Finite Group Databasebeta

← #29834

Subgroups — #29834

Order 256.

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

normal subgroups only

This group has 422 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×#478C211yesmaximal, characteristic, complemented
2C2×#459C211yesmaximal, characteristic
3#2641C211yesmaximal, characteristic, complemented
4#2319C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2054C212yesmaximal, complemented
8#2054C212yesmaximal, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1982C211yesmaximal, characteristic, complemented
11#1843C211yesmaximal, characteristic, complemented
12C22×D8C2212yes—
13C2×#127C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#478—24no—
17#459—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29#394—21no—
30#394—21no—
31#394—22no—
32#390C2212yes—
33#390C2212yes—
34#390C2212yes—
35#389C2212yes—
36#388—22no—
37#381—21no—
38#380—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×D4—21no—
44C22×D4—21no—
45C22×D4C2312yes—
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×D8—24no—
49C2×D8—42no—
50C2×D8—42no—
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54#127—22no—
55#127—22no—
56#125—22no—
57#125—22no—
58C4×D4—22no—
59C4×D4D414yes—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—22no—
78#106—22no—
79C4⋊C8—22no—
80C4⋊C8—22no—
81C4⋊C8—42no—
82C4⋊C8—42no—
83#102—22no—
84#102—22no—
85#102—24no—
86#102—42no—
87#102—42no—
88#95—21no—
89#95—22no—
90#95D411yescharacteristic
91#95D411yescharacteristic
92#95D412yes—
93C24—21no—
94C24—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—81no—
107C2×D4—82no—
108C2×D4—81no—
109C2×D4C2×D412yes—
110C22×C4—21no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—82no—
116C22×C4C2×D412yes—
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121D8—48no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—82no—
139C4⋊C4C2×D412yes—
140C22⋊C4—24no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—22no—
148C42—22no—
149C42—81no—
150C42C2×D412yes—
151C42#4112yes—
152C23—22no—
153C23—42no—
154C23—41no—
155C23—42no—
156C23—41no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23C22×D411yescharacteristic
165D4—22no—
166D4—22no—
167D4—42no—
168D4—42no—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—84no—
173D4—84no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—84no—
194C2×C4#12112yes—
195C2×C4#12112yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4H(5,𝔽2)12yes—
199C2×C42−1+412yes—
200C8—41no—
201C8—41no—
202C8—42no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22C2×#12111yescharacteristic
216C22#53712yes—
217C22#57612yes—
218C22#57611yescharacteristic
219C22#57711yescharacteristic
220C4—22no—
221C4—22no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—82no—
226C4—81no—
227C4—81no—
228C4#53511yescharacteristic
229C4#53611yescharacteristic
230C4#54212yes—
231C2—82no—
232C2—162no—
233C2#262011yescharacteristic
234C2#304111yescharacteristic
235C2#304511yescharacteristic
236C2#325712yes—
237C2#325912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
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×#478C2semidirect
3. #2641C2semidirect
4. #2319C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2054C2semidirect
8. #2054C2semidirect
9. #1989C2semidirect
10. #1982C2semidirect
11. #1843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4592
#26412
#23192
#20672
#20652
#20542
#19892
#19822
#18432