Finite Group Databasebeta

← #34294

Subgroups — #34294

Order 256.

1556 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×#520C211yesmaximal, characteristic, complemented
3C22×C4×Q8C211yesmaximal, characteristic, complemented
4#2367C213yesmaximal, complemented
5#2361C213yesmaximal, complemented
6#2360C213yesmaximal, complemented
7Q8×C4⋊C4C211yesmaximal, characteristic, complemented
8D4×C4⋊C4C213yesmaximal, complemented
9#2341C211yesmaximal, characteristic, complemented
10Q8×C22⋊C4C211yesmaximal, characteristic, complemented
11#2326C213yesmaximal, complemented
12#2323C213yesmaximal, complemented
13C4×#121C213yesmaximal, complemented
14#2290C213yesmaximal, complemented
15#2276C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18#548C4116yescomplemented
19C2×#124C2213yes—
20C2×C22⋊Q8C2213yes—
21C2×#121C2213yes—
22#520—28no—
23C2×C4×Q8—28no—
24C2×C4×Q8C2213yes—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2213yes—
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2213yes—
32C2×C4×D4C2213yes—
33C2×C4×D4C2216yes—
34C2×C4×D4C2213yes—
35C2×#117C2213yes—
36C22×C4⋊C4C2213yes—
37C22×C4⋊C4C2213yes—
38C22×C42C2213yes—
39#391C2213yes—
40#391C2213yes—
41#390C2213yes—
42#390C2213yes—
43#389C2216yes—
44#389C2213yes—
45#389C2213yes—
46#388C2216yes—
47#388C2216yes—
48#388C2213yes—
49#388C2213yes—
50#387C2213yes—
51#387C2213yes—
52#386C2213yes—
53#386C2213yes—
54#385C2216yes—
55#385C2216yes—
56#385C2213yes—
57#385C2211yescharacteristic
58#384C2213yes—
59#380C2213yes—
60#380C2213yes—
61C4×C4⋊C4C2213yes—
62C4×C4⋊C4C2213yes—
63C4×C4⋊C4C2213yes—
64C4×C4⋊C4C2213yes—
65C4×C22⋊C4C2213yes—
66C4×C22⋊C4C2213yes—
67C4×C22⋊C4C2216yes—
68#376C2213yes—
69C2×#41—24no—
70C2×#41—24no—
71C2×#41C2×C418yescomplemented
72C22×Q8—24no—
73C22×Q8—24no—
74C22×Q8C2×C418yescomplemented
75C22×Q8C2313yes—
76C22×Q8C2311yescharacteristic
77C22×D4C2313yes—
78C22×D4C2313yes—
79C23×C4C2311yescharacteristic
80C23×C4C2313yes—
81C23×C4C2313yes—
82C23×C4C2313yes—
83C23×C4C2316yes—
84#124C2×C4124yescomplemented
85C22⋊Q8C2×C4124yescomplemented
86#121C2×C4124yescomplemented
87C4×Q8—24no—
88C4×Q8—24no—
89C4×Q8—24no—
90C4×Q8—212no—
91C4×Q8—212no—
92C4×Q8C2×C418yescomplemented
93C4×Q8D418yescomplemented
94C4×D4—212no—
95C4×D4—212no—
96C4×D4—412no—
97C4×D4C2×C4124yescomplemented
98#117—212no—
99C2×C4⋊C4—212no—
100C2×C4⋊C4—212no—
101C2×C4⋊C4C2313yes—
102C2×C4⋊C4C2313yes—
103C2×C4⋊C4C2313yes—
104C2×C4⋊C4C2316yes—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2313yes—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2316yes—
109C2×C4⋊C4C2313yes—
110C2×C4⋊C4C2313yes—
111C2×C4⋊C4C2313yes—
112C2×C4⋊C4C2313yes—
113C2×C22⋊C4C2313yes—
114C2×C22⋊C4C2313yes—
115C2×C22⋊C4C2313yes—
116C2×C22⋊C4C2311yescharacteristic
117C2×C22⋊C4C2313yes—
118C2×C22⋊C4C2316yes—
119C2×C22⋊C4C2313yes—
120C2×C22⋊C4C2311yescharacteristic
121C2×C22⋊C4C2313yes—
122C2×C22⋊C4C2316yes—
123C2×C42—212no—
124C2×C42C2313yes—
125C2×C42C2313yes—
126C2×C42C2313yes—
127C2×C42C2313yes—
128C2×C42C2313yes—
129C2×C42C2316yes—
130C2×C42C2313yes—
131C2×C42C2313yes—
132#95C2311yescharacteristic
133#95C2313yes—
134#95C2313yes—
135#95C2313yes—
136#95C2311yescharacteristic
137#95C2313yes—
138#95C2313yes—
139#95C2316yes—
140#95C2313yes—
141#95C2313yes—
142#95C2313yes—
143C24C2413yes—
144C24C2411yescharacteristic
145#41—232no—
146C2×Q8—232no—
147C2×Q8—26no—
148C2×Q8—26no—
149C2×Q8C22×C4112yes—
150C2×Q8C22×C414yes—
151C2×Q8C2×D414yes—
152C2×Q8#4114yes—
153C2×D4—26no—
154C2×D4—26no—
155C2×D4—26no—
156C2×D4—26no—
157C2×D4C22×C4112yes—
158C2×D4C22×C4112yes—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—26no—
162C22×C4—26no—
163C22×C4—26no—
164C22×C4—26no—
165C22×C4—22no—
166C22×C4—26no—
