Finite Group Databasebeta

← #29835

Subgroups — #29835

Order 256.

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

normal subgroups only

This group has 398 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×#479C211yesmaximal, characteristic, complemented
2C2×#460C211yesmaximal, characteristic
3#2641C211yesmaximal, characteristic
4#2320C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2055C212yesmaximal, complemented
8#2055C212yesmaximal, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1982C211yesmaximal, characteristic
11#1843C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×#127C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#479—24no—
17#460—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29#394—21no—
30#394—21no—
31#394—22no—
32#391C2212yes—
33#390C2212yes—
34#390C2212yes—
35#389C2212yes—
36#387—22no—
37#383—21no—
38#382—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4—21no—
46C22×D4C2312yes—
47C2×SD16—24no—
48C2×SD16—42no—
49C2×SD16—42no—
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53#127—22no—
54#127—22no—
55#125—22no—
56#125—22no—
57C4×Q8—22no—
58C4×Q8D414yes—
59C2×C4⋊C4—21no—
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—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C2312yes—
74C2×C42C2311yescharacteristic
75#107—22no—
76#106—22no—
77C4⋊C8—22no—
78C4⋊C8—22no—
79C4⋊C8—42no—
80C4⋊C8—42no—
81#103—22no—
82#103—22no—
83#102—24no—
84#102—42no—
85#102—42no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D411yescharacteristic
91#95D411yescharacteristic
92#95D412yes—
93C24—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8C2×D412yes—
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—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119SD16—48no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C4⋊C4—24no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—82no—
138C4⋊C4C2×D412yes—
139C22⋊C4—82no—
140C22⋊C4—82no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C42—24no—
144C42—22no—
145C42—22no—
146C42—81no—
147C42C2×D412yes—
148C42#4112yes—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22×D411yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—42no—
160D4—42no—
161D4—42no—
162D4—42no—
163D4—42no—
164D4—84no—
165D4—84no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4#12112yes—
189C2×C4#12112yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4H(5,𝔽2)12yes—
193C2×C42−1+412yes—
194C8—41no—
195C8—41no—
196C8—42no—
197C22—82no—
198C22—82no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22C2×#12111yescharacteristic
205C22#53712yes—
206C22#57611yescharacteristic
207C22#57712yes—
208C22#57711yescharacteristic
209C4—22no—
210C4—22no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4#53511yescharacteristic
219C4#53611yescharacteristic
220C4#54212yes—
221C2—162no—
222C2#262011yescharacteristic
223C2#304311yescharacteristic
224C2#304511yescharacteristic
225C2#325712yes—
226C2#326012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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×#479C2semidirect
4. #2320C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2055C2semidirect
8. #2055C2semidirect
9. #1989C2semidirect
11. #1843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C2×#4602
#26412
#23202
#20672
#20652
#20552
#19892
#19822
#18432