Finite Group Databasebeta

← #18831

Subgroups — #18831

Order 256.

419 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 113 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#3509C211yesmaximal, characteristic
2#3237C211yesmaximal, characteristic, complemented
3#3230C211yesmaximal, characteristic, complemented
4#3116C211yesmaximal, characteristic
5#1650C212yesmaximal, complemented
6#1648C212yesmaximal, complemented
7#1638C212yesmaximal, complemented
8#1636C212yesmaximal, complemented
9#1556C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, characteristic
12#557—21no—
13#557C2212yes—
14#556—21no—
15#556C2212yes—
16#552—22no—
17#552—22no—
18#541—22no—
19#536—21no—
20#533—21no—
21#533—21no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25#519—21no—
26C4×#41C2211yescharacteristic
27#501—21no—
28#500C2212yes—
29#491—21no—
30#487C2212yes—
31#485—22no—
32#485—21no—
33#484—22no—
34#477—22no—
35#477C2211yescharacteristic
36#477C2212yes—
37#476C2211yescharacteristic
38#475C2212yes—
39#462C2211yescharacteristic
40#461C2211yescharacteristic
41#452—21no—
42#441—21no—
43#440—21no—
44#432C2211yescharacteristic
45#432C2211yescharacteristic
46#431C2211yescharacteristic
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41D411yescharacteristic
54C22×Q8—21no—
55C2×Q16—21no—
56C2×SD16—21no—
57C4⋊Q8—22no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8D412yes—
62C4⋊Q8D411yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2312yes—
66#126—24no—
67#126—24no—
68#125—22no—
69#125—24no—
70#125—22no—
71#125—24no—
72#125—21no—
73#125—21no—
74#124—22no—
75#123—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83C22⋊Q8D412yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8D411yescharacteristic
94C4×Q8C2311yescharacteristic
95C4×Q8C2312yes—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4D411yescharacteristic
99C4×D4C2311yescharacteristic
100#117—24no—
101#117—22no—
102#117—21no—
103#117—22no—
104#117—21no—
105#117—21no—
106#117D411yescharacteristic
107C2×C4⋊C4—21no—
108C2×C4⋊C4D412yes—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#107—21no—
112#107—24no—
113#107—22no—
114#106—21no—
115#106—22no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—22no—
122#103—22no—
123#103—21no—
124#103—21no—
125#103—24no—
126#102—21no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130#97—21no—
131#97—21no—
132C4×C8C2312yes—
133#41—41no—
134#41—41no—
135C2×Q8—21no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8C2×D412yes—
143C2×Q8C2×D411yescharacteristic
144C2×D4—21no—
145C2×D4C2×D411yescharacteristic
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153Q16—42no—
154SD16—42no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—21no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4C2×D411yescharacteristic
180C4⋊C4C2×D414yes—
181C4⋊C4C2×D411yescharacteristic
182C4⋊C4C2×D412yes—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—21no—
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—41no—
190C42—24no—
191C42—24no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—22no—
196C42—41no—
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2411yescharacteristic
201C23—21no—
202C23#12011yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—24no—
206Q8—42no—
207Q8—44no—
208Q8—42no—
209Q8—42no—
210Q8—42no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—42no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4#12011yescharacteristic
233C2×C4#12011yescharacteristic
234C2×C4#12011yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C8—42no—
239C8—41no—
240C8—41no—
241C8—42no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22C2×#12011yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—41no—
254C4—44no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—42no—
259C4—81no—
260C4—81no—
261C4#57511yescharacteristic
262C4#57511yescharacteristic
263C4#57812yes—
264C2—41no—
265C2—81no—
266C2#300511yescharacteristic
267C2#301111yescharacteristic
268C2#302011yescharacteristic
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. #3230C2semidirect
5. #1650C2semidirect
6. #1648C2semidirect
7. #1638C2semidirect
8. #1636C2semidirect
9. #1556C2semidirect
10. #1497C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#32372
#32302
#31162
#16502
#16482
#16382
#16362
#15562
#14972
#14882