Finite Group Databasebeta

← #82194

Subgroups — #82194

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
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic
3C2×#924C211yesmaximal, characteristic, complemented
4C2×#897C211yesmaximal, characteristic, complemented
5#5213C212yesmaximal
6#5212C212yesmaximal, complemented
7#4994C212yesmaximal, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4984C211yesmaximal, characteristic, complemented
10#4981C212yesmaximal, complemented
11#4973C211yesmaximal, characteristic, complemented
12#2417—31nomaximal, Sylow
13C23×C3×C4C2212yes—
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C4⋊C4—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24C4×#67—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2212yes—
31#924—24no—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×C4×Dic3C2211yescharacteristic
35#897—24no—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8—31no—
42C2×C4×D4—31no—
43C22×C4⋊C4D311yescharacteristic, complemented
44#398—32no—
45#396—32no—
46#391—31no—
47#389—31no—
48#386—32no—
49#380—32no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—24no—
56C22×C3×C4—24no—
57C22×C3×C4—42no—
58C22×C3×C4D414yes—
59C22×C3×C4C2312yes—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—22no—
64C2×#67—22no—
65C2×#67Q814yescomplemented
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C2×C4×D3—22no—
72C2×C4×D3—22no—
73C2×C4×D3—22no—
74C2×Dic6—22no—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C3×C4⋊C4—24no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4D414yescomplemented
80#217—24no—
81#217—22no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—24no—
86C4⋊Dic3—22no—
87C4⋊Dic3—42no—
88Dic3⋊C4—24no—
89Dic3⋊C4—22no—
90C4×Dic3—22no—
91C22×Q8—31no—
92C22×D4—31no—
93C23×C4—31no—
94C23×C4C2×D312yes—
95C23×C4C2×D311yescharacteristic
96C22⋊Q8—64no—
97C22⋊Q8—64no—
98C4×D4—64no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4C2×D311yescharacteristic
106C2×C4⋊C4C2×D312yes—
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C42—31no—
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×C3C2×D412yes—
123C23×C3C2×Q812yes—
124C23×C3#4112yes—
125C23×C3C2411yescharacteristic
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—21no—
132C22×D3C2×Q812yes—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—28no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—48no—
142C2×C3×C4C2×D412yes—
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—21no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—22no—
160C2×Dic3—21no—
161C2×Dic3—22no—
162C2×Dic3C2×Q812yes—
163C4×D3—44no—
164Dic6—44no—
165C24—31no—
166C24C22×D311yescharacteristic
167C2×Q8—62no—
168C2×Q8—62no—
169C2×Q8—62no—
170C2×D4—34no—
171C2×D4—62no—
172C2×D4—62no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—32no—
181C22×C4—31no—
182C22×C4—42no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4D1214yes—
187C22×C4C22×D312yes—
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D312yes—
190C22×C4C22×D311yescharacteristic
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—62no—
194C4⋊C4—64no—
195C4⋊C4—62no—
196C4⋊C4—64no—
197C4⋊C4—122no—
198C4⋊C4#6714yescomplemented
199C22⋊C4—62no—
200C22⋊C4—64no—
201C22⋊C4—62no—
202C22⋊C4—64no—
203C22⋊C4—122no—
204C42—62no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3C22⋊Q814yescomplemented
212C22×C3C22×D411yescharacteristic
213C22×C3C22×D411yescharacteristic
214C22×C3C22×Q811yescharacteristic
215C22×C3C2×#4111yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3H(5,𝔽2)11yescharacteristic
218C22×C32−1+411yescharacteristic
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—41no—
223C2×D3—41no—
224C2×D3—41no—
225C2×D3—41no—
226C3×C4—22no—
227C3×C4—24no—
228C3×C4—42no—
229C3×C4#12114yes—
230Dic3—22no—
231Dic3—42no—
232Dic3—41no—
233Dic3—41no—
234Dic3—42no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—32no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23—61no—
245C23—61no—
246C23C2×D1212yes—
247C23#23712yes—
248C23D3×Q812yes—
249C23C23×D311yescharacteristic
250Q8—124no—
251D4—68no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—28no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—32no—
261C2×C4—48no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—61no—
276C2×C4—124no—
277C2×C4C2×D1212yes—
278C2×C4C2×D1212yes—
279C2×C4#23512yes—
280C2×C4C2×#6714yes—
281C2×C4C2×#6712yes—
282C2×C3—24no—
283C2×C3C2×#12111yescharacteristic
284C2×C3C2×C22⋊Q811yescharacteristic
285C2×C3#53711yescharacteristic
286C2×C3#54611yescharacteristic
287C2×C3D4×Q811yescharacteristic
288C2×C3#55211yescharacteristic
289C2×C3#55411yescharacteristic
290D3—42no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—62no—
298C22—62no—
299C22—62no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22—121no—
304C22#89714yescomplemented
305C22C22×D1211yescharacteristic
306C22C2×#23511yescharacteristic
307C22C2×#23711yescharacteristic
308C22#100411yescharacteristic
309C22C2×D3×Q811yescharacteristic
310C22#100711yescharacteristic
311C22C22×#6711yescharacteristic
312C4—22no—
313C4—24no—
314C4—42no—
315C4—62no—
316C4—121no—
317C4—122no—
318C4—121no—
319C4—122no—
320C4#93014yes—
321C3#241711yescharacteristic, complemented, Sylow
322C2—24no—
323C2—122no—
324C2C2×#89711yescharacteristic
325C2#551011yescharacteristic
326C2#551111yescharacteristic
327C2#551311yescharacteristic
328C2C2×#93011yescharacteristic
329C2#580611yescharacteristic
330C2Q8×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC22×#67C22
hypercenterC22×C4⋊C4D3
hypocenterC3#2417
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
3. C2×#924C2semidirect
4. C2×#897C2semidirect
6. #5212C2semidirect
7. #4994C2semidirect
8. #4987C2semidirect
9. #4984C2semidirect
10. #4981C2semidirect
11. #4973C2semidirect
43. C22×C4⋊C4D3semidirect
65. C2×#67Q8semidirect
79. C3×C4⋊C4D4semidirect
198. C4⋊C4#67semidirect
211. C22×C3C22⋊Q8semidirect
304. C22#897semidirect
321. C3#2417semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×C4×#672
C2×#9242
C2×#8972
#52132
#52122
#49942
#49872
#49842
#49812
#49732
#24173