Finite Group Databasebeta

← #73396

Subgroups — #73396

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#876C211yesmaximal, characteristic
2C3×#376C211yesmaximal, characteristic, complemented
3#4940C211yesmaximal, characteristic
4C4×(C2×D3)⋊C4C211yesmaximal, characteristic
5C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#4933C211yesmaximal, characteristic, complemented
7C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
8#4670C211yesmaximal, characteristic
9#4669C211yesmaximal, characteristic
10#4668C211yesmaximal, characteristic
11#4666C211yesmaximal, characteristic
12#4655C211yesmaximal, characteristic, complemented
13#4652C211yesmaximal, characteristic, complemented
14#4649C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2373—31nomaximal, Sylow
17C22×C4×D3C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40#876C418yescomplemented
41C3×#95C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3×#95C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C2×#126—31no—
54#389—31no—
55#388—31no—
56#388—31no—
57#388—31no—
58#387—31no—
59#387—31no—
60#387—31no—
61#385—31no—
62C4×C4⋊C4—31no—
63C4×C4⋊C4—31no—
64C4×C4⋊C4—31no—
65C4×C22⋊C4—31no—
66C4×C22⋊C4—31no—
67#376D311yescharacteristic, complemented
68C23×D3C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C2×C4×D3—22no—
84C2×C4×D3—22no—
85C2×C4×D3—41no—
86C3×C42—22no—
87C3×C42—22no—
88C3×C42C2×C414yescomplemented
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—41no—
92(C2×D3)⋊C4C2×C414yes—
93(C2×D3)⋊C4C2×C414yes—
94(C2×D3)⋊C4C2×C414yes—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97Dic3⋊C4—22no—
98Dic3⋊C4—22no—
99Dic3⋊C4—41no—
100Dic3⋊C4C2×C414yescomplemented
101Dic3⋊C4C2×C414yescomplemented
102Dic3⋊C4C2×C414yescomplemented
103C4×Dic3—22no—
104C4×Dic3—22no—
105C4×Dic3—41no—
106C4×Dic3—41no—
107C4×Dic3—41no—
108C23×C4—31no—
109#126—38no—
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×C22⋊C4—31no—
119C2×C22⋊C4—31no—
120C2×C22⋊C4—31no—
121C2×C22⋊C4—31no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C42—31no—
125C2×C42—31no—
126C2×C42—31no—
127C2×C42—31no—
128C2×C42—31no—
129C2×C42C2×D311yescharacteristic
130C2×C42C2×D311yescharacteristic
131C2×C42C2×D311yescharacteristic
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
142#95C2×D311yescharacteristic
143#95C2×D311yescharacteristic
144C23×C3C2411yescharacteristic
145C22×D3—21no—
146C22×D3—21no—
147C22×D3—21no—
148C22×D3—21no—
149C22×D3—21no—
150C22×D3—21no—
151C22×D3C22×C412yes—
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×C4C22×C412yes—
169C2×C3×C4C22×C412yes—
170C2×C3×C4C22×C412yes—
171C2×C3×C4#4112yes—
172C2×C3×C4#4112yes—
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×Dic3C22×C412yes—
190C2×Dic3C22×C412yes—
191C2×Dic3C22×C412yes—
192C2×Dic3#4112yes—
193C2×Dic3#4112yes—
194C4×D3—44no—
195C24—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—121no—
206C22×C4C22×D311yescharacteristic
207C22×C4C22×D311yescharacteristic
208C22×C4C22×D311yescharacteristic
209C22×C4C22×D311yescharacteristic
210C22×C4C22×D311yescharacteristic
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D311yescharacteristic
213C4⋊C4—34no—
214C4⋊C4—34no—
215C4⋊C4—34no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—121no—
221C22⋊C4—34no—
222C22⋊C4—34no—
223C22⋊C4—34no—
224C22⋊C4—62no—
225C22⋊C4—62no—
226C22⋊C4—121no—
227C42—22no—
228C42—22no—
229C42—62no—
230C42—62no—
231C42—121no—
232C42—121no—
233C42—121no—
234C42C4×D314yescomplemented
235C22×C3C23×C411yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3C2×#4111yescharacteristic
238C22×C3C2×#4111yescharacteristic
239C22×C3C2×#4111yescharacteristic
240C22×C3H(5,𝔽2)11yescharacteristic
241C22×C32−1+411yescharacteristic
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—41no—
246C2×D3—41no—
247C2×D3—41no—
248C2×D3—41no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256Dic3—22no—
257Dic3—22no—
258Dic3—22no—
259Dic3—22no—
260Dic3—22no—
261Dic3—41no—
262Dic3—41no—
263C23—32no—
264C23—61no—
265C23—61no—
266C23—61no—
267C23—61no—
268C23—61no—
269C23—61no—
270C23C23×D311yescharacteristic
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—32no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—32no—
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—124no—
309C2×C4C2×C4×D312yes—
310C2×C4C2×C4×D312yes—
311C2×C4C2×C4×D312yes—
312C2×C4#23512yes—
313C2×C4#23512yes—
314C2×C3C4×#4111yescharacteristic
315C2×C3C4×#4111yescharacteristic
316C2×C3#52011yescharacteristic
317C2×C3#55111yescharacteristic
318C2×C3#55211yescharacteristic
319C2×C3#55311yescharacteristic
320C2×C3#55611yescharacteristic
321D3—42no—
322C22—62no—
323C22—62no—
324C22—62no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22—121no—
329C22C22×C4×D311yescharacteristic
330C22C2×#23511yescharacteristic
331C22C2×#23511yescharacteristic
332C22D3×#4111yescharacteristic
333C22D3×#4111yescharacteristic
334C22#100911yescharacteristic
335C22#101011yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—62no—
344C4—62no—
345C4—62no—
346C4—62no—
347C4—62no—
348C4—121no—
349C4—121no—
350C3#237311yescharacteristic, complemented, Sylow
351C2—122no—
352C2C4×#23511yescharacteristic
353C2#552211yescharacteristic
354C2#552311yescharacteristic
355C2#552711yescharacteristic
356C2#553011yescharacteristic
357C2#553111yescharacteristic
358C2#553311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#376C2
socle, cosocleC23×C3C24
prime radicalC23×D3C23
hypercenter#376D3
hypocenterC3#2373
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C3×#376C2semidirect
5. C4×C4⋊Dic3C2semidirect
6. #4933C2semidirect
7. C4×Dic3⋊C4C2semidirect
12. #4655C2semidirect
13. #4652C2semidirect
14. #4649C2semidirect
15. #4647C2semidirect
40. #876C4semidirect
67. #376D3semidirect
88. C3×C42C2×C4semidirect
100. Dic3⋊C4C2×C4semidirect
101. Dic3⋊C4C2×C4semidirect
102. Dic3⋊C4C2×C4semidirect
234. C42C4×D3semidirect
350. C3#2373semidirect

Maximal subgroups

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

subgroupindex
C2×#8762
C3×#3762
#49402
C4×(C2×D3)⋊C42
C4×C4⋊Dic32
#49332
C4×Dic3⋊C42
#46702
#46692
#46682
#46662
#46552
#46522
#46492
#46472
#23733