Finite Group Databasebeta

← #82778

Subgroups — #82778

Order 384.

620 conjugacy classes of subgroups.

The subgroup lattice has 330 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#4994C212yesmaximal, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4976C212yesmaximal, complemented
9#4975C211yesmaximal, characteristic, complemented
10#4957C212yesmaximal
11#4948C212yesmaximal, complemented
12#2538—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)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2212yes—
36C2×C4×Dic3C2212yes—
37C2×C4×Dic3C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8D311yescharacteristic, complemented
42C2×C4×Q8—31no—
43C2×C4×D4—31no—
44#396—32no—
45#396—32no—
46#390—32no—
47#389—32no—
48#384—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×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67—22no—
64C2×#67—22no—
65C2×#67Q814yescomplemented
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
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—22no—
78#217—42no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—42no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—24no—
85Dic3⋊C4—24no—
86Dic3⋊C4—22no—
87Dic3⋊C4—24no—
88Dic3⋊C4—42no—
89C4×Dic3—24no—
90C4×Dic3D414yes—
91C22×Q8C2×D311yescharacteristic
92C22×D4—31no—
93C23×C4—31no—
94C23×C4C2×D311yescharacteristic
95C22⋊Q8—24no—
96C22⋊Q8—64no—
97C4×Q8—64no—
98C4×D4—64no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4C2×D311yescharacteristic
107C2×C4⋊C4C2×D312yes—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4C2×D312yes—
114C2×C42—31no—
115C2×C42—32no—
116#95—32no—
117#95—32no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3C2×Q812yes—
125C23×C3C2411yescharacteristic
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—21no—
132C22×D3C2×Q812yes—
133C3×Q8—28no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—44no—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D414yes—
145C2×C3×C4C2×D412yes—
146C2×C3×C4#4112yes—
147#67—28no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3C2×D412yes—
160C2×Dic3C2×D412yes—
161C2×Dic3C2×Q812yes—
162C2×Dic3#4112yes—
163C4×D3—44no—
164C24—31no—
165C24C22×D311yescharacteristic
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8#6714yescomplemented
169C2×D4—34no—
170C2×D4—62no—
171C2×D4—62no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D312yes—
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—62no—
190C4⋊C4—64no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C4⋊C4—64no—
194C4⋊C4—62no—
195C4⋊C4—122no—
196C22⋊C4—24no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—64no—
200C22⋊C4—122no—
201C22⋊C4—122no—
202C42—34no—
203C42—64no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3C22×D411yescharacteristic
212C22×C3C22×D411yescharacteristic
213C22×C3C22×Q811yescharacteristic
214C22×C3C2×#4111yescharacteristic
215C22×C3C2×#4111yescharacteristic
216C22×C3H(5,𝔽2)11yescharacteristic
217C22×C32−1+411yescharacteristic
218C2×D3—22no—
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—41no—
222C2×D3—41no—
223C2×D3—41no—
224C2×D3—41no—
225C3×C4—24no—
226C3×C4—41no—
227C3×C4—42no—
228C3×C4#12114yes—
229Dic3—22no—
230Dic3—41no—
231Dic3—42no—
232Dic3—42no—
233Dic3C22⋊Q814yescomplemented
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—32no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23—61no—
245C23—61no—
246C23—61no—
247C23D3×Q812yes—
248C23C23×D311yescharacteristic
249Q8—28no—
250D4—68no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—44no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—124no—
276C2×C4D3×D412yes—
277C2×C4#23712yes—
278C2×C4C2×#6714yes—
279C2×C4C2×#6712yes—
280C2×C3—42no—
281C2×C3C2×#12111yescharacteristic
282C2×C3C2×C22⋊Q811yescharacteristic
283C2×C3#53711yescharacteristic
284C2×C3D4×Q811yescharacteristic
285C2×C3#55011yescharacteristic
286C2×C3#55411yescharacteristic
287C2×C3#55611yescharacteristic
288D3—42no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—62no—
297C22—62no—
298C22—62no—
299C22—121no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22C2×D3×D411yescharacteristic
304C22C2×#23711yescharacteristic
305C22C2×D3×Q811yescharacteristic
306C22#100711yescharacteristic
307C22D3×#4111yescharacteristic
308C22#100911yescharacteristic
309C22C22×#6711yescharacteristic
310C4—24no—
311C4—34no—
312C4—41no—
313C4—42no—
314C4—62no—
315C4—121no—
316C4—122no—
317C4—122no—
318C4#93914yes—
319C3#253811yescharacteristic, complemented, Sylow
320C2—42no—
321C2—122no—
322C2D3×C22⋊Q811yescharacteristic
323C2#562911yescharacteristic
324C2#563511yescharacteristic
325C2#564111yescharacteristic
326C2C2×#93911yescharacteristic
327C2Q8×#6711yescharacteristic
328C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC22×#67C22
hypercenterC2×C22⋊Q8D3
hypocenterC3#2538
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. #4994C2semidirect
7. #4987C2semidirect
8. #4976C2semidirect
9. #4975C2semidirect
11. #4948C2semidirect
41. C2×C22⋊Q8D3semidirect
55. C2×C3×Q8D4semidirect
65. C2×#67Q8semidirect
168. C2×Q8#67semidirect
233. Dic3C22⋊Q8semidirect
319. C3#2538semidirect

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
#49942
#49872
#49762
#49752
#49572
#49482
#25383