Finite Group Databasebeta

← #18677

Subgroups — #18677

Order 256.

429 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 129 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#3502C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic
3#3170C211yesmaximal, characteristic
4#3078C211yesmaximal, characteristic
5#1648C212yesmaximal, complemented
6#1645C212yesmaximal, complemented
7#1636C212yesmaximal, complemented
8#1633C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic
10#1493C211yesmaximal, characteristic
11C2×Q8⋊C8C211yesmaximal, characteristic
12Q82—21no—
13Q82C2212yes—
14#557—21no—
15#557—22no—
16#557—22no—
17#557C2212yes—
18#544—22no—
19#544—21no—
20#533—22no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#519—21no—
25C2×C4×Q8C2211yescharacteristic
26#500C2211yescharacteristic
27#494C2211yescharacteristic
28#487C2211yescharacteristic
29#487C2211yescharacteristic
30#479—22no—
31#477—21no—
32#477C2212yes—
33#477C2212yes—
34#475C2212yes—
35#467—22no—
36#467—22no—
37#462—21no—
38#462C2212yes—
39C4×Q16—22no—
40#428—21no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#419—21no—
44C2×C4×C8C2211yescharacteristic
45#333C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C22×Q8D411yescharacteristic
52C2×Q16—22no—
53C2×Q16—22no—
54C2×M16—21no—
55C22×C8—21no—
56C22×C8—21no—
57C4⋊Q8—24no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—24no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8D411yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68C4⋊Q8C2311yescharacteristic
69#125—24no—
70#125—24no—
71#125—22no—
72#125—24no—
73#125—22no—
74#125—21no—
75#125—21no—
76#125—21no—
77C22⋊Q8—24no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8D412yes—
83C22⋊Q8D412yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—24no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8D412yes—
98C4×Q8D412yes—
99C4×Q8C2312yes—
100C4×Q8C2312yes—
101#117—24no—
102#117—21no—
103#117—21no—
104#117—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#107—22no—
112#106—22no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2312yes—
116C4⋊C8C2312yes—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C2×Q8—24no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8Q1614yescomplemented
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×Q8C2×D412yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142Q16—44no—
143Q16—82no—
144M16—42no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—81no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D412yes—
177C4⋊C4C2×D412yes—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C42—24no—
182C42—24no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—41no—
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2411yescharacteristic
192C23#12011yescharacteristic
193Q8—22no—
194Q8—24no—
195Q8—22no—
196Q8—22no—
197Q8—42no—
198Q8—42no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—81no—
204Q8—84no—
205Q8C2×Q1614yes—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4C2×Q1612yes—
223C2×C4#13712yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—22no—
228C8—41no—
229C8—42no—
230C8—42no—
231C22—21no—
232C22—21no—
233C22#45112yes—
234C22C2×#12011yescharacteristic
235C4—24no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—42no—
244C4—81no—
245C4—81no—
246C4#45112yes—
247C4C22×Q1611yescharacteristic
248C4C2×#13711yescharacteristic
249C4#57711yescharacteristic
250C4#57811yescharacteristic
251C2—22no—
252C2C2×#45111yescharacteristic
253C2#301311yescharacteristic
254C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC23#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #1648C2semidirect
6. #1645C2semidirect
7. #1636C2semidirect
8. #1633C2semidirect
133. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#35022
#32492
#31702
#30782
#16482
#16452
#16362
#16332
#15502
#14932
C2×Q8⋊C82