Finite Group Databasebeta

← C22×#4839 #11658801

Subgroups — C22×#4839 #11658801

Order 768.

3006 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C23×D3×C8C211yesmaximal, characteristic
3C22×#872C211yesmaximal, characteristic
4C22×C3×C4⋊C8C211yesmaximal, characteristic, complemented
5C22×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C22×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C22×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C2×#4839C2124yesmaximal, direct product factor
9C22×#424—31nomaximal, Sylow
10C23×C4×D3C2211yescharacteristic
11C23×C3×C8C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C22×(C2×D3)⋊C4C411yescharacteristic
15C22×(C2×D3)⋊C4C411yescharacteristic
16C22×Dic3⋊C4C411yescharacteristic
17C22×Dic3⋊C4C411yescharacteristic
18C22×C4×Dic3C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C22×ΓL(1,𝔽25)—28no—
22C22×ΓL(1,𝔽25)C22112yes—
23C22×D3×C8—28no—
24C22×D3×C8C22112yes—
25C2×#872C4112yes—
26C2×#872C22112yes—
27C2×C3×C4⋊C8C22112yescomplemented
28C2×Dic3⋊C8C22112yescomplemented
29C2×Dic3⋊C8C22112yescomplemented
30C2×C4⋊(C3⋊C8)C22112yescomplemented
31#4839C22164yesdirect product factor
32C23×M16—31no—
33C24×C8—31no—
34C22×#117—31no—
35C2×#424—324no—
36C22×C4⋊C8—31no—
37C22×C4⋊C8—31no—
38C22×C4⋊C8—31no—
39C22×C4⋊C8D311yescharacteristic, complemented
40C24×D3C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2×C411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C23×Dic3C2×C411yescharacteristic
47C22×C4×D3C2×C416yes—
48C22×C4×D3D418yes—
49C22×C4×D3Q818yes—
50C22×C4×D3C2316yes—
51C22×C3×C8—24no—
52C22×C3×C8—24no—
53C22×C3×C8C2316yes—
54C22×C3×C8C2316yes—
55C2×C3×C42C2×C416yes—
56C2×C3×C42C2316yes—
57C2×(C2×D3)⋊C4C2×C4112yes—
58C2×(C2×D3)⋊C4C2×C4112yes—
59C2×Dic3⋊C4C2×C4112yes—
60C2×Dic3⋊C4C2×C4112yes—
61C2×C4×Dic3C2×C416yes—
62C2×C4×Dic3C2316yes—
63C22×C3⋊C8—24no—
64C22×C3⋊C8—24no—
65C22×C3⋊C8C2316yes—
66C22×C3⋊C8C2316yes—
67C2×ΓL(1,𝔽25)—248no—
68C2×ΓL(1,𝔽25)C23116yes—
69C2×D3×C8—248no—
70C2×D3×C8C23116yes—
71#872C2×C4148yes—
72#872C23116yes—
73C3×C4⋊C8C23116yescomplemented
74Dic3⋊C8C23116yescomplemented
75Dic3⋊C8C23116yescomplemented
76C4⋊(C3⋊C8)C23116yescomplemented
77C24×C4—31no—
78C22×M16—312no—
79C22×M16—68no—
80C23×C8—31no—
81C23×C8—31no—
82C23×C8—312no—
83C23×C8—68no—
84C23×C8C2×D311yescharacteristic
85C23×C8C2×D311yescharacteristic
86C2×#117—312no—
87C2×#117—312no—
88C22×C4⋊C4—31no—
89C22×C4⋊C4—31no—
90C22×C22⋊C4—31no—
91C22×C22⋊C4—31no—
92C22×C42—31no—
93C22×C42C2×D311yescharacteristic
94#424—364no—
95C2×C4⋊C8—312no—
96C2×C4⋊C8—312no—
97C2×C4⋊C8—312no—
98C2×C4⋊C8C2×D3112yescomplemented
99C24×C3C22×C411yescharacteristic
100C23×D3—24no—
101C23×D3—24no—
102C23×D3C4⋊C418yes—
103C23×D3C22×C416yes—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4C22×C416yes—
