Finite Group Databasebeta

← #77271

Subgroups — #77271

Order 384.

400 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#395C211yesmaximal, characteristic, complemented
2#5231C211yesmaximal, characteristic
3#5229C211yesmaximal, characteristic, complemented
4#4990C211yesmaximal, characteristic
5#4990C211yesmaximal, characteristic
6#4987C211yesmaximal, characteristic
7#4985C211yesmaximal, characteristic, complemented
8#4980C211yesmaximal, characteristic, complemented
9#4978C211yesmaximal, characteristic, complemented
10#4675C211yesmaximal, characteristic
11#4668C211yesmaximal, characteristic
12#4664C211yesmaximal, characteristic
13C3⋊#401C211yesmaximal, characteristic, complemented
14#4659C211yesmaximal, characteristic, complemented
15#4654C211yesmaximal, characteristic, complemented
16#2745—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊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×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C3×#95C2211yescharacteristic
35C3×#95C2211yescharacteristic
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#401—31no—
53#400—31no—
54#400—31no—
55#397—31no—
56#395D311yescharacteristic, complemented
57#393—31no—
58#391—31no—
59#388—31no—
60#388—31no—
61#387—31no—
62#387—31no—
63#387—31no—
64#385—31no—
65#381—31no—
66#380—31no—
67C23×D3C2311yescharacteristic
68C2×C3×Q8—22no—
69C2×C3×Q8—41no—
70C2×C3×Q8—41no—
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
84C22×Dic3C2311yescharacteristic
85C2×C4×D3—22no—
86C2×C4×D3—22no—
87C2×C4×D3—41no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—22no—
94(C2×D3)⋊C4—41no—
95(C2×D3)⋊C4—41no—
96(C2×D3)⋊C4—41no—
97C4⋊Dic3—22no—
98C4⋊Dic3—41no—
99Dic3⋊C4—22no—
100Dic3⋊C4—41no—
101C4×Dic3—41no—
102C4×Dic3—41no—
103C22×Q8C2×D311yescharacteristic
104C23×C4—31no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4C2×D311yescharacteristic
110C2×C4⋊C4C2×D311yescharacteristic
111C2×C4⋊C4C2×D311yescharacteristic
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—31no—
118C2×C42—31no—
119C2×C42—31no—
120#95—31no—
121#95—31no—
122#95—31no—
123#95—31no—
124#95—31no—
125#95—31no—
126#95—31no—
127#95—31no—
128#95—31no—
129#95—31no—
130#95—31no—
131#95—31no—
132#95—31no—
133#95—31no—
134#95—31no—
135#95C2×D311yescharacteristic
136#95C2×D311yescharacteristic
137#95C2×D311yescharacteristic
138C23×C3C2411yescharacteristic
139C22×D3—21no—
140C22×D3—21no—
141C22×D3—21no—
142C22×D3—21no—
143C22×D3—21no—
144C22×D3—21no—
145C22×D3#4112yes—
146C3×Q8—44no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
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#4112yes—
167C2×C3×C4#4112yes—
168C2×Dic3—21no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—21no—
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—
189C4×D3—44no—
190C24—31no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4—121no—
204C22×C4C22×D311yescharacteristic
205C22×C4C22×D311yescharacteristic
206C22×C4C22×D311yescharacteristic
207C22×C4C22×D311yescharacteristic
208C22×C4C22×D311yescharacteristic
209C22×C4C22×D311yescharacteristic
210C22×C4C22×D311yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—121no—
217C4⋊C4—121no—
218C22⋊C4—62no—
219C22⋊C4—62no—
220C22⋊C4—62no—
221C22⋊C4—121no—
222C22⋊C4—121no—
223C22⋊C4—121no—
224C42—121no—
225C42—121no—
226C22×C3C2×#4111yescharacteristic
227C22×C3C2×#4111yescharacteristic
228C22×C3C2×#4111yescharacteristic
229C22×C3H(5,𝔽2)11yescharacteristic
230C22×C3H(5,𝔽2)11yescharacteristic
231C22×C32−1+411yescharacteristic
232C22×C32−1+411yescharacteristic
233C2×D3—22no—
234C2×D3—22no—
235C2×D3—22no—
236C2×D3—41no—
237C2×D3—41no—
238C2×D3—41no—
239C2×D3—41no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247Dic3—41no—
248Dic3—41no—
249Dic3—41no—
250Dic3—41no—
251Dic3—41no—
252Dic3—41no—
253Dic3—41no—
254C23—32no—
255C23—61no—
256C23—61no—
257C23—61no—
258C23—61no—
259C23—61no—
260C23—61no—
261C23C23×D311yescharacteristic
262Q8—44no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
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—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
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—124no—
304C2×C4#23712yes—
305C2×C4#23912yes—
306C2×C3#53811yescharacteristic
307C2×C3#53911yescharacteristic
308C2×C3#54211yescharacteristic
309C2×C3#55111yescharacteristic
310C2×C3#55211yescharacteristic
311C2×C3#55611yescharacteristic
312C2×C3#56311yescharacteristic
313D3—42no—
314C22—62no—
315C22—62no—
316C22—62no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22—121no—
321C22C2×#23711yescharacteristic
322C22#100411yescharacteristic
323C22#100411yescharacteristic
324C22C2×#23911yescharacteristic
325C22#100711yescharacteristic
326C22#100711yescharacteristic
327C22D3×#4111yescharacteristic
328C4—22no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—121no—
336C4—121no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4—121no—
342C3#274511yescharacteristic, complemented, Sylow
343C2—122no—
344C2#559311yescharacteristic
345C2#563711yescharacteristic
346C2#563811yescharacteristic
347C2#563911yescharacteristic
348C2#573211yescharacteristic
349C2#573311yescharacteristic
350C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2745
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C3×#395C2semidirect
3. #5229C2semidirect
7. #4985C2semidirect
8. #4980C2semidirect
9. #4978C2semidirect
13. C3⋊#401C2semidirect
14. #4659C2semidirect
15. #4654C2semidirect
56. #395D3semidirect
342. C3#2745semidirect

Maximal subgroups

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

subgroupindex
C3×#3952
#52312
#52292
#49902
#49872
#49852
#49802
#49782
#46752
#46682
#46642
C3⋊#4012
#46592
#46542
#27453