Finite Group Databasebeta

← #75294

Subgroups — #75294

Order 384.

418 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#387C211yesmaximal, characteristic, complemented
2#4990C211yesmaximal, characteristic
3#4990C211yesmaximal, characteristic
4#4980C211yesmaximal, characteristic, complemented
5#4978C211yesmaximal, characteristic, complemented
6#4940C211yesmaximal, characteristic
7C42×Dic3C211yesmaximal, characteristic, complemented
8#4670C211yesmaximal, characteristic
9#4670C211yesmaximal, characteristic
10#4669C211yesmaximal, characteristic
11#4669C211yesmaximal, characteristic
12#4655C211yesmaximal, characteristic, complemented
13#4654C211yesmaximal, characteristic, complemented
14#4653C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2866—31nomaximal, Sylow
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
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#388—31no—
53#388—31no—
54#388—31no—
55#388—31no—
56#388—31no—
57#388—31no—
58#388—31no—
59#387—31no—
60#387—31no—
61#387—31no—
62#387—31no—
63#387—31no—
64#387—31no—
65#387D311yescharacteristic, complemented
66C43—31no—
67C23×D3C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C3×C4⋊C4—22no—
83C3×C4⋊C4—22no—
84C3×C42—22no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—22no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94Dic3⋊C4—22no—
95Dic3⋊C4—22no—
96Dic3⋊C4—22no—
97C4×Dic3—22no—
98C4×Dic3—22no—
99C4×Dic3—22no—
100C4×Dic3—22no—
101C4×Dic3—22no—
102C4×Dic3—22no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—31no—
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⋊C4—31no—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—31no—
117C2×C42—31no—
118C2×C42—31no—
119C2×C42—31no—
120C2×C42—31no—
121C2×C42—31no—
122C2×C42—31no—
123C2×C42C2×D311yescharacteristic
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#95C2×D311yescharacteristic
135#95C2×D311yescharacteristic
136#95C2×D311yescharacteristic
137#95C2×D311yescharacteristic
138C23×C3C2411yescharacteristic
139C22×D3—21no—
140C22×D3—21no—
141C22×D3—21no—
142C22×D3—21no—
143C22×D3—21no—
144C22×D3—21no—
145C22×D3—21no—
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×C3×C4#4112yes—
165C2×C3×C4#4112yes—
166C2×C3×C4#4112yes—
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#4112yes—
185C2×Dic3#4112yes—
186C2×Dic3#4112yes—
187C2×Dic3#4112yes—
188C24—31no—
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D311yescharacteristic
198C22×C4C22×D311yescharacteristic
199C22×C4C22×D311yescharacteristic
200C22×C4C22×D311yescharacteristic
201C22×C4C22×D311yescharacteristic
202C22×C4C22×D311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—62no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C22⋊C4—62no—
211C22⋊C4—62no—
212C22⋊C4—62no—
213C22⋊C4—62no—
214C22⋊C4—62no—
215C22⋊C4—62no—
216C22⋊C4—62no—
217C42—22no—
218C42—62no—
219C42—62no—
220C42—62no—
221C42—62no—
222C42—62no—
223C42—62no—
224C22×C3C2×#4111yescharacteristic
225C22×C3C2×#4111yescharacteristic
226C22×C3C2×#4111yescharacteristic
227C22×C3C2×#4111yescharacteristic
228C22×C3C2×#4111yescharacteristic
229C22×C3C2×#4111yescharacteristic
230C22×C3C2×#4111yescharacteristic
231C2×D3—41no—
232C2×D3—41no—
233C2×D3—41no—
234C2×D3—41no—
235C2×D3—41no—
236C2×D3—41no—
237C2×D3—41no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—22no—
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—
252C23—61no—
253C23—61no—
254C23—61no—
255C23—61no—
256C23—61no—
257C23—61no—
258C23—61no—
259C23C23×D311yescharacteristic
260C2×C4—21no—
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—32no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—32no—
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#23512yes—
300C2×C4#23712yes—
301C2×C4#23912yes—
302C2×C3#52911yescharacteristic
303C2×C3#52911yescharacteristic
304C2×C3#52911yescharacteristic
305C2×C3#52911yescharacteristic
306C2×C3#52911yescharacteristic
307C2×C3#52911yescharacteristic
308C2×C3#52911yescharacteristic
309D3—81no—
310C22—121no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22—121no—
315C22—121no—
316C22—121no—
317C22C2×#23511yescharacteristic
318C22C2×#23711yescharacteristic
319C22C2×#23911yescharacteristic
320C22D3×#4111yescharacteristic
321C22D3×#4111yescharacteristic
322C22D3×#4111yescharacteristic
323C22D3×#4111yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—62no—
332C4—62no—
333C4—62no—
334C4—62no—
335C4—121no—
336C4—121no—
337C4—121no—
338C3#286611yescharacteristic, complemented, Sylow
339C2—241no—
340C2#552811yescharacteristic
341C2#553811yescharacteristic
342C2#558011yescharacteristic
343C2#562411yescharacteristic
344C2#564911yescharacteristic
345C2#569111yescharacteristic
346C2#570611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#387C2
prime radicalC23×D3C23
hypercenter#387D3
hypocenterC3#2866
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C3×#387C2semidirect
4. #4980C2semidirect
5. #4978C2semidirect
7. C42×Dic3C2semidirect
12. #4655C2semidirect
13. #4654C2semidirect
14. #4653C2semidirect
15. #4652C2semidirect
65. #387D3semidirect
338. C3#2866semidirect

Maximal subgroups

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

subgroupindex
C3×#3872
#49902
#49802
#49782
#49402
C42×Dic32
#46702
#46692
#46552
#46542
#46532
#46522
#28663