Finite Group Databasebeta

← #19475

Subgroups — #19475

Order 256.

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

normal subgroups only

This group has 288 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
2#3113C211yesmaximal, characteristic
3C2×#477C211yesmaximal, characteristic
4#2984C211yesmaximal, characteristic
5#1704C212yesmaximal, complemented
6#1692C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1672C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic
10#1558C211yesmaximal, characteristic
11#1552C211yesmaximal, characteristic
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#498C2212yes—
16#487C2212yes—
17#483—21no—
18#483—21no—
19#483—21no—
20#477—21no—
21#477—21no—
22#477C2212yes—
23#477C2212yes—
24#475—22no—
25#475C2211yescharacteristic
26#475C2211yescharacteristic
27#462C2211yescharacteristic
28#462C2211yescharacteristic
29#451—21no—
30#446—22no—
31#446C2212yes—
32C8×Q8C2212yes—
33#441—21no—
34#441—21no—
35#432—21no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#431—21no—
39#431C2211yescharacteristic
40C2×#107—21no—
41C2×C4⋊C8C2211yescharacteristic
42C2×#103—21no—
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#333C2212yes—
47#332C2212yes—
48#332C2212yes—
49C22×Q8—21no—
50C22×Q8D411yescharacteristic
51C2×Q16—21no—
52C2×Q16—21no—
53C22×C8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8C2312yes—
58C4⋊Q8C2311yescharacteristic
59C4⋊Q8C2311yescharacteristic
60#125—21no—
61C22⋊Q8—21no—
62C22⋊Q8—21no—
63C4×Q8—21no—
64C4×Q8—21no—
65C4×Q8—21no—
66C4×Q8—21no—
67C4×Q8D411yescharacteristic
68C4×Q8D411yescharacteristic
69C4×Q8C2311yescharacteristic
70C4×Q8C2311yescharacteristic
71#117—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4D411yescharacteristic
76C2×C42—21no—
77C2×C42C2311yescharacteristic
78#107—22no—
79#107—22no—
80#106—22no—
81#106—22no—
82#106—21no—
83#106—21no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86C4⋊C8—21no—
87C4⋊C8D412yes—
88C4⋊C8C2312yes—
89C4⋊C8C2312yes—
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2311yescharacteristic
92#103—22no—
93#103—22no—
94#103—21no—
95#103—21no—
96#103—22no—
97#103—22no—
98#103—21no—
99#103—21no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8D411yescharacteristic
103C22⋊C8D411yescharacteristic
104#97—22no—
105#97—21no—
106#97—21no—
107C4×C8—22no—
108C4×C8C2312yes—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—82no—
117C2×Q8C2×D411yescharacteristic
118C2×Q8C2×D411yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—41no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4#4111yescharacteristic
125Q16—42no—
126Q16—42no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—42no—
132C2×C8C2×D412yes—
133C2×C8C2×D412yes—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
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—41no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—41no—
157C42C2×D411yescharacteristic
158C42C2×D411yescharacteristic
159C42#4111yescharacteristic
160C42C2411yescharacteristic
161C23#12111yescharacteristic
162Q8—21no—
163Q8—21no—
164Q8—21no—
165Q8—42no—
166Q8—41no—
167Q8—41no—
168Q8—42no—
169Q8—82no—
170Q8—82no—
171Q8#13712yes—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4#12111yescharacteristic
187C2×C4#12111yescharacteristic
188C2×C4#12111yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C22×D411yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C8—22no—
193C8—22no—
194C8—41no—
195C8—42no—
196C8—41no—
197C8—42no—
198C22—21no—
199C22—21no—
200C22#47112yes—
201C22C2×#12111yescharacteristic
202C4—21no—
203C4—21no—
204C4—22no—
205C4—21no—
206C4—21no—
207C4—41no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—81no—
212C4—82no—
213C4C2×#13711yescharacteristic
214C4#57511yescharacteristic
215C4#57812yes—
216C2—22no—
217C2#304411yescharacteristic
218C2#305611yescharacteristic
219C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC23#121
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
5. #1704C2semidirect
6. #1692C2semidirect
7. #1684C2semidirect
8. #1672C2semidirect

Maximal subgroups

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

subgroupindex
#32292
#31132
C2×#4772
#29842
#17042
#16922
#16842
#16722
#15732
#15582
#15522