Finite Group Databasebeta

← #28671

Subgroups — #28671

Order 256.

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

normal subgroups only

This group has 320 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#3166C211yesmaximal, characteristic, complemented
2#3140C211yesmaximal, characteristic
3C2×#488C211yesmaximal, characteristic
4#2960C211yesmaximal, characteristic, complemented
5#2064C212yesmaximal, complemented
6#2063C212yesmaximal
7#1998C211yesmaximal, characteristic, complemented
8#1967C211yesmaximal, characteristic
9#1931C212yesmaximal
10#1930C212yesmaximal, complemented
11C4×#108C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15#501—21no—
16#501—21no—
17#499—22no—
18#489—21no—
19#489—21no—
20#488—22no—
21#488—21no—
22#488—21no—
23#488—21no—
24#488—21no—
25#446—21no—
26#446—21no—
27C8×Q8—21no—
28C8×Q8—21no—
29C2×#108C2212yes—
30#428C2212yes—
31#428C2212yes—
32#424C2212yes—
33#423C2211yescharacteristic
34#419C2212yes—
35#418C2212yes—
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#101C2212yes—
41C2×#100C2212yes—
42#405C2212yes—
43C4×M16C2211yescharacteristic
44C4×M16C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#343C2212yes—
47#343C2212yes—
48#341C2212yes—
49C22×Q8C2311yescharacteristic
50C22×D4C2311yescharacteristic
51C2×M16C2312yes—
52C2×M16C2312yes—
53C2×M16C2312yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C4⋊Q8D411yescharacteristic
57C4⋊Q8D411yescharacteristic
58#125—21no—
59#125—21no—
60#125—21no—
61#125—21no—
62#125D412yes—
63#124—21no—
64#124—21no—
65#124—21no—
66#124—21no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71#117C2312yes—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C42C2311yescharacteristic
79#108—24no—
80#107—22no—
81#107—22no—
82#106—22no—
83#106—22no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86C4⋊C8—21no—
87C4⋊C8—21no—
88#103—21no—
89#103—21no—
90#103—21no—
91#103—21no—
92#103—22no—
93#103—22no—
94#102—21no—
95#102—21no—
96#102—21no—
97#102—21no—
98#102—22no—
99#102—22no—
100#101—24no—
101#100—24no—
102#97—22no—
103#97—21no—
104#97—21no—
105C4×C8—22no—
106C4×C8—21no—
107C4×C8—21no—
108C4×C8D411yescharacteristic
109C4×C8D411yescharacteristic
110C4×C8D411yescharacteristic
111C4×C8D411yescharacteristic
112C24—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4#4111yescharacteristic
127C22×C4#4111yescharacteristic
128C22×C4C2411yescharacteristic
129M16—22no—
130M16—22no—
131M16—22no—
132M16—22no—
133M16—22no—
134M16#4114yes—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8C2×D411yescharacteristic
141C2×C8C2×D411yescharacteristic
142C2×C8C2×D411yescharacteristic
143C2×C8C2×D411yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—22no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C22⋊C4—41no—
162C22⋊C4—41no—
163C42—22no—
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C23—41no—
169C23—42no—
170C23—81no—
171C23—81no—
172C23C2×#4111yescharacteristic
173Q8—42no—
174Q8—42no—
175Q8—41no—
176Q8—41no—
177Q8—42no—
178D4—41no—
179D4—41no—
180D4—42no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4#12111yescharacteristic
192C2×C4#12111yescharacteristic
193C2×C4#12111yescharacteristic
194C2×C4#12111yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4112yes—
198C2×C42−1+412yes—
199C8—21no—
200C8—21no—
201C8—21no—
202C8—21no—
203C8—24no—
204C8—42no—
205C8—42no—
206C22—21no—
207C22—21no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—162no—
212C22C2×#12111yescharacteristic
213C22#55612yes—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—42no—
219C4—42no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4#53211yescharacteristic
224C4#53611yescharacteristic
225C4#54812yes—
226C2—22no—
227C2—161no—
228C2#254511yescharacteristic
229C2#306711yescharacteristic
230C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3166C2semidirect
4. #2960C2semidirect
5. #2064C2semidirect
7. #1998C2semidirect
10. #1930C2semidirect
11. C4×#108C2semidirect

Maximal subgroups

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

subgroupindex
#31662
#31402
C2×#4882
#29602
#20642
#20632
#19982
#19672
#19312
#19302
C4×#1082