Finite Group Databasebeta

← #29196

Subgroups — #29196

Order 256.

345 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#2864C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3#2068C212yesmaximal
4#2066C212yesmaximal
5#1960C211yesmaximal, characteristic
6#1960C211yesmaximal, characteristic
7#1865C212yesmaximal
8#1863C212yesmaximal
9#1847C211yesmaximal, characteristic
10#1836C211yesmaximal, characteristic
11#1767C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8C2212yes—
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#97C2211yescharacteristic
22C2×#97C2211yescharacteristic
23#391C2212yes—
24#389—21no—
25#389—21no—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#387—21no—
33#387—21no—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36#382C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8Q814yes—
48C4×Q8—42no—
49C4×Q8—42no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C42—22no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#107—42no—
73#106—42no—
74C4⋊C8—24no—
75#103—24no—
76#103—24no—
77#103—42no—
78#103—42no—
79#97—22no—
80#97—22no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8C2×Q814yes—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8#4114yes—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C42—22no—
132C42—22no—
133C42—22no—
134C42—22no—
135C42—41no—
136C42—41no—
137C42—44no—
138C42C2×Q812yes—
139C42#4112yes—
140C23C22×D411yescharacteristic
141Q8—24no—
142Q8—24no—
143Q8—42no—
144Q8—42no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—24no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4C22⋊Q812yes—
178C2×C4C22⋊Q812yes—
179C2×C4#13712yes—
180C2×C4#13712yes—
181C2×C4C22×Q811yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4112yes—
184C2×C4H(5,𝔽2)12yes—
185C8—24no—
186C8—42no—
187C22C2×C22⋊Q811yescharacteristic
188C22#53211yescharacteristic
189C22#53511yescharacteristic
190C22C2×#13711yescharacteristic
191C22C2×#13711yescharacteristic
192C22#57511yescharacteristic
193C22#57511yescharacteristic
194C4—22no—
195C4—22no—
196C4—22no—
197C4—22no—
198C4—44no—
199C4—41no—
200C4—41no—
201C4—81no—
202C4—81no—
203C4—81no—
204C4—82no—
205C4—81no—
206C4#55312yes—
207C4#55412yes—
208C2#256611yescharacteristic
209C2#307811yescharacteristic
210C2#307811yescharacteristic
211C2#317211yescharacteristic
212C2#318111yescharacteristic
213C2#321811yescharacteristic
214C2#321911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#28642
C4×C4⋊Q82
#20682
#20662
#19602
#18652
#18632
#18472
#18362
#17672