Finite Group Databasebeta

← #19500

Subgroups — #19500

Order 256.

295 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 109 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#3229C211yesmaximal, characteristic, complemented
2#3107C211yesmaximal, characteristic
3#3079C211yesmaximal, characteristic
4#2991C211yesmaximal, characteristic
5#1703C212yesmaximal, complemented
6#1691C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1672C212yesmaximal, complemented
9#1574C211yesmaximal, characteristic, complemented
10#1561C211yesmaximal, characteristic, complemented
11#1548C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C4×#41C2211yescharacteristic
15#498C2212yes—
16#487C2212yes—
17#485—21no—
18#483—22no—
19#477C2212yes—
20#476C2212yes—
21#475—21no—
22#475—21no—
23#475—21no—
24#475C2211yescharacteristic
25#474—21no—
26#474C2211yescharacteristic
27#462C2211yescharacteristic
28#461C2211yescharacteristic
29#452—21no—
30C8×Q8C2212yes—
31C4×Q16—21no—
32C4×SD16—21no—
33#436—22no—
34#436C2212yes—
35#431—21no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#428—21no—
39#423—21no—
40#423C2211yescharacteristic
41#423C2211yescharacteristic
42#417—21no—
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#333C2212yes—
47#332C2212yes—
48#332C2212yes—
49C2×#41D411yescharacteristic
50C22×Q8—21no—
51C2×Q16—21no—
52C2×SD16—21no—
53C2×M16—21no—
54C2×M16—21no—
55C2×M16—22no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8C2312yes—
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62#125—21no—
63C22⋊Q8—21no—
64C22⋊Q8—21no—
65C4×Q8—21no—
66C4×Q8—21no—
67C4×Q8D411yescharacteristic
68C4×Q8C2311yescharacteristic
69C4×D4—21no—
70C4×D4—21no—
71C4×D4D411yescharacteristic
72C4×D4C2311yescharacteristic
73#117—21no—
74#117—21no—
75#117D411yescharacteristic
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80#107—22no—
81#107—21no—
82#106—22no—
83#106—22no—
84#106—22no—
85#106—21no—
86C4⋊C8—22no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8D411yescharacteristic
90C4⋊C8D411yescharacteristic
91C4⋊C8C2312yes—
92C4⋊C8C2312yes—
93C4⋊C8C2311yescharacteristic
94C4⋊C8C2311yescharacteristic
95#103—22no—
96#103—21no—
97#103—22no—
98#103—22no—
99#103—21no—
100#103—21no—
101#102—22no—
102#102—21no—
103C22⋊C8—21no—
104C22⋊C8—22no—
105C22⋊C8D412yes—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8C2312yes—
109#41—22no—
110#41—41no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—41no—
115C2×Q8—82no—
116C2×Q8C2×D411yescharacteristic
117C2×D4—21no—
118C2×D4C2×D411yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4#4111yescharacteristic
126Q16—42no—
127SD16—42no—
128M16—42no—
129M16—82no—
130M16—81no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8C2×D412yes—
136C2×C8C2×D412yes—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4C2×D411yescharacteristic
150C4⋊C4C2×D411yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—41no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—41no—
159C42C2×D411yescharacteristic
160C42C2×D411yescharacteristic
161C42#4111yescharacteristic
162C42C2411yescharacteristic
163C23—21no—
164C23#12111yescharacteristic
165Q8—21no—
166Q8—41no—
167Q8—41no—
168Q8—41no—
169Q8—41no—
170Q8—82no—
171Q8—82no—
172Q8#13712yes—
173D4—21no—
174D4—21no—
175D4—42no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4#12111yescharacteristic
193C2×C4#12111yescharacteristic
194C2×C4#12111yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C8—22no—
199C8—22no—
200C8—41no—
201C8—42no—
202C8—41no—
203C8—42no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22C2×#12111yescharacteristic
211C4—21no—
212C4—21no—
213C4—22no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4#47112yes—
221C4C2×#13711yescharacteristic
222C4#57511yescharacteristic
223C4#57711yescharacteristic
224C4#57811yescharacteristic
225C2—41no—
226C2—81no—
227C2#304611yescharacteristic
228C2#305611yescharacteristic
229C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
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
1. #3229C2semidirect
5. #1703C2semidirect
6. #1691C2semidirect
7. #1684C2semidirect
8. #1672C2semidirect
9. #1574C2semidirect
10. #1561C2semidirect
11. #1548C2semidirect

Maximal subgroups

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

subgroupindex
#32292
#31072
#30792
#29912
#17032
#16912
#16842
#16722
#15742
#15612
#15482