Finite Group Databasebeta

← #25866

Subgroups — #25866

Order 256.

619 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536C211yesmaximal, characteristic
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1960C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1788C212yesmaximal
10#1787C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14#536—24no—
15#536—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×#123—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8—21no—
25C2×#117C2211yescharacteristic
26#489—24no—
27#487—24no—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#400—21no—
40#399—21no—
41#395—21no—
42#393—21no—
43#389C2212yes—
44#382C2211yescharacteristic
45#380C2212yes—
46#336C2212yes—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—24no—
50C2×#41—42no—
51C22×Q8—22no—
52C22×Q8—22no—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—24no—
56C22×Q8—42no—
57C22×Q8C2311yescharacteristic
58C22×Q8C2311yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C22×C8C2312yes—
66C22×C8C2312yes—
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8D414yes—
73#124—22no—
74#124—22no—
75#124—22no—
76#124D414yes—
77#123—28no—
78#123—28no—
79C22⋊Q8—28no—
80C22⋊Q8—28no—
81#117—22no—
82#117—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2312yes—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#103—24no—
103#103—24no—
104#103—24no—
105#103—42no—
106#103—42no—
107#103—42no—
108#102—24no—
109#102—42no—
110C22⋊C8—24no—
111#97—22no—
112C4×C8—22no—
113#95—22no—
114#95—21no—
115#95—21no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119#41—48no—
120#41—48no—
121C2×Q8—24no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—48no—
132C2×Q8—48no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—82no—
137C2×Q8C2×D412yes—
138C2×Q8C2×D412yes—
139C2×Q8C2×D412yes—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4C2×D412yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4#4114yes—
167C22×C4C2411yescharacteristic
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—82no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—82no—
193C42—21no—
194C42—21no—
195C42C2×D412yes—
196C42C2×D412yes—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—41no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23#12412yes—
207C23C22×D411yescharacteristic
208Q8—24no—
209Q8—24no—
210Q8—24no—
211Q8—48no—
212Q8—42no—
213Q8—42no—
214Q8—42no—
215Q8—84no—
216D4—24no—
217D4—48no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—48no—
233C2×C4—48no—
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#12012yes—
247C2×C4#12012yes—
248C2×C4#12412yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C2×#4112yes—
252C2×C4H(5,𝔽2)12yes—
253C8—42no—
254C8—42no—
255C22—24no—
256C22—22no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22—82no—
265C22C2×#12011yescharacteristic
266C22C2×#12411yescharacteristic
267C22#53511yescharacteristic
268C22#57712yes—
269C22#57812yes—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4#54612yes—
281C4#54612yes—
282C2—42no—
283C2—82no—
284C2#243511yescharacteristic
285C2#301011yescharacteristic
286C2#301111yescharacteristic
287C2#313712yes—
288C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
2. C2×#489C2semidirect
3. C2×#487C2semidirect
4. #2628C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #1960C2semidirect
8. #1958C2semidirect
10. #1787C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4892
C2×#4872
#26282
#20432
#20422
#19602
#19582
#17882
#17872
#17502