Finite Group Databasebeta

← #77509

Subgroups — #77509

Order 384.

412 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
1C3×#397C211yesmaximal, characteristic
2#4978C211yesmaximal, characteristic, complemented
3#4976C211yesmaximal, characteristic, complemented
4#4952C211yesmaximal, characteristic
5#4951C211yesmaximal, characteristic
6#4946C211yesmaximal, characteristic
7#4945C211yesmaximal, characteristic
8#4942C211yesmaximal, characteristic
9Dic3×C22⋊C4C211yesmaximal, characteristic
10#4658C211yesmaximal, characteristic, complemented
11#4658C211yesmaximal, characteristic, complemented
12#4655C211yesmaximal, characteristic, complemented
13#4654C211yesmaximal, characteristic, complemented
14#4652C211yesmaximal, characteristic, complemented
15#4648C211yesmaximal, characteristic, complemented
16#2530—31nomaximal, Sylow
17C23×Dic3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
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#400—31no—
53#400—31no—
54#397—31no—
55#397D311yescharacteristic
56#389—31no—
57#388—31no—
58#388—31no—
59#387—31no—
60#387—31no—
61#387—31no—
62#387—31no—
63#385—31no—
64#380—31no—
65C4×C22⋊C4—31no—
66#376—31no—
67C24×C3C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×Dic3—22no—
75C22×Dic3—22no—
76C22×Dic3—41no—
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C3×C4⋊C4—41no—
86C3×C22⋊C4—22no—
87C3×C22⋊C4—22no—
88C3×C22⋊C4—41no—
89#217—22no—
90#217—22no—
91#217—22no—
92C4⋊Dic3—22no—
93Dic3⋊C4—22no—
94Dic3⋊C4—22no—
95Dic3⋊C4—22no—
96Dic3⋊C4—22no—
97Dic3⋊C4—41no—
98Dic3⋊C4—41no—
99C4×Dic3—22no—
100C4×Dic3—22no—
101C4×Dic3—22no—
102C4×Dic3—41no—
103C23×C4—31no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4C2×D311yescharacteristic
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4C2×D311yescharacteristic
116C2×C22⋊C4C2×D311yescharacteristic
117C2×C22⋊C4C2×D311yescharacteristic
118C2×C42—31no—
119C2×C42—31no—
120C2×C42—31no—
121C2×C42—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#95C2×D311yescharacteristic
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—21no—
163C2×C3×C4—21no—
164C2×Dic3—21no—
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—44no—
185C2×Dic3#4112yes—
186C2×Dic3#4112yes—
187C2×Dic3#4112yes—
188C2×Dic3#4112yes—
189C24C22×D311yescharacteristic
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—121no—
201C22×C4C22×D311yescharacteristic
202C22×C4C22×D311yescharacteristic
203C22×C4C22×D311yescharacteristic
204C22×C4C22×D311yescharacteristic
205C22×C4C22×D311yescharacteristic
206C22×C4C22×D311yescharacteristic
207C4⋊C4—41no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—121no—
214C4⋊C4—121no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—41no—
218C22⋊C4—62no—
219C22⋊C4—62no—
220C22⋊C4—62no—
221C42—62no—
222C42—62no—
223C42—62no—
224C42—121no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3C2×#4111yescharacteristic
233C22×C3C2×#4111yescharacteristic
234C22×C3C2×#4111yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3H(5,𝔽2)11yescharacteristic
238C22×C32−1+411yescharacteristic
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—41no—
245Dic3—22no—
246Dic3—22no—
247Dic3—22no—
248Dic3—22no—
249Dic3—41no—
250Dic3—41no—
251Dic3—41no—
252Dic3—41no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23#23712yes—
260C23C23×D311yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
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—32no—
282C2×C4—32no—
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—124no—
304C2×C3—42no—
305C2×C3#52911yescharacteristic
306C2×C3#52911yescharacteristic
307C2×C3#54211yescharacteristic
308C2×C3#55111yescharacteristic
309C2×C3#55211yescharacteristic
310C2×C3#55211yescharacteristic
311C2×C3#55311yescharacteristic
312C22—22no—
313C22—22no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22C2×#23711yescharacteristic
320C22#100411yescharacteristic
321C22D3×#4111yescharacteristic
322C22D3×#4111yescharacteristic
323C22D3×#4111yescharacteristic
324C22D3×#4111yescharacteristic
325C22#101011yescharacteristic
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—62no—
333C4—62no—
334C4—62no—
335C4—62no—
336C4—121no—
337C4—121no—
338C4—121no—
339C4—121no—
340C3#253011yescharacteristic, complemented, Sylow
341C2—42no—
342C2#560111yescharacteristic
343C2#564911yescharacteristic
344C2#565011yescharacteristic
345C2#565911yescharacteristic
346C2#566911yescharacteristic
347C2#570611yescharacteristic
348C2#570811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC24×C3C23
hypercenter#397D3
hypocenterC3#2530
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #4978C2semidirect
3. #4976C2semidirect
10. #4658C2semidirect
11. #4658C2semidirect
12. #4655C2semidirect
13. #4654C2semidirect
14. #4652C2semidirect
15. #4648C2semidirect
340. C3#2530semidirect

Maximal subgroups

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

subgroupindex
C3×#3972
#49782
#49762
#49522
#49512
#49462
#49452
#49422
Dic3×C22⋊C42
#46582
#46552
#46542
#46522
#46482
#25303