Finite Group Databasebeta

← #19365

Subgroups — #19365

Order 256.

292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 107 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#3239C211yesmaximal, characteristic
2#3238C211yesmaximal, characteristic
3C2×#487C211yesmaximal, characteristic, complemented
4#2966C211yesmaximal, characteristic
5#1704C212yesmaximal, complemented
6#1703C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1683C212yesmaximal, complemented
9#1568C211yesmaximal, characteristic, complemented
10#1558C211yesmaximal, characteristic, complemented
11#1556C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14#529C2211yescharacteristic
15#501—21no—
16#501—21no—
17#489—21no—
18#489—21no—
19#487—21no—
20#487—21no—
21#487C2212yes—
22#487C2212yes—
23#484—21no—
24#483—21no—
25#481—21no—
26#480—21no—
27#477C2212yes—
28#476C2212yes—
29#475C2212yes—
30#474C2212yes—
31#446C2211yescharacteristic
32C8×Q8C2211yescharacteristic
33#436C2211yescharacteristic
34#435—21no—
35C8×D4—21no—
36C8×D4C2211yescharacteristic
37#433—21no—
38#432C2211yescharacteristic
39#432C2211yescharacteristic
40#432C2211yescharacteristic
41C2×#103—21no—
42C2×#103—21no—
43#405—21no—
44C2×C4×C8C2211yescharacteristic
45#335C2212yes—
46#334C2212yes—
47#333C2212yes—
48#333C2212yes—
49#332C2212yes—
50#332C2212yes—
51C22×Q8—21no—
52C22×Q8—21no—
53C2×M16—21no—
54C22×C8—21no—
55C22×C8D411yescharacteristic
56C22×C8D411yescharacteristic
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8C2312yes—
64C4⋊Q8C2312yes—
65#126—21no—
66#125D411yescharacteristic
67#124D411yescharacteristic
68#123—21no—
69C22⋊Q8D411yescharacteristic
70#121D411yescharacteristic
71C4×Q8C2311yescharacteristic
72C4×D4—21no—
73C4×D4C2311yescharacteristic
74#117—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C42C2311yescharacteristic
80#107—22no—
81#107—22no—
82#106—22no—
83#106—22no—
84C4⋊C8—21no—
85C4⋊C8—21no—
86C4⋊C8C2311yescharacteristic
87C4⋊C8C2311yescharacteristic
88C4⋊C8C2312yes—
89C4⋊C8C2312yes—
90#103—22no—
91#103—22no—
92#103—22no—
93#103—22no—
94#103—22no—
95#103—22no—
96#102—22no—
97#102—22no—
98C22⋊C8—21no—
99C22⋊C8—21no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8—21no—
103#97—21no—
104#97—21no—
105#97—21no—
106#97—21no—
107C4×C8—21no—
108C4×C8D411yescharacteristic
109C4×C8D411yescharacteristic
110C4×C8C2311yescharacteristic
111C4×C8C2311yescharacteristic
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—82no—
119C2×Q8C2×D411yescharacteristic
120C2×D4—21no—
121C2×D4C2×D411yescharacteristic
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4#4111yescharacteristic
128M16—42no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—42no—
136C2×C8C2×D411yescharacteristic
137C2×C8C2×D411yescharacteristic
138C2×C8C2×D411yescharacteristic
139C2×C8C2×D411yescharacteristic
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4C2×D411yescharacteristic
153C4⋊C4C2×D411yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C42—21no—
160C42C2×D411yescharacteristic
161C42C2×D411yescharacteristic
162C42#4111yescharacteristic
163C42C2411yescharacteristic
164C23—21no—
165C23#12111yescharacteristic
166Q8—22no—
167Q8—42no—
168Q8—42no—
169Q8—82no—
170Q8—82no—
171Q8—82no—
172D4—22no—
173D4—42no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4#12111yescharacteristic
188C2×C4#12111yescharacteristic
189C2×C4#12111yescharacteristic
190C2×C4C22×D411yescharacteristic
191C2×C4C22×D411yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C8—21no—
194C8—21no—
195C8—21no—
196C8—21no—
197C8—41no—
198C8—41no—
199C8—42no—
200C8—42no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22#47112yes—
207C22C2×#12111yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—82no—
216C4—82no—
217C4#57511yescharacteristic
218C4#57511yescharacteristic
219C4#57812yes—
220C2—22no—
221C2—81no—
222C2#304011yescharacteristic
223C2#305911yescharacteristic
224C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#121D4
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. C2×#487C2semidirect
5. #1704C2semidirect
6. #1703C2semidirect
7. #1684C2semidirect
8. #1683C2semidirect
9. #1568C2semidirect
10. #1558C2semidirect
11. #1556C2semidirect

Maximal subgroups

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

subgroupindex
#32392
#32382
C2×#4872
#29662
#17042
#17032
#16842
#16832
#15682
#15582
#15562