Finite Group Databasebeta

← #18817

Subgroups — #18817

Order 256.

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

normal subgroups only

This group has 442 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#3503C211yesmaximal, characteristic
2#3237C211yesmaximal, characteristic, complemented
3#3209C211yesmaximal, characteristic, complemented
4#3116C211yesmaximal, characteristic
5#1645C212yesmaximal, complemented
6#1643C212yesmaximal, complemented
7#1633C212yesmaximal, complemented
8#1632C212yesmaximal, complemented
9#1558C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, characteristic
12Q82C2212yes—
13#557—21no—
14#554—22no—
15#554C2212yes—
16D4×Q8—22no—
17D4×Q8—21no—
18#544—22no—
19#544—21no—
20#533—21no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#501—21no—
28#494C2212yes—
29#491—21no—
30#487C2212yes—
31#485—22no—
32#485—21no—
33#477C2211yescharacteristic
34#477C2212yes—
35#476C2211yescharacteristic
36#475C2212yes—
37#462—22no—
38#462C2211yescharacteristic
39#461C2211yescharacteristic
40#452—21no—
41#451—22no—
42#441—21no—
43#440—21no—
44#432C2211yescharacteristic
45#432C2211yescharacteristic
46#431C2211yescharacteristic
47#333C2212yes—
48#333C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41D411yescharacteristic
54C22×Q8—22no—
55C22×Q8D412yes—
56C2×Q16—21no—
57C2×Q16—24no—
58C2×SD16—21no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D412yes—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2312yes—
72#125—22no—
73#125—24no—
74#125—22no—
75#125—21no—
76#125—21no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C22⋊Q8—24no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8D412yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D411yescharacteristic
93C4×Q8C2311yescharacteristic
94C4×Q8C2312yes—
95C4×D4—24no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4D411yescharacteristic
100C4×D4C2311yescharacteristic
101#117—22no—
102#117—21no—
103#117—21no—
104#117—21no—
105#117D411yescharacteristic
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—21no—
113#107—22no—
114#106—21no—
115#106—22no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—24no—
122#103—22no—
123#103—22no—
124#103—21no—
125#103—21no—
126#102—21no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130#97—21no—
131#97—21no—
132C4×C8C2312yes—
133#41—22no—
134#41—22no—
135C2×Q8—24no—
136C2×Q8—22no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—84no—
141C2×Q8—82no—
142C2×Q8C2×D412yes—
143C2×Q8C2×D411yescharacteristic
144C2×Q8C2×D414yes—
145C2×D4—21no—
146C2×D4C2×D411yescharacteristic
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155Q16—42no—
156Q16—84no—
157SD16—42no—
158C2×C8—22no—
159C2×C8—21no—
160C2×C8—22no—
161C2×C8—21no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4C2×D411yescharacteristic
184C4⋊C4C2×D411yescharacteristic
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—41no—
192C42—24no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—22no—
197C42—41no—
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2×D411yescharacteristic
201C42C2411yescharacteristic
202C23—21no—
203C23#12011yescharacteristic
204Q8—21no—
205Q8—24no—
206Q8—44no—
207Q8—44no—
208Q8—42no—
209Q8—42no—
210Q8—41no—
211Q8—41no—
212Q8—84no—
213Q8#13712yes—
214D4—24no—
215D4—21no—
216D4#13712yes—
217C2×C4—24no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—82no—
233C2×C4—84no—
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4#12011yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4C22×D411yescharacteristic
240C8—42no—
241C8—41no—
242C8—41no—
243C8—42no—
244C22—21no—
245C22—22no—
246C22—21no—
247C22—21no—
248C22—41no—
249C22—41no—
250C22C2×#12011yescharacteristic
251C4—21no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—42no—
258C4—41no—
259C4—44no—
260C4—42no—
261C4—81no—
262C4—81no—
263C4C2×#13711yescharacteristic
264C4C2×#13711yescharacteristic
265C4#57812yes—
266C2—41no—
267C2—42no—
268C2#300411yescharacteristic
269C2#301111yescharacteristic
270C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
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
2. #3237C2semidirect
3. #3209C2semidirect
5. #1645C2semidirect
6. #1643C2semidirect
7. #1633C2semidirect
8. #1632C2semidirect
9. #1558C2semidirect
10. #1497C2semidirect

Maximal subgroups

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

subgroupindex
#35032
#32372
#32092
#31162
#16452
#16432
#16332
#16322
#15582
#14972
#14882