Finite Group Databasebeta

← #25455

Subgroups — #25455

Order 256.

373 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 147 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×#500C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic
3#2567C211yesmaximal, characteristic
4#2369C211yesmaximal, characteristic
5#2083C212yesmaximal
6#2068C212yesmaximal
7#1987C211yesmaximal, characteristic
8#1947C212yesmaximal
9#1939C211yesmaximal, characteristic
10#1928C212yesmaximal
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#500—24no—
16#477—24no—
17C2×#107C2212yes—
18C2×#107C2212yes—
19C2×#107C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#395—22no—
27#391—22no—
28#391C2212yes—
29#389—21no—
30#389—21no—
31#389—21no—
32#389C2212yes—
33#384C2212yes—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8D414yes—
53C4×Q8—22no—
54C4×Q8—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73#107—24no—
74#107—24no—
75#107—24no—
76C4⋊C8—24no—
77C4⋊C8—22no—
78#103—22no—
79#103—24no—
80#103Q814yescomplemented
81C4×C8—22no—
82#95—22no—
83#95—22no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95D412yes—
88C2×Q8—24no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—42no—
95C2×Q8—81no—
96C2×Q8—81no—
97C2×Q8—81no—
98C2×Q8—81no—
99C2×Q8C2×Q812yes—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8C2×Q812yes—
119C2×C8#4112yes—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—82no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C4⋊C4Q1614yes—
136C4⋊C4C2×D414yes—
137C4⋊C4C2×Q812yes—
138C42—22no—
139C42—22no—
140C42—42no—
141C42—42no—
142C42—82no—
143C42C2×D412yes—
144C42#4112yes—
145C23C22×D411yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—41no—
149Q8—41no—
150Q8—84no—
151Q8—84no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4#12112yes—
178C2×C4#12112yes—
179C2×C4C2×Q1614yes—
180C2×C4C2×Q1612yes—
181C2×C4#13712yes—
182C2×C4C22×D411yescharacteristic
183C2×C4C22×Q811yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C42−1+411yescharacteristic
187C2×C42−1+411yescharacteristic
188C8—22no—
189C8—22no—
190C8—42no—
191C22C2×#12111yescharacteristic
192C22#54711yescharacteristic
193C22D4×Q811yescharacteristic
194C22C22×Q1611yescharacteristic
195C22C2×#13711yescharacteristic
196C22#57611yescharacteristic
197C22#57611yescharacteristic
198C4—24no—
199C4—22no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—44no—
204C4—81no—
205C4—81no—
206C4—82no—
207C4#46214yes—
208C4#53311yescharacteristic
209C4#54111yescharacteristic
210C4#54811yescharacteristic
211C4D4×Q811yescharacteristic
212C2#242911yescharacteristic
213C2C2×#46211yescharacteristic
214C2#304111yescharacteristic
215C2#333811yescharacteristic
216C2#335711yescharacteristic
217C2Q8×Q1611yescharacteristic
218C2#338511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
80. #103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5002
C2×#4772
#25672
#23692
#20832
#20682
#19872
#19472
#19392
#19282
C4×#1032