Finite Group Databasebeta

← #29763

Subgroups — #29763

Order 256.

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

normal subgroups only

This group has 433 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×#481C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3#2619C211yesmaximal, characteristic
4#2316C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#2065C212yesmaximal, complemented
8#2041C212yesmaximal, complemented
9#1875C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12C2×#127C2211yescharacteristic
13C2×#121C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#481—24no—
19#480—24no—
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#394—22no—
30#392—22no—
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#389—22no—
34#389C2212yes—
35#389C2212yes—
36#382C2211yescharacteristic
37#380—22no—
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×D4—21no—
42C22×D4—21no—
43C22×D4—22no—
44C22×D4C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#127—42no—
51#121—22no—
52#121—22no—
53#121—22no—
54#121—22no—
55#117—42no—
56#117—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D412yes—
62C2×C4⋊C4D414yescomplemented
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4D412yes—
73C2×C42—21no—
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—24no—
78C4⋊C8—42no—
79#102—24no—
80#102—24no—
81#102—42no—
82#102—42no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88C24—21no—
89C24—21no—
90C24C2×D411yescharacteristic
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—44no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—82no—
103C2×D4—82no—
104C2×D4—82no—
105C2×D4—82no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—82no—
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4#4112yes—
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C4⋊C4C2×D414yes—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C42—22no—
145C42—22no—
146C42—42no—
147C23—21no—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23#12112yes—
162C23C22×D411yescharacteristic
163D4—42no—
164D4—42no—
165D4—42no—
166D4—42no—
167D4—88no—
168D4—84no—
169D4—84no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—84no—
191C2×C4#12112yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)12yes—
195C2×C42−1+412yes—
196C8—42no—
197C8—42no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22C2×#12111yescharacteristic
214C22#53411yescharacteristic
215C22#53611yescharacteristic
216C22#57612yes—
217C22#57712yes—
218C4—22no—
219C4—42no—
220C4—42no—
221C4—44no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4#53712yes—
226C4#53912yes—
227C2—42no—
228C2—161no—
229C2—161no—
230C2#261811yescharacteristic
231C2#304111yescharacteristic
232C2#304311yescharacteristic
233C2#319812yes—
234C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2619C2
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×#481C2semidirect
2. C2×#480C2semidirect
4. #2316C2semidirect
5. #2085C2semidirect
6. #2067C2semidirect
7. #2065C2semidirect
8. #2041C2semidirect
9. #1875C2semidirect
10. #1875C2semidirect
11. #1798C2semidirect
62. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4802
#26192
#23162
#20852
#20672
#20652
#20412
#18752
#17982