Finite Group Databasebeta

← #78431

Subgroups — #78431

Order 384.

482 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#401C211yesmaximal, characteristic, complemented
2#4675C211yesmaximal, characteristic, complemented
3#4675C211yesmaximal, characteristic, complemented
4#4670C211yesmaximal, characteristic, complemented
5#4669C211yesmaximal, characteristic, complemented
6#4668C211yesmaximal, characteristic, complemented
7#4667C211yesmaximal, characteristic, complemented
8#4666C211yesmaximal, characteristic, complemented
9#4665C211yesmaximal, characteristic
10#4665C211yesmaximal, characteristic
11#4665C211yesmaximal, characteristic
12D3×#95C211yesmaximal, characteristic, complemented
13#4659C211yesmaximal, characteristic, complemented
14#4652C211yesmaximal, characteristic, complemented
15#4651C211yesmaximal, characteristic, complemented
16#2510—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C3×#95C2211yescharacteristic
37C3×#95C2211yescharacteristic
38C3×#95C2211yescharacteristic
39C3×#95C2211yescharacteristic
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#401D311yescharacteristic, complemented
53#400—31no—
54#397—31no—
55#397—31no—
56#388—31no—
57#388—31no—
58#387—31no—
59#386—31no—
60#386—31no—
61#386—31no—
62#385—31no—
63#383—31no—
64#380—31no—
65C4×C22⋊C4—31no—
66C2×#95—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—22no—
83C2×D12—22no—
84C2×D12—22no—
85C2×C4×D3—22no—
86C2×C4×D3—22no—
87C2×C4×D3—22no—
88C2×C4×D3—22no—
89C2×C4×D3—22no—
90C2×C4×D3—41no—
91C2×C4×D3—41no—
92C2×C4×D3—41no—
93C2×C4×D3—41no—
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—41no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102(C2×D3)⋊C4—41no—
103C4⋊Dic3—41no—
104Dic3⋊C4—22no—
105Dic3⋊C4—41no—
106Dic3⋊C4—41no—
107C4×Dic3—22no—
108C4×Dic3—41no—
109C22×D4—31no—
110C23×C4—31no—
111C23×C4—31no—
112C23×C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C22⋊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×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C42—31no—
127C2×C42—31no—
128#95—31no—
129#95—31no—
130#95—31no—
131#95—31no—
132#95—31no—
133#95—31no—
134#95—31no—
135#95—31no—
136#95—31no—
137#95C2×D311yescharacteristic
138#95C2×D311yescharacteristic
139#95C2×D311yescharacteristic
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×D3—21no—
152C22×D3—21no—
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3#4112yes—
157C22×D3#4112yes—
158C22×D3#4112yes—
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×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×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×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3#4112yes—
197C2×Dic3#4112yes—
198D12—44no—
199C4×D3—44no—
200C4×D3—44no—
201C4×D3—44no—
202C24—31no—
203C24—31no—
204C2×D4—62no—
205C2×D4—62no—
206C2×D4—62no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4—62no—
217C22×C4—62no—
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—62no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C22⋊C4—62no—
234C22⋊C4—62no—
235C22⋊C4—62no—
236C22⋊C4—62no—
237C22⋊C4—62no—
238C22⋊C4—121no—
239C22⋊C4—121no—
240C22⋊C4—121no—
241C22⋊C4—121no—
242C42—62no—
243C42—121no—
244C22×C3C2×#4111yescharacteristic
245C22×C3C2×#4111yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3C2×#4111yescharacteristic
248C22×C3C2×#4111yescharacteristic
249C22×C3H(5,𝔽2)11yescharacteristic
250C22×C3H(5,𝔽2)11yescharacteristic
251C2×D3—22no—
252C2×D3—22no—
253C2×D3—22no—
254C2×D3—22no—
255C2×D3—22no—
256C2×D3—22no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3#12614yes—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272Dic3—22no—
273Dic3—22no—
274Dic3—41no—
275Dic3—41no—
276Dic3—41no—
277Dic3—41no—
278C23—32no—
279C23—32no—
280C23—32no—
281C23—61no—
282C23—61no—
283C23—61no—
284C23—61no—
285C23—61no—
286C23—61no—
287C23—61no—
288C23—61no—
289C23—61no—
290C23—61no—
291C23—61no—
292C23C23×D311yescharacteristic
293D4—124no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
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—32no—
316C2×C4—32no—
317C2×C4—61no—
318C2×C4—61no—
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—124no—
334C2×C4—124no—
335C2×C4—124no—
336C2×C3C2×#12611yescharacteristic
337C2×C3#52911yescharacteristic
338C2×C3#53811yescharacteristic
339C2×C3#55111yescharacteristic
340C2×C3#55111yescharacteristic
341C2×C3#55311yescharacteristic
342C2×C3#55311yescharacteristic
343D3—24no—
344D3—81no—
345C22—34no—
346C22—62no—
347C22—62no—
348C22—62no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22D3×#4111yescharacteristic
360C22D3×#4111yescharacteristic
361C22D3×#4111yescharacteristic
362C22D3×#4111yescharacteristic
363C22D3×#4111yescharacteristic
364C22#100911yescharacteristic
365C22#100911yescharacteristic
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—62no—
374C4—62no—
375C4—121no—
376C4—121no—
377C4—121no—
378C4—121no—
379C3#251011yescharacteristic, complemented, Sylow
380C2—64no—
381C2—241no—
382C2D3×#12611yescharacteristic
383C2#570411yescharacteristic
384C2#570511yescharacteristic
385C2#570511yescharacteristic
386C2#570611yescharacteristic
387C2#570911yescharacteristic
388C2#570911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#401C2
prime radicalC22×D12C22
hypercenter#401D3
hypocenterC3#2510
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #401

Complements

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

NH ≅ G/Ntype
1. C3×#401C2semidirect
2. #4675C2semidirect
3. #4675C2semidirect
4. #4670C2semidirect
5. #4669C2semidirect
6. #4668C2semidirect
7. #4667C2semidirect
8. #4666C2semidirect
12. D3×#95C2semidirect
13. #4659C2semidirect
14. #4652C2semidirect
15. #4651C2semidirect
52. #401D3semidirect
379. C3#2510semidirect

Maximal subgroups

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

subgroupindex
C3×#4012
#46752
#46702
#46692
#46682
#46672
#46662
#46652
D3×#952
#46592
#46522
#46512
#25103