Finite Group Databasebeta

← #19447

Subgroups — #19447

Order 256.

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

normal subgroups only

This group has 299 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#3209C211yesmaximal, characteristic, complemented
2#3079C211yesmaximal, characteristic
3#3076C211yesmaximal, characteristic
4#2989C211yesmaximal, characteristic
5#1704C212yesmaximal, complemented
6#1703C212yesmaximal, complemented
7#1690C212yesmaximal, complemented
8#1689C212yesmaximal, complemented
9#1570C211yesmaximal, characteristic, complemented
10#1550C211yesmaximal, characteristic, complemented
11C2×#332C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C4×#41C2211yescharacteristic
15#494C2212yes—
16#487C2212yes—
17#479—21no—
18#478—21no—
19#477—21no—
20#477C2211yescharacteristic
21#476—21no—
22#476C2211yescharacteristic
23#475C2211yescharacteristic
24#475C2212yes—
25#474C2211yescharacteristic
26#474C2212yes—
27#462—21no—
28#462—21no—
29#451—22no—
30#446C2212yes—
31#436C2212yes—
32#435—22no—
33#432—21no—
34#431C2211yescharacteristic
35#428—21no—
36#427—21no—
37#423—21no—
38#423C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41#417—21no—
42#416—21no—
43#334C2212yes—
44#334C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#332C2212yes—
49C2×#41D411yescharacteristic
50C22×Q8—21no—
51C2×Q16—22no—
52C2×Q16—22no—
53C2×M16—21no—
54C2×M16—22no—
55C22×C8—21no—
56C22×C8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8C2312yes—
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#125—21no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8D411yescharacteristic
69C4×Q8C2311yescharacteristic
70C4×D4—21no—
71C4×D4—21no—
72C4×D4D411yescharacteristic
73C4×D4C2311yescharacteristic
74#117—21no—
75#117—21no—
76#117D411yescharacteristic
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81#107—21no—
82#107—21no—
83#107—22no—
84#106—21no—
85#106—21no—
86#106—22no—
87C4⋊C8—22no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8D411yescharacteristic
92C4⋊C8D411yescharacteristic
93C4⋊C8C2312yes—
94C4⋊C8C2311yescharacteristic
95C4⋊C8C2311yescharacteristic
96C4⋊C8C2312yes—
97#103—21no—
98#103—21no—
99#103—22no—
100#103—22no—
101#103—22no—
102#102—21no—
103#102—21no—
104#102—22no—
105C22⋊C8—22no—
106C22⋊C8D412yes—
107#97—22no—
108C4×C8C2312yes—
109#41—22no—
110#41—22no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×Q8C2×D411yescharacteristic
118C2×D4—21no—
119C2×D4C2×D411yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4#4111yescharacteristic
127Q16—82no—
128Q16—82no—
129M16—42no—
130M16—82no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—42no—
136C2×C8—81no—
137C2×C8C2×D412yes—
138C2×C8C2×D412yes—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4C2×D411yescharacteristic
152C4⋊C4C2×D411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—41no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—41no—
161C42C2×D411yescharacteristic
162C42C2×D411yescharacteristic
163C42#4111yescharacteristic
164C42C2411yescharacteristic
165C23—21no—
166C23#12111yescharacteristic
167Q8—21no—
168Q8—21no—
169Q8—41no—
170Q8—42no—
171Q8—41no—
172Q8—42no—
173Q8—81no—
174Q8—82no—
175D4—21no—
176D4—21no—
177D4—42no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4#12111yescharacteristic
193C2×C4#12111yescharacteristic
194C2×C4#12111yescharacteristic
195C2×C4#13712yes—
196C2×C4#13712yes—
197C2×C4C22×D411yescharacteristic
198C2×C4C22×D411yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C8—22no—
201C8—22no—
202C8—41no—
203C8—41no—
204C8—42no—
205C8—42no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22#47012yes—
212C22C2×#12111yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—82no—
220C4—81no—
221C4—81no—
222C4#47012yes—
223C4C2×#13711yescharacteristic
224C4C2×#13711yescharacteristic
225C4#57611yescharacteristic
226C4#57711yescharacteristic
227C2—22no—
228C2—81no—
229C2#304511yescharacteristic
230C2C2×#47011yescharacteristic
231C2#306911yescharacteristic
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. #3209C2semidirect
5. #1704C2semidirect
6. #1703C2semidirect
7. #1690C2semidirect
8. #1689C2semidirect
9. #1570C2semidirect
10. #1550C2semidirect
11. C2×#332C2semidirect

Maximal subgroups

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

subgroupindex
#32092
#30792
#30762
#29892
#17042
#17032
#16902
#16892
#15702
#15502
C2×#3322