Finite Group Databasebeta

← #25832

Subgroups — #25832

Order 256.

477 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 143 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#3503C211yesmaximal, characteristic
2#3166C211yesmaximal, characteristic, complemented
3#3080C211yesmaximal, characteristic
4#2660C211yesmaximal, characteristic, complemented
5#2061C212yesmaximal, complemented
6#2060C212yesmaximal, complemented
7#1997C211yesmaximal, characteristic, complemented
8#1995C211yesmaximal, characteristic, complemented
9#1860C212yesmaximal
10#1859C212yesmaximal
11#1759C211yesmaximal, characteristic
12Q82—21no—
13#557—21no—
14#554—21no—
15#554—22no—
16D4×Q8—22no—
17D4×Q8—21no—
18#544—22no—
19#544C2212yes—
20#533—21no—
21#533C2211yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×#125C2211yescharacteristic
25#520C2212yes—
26C4×#41C2211yescharacteristic
27#501—21no—
28#501—21no—
29#499—22no—
30#479—21no—
31#479—21no—
32#478—21no—
33#478—21no—
34#428C2211yescharacteristic
35#428C2211yescharacteristic
36#428C2212yes—
37#428C2212yes—
38#423C2211yescharacteristic
39C2×#104C2212yes—
40#417C2211yescharacteristic
41#417C2211yescharacteristic
42C4×M16C2211yescharacteristic
43#398—22no—
44#395—22no—
45#384C2212yes—
46#382C2212yes—
47#376C2211yescharacteristic
48#343C2212yes—
49#343C2212yes—
50#339C2212yes—
51#339C2212yes—
52#339C2212yes—
53C2×#41C2311yescharacteristic
54C22×Q8—22no—
55C22×Q8D412yes—
56C2×M16C2311yescharacteristic
57C2×M16C2311yescharacteristic
58C2×M16C2312yes—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8D411yescharacteristic
70#125—24no—
71#125—22no—
72#125—22no—
73#125—21no—
74#125—21no—
75#125—21no—
76#125—21no—
77#125D412yes—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—24no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—24no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8D411yescharacteristic
91C4×Q8D411yescharacteristic
92C4×D4—22no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—24no—
96C4×D4D411yescharacteristic
97C4×D4D411yescharacteristic
98#117—22no—
99#117—21no—
100#117C2311yescharacteristic
101#117C2312yes—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4D412yes—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2312yes—
111C2×C42—21no—
112C2×C42C2312yes—
113C2×C42C2311yescharacteristic
114#107—21no—
115#107—21no—
116#107—22no—
117#107—22no—
118#106—21no—
119#106—21no—
120#106—22no—
121#106—22no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124#104Q818yescomplemented
125#103—21no—
126#103—21no—
127#102—21no—
128#102—21no—
129#97—21no—
130C4×C8—21no—
131#95—21no—
132#95—21no—
133#95—24no—
134#41—22no—
135#41C2×Q814yes—
136C2×Q8—24no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×Q8—82no—
142C2×Q8—84no—
143C2×Q8C2×D411yescharacteristic
144C2×D4—21no—
145C2×D4C2×D411yescharacteristic
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4C2×D414yes—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157M16—22no—
158M16—22no—
159M16—22no—
160M16C2×Q814yescomplemented
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—84no—
187C4⋊C4C2×D412yes—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D411yescharacteristic
190C4⋊C4C2×D411yescharacteristic
191C22⋊C4—24no—
192C22⋊C4—22no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C42—24no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—22no—
201C42—22no—
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2×D411yescharacteristic
206C42C2×Q814yescomplemented
207C23—21no—
208C23C22×D411yescharacteristic
209Q8—21no—
210Q8—42no—
211Q8—44no—
212Q8—42no—
213Q8—82no—
214Q8—84no—
215Q8C4⋊Q812yes—
216D4—24no—
217D4—21no—
218D4C4⋊Q812yes—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—84no—
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×Q812yes—
246C2×C42−1+412yes—
247C8—24no—
248C8—41no—
249C8—41no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22C2×#12011yescharacteristic
256C22D4×Q812yes—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—44no—
263C4—41no—
264C4—44no—
265C4—42no—
266C4—41no—
267C4—42no—
268C4—81no—
269C4—81no—
270C4C2×C4⋊Q811yescharacteristic
271C4#53611yescharacteristic
272C4D4×Q812yes—
273C2—22no—
274C2—42no—
275C2#243411yescharacteristic
276C2#301911yescharacteristic
277C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
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
2. #3166C2semidirect
4. #2660C2semidirect
5. #2061C2semidirect
6. #2060C2semidirect
7. #1997C2semidirect
8. #1995C2semidirect
124. #104Q8semidirect
160. M16C2×Q8semidirect
206. C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35032
#31662
#30802
#26602
#20612
#20602
#19972
#19952
#18602
#18592
#17592