Finite Group Databasebeta

← #27810

Subgroups — #27810

Order 256.

341 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 115 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#3163C211yesmaximal, characteristic, complemented
2#3137C212yesmaximal
3#3059C211yesmaximal, characteristic
4#2038C212yesmaximal
5#2028C212yesmaximal, complemented
6#1979C211yesmaximal, characteristic, complemented
7#1971C211yesmaximal, characteristic, complemented
8#1930C212yesmaximal, complemented
9#1815C211yesmaximal, characteristic
10#1773C211yesmaximal, characteristic, complemented
11#1765C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C2×#131C2211yescharacteristic
14#544C2212yes—
15#536C2212yes—
16#501—22no—
17#499—21no—
18#498—21no—
19#491—22no—
20#489—22no—
21#488—22no—
22#487—22no—
23#470—21no—
24#470—21no—
25#467—22no—
26C2×#108C2211yescharacteristic
27#428C2212yes—
28#427C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#104C2211yescharacteristic
31#419C2212yes—
32#419C2212yes—
33#417C2212yes—
34#416C2211yescharacteristic
35C2×#103C2212yes—
36C2×#101C2212yes—
37#405C2212yes—
38#405C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#391C2212yes—
41#343C2212yes—
42#341C2211yescharacteristic
43#339C2211yescharacteristic
44#339C2212yes—
45C2×#41C2311yescharacteristic
46C22×Q8C2312yes—
47C2×Q16—21no—
48C2×Q16—21no—
49C2×Q16—22no—
50C2×Q16—22no—
51#131—22no—
52#131—22no—
53C2×M16—21no—
54C2×M16C2311yescharacteristic
55C2×M16C2312yes—
56C2×M16C2311yescharacteristic
57C22×C8—21no—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63#125—22no—
64#125—22no—
65#124—22no—
66#123—42no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—42no—
70#117C2312yes—
71#117C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C42C2311yescharacteristic
74#108—22no—
75#107—22no—
76#107D412yes—
77#106—21no—
78#106—21no—
79#106—22no—
80#106D411yescharacteristic
81#106D411yescharacteristic
82#104—22no—
83#103—22no—
84#103—22no—
85#103—22no—
86#103—22no—
87#103—22no—
88#103—22no—
89#103—22no—
90#103D412yes—
91#102—22no—
92#102D412yes—
93#101—24no—
94#97—22no—
95#97—21no—
96#97—22no—
97C4×C8—22no—
98C4×C8—21no—
99#95—42no—
100#41—21no—
101#41—21no—
102#41—21no—
103#41—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—42no—
111C2×Q8C2×D411yescharacteristic
112C2×D4—21no—
113C2×D4C2×D411yescharacteristic
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4C2411yescharacteristic
119Q16—42no—
120Q16—42no—
121M16—21no—
122M16—22no—
123M16—22no—
124M16—21no—
125M16—21no—
126M16—42no—
127M16#4112yes—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—42no—
134C2×C8C2×D412yes—
135C2×C8C2×D411yescharacteristic
136C2×C8C2×D411yescharacteristic
137C2×C8#4112yes—
138C2×C8#4112yes—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4C2×D412yes—
147C4⋊C4C2×D412yes—
148C4⋊C4C2×D412yes—
149C22⋊C4—21no—
150C22⋊C4—22no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—22no—
154C42—21no—
155C42—21no—
156C42—21no—
157C23—21no—
158C23H(5,𝔽2)11yescharacteristic
159Q8—21no—
160Q8—21no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165Q8—162no—
166D4—21no—
167D4—21no—
168D4—41no—
169D4—41no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—82no—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×D411yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C42−1+411yescharacteristic
189C8—22no—
190C8—22no—
191C8—41no—
192C8—42no—
193C8#12412yes—
194C8#12412yes—
195C22—22no—
196C22—41no—
197C22—42no—
198C22#54211yescharacteristic
199C22D4211yescharacteristic
200C22#55511yescharacteristic
201C4—41no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—81no—
206C4—82no—
207C4—82no—
208C4C2×#12411yescharacteristic
209C4#53211yescharacteristic
210C4#54711yescharacteristic
211C4#54811yescharacteristic
212C2—21no—
213C2—21no—
214C2—81no—
215C2#249111yescharacteristic
216C2#328912yes—
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3163C2semidirect
5. #2028C2semidirect
6. #1979C2semidirect
7. #1971C2semidirect
8. #1930C2semidirect
10. #1773C2semidirect

Maximal subgroups

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

subgroupindex
#31632
#31372
#30592
#20382
#20282
#19792
#19712
#19302
#18152
#17732
#17652