Finite Group Databasebeta

← #29841

Subgroups — #29841

Order 256.

377 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 119 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×#475C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic
3#2640C211yesmaximal, characteristic
4#2320C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2065C212yesmaximal
7#2056C212yesmaximal, complemented
8#2055C212yesmaximal
9#1987C211yesmaximal, characteristic, complemented
10#1983C211yesmaximal, characteristic
11#1843C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17#475—24no—
18#464—24no—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29#397—22no—
30#393—21no—
31#393—21no—
32#391C2212yes—
33#391C2212yes—
34#390C2212yes—
35#389C2212yes—
36#387—22no—
37#383—21no—
38#382—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C2×Q16—22no—
48C2×Q16—42no—
49C2×SD16—22no—
50C2×SD16—42no—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56#124—22no—
57#124—22no—
58C4×Q8—22no—
59C4×Q8D414yes—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C2312yes—
73C2×C42C2311yescharacteristic
74#106—24no—
75C4⋊C8—22no—
76C4⋊C8—22no—
77C4⋊C8—42no—
78C4⋊C8—42no—
79#103—24no—
80#103—22no—
81#103—42no—
82#102—22no—
83#102—42no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D411yescharacteristic
91#95D411yescharacteristic
92#95D412yes—
93C24—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—42no—
102C2×Q8—82no—
103C2×Q8C2×D412yes—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118Q16—44no—
119SD16—44no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C4⋊C4—24no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—82no—
135C4⋊C4C2×D412yes—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—24no—
141C42—22no—
142C42—22no—
143C42—81no—
144C42C2×D412yes—
145C42#4112yes—
146C23—41no—
147C23—42no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23C22×D411yescharacteristic
152Q8—22no—
153Q8—22no—
154Q8—42no—
155Q8—41no—
156Q8—41no—
157Q8—42no—
158Q8—84no—
159D4—41no—
160D4—41no—
161D4—42no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4#12112yes—
185C2×C4#12112yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C4H(5,𝔽2)12yes—
189C2×C42−1+412yes—
190C8—42no—
191C8—41no—
192C8—41no—
193C22—81no—
194C22—82no—
195C22—82no—
196C22—81no—
197C22—81no—
198C22C2×#12111yescharacteristic
199C22#53712yes—
200C22#57712yes—
201C22#57711yescharacteristic
202C22#57811yescharacteristic
203C4—22no—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—42no—
209C4—81no—
210C4—82no—
211C4—82no—
212C4#53511yescharacteristic
213C4#53611yescharacteristic
214C4#54212yes—
215C2—161no—
216C2#262011yescharacteristic
217C2#304311yescharacteristic
218C2#304611yescharacteristic
219C2#325812yes—
220C2#326312yes—
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. C2×#475C2semidirect
4. #2320C2semidirect
5. #2066C2semidirect
7. #2056C2semidirect
9. #1987C2semidirect
11. #1843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4642
#26402
#23202
#20662
#20652
#20562
#20552
#19872
#19832
#18432