Finite Group Databasebeta

← C22×#4894 #11659483

Subgroups — C22×#4894 #11659483

Order 768.

2710 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C212yesmaximal, complemented
2C22×#904C211yesmaximal, characteristic
3C22×C3×#108C211yesmaximal, characteristic, complemented
4C22×#825C212yesmaximal, complemented
5C22×#815C211yesmaximal, characteristic, complemented
6C2×#4894C2124yesmaximal, direct product factor
7C22×#444—31nomaximal, Sylow
8C22×#235C2212yes—
9C22×C3×M16C2212yes—
10C23×C3×C8C2211yescharacteristic
11C22×C4×Dic3C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C2×#907—212no—
14C2×#907C22124yescomplemented
15C22×Dic12C411yescharacteristic, complemented
16C2×#904C4112yescomplemented
17C2×#904C22112yes—
18C22×D24C411yescharacteristic, complemented
19C22×#204C412yescomplemented
20C22×ΓL(1,𝔽25)—22no—
21C22×D3×C8—22no—
22C2×C3×#108C22112yescomplemented
23C2×#825C22124yescomplemented
24C2×#815C22112yescomplemented
25#4894C22164yesdirect product factor
26C22×#135—31no—
27C22×#131—32no—
28C2×#444—324no—
29C22×#108D311yescharacteristic, complemented
30C22×#104—32no—
31C22×#97—31no—
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C23×Dic3—21no—
35C2×#235—26no—
36C2×#235—26no—
37C2×#235C2×C4112yes—
38C2×#235C23112yes—
39C22×D12C2×C412yes—
40C22×C4×D3—22no—
41C22×Dic6C2×C412yes—
42C2×C3×M16—26no—
43C2×C3×M16C23112yes—
44C22×C3×C8—22no—
45C22×C3×C8C2×C411yescharacteristic
46C22×C3×C8C2×C411yescharacteristic
47C22×C3×C8C2×C416yes—
48C22×C3×C8C2316yes—
49C2×C4×Dic3—23no—
50C2×C4×Dic3C2316yes—
51C22×C3⋊C8D412yes—
52C22×C3⋊C8D416yes—
53C22×C3⋊C8C2316yes—
54#907—248no—
55#907C23132yescomplemented
56C2×Dic12C2×C4112yescomplemented
57#904C2×C4148yescomplemented
58#904C23116yes—
59C2×D24C2×C4112yescomplemented
60C2×#204C2×C4124yescomplemented
61C2×ΓL(1,𝔽25)—224no—
62C2×D3×C8—224no—
63C3×#108C23116yescomplemented
64#825C23132yescomplemented
65#815C23116yescomplemented
66C22×#41—32no—
67C2×#135—312no—
68C2×#135—312no—
69C22×Q16—31no—
70C22×SD16—32no—
71C22×D8—31no—
72C2×#131—324no—
73C2×#131—612no—
74C22×M16—62no—
75C22×M16C2×D312yes—
76C23×C8—31no—
77C23×C8—62no—
78C23×C8C2×D311yescharacteristic
79C22×C42—31no—
80#444—364no—
81C2×#108C2×D3112yescomplemented
82C2×#104—324no—
83C2×#97—312no—
84C24×C3#4111yescharacteristic
85C23×D3—22no—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C22×C413yes—
88C22×C3×C4C2×D413yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4113yes—
91C22×C3×C4C2413yes—
92C2×#67—212no—
93C2×#67—212no—
94C22×Dic3—22no—
95C22×Dic3—26no—
96C22×Dic3—44no—
97#235—224no—
98#235—212no—
