Finite Group Databasebeta

← #82353

Subgroups — #82353

Order 384.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic
2C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#924C211yesmaximal, characteristic, complemented
4C2×#924C211yesmaximal, characteristic, complemented
5#5220C212yesmaximal
6#5212C211yesmaximal, characteristic, complemented
7#4982C212yesmaximal, complemented
8#4948C212yesmaximal, complemented
9#4942C212yesmaximal, complemented
10#4929C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2417—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C22×Dic6C2211yescharacteristic
18C3×C4×D4—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×C4⋊Dic3—24no—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31#924—24no—
32#924—24no—
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2212yes—
35C2×Dic3⋊C4C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8—31no—
42C2×C4×D4D311yescharacteristic
43C22×C4⋊C4—31no—
44#398—32no—
45#396—32no—
46#391—31no—
47#389—31no—
48#386—32no—
49#380—31no—
50#380—32no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4Q814yescomplemented
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4D414yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×Dic3—24no—
68C22×Dic3—24no—
69C22×Dic3—42no—
70C22×Dic3C2312yes—
71C22×Dic3C2312yes—
72C22×Dic3C2312yes—
73C22×Dic3C2312yes—
74C2×Dic6—22no—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C3×C4⋊C4—22no—
78C3×C22⋊C4—22no—
79C3×C22⋊C4—22no—
80C3×C42—22no—
81#217—24no—
82#217—24no—
83#217—42no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—24no—
87C4⋊Dic3D414yescomplemented
88Dic3⋊C4—24no—
89Dic3⋊C4—24no—
90Dic3⋊C4—42no—
91C22×Q8—31no—
92C22×D4C2×D311yescharacteristic
93C23×C4—32no—
94C23×C4C2×D311yescharacteristic
95C23×C4C2×D311yescharacteristic
96C22⋊Q8—64no—
97C22⋊Q8—64no—
98C4×D4—24no—
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⋊C4—64no—
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4C2×D311yescharacteristic
112C2×C22⋊C4C2×D311yescharacteristic
113C2×C42C2×D311yescharacteristic
114#95—32no—
115#95—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×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3C2×D412yes—
129C23×C3C2×Q812yes—
130C23×C3C2×Q812yes—
131C23×C3#4112yes—
132C23×C3C2411yescharacteristic
133C3×D4—28no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—28no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—44no—
146C2×C3×C4C2×D412yes—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D412yes—
149C2×C3×C4C2×Q812yes—
150C2×C3×C4#4112yes—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—48no—
163C2×Dic3C2×D414yes—
164Dic6—44no—
165C24C22×D311yescharacteristic
166C24C22×D311yescharacteristic
167C2×Q8—62no—
168C2×Q8—62no—
169C2×Q8—62no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4Dic614yescomplemented
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—32no—
179C22×C4—32no—
180C22×C4—32no—
181C22×C4—32no—
182C22×C4—64no—
183C22×C4—64no—
184C22×C4—122no—
185C22×C4#6714yes—
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—34no—
193C4⋊C4—64no—
194C4⋊C4—64no—
195C4⋊C4—62no—
196C4⋊C4—64no—
197C4⋊C4—62no—
198C4⋊C4—122no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—64no—
202C22⋊C4—64no—
203C22⋊C4—122no—
204C42—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3C22⋊Q814yescomplemented
219C22×C3C22×D411yescharacteristic
220C22×C3C22×D411yescharacteristic
221C22×C3C22×Q811yescharacteristic
222C22×C3C2×#4111yescharacteristic
223C22×C3C2×#4111yescharacteristic
224C22×C3H(5,𝔽2)11yescharacteristic
225C22×C32−1+411yescharacteristic
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4#12114yes—
231Dic3—24no—
232Dic3—42no—
233Dic3—42no—
234Dic3—42no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C2×Dic612yes—
246C23C2×Dic612yes—
247C23#23512yes—
248C23C2×#6712yes—
249C23C23×D311yescharacteristic
250Q8—124no—
251D4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—28no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—34no—
264C2×C4—44no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—128no—
277C2×C4C2×Dic612yes—
278C2×C4D3×D412yes—
279C2×C4#23712yes—
280C2×C4C2×#6712yes—
281C2×C4C2×#6712yes—
282C2×C3—24no—
283C2×C3—42no—
284C2×C3C2×#12111yescharacteristic
285C2×C3C2×C22⋊Q811yescharacteristic
286C2×C3#53711yescharacteristic
287C2×C3#54611yescharacteristic
288C2×C3D4×Q811yescharacteristic
289C2×C3#55211yescharacteristic
290C2×C3#55411yescharacteristic
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22#92414yescomplemented
305C22C22×Dic611yescharacteristic
306C22C2×#23511yescharacteristic
307C22C2×D3×D411yescharacteristic
308C22C2×#23711yescharacteristic
309C22#100911yescharacteristic
310C22#101011yescharacteristic
311C22C22×#6711yescharacteristic
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—64no—
317C4—122no—
318C4—122no—
319C4—122no—
320C4#93914yes—
321C3#241711yescharacteristic, complemented, Sylow
322C2—24no—
323C2—42no—
324C2D4×Dic611yescharacteristic
325C2#554111yescharacteristic
326C2#554311yescharacteristic
327C2#555011yescharacteristic
328C2C2×#92411yescharacteristic
329C2C2×#93911yescharacteristic
330C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2417
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic3C2semidirect
3. C2×#924C2semidirect
4. C2×#924C2semidirect
6. #5212C2semidirect
7. #4982C2semidirect
8. #4948C2semidirect
9. #4942C2semidirect
10. #4929C2semidirect
11. #4928C2semidirect
55. C2×C3×D4Q8semidirect
87. C4⋊Dic3D4semidirect
172. C2×D4Dic6semidirect
218. C22×C3C22⋊Q8semidirect
304. C22#924semidirect
321. C3#2417semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C22×C4⋊Dic32
C2×#9242
#52202
#52122
#49822
#49482
#49422
#49292
#49282
#24173