Finite Group Databasebeta

← #24672

Subgroups — #24672

Order 256.

377 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 117 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#3039C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C4×#137C211yesmaximal, characteristic
4#2925C211yesmaximal, characteristic
5#2052C212yesmaximal
6#2020C212yesmaximal
7#1979C212yesmaximal, complemented
8#1966C211yesmaximal, characteristic, complemented
9#1950C211yesmaximal, characteristic, complemented
10#1930C212yesmaximal, complemented
11#1801C211yesmaximal, characteristic, complemented
12C2×#137C2211yescharacteristic
13C2×#137C2212yes—
14C22×Q16C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C4×#41C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18#462—21no—
19#462—21no—
20#462—21no—
21#462—21no—
22#462—22no—
23#461—21no—
24#461—21no—
25#441—21no—
26#440—21no—
27C4×Q16—21no—
28C4×SD16—21no—
29#433—22no—
30#432—21no—
31#431—21no—
32C2×#108C2212yes—
33#428C2211yescharacteristic
34#423C2211yescharacteristic
35#423C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37#419C2211yescharacteristic
38#419C2212yes—
39#417C2211yescharacteristic
40C2×#103C2212yes—
41C2×#101C2212yes—
42C2×#101C2212yes—
43C2×#101C2212yes—
44#408C2212yes—
45#405C2212yes—
46C4×M16C2211yescharacteristic
47#343C2212yes—
48#341C2212yes—
49C2×#41C2311yescharacteristic
50C22×Q8C2312yes—
51C22×Q8C2311yescharacteristic
52#137—22no—
53#137—22no—
54#137—42no—
55#137—42no—
56C2×Q16—21no—
57C2×Q16—22no—
58C2×Q16—24no—
59C2×Q16—22no—
60C2×Q16—22no—
61C2×Q16—42no—
62C2×SD16—21no—
63C2×SD16—22no—
64C2×M16C2312yes—
65C2×M16C2312yes—
66C2×M16C2311yescharacteristic
67C2×M16C2311yescharacteristic
68C22×C8C2312yes—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8D411yescharacteristic
78C4×Q8D411yescharacteristic
79C4×D4—21no—
80C4×D4—21no—
81C4×D4D411yescharacteristic
82C4×D4D411yescharacteristic
83#117—21no—
84#117C2311yescharacteristic
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#108—42no—
93#107—21no—
94#106—21no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8D411yescharacteristic
100C4⋊C8D411yescharacteristic
101C4⋊C8D412yes—
102#103—21no—
103#103—22no—
104#103—22no—
105#103—21no—
106#103—21no—
107#103—22no—
108#103—22no—
109#102—21no—
110#101—24no—
111#101—42no—
112#101—42no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115#97—22no—
116#97—21no—
117C4×C8—22no—
118C4×C8—21no—
119#41—21no—
120#41—21no—
121#41—21no—
122#41—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—21no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8C2×D411yescharacteristic
135C2×Q8#4112yes—
136C2×D4—21no—
137C2×D4C2×D411yescharacteristic
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4#4111yescharacteristic
145C22×C4#4111yescharacteristic
146C22×C4C2411yescharacteristic
147Q16—24no—
148Q16—44no—
149Q16—44no—
150Q16—44no—
151SD16—24no—
152SD16—44no—
153M16—21no—
154M16—22no—
155M16—21no—
156M16—22no—
157M16—22no—
158M16—21no—
159M16—22no—
160M16#4112yes—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4C2×D411yescharacteristic
180C4⋊C4C2×D411yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—41no—
188C42—41no—
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C23—21no—
194C23C2×#4111yescharacteristic
195Q8—22no—
196Q8—21no—
197Q8—21no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205D4—21no—
206D4—21no—
207D4—41no—
208D4—41no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4#12111yescharacteristic
227C2×C4#12111yescharacteristic
228C2×C4#12111yescharacteristic
229C2×C4#12111yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C2×C4C2×#4111yescharacteristic
233C2×C4C2×#4111yescharacteristic
234C2×C4H(5,𝔽2)11yescharacteristic
235C2×C4H(5,𝔽2)11yescharacteristic
236C8—22no—
237C8—22no—
238C8—22no—
239C8—41no—
240C8—42no—
241C8—42no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22C2×#12111yescharacteristic
248C22#55311yescharacteristic
249C22#55511yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—41no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—82no—
258C4—81no—
259C4#47312yes—
260C4#52511yescharacteristic
261C4#53511yescharacteristic
262C4#54611yescharacteristic
263C4#55011yescharacteristic
264C2—21no—
265C2—21no—
266C2—81no—
267C2#241411yescharacteristic
268C2C2×#47311yescharacteristic
269C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
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×#462C2semidirect
7. #1979C2semidirect
8. #1966C2semidirect
9. #1950C2semidirect
10. #1930C2semidirect
11. #1801C2semidirect

Maximal subgroups

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

subgroupindex
#30392
C2×#4622
C4×#1372
#29252
#20522
#20202
#19792
#19662
#19502
#19302
#18012