Finite Group Databasebeta

← #76278

Subgroups — #76278

Order 384.

464 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
1C3×#389C211yesmaximal, characteristic, complemented
2#4994C211yesmaximal, characteristic, complemented
3#4994C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic, complemented
5#4993C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4989C211yesmaximal, characteristic, complemented
9#4987C211yesmaximal, characteristic, complemented
10#4939C211yesmaximal, characteristic
11#4933C211yesmaximal, characteristic, complemented
12#4665C211yesmaximal, characteristic
13#4665C211yesmaximal, characteristic
14#4661C211yesmaximal, characteristic, complemented
15#4660C211yesmaximal, characteristic, complemented
16#2780—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C42C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3×#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#400—31no—
53#399—31no—
54#398—31no—
55#396—31no—
56#396—31no—
57#396—31no—
58#390—31no—
59#389D311yescharacteristic, complemented
60#387—31no—
61#386—31no—
62#386—31no—
63#385—31no—
64#385—31no—
65#385—31no—
66#380—31no—
67C23×D3C2311yescharacteristic
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
82C2×D12—41no—
83C2×D12—41no—
84C2×D12Q814yes—
85C2×C4×D3—22no—
86C2×C4×D3—41no—
87C2×C4×D3—41no—
88C2×C4×D3—41no—
89C2×C4×D3—41no—
90C2×C4×D3—41no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C4⋊C4—41no—
95C3×C42—41no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—41no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102(C2×D3)⋊C4—41no—
103(C2×D3)⋊C4—41no—
104(C2×D3)⋊C4—41no—
105(C2×D3)⋊C4—41no—
106C4⋊Dic3—41no—
107C4⋊Dic3—41no—
108C4⋊Dic3—41no—
109Dic3⋊C4—22no—
110Dic3⋊C4—41no—
111Dic3⋊C4—41no—
112Dic3⋊C4—41no—
113C22×D4—31no—
114C23×C4—31no—
115C23×C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4C2×D311yescharacteristic
124C2×C4⋊C4C2×D311yescharacteristic
125C2×C4⋊C4C2×D311yescharacteristic
126C2×C4⋊C4C2×D311yescharacteristic
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
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×C42C2×D311yescharacteristic
138#95—31no—
139#95—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95C2×D311yescharacteristic
147#95C2×D311yescharacteristic
148C23×C3C2411yescharacteristic
149C22×D3—21no—
150C22×D3—21no—
151C22×D3—21no—
152C22×D3—21no—
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×D3C2×Q812yes—
162C22×D3C2×Q812yes—
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×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×C4C2×Q812yes—
182C2×C3×C4#4112yes—
183C2×C3×C4#4112yes—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
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—
202D12—44no—
203C4×D3—44no—
204C4×D3—44no—
205C24—31no—
206C24—31no—
207C2×D4—34no—
208C2×D4—121no—
209C2×D4—121no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—62no—
217C22×C4—121no—
218C22×C4—121no—
219C22×C4—121no—
220C22×C4—121no—
221C22×C4—121no—
222C22×C4C22×D311yescharacteristic
223C22×C4C22×D311yescharacteristic
224C22×C4C22×D311yescharacteristic
225C22×C4C22×D311yescharacteristic
226C22×C4C22×D311yescharacteristic
227C22×C4C22×D311yescharacteristic
228C22×C4C22×D311yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—62no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C22⋊C4—62no—
243C22⋊C4—121no—
244C22⋊C4—121no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C42—41no—
251C22×C3C22×Q811yescharacteristic
252C22×C3C2×#4111yescharacteristic
253C22×C3C2×#4111yescharacteristic
254C22×C3H(5,𝔽2)11yescharacteristic
255C22×C3H(5,𝔽2)11yescharacteristic
256C22×C3H(5,𝔽2)11yescharacteristic
257C22×C3H(5,𝔽2)11yescharacteristic
258C2×D3—22no—
259C2×D3—22no—
260C2×D3—22no—
261C2×D3—22no—
262C2×D3—22no—
263C2×D3—22no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285C23—32no—
286C23—32no—
287C23—61no—
288C23—61no—
289C23—61no—
290C23—61no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23C23×D311yescharacteristic
300D4—124no—
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—21no—
318C2×C4—21no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
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—124no—
338C2×C4—124no—
339C2×C4#23512yes—
340C2×C4D3×Q812yes—
341C2×C4#23912yes—
342C2×C3#53811yescharacteristic
343C2×C3#54011yescharacteristic
344C2×C3#54311yescharacteristic
345C2×C3#55311yescharacteristic
346C2×C3#55411yescharacteristic
347C2×C3#55411yescharacteristic
348C2×C3#56011yescharacteristic
349D3—42no—
350D3—42no—
351C22—62no—
352C22—62no—
353C22—62no—
354C22—62no—
355C22—62no—
356C22—62no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22C2×#23511yescharacteristic
366C22#100411yescharacteristic
367C22#100411yescharacteristic
368C22C2×D3×Q811yescharacteristic
369C22C2×#23911yescharacteristic
370C22#100911yescharacteristic
371C22#100911yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4—121no—
383C4—121no—
384C4—121no—
385C3#278011yescharacteristic, complemented, Sylow
386C2—122no—
387C2—122no—
388C2#551411yescharacteristic
389C2#556011yescharacteristic
390C2#557911yescharacteristic
391C2#562011yescharacteristic
392C2#563411yescharacteristic
393C2#569711yescharacteristic
394C2#573011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2780
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C3×#389C2semidirect
2. #4994C2semidirect
3. #4994C2semidirect
4. #4994C2semidirect
5. #4993C2semidirect
6. #4989C2semidirect
7. #4989C2semidirect
8. #4989C2semidirect
9. #4987C2semidirect
11. #4933C2semidirect
14. #4661C2semidirect
15. #4660C2semidirect
59. #389D3semidirect
385. C3#2780semidirect

Maximal subgroups

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

subgroupindex
C3×#3892
#49942
#49932
#49892
#49872
#49392
#49332
#46652
#46612
#46602
#27803