Finite Group Databasebeta

← C4⋊C4×#67 #82187

Subgroups — C4⋊C4×#67 #82187

Order 384.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#67C212yesmaximal
3C2×C4×#67C211yesmaximal, characteristic
4C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
5#5212C211yesmaximal, characteristic, complemented
6#5210C212yesmaximal, complemented
7#4989C212yesmaximal, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4976C212yesmaximal, complemented
10#4975C211yesmaximal, characteristic, complemented
11Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
12D4×C4⋊C4—31nomaximal, Sylow
13C23×C3×C4C2212yes—
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×C4×D3C2212yes—
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C4×#67—28no—
25C4×#67C418yescomplemented
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2211yescharacteristic
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36D3×C4⋊C4—24no—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C4×D4—31no—
41C2×C4×D4—32no—
42C22×C4⋊C4—31no—
43C22×C4⋊C4D311yescharacteristic, complemented
44#389—32no—
45#385—32no—
46#385—32no—
47#384—31no—
48#380—31no—
49#380—31no—
50C4×C4⋊C4—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×C4C2×C414yescomplemented
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67C2×C414yes—
64C2×#67D414yes—
65C2×#67Q814yes—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C2×C4×D3—22no—
72C2×C4×D3—22no—
73C2×C4×D3—24no—
74C2×C4×D3—24no—
75C2×C4×D3—42no—
76C2×C4×D3C2×C414yescomplemented
77C3×C4⋊C4—24no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4D414yescomplemented
80#217—24no—
81#217C2×C414yescomplemented
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4C2×C414yescomplemented
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86Dic3⋊C4—24no—
87Dic3⋊C4—24no—
88Dic3⋊C4—42no—
89Dic3⋊C4C2×C414yescomplemented
90C4×Dic3—24no—
91C4×Dic3C2×C414yescomplemented
92C22×D4—31no—
93C23×C4—31no—
94C23×C4—32no—
95C23×C4C2×D312yes—
96C23×C4C2×D311yescharacteristic
97C4×D4—38no—
98C4×D4—68no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—64no—
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C4⋊C4C2×D312yes—
109C2×C4⋊C4C2×D311yescharacteristic
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C42—31no—
115C2×C42—32no—
116#95—32no—
117#95—32no—
118#95—32no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3C22×C412yes—
124C23×C3C2×D412yes—
125C23×C3C2×Q812yes—
126C23×C3C2411yescharacteristic
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3C22×C412yes—
132C22×D3C2×D412yes—
133C22×D3C2×Q812yes—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—28no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—48no—
143C2×C3×C4C22×C412yes—
144C2×C3×C4C22×C412yes—
145C2×C3×C4C2×D414yes—
146C2×C3×C4C2×D412yes—
147C2×C3×C4#4112yes—
148#67C4⋊C4116yesdirect product factor
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3C22×C412yes—
160C2×Dic3C22×C412yes—
161C2×Dic3C22×C412yes—
162C2×Dic3C2×D412yes—
163C2×Dic3C2×Q812yes—
164C4×D3—28no—
165C4×D3—48no—
166C24—31no—
167C24C22×D311yescharacteristic
168C2×D4—34no—
169C2×D4—34no—
170C2×D4—34no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—34no—
179C22×C4—32no—
180C22×C4—32no—
181C22×C4—42no—
182C22×C4—64no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—64no—
186C22×C4—122no—
187C22×C4C4×D314yescomplemented
188C22×C4C22×D312yes—
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D312yes—
191C22×C4C22×D311yescharacteristic
192C4⋊C4—24no—
193C4⋊C4—22no—
194C4⋊C4—34no—
