Finite Group Databasebeta

← #9786

Subgroups — #9786

Order 256.

276 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 71 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×#465C211yesmaximal, characteristic
2#1996C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1808C211yesmaximal, characteristic, complemented
5#1752C211yesmaximal, characteristic, complemented
6#1520C211yesmaximal, characteristic, complemented
7#1519C211yesmaximal, characteristic
8C22×SD16C411yescharacteristic, complemented
9C23×C8C2211yescharacteristic
10C2×C22⋊Q8C2211yescharacteristic
11C2×#121C2211yescharacteristic
12#465—21no—
13#465—21no—
14#465—21no—
15#465—21no—
16C2×#107—21no—
17C2×#106C411yescharacteristic, complemented
18C2×#103C411yescharacteristic, complemented
19C2×#102C411yescharacteristic, complemented
20C2×#99—22no—
21C2×C22⋊C8C2211yescharacteristic
22C2×#97—21no—
23#380C2211yescharacteristic
24#380C2211yescharacteristic
25C4×C22⋊C4C2211yescharacteristic
26#342—22no—
27#340—21no—
28#336—21no—
29#328—22no—
30#327—22no—
31C22×Q8C2×C411yescharacteristic
32C22×D4—21no—
33C22×D4C2×C411yescharacteristic
34C23×C4C2311yescharacteristic
35C2×SD16—21no—
36C2×SD16—21no—
37C2×SD16—42no—
38C22×C8—21no—
39C22×C8—42no—
40C22×C8C2×C411yescharacteristic
41C22×C8C2×C411yescharacteristic
42C22×C8D411yescharacteristic
43C22×C8D411yescharacteristic
44C22×C8D411yescharacteristic
45C22×C8D411yescharacteristic
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50#121—21no—
51#121—21no—
52#121—21no—
53#121—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4D412yes—
64C2×C42—21no—
65#107—42no—
66#106—21no—
67#106—21no—
68#103—21no—
69#103—21no—
70#102—21no—
71#102—21no—
72#99—44no—
73C22⋊C8—21no—
74C22⋊C8D412yes—
75#97—42no—
76#95—21no—
77#95—21no—
78#95—21no—
79C24—21no—
80C24C2×D411yescharacteristic
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—42no—
92C2×D4—42no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4C22×C411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4#4111yescharacteristic
106C22×C4#4111yescharacteristic
107SD16—42no—
108SD16—42no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8C22⋊C411yescharacteristic
119C2×C8C22⋊C411yescharacteristic
120C2×C8C22⋊C411yescharacteristic
121C2×C8C22⋊C411yescharacteristic
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—41no—
141C42—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—42no—
146C23—81no—
147C23—81no—
148C23C4×D411yescharacteristic
149C23#12111yescharacteristic
150C23#12711yescharacteristic
151C23AGL(1,ℤ/8)12yes—
152C23#13712yes—
153Q8—41no—
154Q8—41no—
155Q8—81no—
156D4—41no—
157D4—41no—
158D4—81no—
159D4—82no—
160D4—81no—
161D4—81no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4C2×C22⋊C411yescharacteristic
187C2×C4C4×D411yescharacteristic
188C2×C4#12111yescharacteristic
189C2×C4#12411yescharacteristic
190C8—21no—
191C8—21no—
192C8—21no—
193C8—42no—
194C8#9912yescomplemented
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—162no—
207C22#39011yescharacteristic
208C22#44012yes—
209C22#46812yes—
210C22#49611yescharacteristic
211C22#49711yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4—81no—
219C4—161no—
220C4—161no—
221C4C2×#9911yescharacteristic
222C4#41011yescharacteristic
223C2—22no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—161no—
228C2#196411yescharacteristic
229C2#197311yescharacteristic
230C2#197611yescharacteristic
subgroupG/H
center, socleC22#390
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×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
2. #1996C2semidirect
4. #1808C2semidirect
5. #1752C2semidirect
6. #1520C2semidirect
8. C22×SD16C4semidirect
17. C2×#106C4semidirect
18. C2×#103C4semidirect
19. C2×#102C4semidirect
194. C8#99semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
#19962
#19642
#18082
#17522
#15202
#15192