Finite Group Databasebeta

← #27921

Subgroups — #27921

Order 256.

442 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#482C211yesmaximal, characteristic
4C2×#480C211yesmaximal, characteristic
5#2620C211yesmaximal, characteristic
6#2317C211yesmaximal, characteristic, complemented
7#2097C211yesmaximal, characteristic, complemented
8#2095C211yesmaximal, characteristic, complemented
9#2093C211yesmaximal, characteristic, complemented
10#2092C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1958C211yesmaximal, characteristic, complemented
13#1847C211yesmaximal, characteristic, complemented
14#1845C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#485—24no—
23#484—24no—
24#482—24no—
25#480—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#396—22no—
47#394—22no—
48#391C2211yescharacteristic
49#390C2211yescharacteristic
50#389—22no—
51#385—22no—
52#384C2211yescharacteristic
53#384C2211yescharacteristic
54#384C2211yescharacteristic
55#382C2211yescharacteristic
56#382C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#124—22no—
69#124—22no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—22no—
73#121—22no—
74#117—22no—
75#117—22no—
76#117—22no—
77#117—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4D412yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D412yes—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#107—22no—
107#106—22no—
108#106—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117C22⋊C8D414yes—
118C22⋊C8D414yes—
119#97—22no—
120C4×C8—22no—
121#95—22no—
122#95—22no—
123#95—22no—
124C24—21no—
125C24C2×D411yescharacteristic
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—42no—
139C2×D4—82no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—82no—
153C22×C4C2×D412yes—
154C22×C4C2×D412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
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×C8C2×D412yes—
168C2×C8C2×D412yes—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C4⋊C4—24no—
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—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C22⋊C4—24no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—44no—
193C22⋊C4—42no—
194C22⋊C4—82no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42#4112yes—
199C42#4112yes—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23#12712yes—
210C23C22×D411yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—84no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4#12314yes—
248C2×C4#12712yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4C2×#4111yescharacteristic
253C2×C42−1+411yescharacteristic
254C2×C42−1+411yescharacteristic
255C8—22no—
256C8—22no—
257C8—22no—
258C8—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—42no—
263C22—42no—
264C22—82no—
265C22—81no—
266C22—81no—
267C22—82no—
268C22—81no—
269C22C2×#12311yescharacteristic
270C22C2×#12711yescharacteristic
271C22#53611yescharacteristic
272C22#57612yes—
273C22#57812yes—
274C4—24no—
275C4—22no—
276C4—42no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#54711yescharacteristic
285C4#54711yescharacteristic
286C4#54711yescharacteristic
287C4#54711yescharacteristic
288C2—42no—
289C2—161no—
290C2#250111yescharacteristic
291C2#309812yes—
292C2#315511yescharacteristic
293C2#315611yescharacteristic
294C2#324312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
2. C2×#485C2semidirect
6. #2317C2semidirect
7. #2097C2semidirect
8. #2095C2semidirect
9. #2093C2semidirect
10. #2092C2semidirect
11. #1959C2semidirect
12. #1958C2semidirect
13. #1847C2semidirect
14. #1845C2semidirect
15. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4852
C2×#4822
C2×#4802
#26202
#23172
#20972
#20952
#20932
#20922
#19592
#19582
#18472
#18452
#17502