Finite Group Databasebeta

← #76256

Subgroups — #76256

Order 384.

408 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
1C3×#389C211yesmaximal, characteristic, complemented
2#4994C211yesmaximal, characteristic
3#4994C211yesmaximal, characteristic
4#4993C211yesmaximal, characteristic
5#4993C211yesmaximal, characteristic
6#4983C211yesmaximal, characteristic, complemented
7#4983C211yesmaximal, characteristic, complemented
8#4982C211yesmaximal, characteristic, complemented
9#4981C211yesmaximal, characteristic, complemented
10C4×(C2×D3)⋊C4C211yesmaximal, characteristic
11#4933C211yesmaximal, characteristic, complemented
12#4667C211yesmaximal, characteristic
13#4664C211yesmaximal, characteristic
14#4654C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2667—31nomaximal, Sylow
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
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#399—31no—
53#399—31no—
54#398—31no—
55#398—31no—
56#398—31no—
57#398—31no—
58#396—31no—
59#396—31no—
60#389—31no—
61#389D311yescharacteristic, complemented
62#387—31no—
63#387—31no—
64#381—31no—
65#380—31no—
66C4×C22⋊C4—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
82C2×C4×D3—41no—
83C2×C4×D3—41no—
84C2×C4×D3Q814yes—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—41no—
89C3×C42—41no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—41no—
93(C2×D3)⋊C4—41no—
94(C2×D3)⋊C4—41no—
95(C2×D3)⋊C4—41no—
96C4⋊Dic3—22no—
97C4⋊Dic3—41no—
98Dic3⋊C4—22no—
99Dic3⋊C4—22no—
100Dic3⋊C4—41no—
101Dic3⋊C4—41no—
102Dic3⋊C4—41no—
103Dic3⋊C4—41no—
104Dic3⋊C4—41no—
105Dic3⋊C4—41no—
106C4×Dic3—41no—
107C23×C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4C2×D311yescharacteristic
119C2×C4⋊C4C2×D311yescharacteristic
120C2×C4⋊C4C2×D311yescharacteristic
121C2×C4⋊C4C2×D311yescharacteristic
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C42—31no—
129C2×C42C2×D311yescharacteristic
130#95—31no—
131#95—31no—
132#95—31no—
133#95—31no—
134#95—31no—
135#95—31no—
136#95—31no—
137#95—31no—
138#95—31no—
139#95—31no—
140#95C2×D311yescharacteristic
141#95C2×D311yescharacteristic
142C23×C3C2411yescharacteristic
143C22×D3—21no—
144C22×D3—21no—
145C22×D3—21no—
146C22×D3—21no—
147C22×D3—21no—
148C22×D3—21no—
149C22×D3C2×Q812yes—
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—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4C2×Q812yes—
170C2×C3×C4#4112yes—
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—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×Q812yes—
191C2×Dic3#4112yes—
192C4×D3—44no—
193C24—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—34no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—121no—
203C22×C4—121no—
204C22×C4C22×D311yescharacteristic
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—41no—
214C4⋊C4—41no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C4⋊C4—121no—
219C4⋊C4—121no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C22⋊C4—62no—
226C22⋊C4—62no—
227C22⋊C4—121no—
228C22⋊C4—121no—
229C22⋊C4—121no—
230C22⋊C4—121no—
231C42—41no—
232C42—121no—
233C22×C3C22×Q811yescharacteristic
234C22×C3C2×#4111yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3H(5,𝔽2)11yescharacteristic
237C22×C3H(5,𝔽2)11yescharacteristic
238C22×C3H(5,𝔽2)11yescharacteristic
239C22×C32−1+411yescharacteristic
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—41no—
244C2×D3—41no—
245C2×D3—41no—
246C2×D3—41no—
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254Dic3—22no—
255Dic3—22no—
256Dic3—41no—
257Dic3—41no—
258Dic3—41no—
259Dic3—41no—
260Dic3—41no—
261C23—32no—
262C23—61no—
263C23—61no—
264C23—61no—
265C23—61no—
266C23—61no—
267C23—61no—
268C23C23×D311yescharacteristic
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—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—32no—
289C2×C4—32no—
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—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—124no—
310C2×C4#23512yes—
311C2×C4D3×Q812yes—
312C2×C3#53311yescharacteristic
313C2×C3#53911yescharacteristic
314C2×C3#53911yescharacteristic
315C2×C3#54011yescharacteristic
316C2×C3#54011yescharacteristic
317C2×C3#54311yescharacteristic
318C2×C3#54411yescharacteristic
319D3—42no—
320C22—62no—
321C22—62no—
322C22—62no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22—121no—
327C22C2×#23511yescharacteristic
328C22#100411yescharacteristic
329C22#100411yescharacteristic
330C22C2×D3×Q811yescharacteristic
331C22#100711yescharacteristic
332C22D3×#4111yescharacteristic
333C22#100911yescharacteristic
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—62no—
342C4—62no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4—121no—
348C3#266711yescharacteristic, complemented, Sylow
349C2—122no—
350C2#551511yescharacteristic
351C2#556211yescharacteristic
352C2#557811yescharacteristic
353C2#561511yescharacteristic
354C2#563411yescharacteristic
355C2#569211yescharacteristic
356C2#572911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC23×D3C23
hypercenter#389D3
hypocenterC3#2667
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C3×#389C2semidirect
6. #4983C2semidirect
7. #4983C2semidirect
8. #4982C2semidirect
9. #4981C2semidirect
11. #4933C2semidirect
14. #4654C2semidirect
15. #4647C2semidirect
61. #389D3semidirect
348. C3#2667semidirect

Maximal subgroups

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

subgroupindex
C3×#3892
#49942
#49932
#49832
#49822
#49812
C4×(C2×D3)⋊C42
#49332
#46672
#46642
#46542
#46472
#26673