107C22×C3×C4C22×C416yes—
108C22×C3×C4C22×C413yes—
109C22×C3×C4C2×D414yes—
110C22×C3×C4C2×Q814yes—
111C22×C3×C4C2413yes—
112C22×Dic3—24no—
113C22×Dic3—24no—
114C22×Dic3C4⋊C418yes—
115C22×Dic3C22×C416yes—
116C22×Dic3C22×C416yes—
117C22×Dic3C22×C416yes—
118C2×C4×D3C4⋊C4148yes—
119C2×C4×D3C22×C4112yes—
120C2×C4×D3C2×D4124yes—
121C2×C4×D3C2×Q8124yes—
122C2×C4×D3C2414yes—
123C2×C3×C8—212no—
124C2×C3×C8—212no—
125C2×C3×C8C2414yes—
126C2×C3×C8C2414yes—
127C3×C42C22×C4112yes—
128C3×C42C2414yes—
129(C2×D3)⋊C4C22×C4116yes—
130(C2×D3)⋊C4C22×C4116yes—
131Dic3⋊C4C22×C4116yes—
132Dic3⋊C4C22×C4116yes—
133C4×Dic3C22×C4112yes—
134C4×Dic3C2414yes—
135C2×C3⋊C8—212no—
136C2×C3⋊C8—212no—
137C2×C3⋊C8C2414yes—
138C2×C3⋊C8C2414yes—
139ΓL(1,𝔽25)—232no—
140D3×C8—232no—
141C25—31no—
142C23×C4—31no—
143C23×C4—36no—
144C23×C4—38no—
145C23×C4—38no—
146C23×C4—36no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4C4×D311yescharacteristic
150C23×C4C4×D311yescharacteristic
151C23×C4C22×D311yescharacteristic
152C2×M16—316no—
153C2×M16—648no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—36no—
157C22×C8—36no—
158C22×C8—316no—
159C22×C8—648no—
160C22×C8—64no—
161C22×C8—64no—
162C22×C8C22×D316yes—
163C22×C8C22×D316yes—
164#117—348no—
165#117—316no—
166C2×C4⋊C4—312no—
167C2×C4⋊C4—312no—
168C2×C22⋊C4—312no—
169C2×C22⋊C4—312no—
170C2×C42—36no—
171C2×C42—36no—
172C2×C42C4×D316yes—
173C2×C42C22×D316yes—
174C4⋊C8—316no—
175C4⋊C8—316no—
176C4⋊C8—316no—
177C4⋊C8C22×D3116yescomplemented
178C23×C3C2×C4⋊C414yes—
179C23×C3#13114yes—
180C23×C3#13114yes—
181C23×C3C23×C413yes—
182C22×D3—212no—
183C22×D3—212no—
184C22×D3—232no—
185C22×D3C2×C4⋊C4124yes—
186C22×D3C23×C414yes—
187C2×C3×C4—212no—
188C2×C3×C4—212no—
189C2×C3×C4C2×C4⋊C4112yes—
190C2×C3×C4C23×C414yes—
191C2×C3×C4C23×C414yes—
192C2×C3×C4C23×C413yes—
193C2×C3×C4C22×D416yes—
194C2×C3×C4C22×Q816yes—
195C2×C3×C4C2511yescharacteristic
196C2×Dic3—212no—
197C2×Dic3—212no—
198C2×Dic3C2×C4⋊C4124yes—
199C2×Dic3C23×C414yes—
200C2×Dic3C23×C414yes—
201C2×Dic3C23×C414yes—
202C4×D3C2×C4⋊C4148yes—
203C4×D3C22×D418yes—
204C4×D3C22×Q818yes—
205C3×C8—24no—
206C3×C8—24no—
207C3⋊C8—24no—
208C3⋊C8—24no—
209C24—38no—
210C24—36no—
211C24—64no—
212C24—64no—
213C24C2×C4×D311yescharacteristic
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—38no—
217C22×C4—36no—
218C22×C4—348no—
219C22×C4—312no—
220C22×C4—324no—
221C22×C4—324no—
222C22×C4—34no—
223C22×C4—36no—
224C22×C4—36no—
225C22×C4—64no—
226C22×C4—64no—
227C22×C4C2×C4×D316yes—
228C22×C4C2×C4×D316yes—
229C22×C4C2×C4×D313yes—
230C22×C4D3×D414yes—
231C22×C4D3×Q814yes—
232C22×C4C23×D313yes—
233M16—632no—
234C2×C8—212no—
235C2×C8—212no—
236C2×C8—34no—
237C2×C8—34no—
238C2×C8—632no—
239C2×C8—612no—
240C2×C8—612no—
241C2×C8C23×D314yes—
242C2×C8C23×D314yes—
243C4⋊C4—316no—
244C4⋊C4—316no—
