Finite Group Databasebeta

← #28666

Subgroups — #28666

Order 256.

349 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 123 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×#501C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic
3#2642C211yesmaximal, characteristic
4#2331C211yesmaximal, characteristic
5#2068C212yesmaximal
6#2066C212yesmaximal
7#1991C211yesmaximal, characteristic
8#1960C211yesmaximal, characteristic
9#1928C212yesmaximal
10#1927C212yesmaximal
11#1768C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15#501—24no—
16#487—24no—
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#97C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#400—22no—
27#400—22no—
28#391—22no—
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#391C2211yescharacteristic
32#391C2211yescharacteristic
33#389C2212yes—
34#387—22no—
35#382C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37#376C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8D414yes—
50#125—22no—
51#125—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C42—22no—
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—24no—
67#106—24no—
68C4⋊C8—24no—
69#103—24no—
70#103—24no—
71#103—24no—
72#103—24no—
73#97—22no—
74C4×C8—22no—
75#95—22no—
76#95—21no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95D412yes—
84#95D412yes—
85C2×Q8—24no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—82no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2411yescharacteristic
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8#4114yes—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—82no—
125C4⋊C4C2×D414yes—
126C42—22no—
127C42—22no—
128C42—82no—
129C42C2×D412yes—
130C42#4112yes—
131C23C22×D411yescharacteristic
132Q8—42no—
133Q8—42no—
134Q8—42no—
135Q8—42no—
136Q8—84no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4#12112yes—
162C2×C4#12112yes—
163C2×C4C22×D411yescharacteristic
164C2×C4C2×#4112yes—
165C2×C4C2×#4111yescharacteristic
166C2×C42−1+412yes—
167C8—24no—
168C8—42no—
169C22C2×#12111yescharacteristic
170C22#53211yescharacteristic
171C22#53611yescharacteristic
172C22#57712yes—
173C22#57812yes—
174C4—22no—
175C4—22no—
176C4—42no—
177C4—42no—
178C4—44no—
179C4—82no—
180C4—81no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4#54812yes—
185C4#55612yes—
186C2#254511yescharacteristic
187C2#304211yescharacteristic
188C2#304411yescharacteristic
189C2#317812yes—
190C2#324212yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4872
#26422
#23312
#20682
#20662
#19912
#19602
#19282
#19272
#17682