Finite Group Databasebeta

← #29957

Subgroups — #29957

Order 256.

432 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#2851C211yesmaximal, characteristic
2C2×#391C211yesmaximal, characteristic
3#2097C212yesmaximal, complemented
4#2068C212yesmaximal, complemented
5#2066C212yesmaximal, complemented
6#2026C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic
8#1852C211yesmaximal, characteristic
9#1797C211yesmaximal, characteristic
10#1787C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25#391—21no—
26#391—21no—
27#391C2212yes—
28#391C2212yes—
29#389C2211yescharacteristic
30#389C2211yescharacteristic
31#387—21no—
32#387—21no—
33#385—21no—
34#385—21no—
35#382C2211yescharacteristic
36#380C2211yescharacteristic
37#380C2211yescharacteristic
38C2×#95—21no—
39C2×#95—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—22no—
45C22×Q8—42no—
46C22×Q8—42no—
47C22×Q8Q814yes—
48C22×Q8C2312yes—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56#125—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—42no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—42no—
76#106—42no—
77C4⋊C8—42no—
78#103—42no—
79#103—42no—
80#103—42no—
81#103—42no—
82C22⋊C8—22no—
83C22⋊C8—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24C2×D411yescharacteristic
95C2×Q8—24no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—48no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—48no—
105C2×Q8C2×Q814yes—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×Q812yes—
132C22×C4#4112yes—
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—41no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23C22⋊Q812yes—
162C23#13512yes—
163C23#13712yes—
164C23C22×D411yescharacteristic
165Q8—24no—
166Q8—24no—
167Q8—48no—
168Q8—42no—
169Q8—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—48no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—84no—
207C2×C4C22⋊Q812yes—
208C2×C4#13712yes—
209C2×C4C22×Q811yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4H(5,𝔽2)12yes—
212C2×C4H(5,𝔽2)12yes—
213C8—42no—
214C8—42no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22#47914yes—
222C22C2×C22⋊Q811yescharacteristic
223C22#53411yescharacteristic
224C22#53511yescharacteristic
225C22C2×#13511yescharacteristic
226C22C2×#13711yescharacteristic
227C22C2×#13711yescharacteristic
228C22#57511yescharacteristic
229C4—22no—
230C4—22no—
231C4—44no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4—81no—
240C4#53812yes—
241C4#54312yes—
242C2—24no—
243C2#262611yescharacteristic
244C2C2×#47911yescharacteristic
245C2#307811yescharacteristic
246C2#319011yescharacteristic
247C2#319611yescharacteristic
248C2#321111yescharacteristic
249C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2097C2semidirect
4. #2068C2semidirect
5. #2066C2semidirect
6. #2026C2semidirect

Maximal subgroups

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

subgroupindex
#28512
C2×#3912
#20972
#20682
#20662
#20262
#18532
#18522
#17972
#17872