Finite Group Databasebeta

← #25614

Subgroups — #25614

Order 256.

608 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 131 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×#542C211yesmaximal, characteristic, complemented
2C2×#496C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#2619C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14#542—24no—
15#542—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#126—21no—
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2212yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#496—24no—
30#486—24no—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#394—21no—
39#394—21no—
40#392—21no—
41#392—21no—
42#390C2212yes—
43#382C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2312yes—
54C22×D4C2311yescharacteristic
55C23×C4—21no—
56C23×C4C2311yescharacteristic
57C2×SD16—24no—
58C2×SD16—24no—
59C2×SD16—42no—
60C2×D8—24no—
61C2×D8—24no—
62C2×D8—42no—
63C22×C8C2312yes—
64C22×C8C2312yes—
65C4⋊Q8—22no—
66C4⋊Q8D414yescomplemented
67#127—22no—
68#127—22no—
69#126—28no—
70#124—28no—
71#124—22no—
72#124D414yescomplemented
73#123—28no—
74C22⋊Q8—28no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#121—22no—
78#121—22no—
79#121—42no—
80#121—42no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×D4—22no—
84C4×D4—22no—
85#117—22no—
86#117—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#103—42no—
109#102—24no—
110#102—24no—
111#102—42no—
112C22⋊C8—24no—
113#97—22no—
114C4×C8—22no—
115#95—21no—
116#95—21no—
117C24—21no—
118C24—22no—
119C24C2×D411yescharacteristic
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—82no—
125C2×Q8C2×D412yes—
126C2×D4—24no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—44no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—82no—
140C2×D4—82no—
141C2×D4—82no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4C2×D412yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—82no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161SD16—48no—
162D8—48no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8#4114yes—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4C2×D412yes—
179C4⋊C4C2×D412yes—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—44no—
188C22⋊C4—82no—
189C22⋊C4—82no—
190C42—24no—
191C42—21no—
192C42—21no—
193C42C2×D412yes—
194C42C2×D412yes—
195C23—21no—
196C23—22no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23C22×D411yescharacteristic
213Q8—24no—
214Q8—42no—
215Q8—84no—
216D4—24no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—84no—
223D4—84no—
224D4—84no—
225D4—88no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
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—42no—
247C2×C4—84no—
248C2×C4#12012yes—
249C2×C4#12012yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C2×#4112yes—
253C2×C4H(5,𝔽2)12yes—
254C8—24no—
255C8—42no—
256C22—42no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—42no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22C2×#12011yescharacteristic
274C22#53211yescharacteristic
275C22#53511yescharacteristic
276C22#57612yes—
277C22#57712yes—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—42no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—41no—
286C4—82no—
287C4#54612yes—
288C4#55012yes—
289C2—81no—
290C2—82no—
291C2—162no—
292C2#243111yescharacteristic
293C2#300811yescharacteristic
294C2#301011yescharacteristic
295C2#317912yes—
296C2#322412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#542C2semidirect
2. C2×#496C2semidirect
3. C2×#486C2semidirect
4. #2619C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1993C2semidirect
8. #1958C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
11. #1750C2semidirect
66. C4⋊Q8D4semidirect
72. #124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4962
C2×#4862
#26192
#20002
#19992
#19932
#19582
#18802
#18792
#17502