Finite Group Databasebeta

← #82777

Subgroups — #82777

Order 384.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 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×Q8×Dic3C211yesmaximal, characteristic, complemented
4C2×C4×#67C211yesmaximal, characteristic
5#5212C211yesmaximal, characteristic, complemented
6#4990C212yesmaximal, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4982C212yesmaximal, complemented
9#4977C211yesmaximal, characteristic, complemented
10#4961C212yesmaximal
11#4945C212yesmaximal, complemented
12#2536—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C3×C22⋊Q8—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2212yes—
23#946—24no—
24Q8×Dic3—24no—
25C4×#67—24no—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2211yescharacteristic
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8D311yescharacteristic, complemented
42C2×C4×Q8—31no—
43C2×C4×D4—31no—
44#398—32no—
45#394—32no—
46#388—32no—
47#388—32no—
48#382—31no—
49#380—31no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C2×C3×Q8—22no—
54C2×C3×Q8—22no—
55C2×C3×Q8D414yescomplemented
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C2×C4×D3—22no—
72C2×C4×D3—22no—
73C2×C4×D3—22no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—24no—
76C3×C22⋊C4—24no—
77#217—24no—
78#217—22no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—22no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—24no—
85Dic3⋊C4—24no—
86Dic3⋊C4—22no—
87C4×Dic3—24no—
88C4×Dic3D414yes—
89C22×Q8C2×D311yescharacteristic
90C22×D4—31no—
91C23×C4—31no—
92C23×C4C2×D311yescharacteristic
93C22⋊Q8—24no—
94C22⋊Q8—64no—
95C4×Q8—64no—
96C4×D4—64no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C4⋊C4C2×D312yes—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4C2×D312yes—
110C2×C42—31no—
111C2×C42—32no—
112#95—32no—
113#95—32no—
114#95—32no—
115#95—32no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3#4112yes—
123C23×C3C2411yescharacteristic
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3#4112yes—
131C3×Q8—28no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—44no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D414yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yes—
145#67—28no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3C2×D412yes—
160C2×Dic3C2×D412yes—
161C4×D3—44no—
162C24—31no—
163C24C22×D311yescharacteristic
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8#6714yescomplemented
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—31no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4C22×D312yes—
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D312yes—
184C22×C4C22×D311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—62no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C4⋊C4—64no—
191C4⋊C4—62no—
192C22⋊C4—24no—
193C22⋊C4—62no—
194C22⋊C4—64no—
195C22⋊C4—62no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C42—34no—
199C42—64no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3C22×D411yescharacteristic
208C22×C3C22×D411yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3C2×#4111yescharacteristic
211C22×C3C2×#4111yescharacteristic
212C22×C32−1+411yescharacteristic
213C22×C32−1+411yescharacteristic
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C3×C4—24no—
222C3×C4—41no—
223C3×C4—42no—
224C3×C4#12114yes—
225Dic3—22no—
226Dic3—22no—
227Dic3—41no—
228Dic3—42no—
229Dic3—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—32no—
237C23—61no—
238C23—61no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23#23912yes—
244C23C23×D311yescharacteristic
245Q8—28no—
246D4—68no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—44no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—61no—
271C2×C4—124no—
272C2×C4D3×D412yes—
273C2×C4#23712yes—
274C2×C4C2×#6714yes—
275C2×C4C2×#6712yes—
276C2×C3—42no—
277C2×C3C2×#12111yescharacteristic
278C2×C3#52511yescharacteristic
279C2×C3#53611yescharacteristic
280C2×C3#54811yescharacteristic
281C2×C3#54811yescharacteristic
282C2×C3#55211yescharacteristic
283C2×C3#55211yescharacteristic
284D3—42no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—62no—
293C22—62no—
294C22—62no—
295C22—121no—
296C22—121no—
297C22—121no—
298C22—121no—
299C22C2×D3×D411yescharacteristic
300C22C2×#23711yescharacteristic
301C22C2×#23911yescharacteristic
302C22#100711yescharacteristic
303C22D3×#4111yescharacteristic
304C22#101011yescharacteristic
305C22C22×#6711yescharacteristic
306C4—24no—
307C4—41no—
308C4—42no—
309C4—62no—
310C4—62no—
311C4—121no—
312C4—122no—
313C4—122no—
314C4#93914yes—
315C3#253611yescharacteristic, complemented, Sylow
316C2—42no—
317C2—122no—
318C2#562711yescharacteristic
319C2#562811yescharacteristic
320C2#563911yescharacteristic
321C2#564211yescharacteristic
322C2C2×#93911yescharacteristic
323C2#581711yescharacteristic
324C2#583111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC22×#67C22
hypercenterC2×C22⋊Q8D3
hypocenterC3#2536
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×Q8×Dic3C2semidirect
5. #5212C2semidirect
6. #4990C2semidirect
7. #4987C2semidirect
8. #4982C2semidirect
9. #4977C2semidirect
11. #4945C2semidirect
41. C2×C22⋊Q8D3semidirect
55. C2×C3×Q8D4semidirect
166. C2×Q8#67semidirect
315. C3#2536semidirect

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×Q8×Dic32
C2×C4×#672
#52122
#49902
#49872
#49822
#49772
#49612
#49452
#25363