245C22⋊C4—316no—
246C22⋊C4—316no—
247C42—312no—
248C42—34no—
249C42C2×C4×D3112yes—
250C42C23×D314yes—
251C22×C3#424116yescomplemented
252C22×C3C22×C4⋊C416yes—
253C22×C3C2×#13116yes—
254C22×C3C2×#13116yes—
255C22×C3C24×C411yescharacteristic
256C2×D3—248no—
257C2×D3—24no—
258C2×D3—24no—
259C2×D3C22×C4⋊C418yes—
260C3×C4—24no—
261C3×C4—24no—
262C3×C4C22×C4⋊C416yes—
263C3×C4C23×D411yescharacteristic
264C3×C4C23×Q811yescharacteristic
265Dic3—24no—
266Dic3—24no—
267Dic3C22×C4⋊C418yes—
268C23—34no—
269C23—324no—
270C23—612no—
271C23—612no—
272C23—632no—
273C23D3×C4⋊C414yes—
274C23#90114yes—
275C23#90714yes—
276C23C22×C4×D313yes—
277C2×C4—212no—
278C2×C4—212no—
279C2×C4—34no—
280C2×C4—324no—
281C2×C4—38no—
282C2×C4—38no—
283C2×C4—348no—
284C2×C4—34no—
285C2×C4—34no—
286C2×C4—612no—
287C2×C4—612no—
288C2×C4D3×C4⋊C4112yes—
289C2×C4C22×C4×D314yes—
290C2×C4C22×C4×D314yes—
291C2×C4C22×C4×D313yes—
292C2×C4C2×D3×D416yes—
293C2×C4C2×D3×Q816yes—
294C2×C4C24×D311yescharacteristic
295C8—24no—
296C8—24no—
297C8—64no—
298C8—64no—
299C2×C3C2×#424112yescomplemented
300C2×C3C23×C4⋊C411yescharacteristic
301C2×C3C22×#13111yescharacteristic
302C2×C3C22×#13111yescharacteristic
303D3—28no—
304C22—38no—
305C22—64no—
306C22—64no—
307C22—648no—
308C22#4839116yesdirect product factor
309C22C2×D3×C4⋊C416yes—
310C22C2×#90116yes—
311C22C2×#90716yes—
312C22C23×C4×D311yescharacteristic
313C4—24no—
314C4—24no—
315C4—38no—
316C4—64no—
317C4—64no—
318C4C2×D3×C4⋊C416yes—
319C4C22×D3×D411yescharacteristic
320C4C22×D3×Q811yescharacteristic
321C3C22×#42411yescharacteristic, complemented, Sylow
322C2—68no—
323C2C2×#4839112yesdirect product factor
324C2C22×D3×C4⋊C411yescharacteristic
325C2C22×#90111yescharacteristic
326C2C22×#90711yescharacteristic
subgroupG/H
centerC23×C4C22×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC22×C3×C4⋊C8C2
socleC24×C3C22×C4
prime radicalC24×D3C2×C4
cosocleC2×C3×C4C25
hypercenterC22×C4⋊C8D3
hypocenterC3C22×#424
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
4. C22×C3×C4⋊C8C2semidirect
5. C22×Dic3⋊C8C2semidirect
6. C22×Dic3⋊C8C2semidirect
7. C22×C4⋊(C3⋊C8)C2semidirect
8. C2×#4839C2direct (N × H)
27. C2×C3×C4⋊C8C22semidirect
28. C2×Dic3⋊C8C22semidirect
29. C2×Dic3⋊C8C22semidirect
30. C2×C4⋊(C3⋊C8)C22semidirect
31. #4839C22direct (N × H)
39. C22×C4⋊C8D3semidirect
73. C3×C4⋊C8C23semidirect
74. Dic3⋊C8C23semidirect
75. Dic3⋊C8C23semidirect
76. C4⋊(C3⋊C8)C23semidirect
98. C2×C4⋊C8C2×D3semidirect
177. C4⋊C8C22×D3semidirect
251. C22×C3#424semidirect
299. C2×C3C2×#424semidirect
308. C22#4839direct (N × H)
321. C3C22×#424semidirect
323. C2C2×#4839direct (N × H)

Maximal subgroups

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

subgroupindex
C23×ΓL(1,𝔽25)2
C23×D3×C82
C22×#8722
C22×C3×C4⋊C82
C22×Dic3⋊C82
C22×C4⋊(C3⋊C8)2
C2×#48392
C22×#4243