Finite Group Databasebeta

← #76821

Subgroups — #76821

Order 384.

424 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#393C211yesmaximal, characteristic
2#5230C211yesmaximal, characteristic, complemented
3#5229C211yesmaximal, characteristic, complemented
4#4948C211yesmaximal, characteristic
5#4948C211yesmaximal, characteristic
6#4947C211yesmaximal, characteristic
7#4943C211yesmaximal, characteristic
8#4942C211yesmaximal, characteristic
9Dic3×C22⋊C4C211yesmaximal, characteristic
10#4658C211yesmaximal, characteristic, complemented
11#4658C211yesmaximal, characteristic, complemented
12#4656C211yesmaximal, characteristic, complemented
13#4653C211yesmaximal, characteristic, complemented
14#4652C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2665—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
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#400—31no—
53#400—31no—
54#396—31no—
55#396—31no—
56#395—31no—
57#395—31no—
58#393—31no—
59#393D311yescharacteristic
60#391—31no—
61#391—31no—
62#387—31no—
63#387—31no—
64#381—31no—
65#380—31no—
66C4×C22⋊C4—31no—
67C24×C3C2311yescharacteristic
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×Dic3—41no—
78C22×Dic3—41no—
79C22×Dic3Q814yescomplemented
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C2×Dic6—22no—
89C2×Dic6—41no—
90C2×Dic6—41no—
91C3×C22⋊C4—22no—
92C3×C22⋊C4—41no—
93C3×C22⋊C4—41no—
94#217—22no—
95#217—41no—
96#217—41no—
97C4⋊Dic3—22no—
98C4⋊Dic3—41no—
99Dic3⋊C4—22no—
100Dic3⋊C4—41no—
101Dic3⋊C4—41no—
102Dic3⋊C4—41no—
103Dic3⋊C4—41no—
104Dic3⋊C4—41no—
105C4×Dic3—41no—
106C4×Dic3—41no—
107C22×Q8—31no—
108C22×Q8C2×D311yescharacteristic
109C23×C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—31no—
120C2×C22⋊C4—31no—
121C2×C22⋊C4C2×D311yescharacteristic
122C2×C22⋊C4C2×D311yescharacteristic
123C2×C22⋊C4C2×D311yescharacteristic
124C2×C42—31no—
125C2×C42—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#95—31no—
136#95—31no—
137#95—31no—
138#95—31no—
139#95C2×D311yescharacteristic
140#95C2×D311yescharacteristic
141#95C2×D311yescharacteristic
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3C2×Q812yes—
149C23×C3C2411yescharacteristic
150C3×Q8—44no—
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—21no—
167C2×C3×C4—21no—
168C2×C3×C4#4112yes—
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—
189C2×Dic3—21no—
190C2×Dic3—44no—
191C2×Dic3C2×Q812yes—
192C2×Dic3C2×Q812yes—
193C2×Dic3#4112yes—
194Dic6—44no—
195C24C22×D311yescharacteristic
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—62no—
200C2×Q8—121no—
201C2×Q8—121no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—34no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—121no—
212C22×C4—121no—
213C22×C4C22×D311yescharacteristic
214C22×C4C22×D311yescharacteristic
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C4⋊C4—62no—
220C4⋊C4—62no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C22⋊C4—22no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—62no—
231C22⋊C4—121no—
232C22⋊C4—121no—
233C42—121no—
234C42—121no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3C22×Q811yescharacteristic
243C22×C3C2×#4111yescharacteristic
244C22×C3C2×#4111yescharacteristic
245C22×C3H(5,𝔽2)11yescharacteristic
246C22×C3H(5,𝔽2)11yescharacteristic
247C22×C3H(5,𝔽2)11yescharacteristic
248C22×C32−1+411yescharacteristic
249C3×C4—22no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255Dic3—22no—
256Dic3—22no—
257Dic3—22no—
258Dic3—41no—
259Dic3—41no—
260Dic3—41no—
261Dic3—41no—
262Dic3—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23D3×Q812yes—
270C23C23×D311yescharacteristic
271Q8—44no—
272Q8—124no—
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—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—32no—
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—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—124no—
315C2×C4#23712yes—
316C2×C3—42no—
317C2×C3#53311yescharacteristic
318C2×C3#53811yescharacteristic
319C2×C3#53811yescharacteristic
320C2×C3#54211yescharacteristic
321C2×C3#54211yescharacteristic
322C2×C3#54311yescharacteristic
323C2×C3#54411yescharacteristic
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22C2×#23711yescharacteristic
332C22#100411yescharacteristic
333C22#100411yescharacteristic
334C22C2×D3×Q811yescharacteristic
335C22D3×#4111yescharacteristic
336C22#100911yescharacteristic
337C22#101011yescharacteristic
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—62no—
345C4—62no—
346C4—62no—
347C4—121no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C3#266511yescharacteristic, complemented, Sylow
353C2—42no—
354C2#559311yescharacteristic
355C2#562211yescharacteristic
356C2#562311yescharacteristic
357C2#563411yescharacteristic
358C2#566911yescharacteristic
359C2#567811yescharacteristic
360C2#568211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2665
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #5230C2semidirect
3. #5229C2semidirect
10. #4658C2semidirect
11. #4658C2semidirect
12. #4656C2semidirect
13. #4653C2semidirect
14. #4652C2semidirect
15. #4646C2semidirect
79. C22×Dic3Q8semidirect
352. C3#2665semidirect

Maximal subgroups

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

subgroupindex
C3×#3932
#52302
#52292
#49482
#49472
#49432
#49422
Dic3×C22⋊C42
#46582
#46562
#46532
#46522
#46462
#26653