Finite Group Databasebeta

← #18669

Subgroups — #18669

Order 256.

441 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 121 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#3507C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic
3#3170C211yesmaximal, characteristic, complemented
4#3078C211yesmaximal, characteristic, complemented
5#1650C212yesmaximal, complemented
6#1643C212yesmaximal, complemented
7#1638C212yesmaximal, complemented
8#1632C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1492C211yesmaximal, characteristic
11C2×Q8⋊C8C211yesmaximal, characteristic, complemented
12#559—21no—
13#557—21no—
14#556—22no—
15#556C2212yes—
16#554C2212yes—
17#552—22no—
18#539—22no—
19#536—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#529—22no—
24#520—21no—
25C2×C4×Q8C2211yescharacteristic
26#500C2211yescharacteristic
27#494C2211yescharacteristic
28#487C2211yescharacteristic
29#487C2211yescharacteristic
30#479—22no—
31#477C2212yes—
32#476C2212yes—
33#475—21no—
34#475C2212yes—
35#467—22no—
36#465—22no—
37#461C2212yes—
38#460—21no—
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—
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8D411yescharacteristic
53C2×Q16—22no—
54C2×SD16—22no—
55C2×M16—21no—
56C22×C8—21no—
57C22×C8—21no—
58C4⋊Q8—22no—
59C4⋊Q8D411yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64#127D411yescharacteristic
65#126—24no—
66#126—24no—
67#125—24no—
68#125—22no—
69#125—21no—
70#125—21no—
71#124—22no—
72#124—22no—
73#124—21no—
74#123—24no—
75#123—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8D412yes—
83#121—21no—
84#121D412yes—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D412yes—
93C4×Q8D412yes—
94C4×Q8C2312yes—
95C4×D4—24no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4C2312yes—
100#117—24no—
101#117—21no—
102#117—22no—
103#117—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#106—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#102—22no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126#41—42no—
127#41—42no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8C2×D412yes—
136C2×Q8C2×D412yes—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4C2×D412yes—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148Q16—44no—
149SD16—82no—
150M16—42no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—42no—
156C2×C8—81no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
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—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—41no—
185C42—24no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—41no—
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C42C2411yescharacteristic
194C23—22no—
195C23#12011yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—42no—
199Q8—42no—
200Q8—41no—
201Q8—41no—
202Q8—42no—
203Q8—42no—
204Q8—41no—
205Q8—41no—
206D4—22no—
207D4—22no—
208D4—44no—
209D4—42no—
210D4—81no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—84no—
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4#13512yes—
230C2×C4#13712yes—
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C8—22no—
235C8—41no—
236C8—42no—
237C8—42no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22#45212yes—
244C22C2×#12011yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4#45212yes—
256C4C2×#13511yescharacteristic
257C4C2×#13711yescharacteristic
258C4#57711yescharacteristic
259C4#57811yescharacteristic
260C2—22no—
261C2—82no—
262C2C2×#45211yescharacteristic
263C2#301311yescharacteristic
264C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
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
3. #3170C2semidirect
4. #3078C2semidirect
5. #1650C2semidirect
6. #1643C2semidirect
7. #1638C2semidirect
8. #1632C2semidirect
9. #1550C2semidirect
11. C2×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32462
#31702
#30782
#16502
#16432
#16382
#16322
#15502
#14922
C2×Q8⋊C82