Finite Group Databasebeta

← #10374

Subgroups — #10374

Order 256.

254 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 57 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#2549C211yesmaximal, characteristic, complemented
2#1964C211yesmaximal, characteristic
3#1519C211yesmaximal, characteristic
4C2×#327C211yesmaximal, characteristic
5#1294C211yesmaximal, characteristic, complemented
6#1289C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8—21no—
9C2×#121C2211yescharacteristic
10C2×#99—21no—
11C2×#99—21no—
12C2×C22⋊C8C2211yescharacteristic
13C2×C22⋊C8C2211yescharacteristic
14#398—21no—
15#395—21no—
16#391—21no—
17#389—21no—
18#385—21no—
19#380C2211yescharacteristic
20#380C2211yescharacteristic
21C4×C22⋊C4C2211yescharacteristic
22C2×#95C2211yescharacteristic
23#327—21no—
24#327—21no—
25#327—21no—
26#327—21no—
27C22×Q8—21no—
28C22×D4—21no—
29C22×D4Q811yescharacteristic, complemented
30C23×C4—21no—
31C23×C4C2311yescharacteristic
32C22×C8—21no—
33C22×C8—21no—
34C22⋊Q8—42no—
35C22⋊Q8—42no—
36#121—21no—
37#121—21no—
38#121—21no—
39#121—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4Q811yescharacteristic, complemented
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4D411yescharacteristic
52C2×C22⋊C4D411yescharacteristic
53C2×C42—21no—
54#99—42no—
55#99—42no—
56C22⋊C8—21no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—21no—
65#95D411yescharacteristic
66#95D411yescharacteristic
67#95D411yescharacteristic
68#95D411yescharacteristic
69C24—21no—
70C24C2×D411yescharacteristic
71C2×Q8—42no—
72C2×Q8—42no—
73C2×Q8—81no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—81no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—42no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—81no—
95C22×C4—81no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×Q811yescharacteristic
99C22×C4#4111yescharacteristic
100C22×C4#4111yescharacteristic
101C22×C4#4111yescharacteristic
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—81no—
105C2×C8—81no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—81no—
113C4⋊C4—81no—
114C4⋊C4—81no—
115C22⋊C4—21no—
116C22⋊C4—21no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4D812yes—
124C22⋊C4SD1612yes—
125C42—81no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—81no—
132C23—81no—
133C23#12011yescharacteristic
134C23C22⋊Q811yescharacteristic
135C23#12411yescharacteristic
136C23C2×D811yescharacteristic
137C23C2×SD1611yescharacteristic
138C23AGL(1,ℤ/8)11yescharacteristic
139C23#13711yescharacteristic
140Q8—84no—
141D4—81no—
142D4—81no—
143D4—81no—
144D4—81no—
145D4—81no—
146D4—81no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4C22⋊Q811yescharacteristic
180C2×C4C22⋊Q811yescharacteristic
181C2×C4#12411yescharacteristic
182C2×C4#12411yescharacteristic
183C8—81no—
184C8—81no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—161no—
199C22—161no—
200C22#39311yescharacteristic
201C22#44711yescharacteristic
202C22#44811yescharacteristic
203C22#47411yescharacteristic
204C22#47611yescharacteristic
205C22#48611yescharacteristic
206C22#48911yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4—81no—
215C4—81no—
216C4—81no—
217C4—81no—
218C4—161no—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2—21no—
225C2—161no—
226C2#202311yescharacteristic
227C2#203111yescharacteristic
228C2#203211yescharacteristic
subgroupG/H
center, socleC22#393
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2549C2semidirect
5. #1294C2semidirect
6. #1289C2semidirect
7. #1282C2semidirect
29. C22×D4Q8semidirect
47. C2×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#25492
#19642
#15192
C2×#3272
#12942
#12892
#12822