Finite Group Databasebeta

← #18919

Subgroups — #18919

Order 256.

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

normal subgroups only

This group has 473 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#3506C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3#3209C211yesmaximal, characteristic, complemented
4C2×#476C211yesmaximal, characteristic, complemented
5#1635C214yesmaximal, complemented
6#1634C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10#559—22no—
11#556—21no—
12#555C2212yes—
13#553—22no—
14#550—21no—
15#547C2212yes—
16#538—22no—
17#535—21no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21#525—22no—
22#520—21no—
23C2×C4×D4C2211yescharacteristic
24#498—21no—
25#494C2212yescomplemented
26#487C2212yescomplemented
27#486—21no—
28#481—22no—
29#476—22no—
30#476C2214yes—
31#476C2214yescomplemented
32#462—22no—
33#451—22no—
34#431C2211yescharacteristic
35#431C2211yescharacteristic
36C2×#106—21no—
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#333C2212yescomplemented
40#332C2212yescomplemented
41D4⋊C8C2214yes—
42D4⋊C8C2214yescomplemented
43C2×#41—22no—
44C2×#41—21no—
45C22×Q8—21no—
46C22×D4D411yescharacteristic
47C23×C4—21no—
48C2×Q16—24no—
49C22×C8—21no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D411yescharacteristic
54C4⋊Q8C2312yes—
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57#127—22no—
58#127D411yescharacteristic
59#126—24no—
60#125—21no—
61#124—22no—
62#124—24no—
63#124—21no—
64#123—24no—
65#123—24no—
66C22⋊Q8—21no—
67C22⋊Q8—22no—
68C22⋊Q8D412yes—
69#121—24no—
70#121—22no—
71#121—22no—
72#121—22no—
73#121—21no—
74#121D412yes—
75#120—24no—
76C4×Q8—22no—
77C4×Q8—21no—
78C4×Q8D412yes—
79C4×D4—24no—
80C4×D4—22no—
81C4×D4—24no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4D412yes—
86C4×D4C2312yes—
87C4×D4C2312yes—
88#117—21no—
89#117—22no—
90#117—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C22⋊C4—21no—
96C2×C42C2311yescharacteristic
97#106—24no—
98#106—24no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8C2312yes—
102C4⋊C8C2312yes—
103#103—24no—
104#102—24no—
105#102—24no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—21no—
111#41—42no—
112#41—42no—
113#41—84no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—41no—
118C2×Q8—82no—
119C2×Q8C2×D412yes—
120C2×D4—22no—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4C2×D412yes—
129C2×D4C2×D412yes—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—82no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140Q16—84no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—42no—
145C4⋊C4—24no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C42—24no—
169C42—22no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2411yescharacteristic
174C23—22no—
175C23—22no—
176C23—42no—
177C23—81no—
178C23—81no—
179C23#12011yescharacteristic
180Q8—42no—
181Q8—44no—
182Q8—42no—
183Q8—42no—
184Q8—82no—
185D4—22no—
186D4—22no—
187D4—24no—
188D4—44no—
189D4—42no—
190D4—42no—
191D4—81no—
192D4—84no—
193D4—84no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4#12011yescharacteristic
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4#13512yes—
212C2×C4#13712yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C8—42no—
217C8—42no—
218C8—42no—
219C22—21no—
220C22—21no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—84no—
228C22#45212yes—
229C22C2×#12011yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—82no—
238C4#45212yes—
239C4C2×#13511yescharacteristic
240C4C2×#13711yescharacteristic
241C4#57712yes—
242C2—22no—
243C2—82no—
244C2—82no—
245C2C2×#45211yescharacteristic
246C2#300911yescharacteristic
247C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3506C2
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. #3227C2semidirect
3. #3209C2semidirect
4. C2×#476C2semidirect
5. #1635C2semidirect
6. #1634C2semidirect
7. #1542C2semidirect
8. #1480C2semidirect
9. #1478C2semidirect
25. #494C22semidirect
26. #487C22semidirect
31. #476C22semidirect
39. #333C22semidirect
40. #332C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35062
#32272
#32092
C2×#4762
#16352
#16342
#15422
#14802
#14782