167C22×C4—26no—
168C22×C4—212no—
169C22×C4—43no—
170C22×C4—46no—
171C22×C4—46no—
172C22×C4C22×C4112yes—
173C22×C4C22×C4112yes—
174C22×C4C2411yescharacteristic
175C22×C4C2411yescharacteristic
176C22×C4C2413yes—
177C22×C4C2413yes—
178C22×C4C2413yes—
179C22×C4C2411yescharacteristic
180C22×C4C2413yes—
181C22×C4C2413yes—
182C22×C4C2411yescharacteristic
183C22×C4C2411yescharacteristic
184C22×C4C2413yes—
185C22×C4C2411yescharacteristic
186C22×C4C2413yes—
187C4⋊C4—26no—
188C4⋊C4—26no—
189C4⋊C4—212no—
190C4⋊C4—26no—
191C4⋊C4—22no—
192C4⋊C4—26no—
193C4⋊C4—22no—
194C4⋊C4—43no—
195C4⋊C4—46no—
196C4⋊C4C22×C4112yes—
197C4⋊C4C22×C4112yes—
198C4⋊C4C2×D4112yes—
199C22⋊C4—26no—
200C22⋊C4—22no—
201C22⋊C4—26no—
202C22⋊C4—26no—
203C22⋊C4—26no—
204C22⋊C4—43no—
205C22⋊C4—46no—
206C22⋊C4C22×C414yes—
207C22⋊C4C22×C4112yes—
208C22⋊C4C22×C4124yes—
209C42—26no—
210C42—26no—
211C42—26no—
212C42—26no—
213C42—26no—
214C42—46no—
215C42C22×C4112yes—
216C42C2×D4112yes—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—23no—
222C23—23no—
223C23—23no—
224C23—23no—
225C23—23no—
226C23—23no—
227C23C23×C416yes—
228C23C23×C412yes—
229C23H(5,𝔽2)12yes—
230C232−1+412yes—
231C23C2511yescharacteristic
232Q8—224no—
233Q8C4×D4116yescomplemented
234D4—224no—
235D4—224no—
236C2×C4—28no—
237C2×C4—21no—
238C2×C4—23no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—23no—
242C2×C4—23no—
243C2×C4—23no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—23no—
249C2×C4—21no—
250C2×C4—23no—
251C2×C4—224no—
252C2×C4—224no—
253C2×C4—21no—
254C2×C4—23no—
255C2×C4—23no—
256C2×C4—21no—
257C2×C4—23no—
258C2×C4—23no—
259C2×C4—23no—
260C2×C4—23no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—23no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—23no—
267C2×C4—23no—
268C2×C4—412no—
269C2×C4—812no—
270C2×C4C23×C412yes—
271C2×C4C23×C416yes—
272C2×C4C23×C416yes—
273C2×C4C23×C412yes—
274C2×C4C23×C416yes—
275C2×C4C22×D412yes—
276C2×C4C22×D416yes—
277C2×C4C22×D416yes—
278C2×C4C2×#4116yes—
279C2×C42−1+412yes—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—26no—
287C22—26no—
288C22—26no—
289C22—43no—
290C22—43no—
291C22—43no—
292C22—43no—
293C22#52014yescomplemented
294C22C24×C411yescharacteristic
295C22C23×D411yescharacteristic
296C22C22×#4111yescharacteristic
297C22C2×H(5,𝔽2)11yescharacteristic
298C22C2×2−1+411yescharacteristic
299C22C2×2−1+411yescharacteristic
300C22#58511yescharacteristic
301C4—22no—
302C4—26no—
303C4—26no—
304C4—22no—
305C4—22no—
306C4—22no—
307C4—26no—
308C4—41no—
309C4—43no—
310C4—43no—
311C4—41no—
312C4—43no—
313C4C2×C4×D4112yes—
314C2—24no—
315C2—46no—
316C2C22×C4×D411yescharacteristic
317C2C2×#52011yescharacteristic
318C2C4×2−1+411yescharacteristic
319C2#348611yescharacteristic
320C2#348811yescharacteristic
321C2#351611yescharacteristic
322C2#352211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC23C25
commutatorC22C24×C4
prime radicalC2×#548C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#520C2semidirect
3. C22×C4×Q8C2semidirect
4. #2367C2semidirect
5. #2361C2semidirect
6. #2360C2semidirect
7. Q8×C4⋊C4C2semidirect
8. D4×C4⋊C4C2semidirect
9. #2341C2semidirect
10. Q8×C22⋊C4C2semidirect
11. #2326C2semidirect
12. #2323C2semidirect
13. C4×#121C2semidirect
14. #2290C2semidirect
15. #2276C2semidirect
18. #548C4semidirect
71. C2×#41C2×C4semidirect
74. C22×Q8C2×C4semidirect
84. #124C2×C4semidirect
85. C22⋊Q8C2×C4semidirect
86. #121C2×C4semidirect
92. C4×Q8C2×C4semidirect
93. C4×Q8D4semidirect
97. C4×D4C2×C4semidirect
233. Q8C4×D4semidirect
293. C22#520semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#5202
C22×C4×Q82
#23672
#23612
#23602
Q8×C4⋊C42
D4×C4⋊C42
#23412
Q8×C22⋊C42
#23262
#23232
C4×#1212
#22902
#22762