Finite Group Databasebeta

← #25406

Subgroups — #25406

Order 256.

442 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 125 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#3152C211yesmaximal, characteristic
2C2×#473C212yesmaximal, complemented
3#3037C211yesmaximal, characteristic
4C2×#462C211yesmaximal, characteristic, complemented
5#2935C211yesmaximal, characteristic
6#2052C212yesmaximal, complemented
7#2020C212yesmaximal, complemented
8#1966C211yesmaximal, characteristic, complemented
9#1950C211yesmaximal, characteristic, complemented
10#1908C212yesmaximal, complemented
11C4×#101C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×#137C2212yes—
14C2×#137C2212yes—
15C22×Q16C2212yes—
16C2×#131C2212yes—
17#532C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20#497—22no—
21#494—21no—
22#492—21no—
23#473—28no—
24#464—22no—
25#462—21no—
26#462—21no—
27#462—22no—
28#462—21no—
29#460—21no—
30#436—21no—
31#435—22no—
32C8×D4—21no—
33C2×#108C2212yes—
34#423C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36#417C2212yes—
37C2×#103C2212yes—
38C2×#101C2212yes—
39C2×#101C2211yescharacteristic
40C2×#101C2212yes—
41C2×#101C2211yescharacteristic
42#408C2212yes—
43C4×M16C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#343C2211yescharacteristic
46#343C2211yescharacteristic
47#341C2212yes—
48C2×#41C2312yes—
49C22×Q8C2312yes—
50C22×Q8C2311yescharacteristic
51#137—24no—
52#137—24no—
53#137—24no—
54#137—42no—
55#137—42no—
56#137—42no—
57C2×Q16—24no—
58C2×Q16—22no—
59C2×Q16—22no—
60C2×Q16—22no—
61C2×Q16—22no—
62C2×Q16—22no—
63C2×Q16—42no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67#131—24no—
68#131—42no—
69C2×M16—22no—
70C2×M16C2312yes—
71C2×M16C2311yescharacteristic
72C2×M16C2311yescharacteristic
73C2×M16C2311yescharacteristic
74C2×M16C2311yescharacteristic
75C22×C8—22no—
76C22×C8C2312yes—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8D411yescharacteristic
82#127D411yescharacteristic
83#124D412yes—
84#121—21no—
85#121—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×D4—21no—
91C4×D4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98#108—24no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8D411yescharacteristic
102C4⋊C8D412yes—
103C4⋊C8D411yescharacteristic
104#103—22no—
105#103—22no—
106#103—22no—
107#102—22no—
108#101—24no—
109#101—22no—
110#101—42no—
111#101D414yescomplemented
112C22⋊C8—22no—
113C22⋊C8—22no—
114#97—21no—
115#97—21no—
116C4×C8—21no—
117C4×C8—21no—
118#41—22no—
119#41—22no—
120#41—22no—
121#41—22no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C2×D412yes—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4#4111yescharacteristic
144C22×C4#4111yescharacteristic
145C22×C4C2411yescharacteristic
146Q16—44no—
147Q16—44no—
148Q16—44no—
149Q16—44no—
150Q16—44no—
151SD16—44no—
152SD16—44no—
153SD16—44no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—22no—
158M16—21no—
159M16—21no—
160M16—22no—
161M16—44no—
162M16C2×D412yes—
163M16C2×D412yes—
164M16#4112yes—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—44no—
173C2×C8C2×D412yes—
174C2×C8C2×D412yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C42—41no—
188C42—41no—
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C23—22no—
194C23C2×#4111yescharacteristic
195Q8—22no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205D4—22no—
206D4—22no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—81no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4#12111yescharacteristic
225C2×C4#12111yescharacteristic
226C2×C4#12111yescharacteristic
227C2×C4#12111yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C2×#4111yescharacteristic
232C2×C4H(5,𝔽2)11yescharacteristic
233C2×C4H(5,𝔽2)11yescharacteristic
234C8—22no—
235C8—22no—
236C8—22no—
237C8—22no—
238C8—22no—
239C8—42no—
240C8—41no—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22C2×#12111yescharacteristic
247C22#53211yescharacteristic
248C22#54011yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—42no—
254C4—82no—
255C4—81no—
256C4#47312yescomplemented
257C4D4211yescharacteristic
258C4D4211yescharacteristic
259C4#54611yescharacteristic
260C4#55011yescharacteristic
261C2—21no—
262C2—21no—
263C2—82no—
264C2#242811yescharacteristic
265C2C2×#47311yescharacteristic
266C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#473C2semidirect
4. C2×#462C2semidirect
6. #2052C2semidirect
7. #2020C2semidirect
8. #1966C2semidirect
9. #1950C2semidirect
10. #1908C2semidirect
11. C4×#101C2semidirect
111. #101D4semidirect
256. C4#473semidirect

Maximal subgroups

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

subgroupindex
#31522
C2×#4732
#30372
C2×#4622
#29352
#20522
#20202
#19662
#19502
#19082
C4×#1012