Finite Group Databasebeta

← #25314

Subgroups — #25314

Order 256.

340 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 117 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#3139C211yesmaximal, characteristic
2#3078C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic
4#2962C211yesmaximal, characteristic
5#2064C212yesmaximal
6#2047C212yesmaximal
7#1979C212yesmaximal
8#1966C211yesmaximal, characteristic
9#1950C211yesmaximal, characteristic
10#1930C212yesmaximal
11C4×#101C211yesmaximal, characteristic
12C22×Q16C2212yes—
13#533C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#491—22no—
17#487—21no—
18#487—21no—
19#479—22no—
20#477—21no—
21#475—21no—
22#462—21no—
23#462—22no—
24#462—21no—
25#446—22no—
26#446—21no—
27C8×Q8—21no—
28C2×#108C2212yes—
29#428C2212yes—
30#424C2212yes—
31#423C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33#419C2212yes—
34#419C2212yes—
35#419C2212yes—
36C2×#103C2212yes—
37C2×#101C2212yes—
38C2×#101C2211yescharacteristic
39C2×#101C2211yescharacteristic
40C2×#101C2212yes—
41#405C2212yes—
42C4×M16C2211yescharacteristic
43C4×M16C2211yescharacteristic
44#343C2211yescharacteristic
45#343C2211yescharacteristic
46#341C2212yes—
47C22×Q8C2312yes—
48C22×Q8C2311yescharacteristic
49C2×Q16—24no—
50C2×Q16—22no—
51C2×Q16—22no—
52C2×Q16—42no—
53C2×M16C2311yescharacteristic
54C2×M16C2312yes—
55C2×M16C2311yescharacteristic
56C2×M16C2311yescharacteristic
57C2×M16C2311yescharacteristic
58C22×C8C2312yes—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8D411yescharacteristic
65#125D412yes—
66C22⋊Q8—21no—
67C22⋊Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8—21no—
74#117C2312yes—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2311yescharacteristic
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81#108—42no—
82#107—22no—
83#106—22no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86C4⋊C8—21no—
87C4⋊C8—21no—
88C4⋊C8D411yescharacteristic
89C4⋊C8D411yescharacteristic
90C4⋊C8D412yes—
91#103—22no—
92#103—22no—
93#103—22no—
94#103—22no—
95#103—22no—
96#103—22no—
97#103—22no—
98#103—22no—
99#101—22no—
100#101—22no—
101#101—24no—
102#101—42no—
103#97—22no—
104#97—21no—
105#97—21no—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8#4112yes—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4#4111yescharacteristic
127C22×C4#4111yescharacteristic
128C22×C4C2411yescharacteristic
129Q16—44no—
130Q16—44no—
131M16—21no—
132M16—21no—
133M16—21no—
134M16—22no—
135M16—21no—
136M16—21no—
137M16—22no—
138M16—21no—
139M16—21no—
140M16#4112yes—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8C2×D412yes—
149C2×C8C2×D412yes—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—22no—
165C42—22no—
166C42—41no—
167C42—41no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C23C2×#4111yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—81no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—81no—
195C2×C4#12111yescharacteristic
196C2×C4#12111yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4#12111yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4H(5,𝔽2)11yescharacteristic
204C2×C42−1+411yescharacteristic
205C8—22no—
206C8—22no—
207C8—22no—
208C8—41no—
209C8—41no—
210C8—41no—
211C8—42no—
212C22—21no—
213C22—21no—
214C22C2×#12111yescharacteristic
215C22#52911yescharacteristic
216C22#54211yescharacteristic
217C4—21no—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—42no—
222C4—42no—
223C4—82no—
224C4—81no—
225C4#47312yes—
226C4#54611yescharacteristic
227C4#54811yescharacteristic
228C4#54811yescharacteristic
229C4#55011yescharacteristic
230C2—21no—
231C2—21no—
232C2#242611yescharacteristic
233C2C2×#47311yescharacteristic
234C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC23C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ 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
#31392
#30782
C2×#4622
#29622
#20642
#20472
#19792
#19662
#19502
#19302
C4×#1012