195C4⋊C4—64no—
196C4⋊C4—62no—
197C4⋊C4—64no—
198C4⋊C4—62no—
199C4⋊C4—122no—
200C4⋊C4#6714yesdirect product factor
201C22⋊C4—34no—
202C22⋊C4—34no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C42—34no—
206C42—64no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3C2×C4⋊C414yescomplemented
214C22×C3C23×C411yescharacteristic
215C22×C3C22×D411yescharacteristic
216C22×C3C22×D411yescharacteristic
217C22×C3C22×Q811yescharacteristic
218C22×C3C2×#4111yescharacteristic
219C22×C3H(5,𝔽2)11yescharacteristic
220C22×C32−1+411yescharacteristic
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—22no—
224C2×D3—22no—
225C2×D3—22no—
226C2×D3—22no—
227C2×D3C2×C4⋊C414yescomplemented
228C3×C4—22no—
229C3×C4—24no—
230C3×C4—42no—
231C3×C4C4×D414yescomplemented
232Dic3—22no—
233Dic3—22no—
234Dic3—42no—
235Dic3—42no—
236Dic3C2×C4⋊C414yescomplemented
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—32no—
242C23—32no—
243C23—32no—
244C23—61no—
245C23—61no—
246C23—61no—
247C23—61no—
248C23C2×C4×D312yes—
249C23D3×D412yes—
250C23D3×Q812yes—
251C23C23×D311yescharacteristic
252D4—316no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—48no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—68no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—128no—
279C2×C4C2×C4×D312yes—
280C2×C4C2×C4×D312yes—
281C2×C4#23512yes—
282C2×C4C2×#6714yes—
283C2×C4C2×#6712yes—
284C2×C3—24no—
285C2×C3C22×C4⋊C411yescharacteristic
286C2×C3C2×C4×D411yescharacteristic
287C2×C3#52011yescharacteristic
288C2×C3D4211yescharacteristic
289C2×C3#54711yescharacteristic
290C2×C3D4×Q811yescharacteristic
291C2×C3#55411yescharacteristic
292D3—24no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—34no—
300C22—62no—
301C22—62no—
302C22—62no—
303C22—62no—
304C22—62no—
305C22—62no—
306C22D3×C4⋊C414yescomplemented
307C22C22×C4×D311yescharacteristic
308C22C2×#23511yescharacteristic
309C22C2×D3×D411yescharacteristic
310C22#100411yescharacteristic
311C22C2×D3×Q811yescharacteristic
312C22#100711yescharacteristic
313C22C22×#6711yescharacteristic
314C4—22no—
315C4—24no—
316C4—34no—
317C4—42no—
318C4—62no—
319C4—62no—
320C4—122no—
321C4—122no—
322C4C4×#6714yescomplemented
323C3D4×C4⋊C411yescharacteristic, complemented, Sylow
324C2—24no—
325C2—64no—
326C2C2×D3×C4⋊C411yescharacteristic
327C2#550411yescharacteristic
328C2#550711yescharacteristic
329C2#551111yescharacteristic
330C2C2×C4×#6711yescharacteristic
331C2D4×#6711yescharacteristic
332C2Q8×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC22×C3×C4⋊C4C2
socle, cosocleC23×C3C24
prime radicalC22×#67C22
hypercenterC22×C4⋊C4D3
hypocenterC3D4×C4⋊C4
LCSG ▷ C22×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
4. C2×D3×C4⋊C4C2semidirect
5. #5212C2semidirect
6. #5210C2semidirect
7. #4989C2semidirect
8. #4987C2semidirect
9. #4976C2semidirect
10. #4975C2semidirect
11. Dic3×C4⋊C4C2semidirect
25. C4×#67C4semidirect
43. C22×C4⋊C4D3semidirect
58. C22×C3×C4C2×C4semidirect
76. C2×C4×D3C2×C4semidirect
79. C3×C4⋊C4D4semidirect
81. #217C2×C4semidirect
83. (C2×D3)⋊C4C2×C4semidirect
89. Dic3⋊C4C2×C4semidirect
91. C4×Dic3C2×C4semidirect
148. #67C4⋊C4direct (N × H)
187. C22×C4C4×D3semidirect
200. C4⋊C4#67direct (N × H)
213. C22×C3C2×C4⋊C4semidirect
227. C2×D3C2×C4⋊C4semidirect
231. C3×C4C4×D4semidirect
236. Dic3C2×C4⋊C4semidirect
306. C22D3×C4⋊C4semidirect
322. C4C4×#67semidirect
323. C3D4×C4⋊C4semidirect

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×D3×C4⋊C42
#52122
#52102
#49892
#49872
#49762
#49752
Dic3×C4⋊C42
D4×C4⋊C43