Finite Group Databasebeta

← #29267

Subgroups — #29267

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2331C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#2097C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic
7#1983C211yesmaximal, characteristic
8#1983C211yesmaximal, characteristic
9#1948C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic
15#1832C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34#391—21no—
35#391—21no—
36#391C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#387—21no—
43#387—21no—
44#385—21no—
45#385—21no—
46#384C2211yescharacteristic
47#382C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50#376C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69#124—22no—
70#124Q814yescomplemented
71C4×Q8—41no—
72C4×Q8—41no—
73C4×D4—41no—
74C4×D4—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—41no—
99#107—41no—
100#106—41no—
101#106—41no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104#103—22no—
105#103—22no—
106#103—41no—
107#103—41no—
108#102—22no—
109#102—22no—
110#102—41no—
111#102—41no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95D411yescharacteristic
121#95D411yescharacteristic
122#95D411yescharacteristic
123#95D411yescharacteristic
124C24—21no—
125C2×Q8—24no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—81no—
132C2×Q8—81no—
133C2×Q8C2×Q812yes—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4C2×Q812yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C4⋊C4—24no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—42no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4#4112yes—
186C4⋊C4#4112yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—42no—
197C42—81no—
198C42—81no—
199C42C2×Q812yes—
200C42#4112yes—
201C23—22no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23C22×D411yescharacteristic
209Q8—22no—
210Q8—22no—
211Q8—41no—
212Q8—41no—
213Q8—84no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—84no—
256C2×C4C22⋊Q812yes—
257C2×C4C22⋊Q812yes—
258C2×C4C22×Q811yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22C2×C22⋊Q811yescharacteristic
276C22#52511yescharacteristic
277C22#53511yescharacteristic
278C22#57611yescharacteristic
279C22#57711yescharacteristic
280C22#57711yescharacteristic
281C22#57811yescharacteristic
282C4—22no—
283C4—22no—
284C4—42no—
285C4—41no—
286C4—42no—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4#55411yescharacteristic
296C4#55411yescharacteristic
297C4#55911yescharacteristic
298C4#55911yescharacteristic
299C2—82no—
300C2#258011yescharacteristic
301C2#308311yescharacteristic
302C2#308411yescharacteristic
303C2#311811yescharacteristic
304C2#312111yescharacteristic
305C2#322411yescharacteristic
306C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2331C2semidirect
3. #2097C2semidirect
5. #2024C2semidirect
9. #1948C2semidirect
10. #1947C2semidirect
11. #1865C2semidirect
13. #1863C2semidirect
15. #1832C2semidirect
70. #124Q8semidirect

Maximal subgroups

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

subgroupindex
#23312
#23262
#20972
#20922
#20242
#20232
#19832
#19482
#19472
#18652
#18642
#18632
#18622
#18322