Finite Group Databasebeta

← #29837

Subgroups — #29837

Order 256.

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

normal subgroups only

This group has 356 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
2C2×#462C211yesmaximal, characteristic
3#2642C211yesmaximal, characteristic
4#2320C211yesmaximal, characteristic
5#2068C212yesmaximal
6#2066C212yesmaximal
7#2056C212yesmaximal
8#2056C212yesmaximal
9#1989C211yesmaximal, characteristic
10#1981C211yesmaximal, characteristic
11#1843C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#479—24no—
17#462—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29#400—21no—
30#400—21no—
31#400—22no—
32#391C2212yes—
33#391C2212yes—
34#391C2212yes—
35#389C2212yes—
36#387—22no—
37#383—21no—
38#382—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C2×Q16—24no—
46C2×Q16—42no—
47C2×Q16—42no—
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#125—22no—
54#125—22no—
55C4×Q8—22no—
56C4×Q8D414yes—
57C2×C4⋊C4—21no—
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×C42—21no—
68C2×C42—21no—
69C2×C42C2312yes—
70C2×C42C2311yescharacteristic
71#107—22no—
72#106—22no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75C4⋊C8—42no—
76C4⋊C8—42no—
77#103—22no—
78#103—22no—
79#103—24no—
80#103—42no—
81#103—42no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D411yescharacteristic
91#95D411yescharacteristic
92#95D412yes—
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8C2×D412yes—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4C2×D412yes—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2411yescharacteristic
114Q16—48no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—24no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—82no—
135C4⋊C4C2×D412yes—
136C42—24no—
137C42—22no—
138C42—22no—
139C42—81no—
140C42C2×D412yes—
141C42#4112yes—
142C23C22×D411yescharacteristic
143Q8—22no—
144Q8—22no—
145Q8—42no—
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4#12112yes—
176C2×C4#12112yes—
177C2×C4C22×D411yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C4H(5,𝔽2)12yes—
180C2×C42−1+412yes—
181C8—41no—
182C8—41no—
183C8—42no—
184C22C2×#12111yescharacteristic
185C22#53712yes—
186C22#57711yescharacteristic
187C22#57812yes—
188C22#57811yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—82no—
196C4—82no—
197C4—81no—
198C4—81no—
199C4#53511yescharacteristic
200C4#53611yescharacteristic
201C4#54212yes—
202C2#262011yescharacteristic
203C2#304411yescharacteristic
204C2#304611yescharacteristic
205C2#325812yes—
206C2#326112yes—
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×#4792
C2×#4622
#26422
#23202
#20682
#20662
#20562
#19892
#19812
#18432