99#235—212no—
100#235C22×C4124yes—
101#235C2418yes—
102C2×D12—26no—
103C2×D12C22×C4112yes—
104C2×C4×D3—212no—
105C2×C4×D3—212no—
106C2×Dic6—26no—
107C2×Dic6C22×C4112yes—
108C3×M16—212no—
109C3×M16C2418yes—
110C2×C3×C8—212no—
111C2×C3×C8C22×C416yes—
112C2×C3×C8C22×C416yes—
113C2×C3×C8C22×C4112yes—
114C2×C3×C8C2414yes—
115C4×Dic3—26no—
116C4×Dic3C2414yes—
117C2×C3⋊C8C2×D4112yes—
118C2×C3⋊C8C2×D4112yes—
119C2×C3⋊C8C2414yes—
120Dic12C22×C4116yescomplemented
121D24C22×C4116yescomplemented
122#204C22×C4132yescomplemented
123ΓL(1,𝔽25)—232no—
124D3×C8—232no—
125C2×#41—312no—
126C2×#41—312no—
127C2×#41—66no—
128C2×#41—66no—
129C22×Q8—32no—
130C22×D4—32no—
131C22×D4—62no—
132C23×C4—62no—
133C23×C4—61no—
134C23×C4C22×D311yescharacteristic
135#135—348no—
136#135—316no—
137C2×Q16—312no—
138C2×SD16—324no—
139C2×D8—312no—
140#131—332no—
141#131—648no—
142C2×M16—26no—
143C2×M16—624no—
144C2×M16C22×D3112yes—
145C22×C8—22no—
146C22×C8—36no—
147C22×C8—32no—
148C22×C8—36no—
149C22×C8—624no—
150C22×C8C4×D311yescharacteristic
151C22×C8C4×D311yescharacteristic
152C22×C8C4×D316yes—
153C22×C8C22×D316yes—
154C2×C42—36no—
155C2×C42—63no—
156#108C22×D3116yescomplemented
157#104—332no—
158#97—316no—
159C23×C3—24no—
160C23×C3C4×D413yes—
161C23×C3C4×D411yescharacteristic
162C23×C3C2×#4113yes—
163C22×D3—212no—
164C22×D3—48no—
165C2×C3×C4C4×D413yes—
166C2×C3×C4C4×D413yes—
167C2×C3×C4C4×D416yes—
168C2×C3×C4C23×C413yes—
169C2×C3×C4C23×C413yes—
170C2×C3×C4C22×D413yes—
171C2×C3×C4C22×D413yes—
172C2×C3×C4C2×#4113yes—
173C2×C3×C4C2511yescharacteristic
174#67—224no—
175#67—28no—
176C2×Dic3—212no—
177C2×Dic3—24no—
178C2×Dic3—412no—
179D12—212no—
180D12C23×C418yes—
181C4×D3—224no—
182C4×D3—28no—
183Dic6—212no—
184Dic6C23×C418yes—
185C3×C8—28no—
186C3×C8C23×C414yes—
187C3×C8C23×C414yes—
188C3⋊C8C22×D418yes—
189C24—62no—
190C24#23911yescharacteristic
191#41—324no—
192#41—38no—
193#41—624no—
194#41—612no—
195#41—612no—
196C2×Q8—312no—
197C2×Q8—66no—
198C2×D4—312no—
199C2×D4—612no—
200C2×D4—612no—
201C2×D4—66no—
202C22×C4—62no—
203C22×C4—612no—
204C22×C4—612no—
205C22×C4—66no—
206C22×C4—124no—
207C22×C4C2×C4×D311yescharacteristic
208C22×C4C2×C4×D313yes—
209C22×C4D3×D413yes—
210C22×C4D3×D411yescharacteristic
211C22×C4#23913yes—
212C22×C4C23×D313yes—
213Q16—316no—
214SD16—332no—
215D8—316no—
216M16—212no—
217M16—632no—
218M16C23×D318yes—
219C2×C8—212no—
220C2×C8—312no—
221C2×C8—312no—
222C2×C8—34no—
223C2×C8—632no—
224C2×C8C2×C4×D316yes—
225C2×C8C2×C4×D316yes—
226C2×C8C2×C4×D3112yes—
227C2×C8C23×D314yes—
228C42—34no—
229C42—66no—
230C22×C3—26no—
231C22×C3—26no—
