Finite Group Databasebeta

← #10000

Subgroups — #10000

Order 256.

316 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 59 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#3101C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic, complemented
3#2535C211yesmaximal, characteristic
4#1518C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#121—21no—
10C2×C4×D4C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#483—21no—
13#478—22no—
14#476—21no—
15#476—21no—
16#474—21no—
17#461—21no—
18#450—21no—
19C4×SD16—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#428—21no—
23#423C2211yescharacteristic
24#418—21no—
25#397—21no—
26#392—21no—
27#386—22no—
28#382C2211yescharacteristic
29#380—21no—
30#328—22no—
31D4⋊C8—22no—
32D4⋊C8—22no—
33C22×D4—22no—
34C22×D4D411yescharacteristic
35C22×D4D411yescharacteristic
36C23×C4—21no—
37C23×C4—21no—
38C2×SD16—22no—
39C2×M16—21no—
40C4⋊Q8D411yescharacteristic, complemented
41C4⋊Q8D411yescharacteristic, complemented
42#125—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45#121—42no—
46#121—42no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57#117—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C42C2311yescharacteristic
66#107—22no—
67#106—22no—
68#106—22no—
69C4⋊C8—21no—
70C4⋊C8—21no—
71C4⋊C8—21no—
72C4⋊C8D412yescomplemented
73#103—22no—
74#102—22no—
75#102—22no—
76#102—22no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79C4×C8—21no—
80C4×C8D412yescomplemented
81#95—22no—
82#95—21no—
83#95D412yes—
84C24—21no—
85C24—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4—44no—
97C2×D4—44no—
98C2×D4—82no—
99C2×D4SD1612yescomplemented
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—82no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115SD16—44no—
116M16—41no—
117M16—41no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—41no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C22⋊C4—44no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—82no—
140C42—41no—
141C42C2×D411yescharacteristic
142C42C2×D411yescharacteristic
143C42C2×D411yescharacteristic
144C42#4111yescharacteristic
145C23—22no—
146C23—22no—
147C23—41no—
148C23—42no—
149C23—42no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23#12011yescharacteristic
158Q8—42no—
159Q8—81no—
160D4—22no—
161D4—22no—
162D4—22no—
163D4—22no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—88no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4#12011yescharacteristic
188C2×C4#12011yescharacteristic
189C2×C4#12111yescharacteristic
190C2×C4#12111yescharacteristic
191C2×C4#12111yescharacteristic
192C2×C4#12711yescharacteristic
193C2×C4C2×SD1611yescharacteristic
194C2×C4#13511yescharacteristic
195C2×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C4#13711yescharacteristic
197C8—42no—
198C8—42no—
199C8—81no—
200C22—21no—
201C22—21no—
202C22—42no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—81no—
208C22—82no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22#39211yescharacteristic
215C22#45011yescharacteristic
216C22#45211yescharacteristic
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—82no—
222C4—81no—
223C4—81no—
224C4—81no—
225C4—81no—
226C4#44811yescharacteristic
227C4#44911yescharacteristic
228C4#46511yescharacteristic
229C4#46811yescharacteristic
230C2—21no—
231C2—21no—
232C2—82no—
233C2—82no—
234C2#200011yescharacteristic
235C2#200811yescharacteristic
236C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#2535C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3101C2semidirect
2. #3077C2semidirect
4. #1518C2semidirect
5. #1486C2semidirect
6. #1478C2semidirect
7. #1286C2semidirect
40. C4⋊Q8D4semidirect
41. C4⋊Q8D4semidirect
72. C4⋊C8D4semidirect
80. C4×C8D4semidirect
99. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#31012
#30772
#25352
#15182
#14862
#14782
#12862