Finite Group Databasebeta

← #76130

Subgroups — #76130

Order 384.

476 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#898C211yesmaximal, characteristic, complemented
2C2×#876C211yesmaximal, characteristic, complemented
3C3×#389C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic, complemented
5#4993C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic, complemented
7#4991C211yesmaximal, characteristic
8#4988C211yesmaximal, characteristic
9#4976C211yesmaximal, characteristic, complemented
10#4976C211yesmaximal, characteristic, complemented
11#4940C211yesmaximal, characteristic, complemented
12#4676C211yesmaximal, characteristic, complemented
13#4672C211yesmaximal, characteristic
14#4669C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2727—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46#898—24no—
47#876—24no—
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#126—31no—
55C2×#126—31no—
56#399—31no—
57#399—31no—
58#396—31no—
59#396—31no—
60#394—31no—
61#394—31no—
62#390—31no—
63#389—31no—
64#389—31no—
65#389—31no—
66#389D311yescharacteristic, complemented
67#388—31no—
68#388—31no—
69C23×D3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C2×D12—22no—
85C2×D12—41no—
86C2×D12—41no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C42—22no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—22no—
94(C2×D3)⋊C4—22no—
95(C2×D3)⋊C4—22no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—22no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102(C2×D3)⋊C4—41no—
103(C2×D3)⋊C4—41no—
104C4⋊Dic3—22no—
105Dic3⋊C4—22no—
106Dic3⋊C4—22no—
107Dic3⋊C4—22no—
108Dic3⋊C4—22no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111Dic3⋊C4—41no—
112Dic3⋊C4D414yescomplemented
113C4×Dic3—22no—
114C4×Dic3—41no—
115C22×D4—31no—
116#126—64no—
117#126—64no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4C2×D311yescharacteristic
128C2×C4⋊C4C2×D311yescharacteristic
129C2×C4⋊C4C2×D311yescharacteristic
130C2×C4⋊C4C2×D311yescharacteristic
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42C2×D311yescharacteristic
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95C2×D311yescharacteristic
151#95C2×D311yescharacteristic
152C23×C3C2411yescharacteristic
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3—21no—
158C22×D3—21no—
159C22×D3—21no—
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4C2×D412yes—
187C2×C3×C4#4112yes—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3C2×D412yes—
204C2×Dic3C2×D412yes—
205C2×Dic3#4112yes—
206D12—44no—
207C24—31no—
208C24—31no—
209C2×D4—62no—
210C2×D4—121no—
211C2×D4—121no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C22×C4C22×D311yescharacteristic
221C22×C4C22×D311yescharacteristic
222C22×C4C22×D311yescharacteristic
223C22×C4C22×D311yescharacteristic
224C22×C4C22×D311yescharacteristic
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—34no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C22⋊C4—62no—
239C22⋊C4—62no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C22⋊C4—62no—
243C22⋊C4—62no—
244C22⋊C4—62no—
245C22⋊C4—62no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C42—22no—
251C42—62no—
252C42—121no—
253C22×C3C22×D411yescharacteristic
254C22×C3C2×#4111yescharacteristic
255C22×C3C2×#4111yescharacteristic
256C22×C3H(5,𝔽2)11yescharacteristic
257C22×C3H(5,𝔽2)11yescharacteristic
258C22×C32−1+411yescharacteristic
259C22×C32−1+411yescharacteristic
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281Dic3—22no—
282Dic3—22no—
283Dic3—22no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287C23—61no—
288C23—61no—
289C23—61no—
290C23—61no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23C23×D311yescharacteristic
302D4—124no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4#23512yes—
341C2×C4D3×D412yes—
342C2×C3#53711yescharacteristic
343C2×C3#53811yescharacteristic
344C2×C3#53911yescharacteristic
345C2×C3#54711yescharacteristic
346C2×C3#55011yescharacteristic
347C2×C3#55611yescharacteristic
348C2×C3#56311yescharacteristic
349D3—81no—
350D3—81no—
351C22—121no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22C2×#23511yescharacteristic
366C22C2×D3×D411yescharacteristic
367C22#100411yescharacteristic
368C22#100711yescharacteristic
369C22#100711yescharacteristic
370C22D3×#4111yescharacteristic
371C22#100911yescharacteristic
372C4—22no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—62no—
380C4—62no—
381C4—62no—
382C4—121no—
383C4—121no—
384C4—121no—
385C3#272711yescharacteristic, complemented, Sylow
386C2—241no—
387C2—241no—
388C2#550711yescharacteristic
389C2#556011yescharacteristic
390C2#556811yescharacteristic
391C2#559911yescharacteristic
392C2#562911yescharacteristic
393C2#569211yescharacteristic
394C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2727
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#898C2semidirect
2. C2×#876C2semidirect
3. C3×#389C2semidirect
4. #4994C2semidirect
5. #4993C2semidirect
6. #4993C2semidirect
9. #4976C2semidirect
10. #4976C2semidirect
11. #4940C2semidirect
12. #4676C2semidirect
14. #4669C2semidirect
15. #4647C2semidirect
66. #389D3semidirect
112. Dic3⋊C4D4semidirect
385. C3#2727semidirect

Maximal subgroups

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

subgroupindex
C2×#8982
C2×#8762
C3×#3892
#49942
#49932
#49912
#49882
#49762
#49402
#46762
#46722
#46692
#46472
#27273