Finite Group Databasebeta

← #29836

Subgroups — #29836

Order 256.

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

normal subgroups only

This group has 380 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
2C2×#461C211yesmaximal, characteristic
3#2642C211yesmaximal, characteristic, complemented
4#2319C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2055C212yesmaximal
8#2055C212yesmaximal
9#1989C211yesmaximal, characteristic, complemented
10#1981C211yesmaximal, characteristic, complemented
11#1843C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#478—24no—
17#461—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29#400—21no—
30#400—21no—
31#400—22no—
32#391C2212yes—
33#391C2212yes—
34#390C2212yes—
35#389C2212yes—
36#388—22no—
37#381—21no—
38#380—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C2×SD16—24no—
47C2×SD16—42no—
48C2×SD16—42no—
49C22×C8C2312yes—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54#125—22no—
55#125—22no—
56C4×D4—22no—
57C4×D4D414yes—
58C2×C4⋊C4—21no—
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—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73#107—22no—
74#106—22no—
75C4⋊C8—22no—
76C4⋊C8—22no—
77C4⋊C8—42no—
78C4⋊C8—42no—
79#103—24no—
80#103—42no—
81#103—42no—
82#102—22no—
83#102—22no—
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—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4C2×D412yes—
104C22×C4—21no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—82no—
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116SD16—48no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4C2×D412yes—
137C22⋊C4—24no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—22no—
141C42—22no—
142C42—81no—
143C42C2×D412yes—
144C42#4112yes—
145C23—22no—
146C23—42no—
147C23—41no—
148C23—42no—
149C23—41no—
150C23C22×D411yescharacteristic
151Q8—42no—
152Q8—42no—
153Q8—42no—
154Q8—42no—
155D4—22no—
156D4—22no—
157D4—42no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—84no—
181C2×C4#12112yes—
182C2×C4#12112yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4H(5,𝔽2)12yes—
186C2×C42−1+412yes—
187C8—41no—
188C8—41no—
189C8—42no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—82no—
194C22—82no—
195C22C2×#12111yescharacteristic
196C22#53712yes—
197C22#57712yes—
198C22#57711yescharacteristic
199C22#57811yescharacteristic
200C4—22no—
201C4—22no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—82no—
207C4—81no—
208C4—81no—
209C4#53511yescharacteristic
210C4#53611yescharacteristic
211C4#54212yes—
212C2—82no—
213C2#262011yescharacteristic
214C2#304211yescharacteristic
215C2#304611yescharacteristic
216C2#325812yes—
217C2#326012yes—
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
3. #2642C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
9. #1989C2semidirect
10. #1981C2semidirect
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×#4612
#26422
#23192
#20682
#20662
#20552
#19892
#19812
#18432