3×C22⋊C4—22no—
91C3×C22⋊C4—22no—
92#217—22no—
93#217—22no—
94#217—22no—
95#217—41no—
96C4⋊Dic3—41no—
97Dic3⋊C4—22no—
98Dic3⋊C4—22no—
99C4×Dic3—22no—
100C4×Dic3—22no—
101C4×Dic3—22no—
102C4×Dic3—41no—
103C22×Q8—31no—
104C23×C4C2×D311yescharacteristic
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4C2×D311yescharacteristic
109C2×C4⋊C4C2×D311yescharacteristic
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4C2×D311yescharacteristic
115C2×C22⋊C4C2×D311yescharacteristic
116C2×C42—31no—
117C2×C42—31no—
118C2×C42—31no—
119C2×C42—31no—
120#95—31no—
121#95—31no—
122#95—31no—
123#95—31no—
124#95—31no—
125#95—31no—
126#95—31no—
127#95—31no—
128#95—31no—
129#95—31no—
130#95—31no—
131#95—31no—
132#95—31no—
133#95—31no—
134#95—31no—
135#95—31no—
136#95C2×D311yescharacteristic
137#95C2×D311yescharacteristic
138C23×C3—21no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3#4112yes—
145C23×C3C2411yescharacteristic
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—44no—
163C2×C3×C4#4112yes—
164C2×C3×C4#4112yes—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—21no—
168C2×Dic3—21no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—21no—
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3—21no—
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3#4112yes—
188C2×Dic3#4112yes—
189Dic6—44no—
190C24C22×D311yescharacteristic
191C2×Q8—62no—
192C2×Q8—62no—
193C2×Q8—121no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—41no—
205C22×C4C22×D311yescharacteristic
206C22×C4C22×D311yescharacteristic
207C22×C4C22×D311yescharacteristic
208C22×C4C22×D311yescharacteristic
209C22×C4C22×D311yescharacteristic
210C22×C4C22×D311yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—62no—
214C4⋊C4—62no—
215C4⋊C4—121no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—62no—
219C22⋊C4—62no—
220C22⋊C4—62no—
221C22⋊C4—121no—
222C42—62no—
223C42—62no—
224C42—62no—
225C42—121no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3C2×#4111yescharacteristic
234C22×C3C2×#4111yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3C2×#4111yescharacteristic
238C22×C3H(5,𝔽2)11yescharacteristic
239C22×C32−1+411yescharacteristic
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—41no—
246Dic3—22no—
247Dic3—22no—
248Dic3—41no—
249Dic3—41no—
250Dic3—41no—
251Dic3—41no—
252Dic3—41no—
253Dic3—41no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#23712yes—
261C23C23×D311yescharacteristic
262Q8—124no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—44no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4#23512yes—
305C2×C4#23712yes—
306C2×C3—42no—
307C2×C3#52911yescharacteristic
308C2×C3#52911yescharacteristic
309C2×C3#54211yescharacteristic
310C2×C3#55111yescharacteristic
311C2×C3#55111yescharacteristic
312C2×C3#55211yescharacteristic
313C2×C3#55611yescharacteristic
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C2×#23511yescharacteristic
322C22C2×#23711yescharacteristic
323C22C2×#23711yescharacteristic
324C22#100411yescharacteristic
325C22#100711yescharacteristic
326C22D3×#4111yescharacteristic
327C22D3×#4111yescharacteristic
328C4—22no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—62no—
335C4—62no—
336C4—121no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4—121no—
342C3#252711yescharacteristic, complemented, Sylow
343C2—42no—
344C2#551311yescharacteristic
345C2#555611yescharacteristic
346C2#555711yescharacteristic
347C2#558711yescharacteristic
348C2#564011yescharacteristic
349C2#564111yescharacteristic
350C2#564911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2527
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #4986C2semidirect
5. #4980C2semidirect
6. #4978C2semidirect
7. #4973C2semidirect
12. C3⋊#401C2semidirect
13. #4652C2semidirect
14. #4648C2semidirect
15. #4646C2semidirect
342. C3#2527semidirect

Maximal subgroups

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

subgroupindex
C3×#3852
#52112
C4×#2172
#49862
#49802
#49782
#49732
#49522
#49512
#49472
#49442
C3⋊#4012
#46522
#46482
#46462
#25273