Finite Group Databasebeta

← #28288

Subgroups — #28288

Order 256.

397 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 135 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×#477C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic
3#2331C211yesmaximal, characteristic, complemented
4#2326C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2067C212yesmaximal
7#1983C212yesmaximal
8#1947C212yesmaximal, complemented
9#1843C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#477—24no—
17#474—24no—
18C2×#107C2212yes—
19C2×#107C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28#391—22no—
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#389C2212yes—
32#388—22no—
33#387—22no—
34#385—22no—
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37C4×C4⋊C4C2212yes—
38#376C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8D414yes—
49#124—22no—
50#124Q814yescomplemented
51C4×Q8—22no—
52C4×Q8—22no—
53C4×D4—22no—
54C4×D4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C42—21no—
67C2×C42—22no—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—24no—
72#107—24no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75C4⋊C8—24no—
76C4⋊C8—42no—
77#103—24no—
78#103—42no—
79#102—24no—
80#102—42no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—22no—
86#95D412yes—
87#95D412yes—
88C24—21no—
89C2×Q8—24no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—82no—
96C2×Q8C2×Q812yes—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4C2×Q812yes—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—82no—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2411yescharacteristic
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C4⋊C4C2×D414yes—
133C22⋊C4—28no—
134C22⋊C4—42no—
135C22⋊C4—82no—
136C42—21no—
137C42—21no—
138C42—42no—
139C42—82no—
140C42C2×D412yes—
141C42C2×Q812yes—
142C42#4112yes—
143C42#4112yes—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—42no—
149C23C22×D411yescharacteristic
150Q8—22no—
151Q8—22no—
152Q8—41no—
153Q8—41no—
154Q8—84no—
155D4—22no—
156D4—22no—
157D4—41no—
158D4—41no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4#12112yes—
181C2×C4#12112yes—
182C2×C4C22⋊Q814yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×Q811yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C4H(5,𝔽2)11yescharacteristic
188C2×C42−1+411yescharacteristic
189C8—42no—
190C8—42no—
191C22—44no—
192C22—42no—
193C22—82no—
194C22—81no—
195C22—81no—
196C22C2×#12111yescharacteristic
197C22C2×C22⋊Q811yescharacteristic
198C22#53711yescharacteristic
199C22#57612yes—
200C22#57812yes—
201C4—24no—
202C4—22no—
203C4—22no—
204C4—42no—
205C4—44no—
206C4—42no—
207C4—82no—
208C4—81no—
209C4—81no—
210C4D4×Q811yescharacteristic
211C4#55011yescharacteristic
212C4#55411yescharacteristic
213C4#55611yescharacteristic
214C2—82no—
215C2#253811yescharacteristic
216C2#304111yescharacteristic
217C2#304411yescharacteristic
218C2#308312yes—
219C2#326212yes—
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
1. C2×#477C2semidirect
3. #2331C2semidirect
5. #2068C2semidirect
8. #1947C2semidirect
9. #1843C2semidirect
10. #1805C2semidirect
50. #124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4742
#23312
#23262
#20682
#20672
#19832
#19472
#18432
#18052
#18042