Finite Group Databasebeta

← #82780

Subgroups — #82780

Order 384.

676 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#946C211yesmaximal, characteristic, complemented
3C2×#944C211yesmaximal, characteristic, complemented
4C2×C4×#67C211yesmaximal, characteristic
5C2×#924C211yesmaximal, characteristic, complemented
6C2×#897C211yesmaximal, characteristic, complemented
7#4993C212yesmaximal, complemented
8#4975C212yesmaximal, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4957C212yesmaximal
11#4949C212yesmaximal, complemented
12#2542—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C3×C22⋊Q8—24no—
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24#946—24no—
25#944—24no—
26C4×#67—24no—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×C4⋊Dic3C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32#924—24no—
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2212yes—
35C2×Dic3⋊C4C2212yes—
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2212yes—
38C2×C4×Dic3C2212yes—
39C2×C4×Dic3C2211yescharacteristic
40#897—24no—
41C3⋊#95C2212yes—
42C2×C4⋊Q8—31no—
43C2×C22⋊Q8—31no—
44C2×C22⋊Q8—31no—
45C2×C22⋊Q8—31no—
46C2×C22⋊Q8D311yescharacteristic, complemented
47C2×C4×D4—31no—
48#399—32no—
49#399—32no—
50#390—32no—
51#384—32no—
52C4×C4⋊C4—31no—
53C24×C3C2311yescharacteristic
54C23×D3C2311yescharacteristic
55C2×C3×Q8—22no—
56C2×C3×Q8—22no—
57C2×C3×Q8—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C2×#67—22no—
66C2×#67—22no—
67C2×#67Q814yescomplemented
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73C2×C4×D3—22no—
74C2×C4×D3—22no—
75C2×C4×D3—22no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—22no—
79C3×C4⋊C4—24no—
80C3×C4⋊C4D414yescomplemented
81C3×C22⋊C4—24no—
82#217—24no—
83#217—22no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—24no—
87C4⋊Dic3—24no—
88C4⋊Dic3D414yescomplemented
89Dic3⋊C4—24no—
90Dic3⋊C4—22no—
91Dic3⋊C4—24no—
92Dic3⋊C4—24no—
93Dic3⋊C4—24no—
94C4×Dic3—24no—
95C4×Dic3D414yes—
96C22×Q8—31no—
97C22×Q8C2×D311yescharacteristic
98C22×D4—31no—
99C23×C4—31no—
100C23×C4C2×D311yescharacteristic
101C4⋊Q8—64no—
102C22⋊Q8—24no—
103C22⋊Q8—64no—
104C22⋊Q8—64no—
105C22⋊Q8—64no—
106C4×D4—64no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4C2×D312yes—
115C2×C4⋊C4C2×D311yescharacteristic
116C2×C22⋊C4—31no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4C2×D312yes—
122C2×C42—31no—
123C2×C42—32no—
124#95—32no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3C2×Q812yes—
132C23×C3C2411yescharacteristic
133C22×D3—21no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—21no—
139C22×D3C2×Q812yes—
140C3×Q8—44no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—21no—
149C2×C3×C4—44no—
150C2×C3×C4C2×D412yes—
151C2×C3×C4C2×D412yes—
152C2×C3×C4C2×D414yes—
153C2×C3×C4C2×D412yes—
154#67—28no—
155C2×Dic3—22no—
156C2×Dic3—21no—
157C2×Dic3—22no—
158C2×Dic3—21no—
159C2×Dic3—22no—
160C2×Dic3—21no—
161C2×Dic3—22no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—22no—
165C2×Dic3C2×D412yes—
166C2×Dic3C2×D414yes—
167C2×Dic3C2×D412yes—
168C2×Dic3C2×Q812yes—
169C2×Dic3#4112yes—
170C4×D3—44no—
171Dic6—44no—
172C24—31no—
173C24C22×D311yescharacteristic
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—62no—
178C2×Q8—62no—
179C2×Q8—62no—
180C2×D4—34no—
181C2×D4—62no—
