Finite Group Databasebeta

← #29855

Subgroups — #29855

Order 256.

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

normal subgroups only

This group has 416 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×#490C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2629C211yesmaximal, characteristic, complemented
4#2619C211yesmaximal, characteristic
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2034C212yesmaximal, complemented
8#2033C212yesmaximal, complemented
9#1991C211yesmaximal, characteristic, complemented
10#1957C211yesmaximal, characteristic, complemented
11#1847C211yesmaximal, characteristic, complemented
12C2×#127C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#490—24no—
18#433—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2212yes—
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#398—22no—
30#396—22no—
31#394—22no—
32#392—22no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389C2212yes—
36#384C2211yescharacteristic
37#382C2211yescharacteristic
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41C22×D4—21no—
42C22×D4—21no—
43C22×D4—22no—
44C22×D4C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49#127—22no—
50#127—22no—
51#125—22no—
52#125—22no—
53#121—42no—
54#121—42no—
55#117—22no—
56#117—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D412yes—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#106—42no—
75C4⋊C8—24no—
76#102—24no—
77#102—24no—
78#102—42no—
79#102—42no—
80C22⋊C8—24no—
81#97D414yescomplemented
82C4×C8—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86C24—21no—
87C24—21no—
88C24C2×D411yescharacteristic
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—44no—
100C2×D4—82no—
101C2×D4—82no—
102C2×D4—82no—
103C2×D4—82no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2411yescharacteristic
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8C2×D414yes—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—44no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C22⋊C4—24no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—82no—
136C22⋊C4—82no—
137C22⋊C4—82no—
138C42—21no—
139C42—21no—
140C42C2×D412yes—
141C42#4112yes—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—22no—
146C23—22no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23C22×D411yescharacteristic
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—88no—
163D4—84no—
164D4—84no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4#12114yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C4H(5,𝔽2)12yes—
189C2×C42−1+412yes—
190C8—24no—
191C8—42no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22—42no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—82no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22C2×#12111yescharacteristic
208C22#53511yescharacteristic
209C22#53611yescharacteristic
210C22#57612yes—
211C22#57712yes—
212C4—24no—
213C4—41no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4—82no—
219C4#53712yes—
220C4#54212yes—
221C2—81no—
222C2—161no—
223C2—161no—
224C2#262011yescharacteristic
225C2#306212yes—
226C2#322011yescharacteristic
227C2#322211yescharacteristic
228C2#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×#490C2semidirect
2. C2×#433C2semidirect
3. #2629C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1991C2semidirect
10. #1957C2semidirect
11. #1847C2semidirect
81. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4332
#26292
#26192
#20672
#20652
#20342
#20332
#19912
#19572
#18472