Finite Group Databasebeta

← #77261

Subgroups — #77261

Order 384.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#946C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic, complemented
3C2×#896C211yesmaximal, characteristic
4C2×#888C211yesmaximal, characteristic, complemented
5C3×#395C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic
7#4990C211yesmaximal, characteristic
8#4982C211yesmaximal, characteristic, complemented
9#4979C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic
11#4669C211yesmaximal, characteristic
12#4668C211yesmaximal, characteristic
13#4657C211yesmaximal, characteristic, complemented
14#4655C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2677—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24#946—24no—
25#944—24no—
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×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44#896—24no—
45#888—24no—
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3×#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×C4⋊Q8—31no—
57C2×C4⋊Q8—31no—
58C2×C22⋊Q8—31no—
59C2×C22⋊Q8—31no—
60#399—31no—
61#398—31no—
62#398—31no—
63#395—31no—
64#395D311yescharacteristic, complemented
65#393—31no—
66#388—31no—
67#388—31no—
68#387—31no—
69#387—31no—
70#385—31no—
71C23×D3C2311yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—41no—
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—41no—
92C2×Dic6—22no—
93C2×Dic6—22no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—41no—
97C2×Dic6—41no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—41no—
101(C2×D3)⋊C4—22no—
102(C2×D3)⋊C4—22no—
103(C2×D3)⋊C4—22no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—41no—
106(C2×D3)⋊C4Q814yescomplemented
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109Dic3⋊C4—22no—
110Dic3⋊C4—22no—
111Dic3⋊C4—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4D414yes—
116Dic3⋊C4D414yes—
117C4×Dic3—22no—
118C4×Dic3—22no—
119C22×Q8—31no—
120C22×Q8—31no—
121C22×Q8C2×D311yescharacteristic
122C23×C4—31no—
123C4⋊Q8—64no—
124C4⋊Q8—64no—
125C22⋊Q8—64no—
126C22⋊Q8—64no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4C2×D311yescharacteristic
138C2×C4⋊C4C2×D311yescharacteristic
139C2×C4⋊C4C2×D311yescharacteristic
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C42—31no—
147C2×C42—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95C2×D311yescharacteristic
156#95C2×D311yescharacteristic
157#95C2×D311yescharacteristic
158C23×C3C2411yescharacteristic
159C22×D3—21no—
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3C2×Q812yes—
166C3×Q8—44no—
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×C4C2×D412yes—
186C2×C3×C4C2×D412yes—
187C2×C3×C4C2×Q812yes—
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×Dic3—21no—
204C2×Dic3C2×D412yes—
205C2×Dic3C2×D412yes—
206C2×Dic3C2×D412yes—
207C2×Dic3C2×D412yes—
208C2×Dic3C2×Q812yes—
209C4×D3—44no—
210Dic6—44no—
211Dic6—44no—
212C24—31no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—62no—
217C2×Q8—62no—
218C2×Q8—62no—
219C2×Q8—62no—
220C2×Q8—121no—
221C2×Q8—121no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—62no—
230C22×C4—62no—
231C22×C4—121no—
232C22×C4C22×D311yescharacteristic
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—34no—
242C4⋊C4—34no—
243C4⋊C4—41no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C22⋊C4—34no—
253C22⋊C4—62no—
254C22⋊C4—62no—
255C22⋊C4—62no—
256C22⋊C4—62no—
257C22⋊C4—121no—
258C42—62no—
259C42—62no—
260C22×C3C22×D411yescharacteristic
261C22×C3C22×D411yescharacteristic
262C22×C3C22×Q811yescharacteristic
263C22×C3H(5,𝔽2)11yescharacteristic
264C22×C32−1+411yescharacteristic
265C22×C32−1+411yescharacteristic
266C22×C32−1+411yescharacteristic
267C2×D3—22no—
268C2×D3—22no—
269C2×D3—22no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281Dic3—22no—
282Dic3—22no—
283Dic3—22no—
284Dic3—22no—
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288C23—32no—
289C23—61no—
290C23—61no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23C23×D311yescharacteristic
296Q8—44no—
297Q8—124no—
298Q8—124no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
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—32no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
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—124no—
339C2×C4D3×D412yes—
340C2×C4D3×D412yes—
341C2×C4D3×Q812yes—
342C2×C3#53611yescharacteristic
343C2×C3#53611yescharacteristic
344C2×C3#54411yescharacteristic
345C2×C3D4211yescharacteristic
346C2×C3D4×Q811yescharacteristic
347C2×C3D4×Q811yescharacteristic
348C2×C3#56311yescharacteristic
349D3—42no—
350C22—62no—
351C22—62no—
352C22—62no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22C2×D3×D411yescharacteristic
358C22C2×D3×D411yescharacteristic
359C22#100411yescharacteristic
360C22C2×D3×Q811yescharacteristic
361C22#100711yescharacteristic
362C22#100711yescharacteristic
363C22#101011yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—62no—
372C4—62no—
373C4—62no—
374C4—62no—
375C4—62no—
376C4—121no—
377C4—121no—
378C3#267711yescharacteristic, complemented, Sylow
379C2—122no—
380C2#559011yescharacteristic
381C2#562811yescharacteristic
382C2#563211yescharacteristic
383C2#564611yescharacteristic
384C2#572411yescharacteristic
385C2#572711yescharacteristic
386C2#572911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2677
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
4. C2×#888C2semidirect
5. C3×#395C2semidirect
8. #4982C2semidirect
9. #4979C2semidirect
13. #4657C2semidirect
14. #4655C2semidirect
15. #4652C2semidirect
64. #395D3semidirect
106. (C2×D3)⋊C4Q8semidirect
378. C3#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#9442
C2×#8962
C2×#8882
C3×#3952
#49932
#49902
#49822
#49792
#46732
#46692
#46682
#46572
#46552
#46522
#26773