232C22×C3#44414yescomplemented
233C22×C3C2×C4×D413yes—
234C22×C3C2×C4×D413yes—
235C22×C3C22×#4111yescharacteristic
236C2×D3—28no—
237C2×D3—424no—
238C3×C4C2×C4×D413yes—
239C3×C4C2×C4×D413yes—
240C3×C4C24×C411yescharacteristic
241C3×C4C23×D411yescharacteristic
242Dic3—28no—
243Dic3—44no—
244C23—24no—
245C23—612no—
246C23—128no—
247C23#89313yes—
248C23#89311yescharacteristic
249C23C2×#23913yes—
250Q8—38no—
251Q8—612no—
252D4—38no—
253D4—68no—
254D4—624no—
255D4—612no—
256C2×C4—612no—
257C2×C4—68no—
258C2×C4—624no—
259C2×C4—64no—
260C2×C4—1212no—
261C2×C4#89313yes—
262C2×C4#89313yes—
263C2×C4#89316yes—
264C2×C4C22×C4×D313yes—
265C2×C4C22×C4×D313yes—
266C2×C4C2×D3×D413yes—
267C2×C4C2×D3×D413yes—
268C2×C4C2×#23913yes—
269C2×C4C24×D311yescharacteristic
270C8—28no—
271C8—38no—
272C8C22×C4×D314yes—
273C8C22×C4×D314yes—
274C2×C3—21no—
275C2×C3—23no—
276C2×C3C2×#44416yescomplemented
277C2×C3C22×C4×D411yescharacteristic
278D3—48no—
279C22—26no—
280C22—26no—
281C22—68no—
282C22—1224no—
283C22#489414yesdirect product factor
284C22C2×#89313yes—
285C22C2×#89313yes—
286C22C22×#23911yescharacteristic
287C4—68no—
288C4—124no—
289C4C2×#89313yes—
290C4C2×#89313yes—
291C4C23×C4×D311yescharacteristic
292C4C22×D3×D411yescharacteristic
293C3C22×#44411yescharacteristic, complemented, Sylow
294C2—21no—
295C2—23no—
296C2—128no—
297C2C2×#489416yesdirect product factor
298C2C22×#89311yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC22×C3×#108C2
socleC23×C3C4×D4
prime radicalC22×#904C2
cosocleC2×C3×C4C25
hypercenterC22×#108D3
hypocenterC3C22×#444
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
1. C22×#907C2semidirect
3. C22×C3×#108C2semidirect
4. C22×#825C2semidirect
5. C22×#815C2semidirect
6. C2×#4894C2direct (N × H)
14. C2×#907C22semidirect
15. C22×Dic12C4semidirect
16. C2×#904C4semidirect
18. C22×D24C4semidirect
19. C22×#204C4semidirect
22. C2×C3×#108C22semidirect
23. C2×#825C22semidirect
24. C2×#815C22semidirect
25. #4894C22direct (N × H)
29. C22×#108D3semidirect
55. #907C23semidirect
56. C2×Dic12C2×C4semidirect
57. #904C2×C4semidirect
59. C2×D24C2×C4semidirect
60. C2×#204C2×C4semidirect
63. C3×#108C23semidirect
64. #825C23semidirect
65. #815C23semidirect
81. C2×#108C2×D3semidirect
120. Dic12C22×C4semidirect
121. D24C22×C4semidirect
122. #204C22×C4semidirect
156. #108C22×D3semidirect
232. C22×C3#444semidirect
276. C2×C3C2×#444semidirect
283. C22#4894direct (N × H)
293. C3C22×#444semidirect
297. C2C2×#4894direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9072
C22×#9042
C22×C3×#1082
C22×#8252
C22×#8152
C2×#48942
C22×#4443