182C2×D4—62no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—32no—
189C22×C4—32no—
190C22×C4—31no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4C22×D312yes—
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D312yes—
197C22×C4C22×D311yescharacteristic
198C4⋊C4—24no—
199C4⋊C4—34no—
200C4⋊C4—62no—
201C4⋊C4—64no—
202C4⋊C4—64no—
203C4⋊C4—64no—
204C4⋊C4—64no—
205C4⋊C4—64no—
206C4⋊C4#6714yescomplemented
207C22⋊C4—24no—
208C22⋊C4—62no—
209C22⋊C4—64no—
210C22⋊C4—62no—
211C22⋊C4—64no—
212C22⋊C4—64no—
213C42—34no—
214C42—64no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3C22×D411yescharacteristic
223C22×C3C22×D411yescharacteristic
224C22×C3C22×D411yescharacteristic
225C22×C3C22×Q811yescharacteristic
226C22×C3C2×#4111yescharacteristic
227C22×C32−1+411yescharacteristic
228C22×C32−1+411yescharacteristic
229C2×D3—22no—
230C2×D3—22no—
231C2×D3—22no—
232C2×D3—41no—
233C2×D3—41no—
234C2×D3—41no—
235C2×D3—41no—
236C3×C4—24no—
237C3×C4—41no—
238C3×C4—42no—
239C3×C4#12714yes—
240Dic3—22no—
241Dic3—24no—
242Dic3—42no—
243Dic3—41no—
244Dic3C22⋊Q814yescomplemented
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—32no—
252C23—61no—
253C23—61no—
254C23—61no—
255C23—61no—
256C23—61no—
257C23—61no—
258C23D3×Q812yes—
259C23C23×D311yescharacteristic
260Q8—44no—
261Q8—124no—
262D4—68no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—34no—
276C2×C4—44no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—62no—
283C2×C4—61no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—61no—
287C2×C4—124no—
288C2×C4D3×D412yes—
289C2×C4D3×D412yes—
290C2×C4C2×#6714yes—
291C2×C4C2×#6712yes—
292C2×C3—42no—
293C2×C3C2×C22⋊Q811yescharacteristic
294C2×C3C2×#12711yescharacteristic
295C2×C3#53611yescharacteristic
296C2×C3#54711yescharacteristic
297C2×C3#54711yescharacteristic
298C2×C3D4×Q811yescharacteristic
299C2×C3D4×Q811yescharacteristic
300D3—42no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—62no—
309C22—62no—
310C22—62no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22—121no—
315C22C2×D3×D411yescharacteristic
316C22C2×D3×D411yescharacteristic
317C22C2×D3×Q811yescharacteristic
318C22#100711yescharacteristic
319C22D3×#4111yescharacteristic
320C22#101011yescharacteristic
321C22C22×#6711yescharacteristic
322C4—24no—
323C4—34no—
324C4—41no—
325C4—42no—
326C4—62no—
327C4—64no—
328C4—121no—
329C4—122no—
330C4#94014yes—
331C3#254211yescharacteristic, complemented, Sylow
332C2—42no—
333C2—122no—
334C2D3×C22⋊Q811yescharacteristic
335C2#562811yescharacteristic
336C2#563111yescharacteristic
337C2#563211yescharacteristic
338C2C2×#94011yescharacteristic
339C2Q8×#6711yescharacteristic
340C2#582411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC22×#67C22
hypercenterC2×C22⋊Q8D3
hypocenterC3#2542
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
2. C2×#946C2semidirect
3. C2×#944C2semidirect
5. C2×#924C2semidirect
6. C2×#897C2semidirect
7. #4993C2semidirect
8. #4975C2semidirect
9. Dic3×C4⋊C4C2semidirect
11. #4949C2semidirect
46. C2×C22⋊Q8D3semidirect
67. C2×#67Q8semidirect
80. C3×C4⋊C4D4semidirect
88. C4⋊Dic3D4semidirect
206. C4⋊C4#67semidirect
244. Dic3C22⋊Q8semidirect
331. C3#2542semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#9462
C2×#9442
C2×C4×#672
C2×#9242
C2×#8972
#49932
#49752
Dic3×C4⋊C42
#49572
#49492
#25423