Finite Group Databasebeta

← #24722

Subgroups — #24722

Order 256.

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

normal subgroups only

This group has 337 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#3079C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3C4×#137C211yesmaximal, characteristic
4#2925C211yesmaximal, characteristic
5#2078C212yesmaximal
6#2052C212yesmaximal
7#1997C211yesmaximal, characteristic, complemented
8#1952C212yesmaximal, complemented
9#1950C211yesmaximal, characteristic, complemented
10#1930C212yesmaximal, complemented
11#1801C211yesmaximal, characteristic, complemented
12C2×#137C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C4×#41C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#477—21no—
17#477—21no—
18#477—21no—
19#477—22no—
20#476—21no—
21#475—21no—
22#474—21no—
23#441—21no—
24#440—21no—
25C4×Q16—21no—
26C4×SD16—21no—
27#433—22no—
28#432—21no—
29#431—21no—
30C2×#108C2212yes—
31#428C2212yes—
32#428C2211yescharacteristic
33C2×#107C2212yes—
34#423C2211yescharacteristic
35#423C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37#419C2211yescharacteristic
38#417C2212yes—
39#417C2211yescharacteristic
40C2×#103C2212yes—
41C2×#101C2212yes—
42#408C2212yes—
43#405C2212yes—
44C4×M16C2211yescharacteristic
45#343C2212yes—
46#343C2212yes—
47#343C2212yes—
48#341C2212yes—
49C2×#41C2311yescharacteristic
50C22×Q8—21no—
51C22×Q8C2311yescharacteristic
52#137—22no—
53#137—22no—
54C2×Q16—21no—
55C2×SD16—21no—
56C2×M16C2311yescharacteristic
57C2×M16C2312yes—
58C2×M16C2312yes—
59C2×M16C2311yescharacteristic
60C22×C8C2312yes—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4×Q8—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8D411yescharacteristic
70C4×Q8D411yescharacteristic
71C4×D4—21no—
72C4×D4—21no—
73C4×D4D411yescharacteristic
74C4×D4D411yescharacteristic
75#117—21no—
76#117C2311yescharacteristic
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42C2311yescharacteristic
84#108—24no—
85#107—22no—
86#107—22no—
87#107—21no—
88#107—22no—
89#106—21no—
90#106—22no—
91C4⋊C8—21no—
92C4⋊C8—21no—
93C4⋊C8—21no—
94C4⋊C8—21no—
95C4⋊C8D411yescharacteristic
96C4⋊C8D411yescharacteristic
97C4⋊C8D412yes—
98#103—21no—
99#103—22no—
100#103—22no—
101#103—22no—
102#103—21no—
103#103—21no—
104#102—21no—
105#102—22no—
106#101—24no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109#97—22no—
110#97—21no—
111C4×C8—22no—
112C4×C8—21no—
113#41—21no—
114#41—21no—
115#41—21no—
116#41—21no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—82no—
125C2×Q8C2×D411yescharacteristic
126C2×Q8#4112yes—
127C2×D4—21no—
128C2×D4C2×D411yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4#4111yescharacteristic
136C22×C4#4111yescharacteristic
137C22×C4C2411yescharacteristic
138Q16—24no—
139SD16—24no—
140M16—21no—
141M16—22no—
142M16—22no—
143M16—22no—
144M16—21no—
145M16—21no—
146M16—22no—
147M16#4112yes—
148C2×C8—21no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8C2×D412yes—
155C2×C8C2×D412yes—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4C2×D411yescharacteristic
167C4⋊C4C2×D411yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—41no—
175C42—41no—
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2×D411yescharacteristic
180C23—21no—
181C23C2×#4111yescharacteristic
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8—82no—
189Q8—82no—
190D4—21no—
191D4—21no—
192D4—41no—
193D4—41no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4#12111yescharacteristic
211C2×C4#12111yescharacteristic
212C2×C4#12111yescharacteristic
213C2×C4#12111yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C4C2×#4111yescharacteristic
218C2×C42−1+411yescharacteristic
219C2×C42−1+411yescharacteristic
220C8—22no—
221C8—22no—
222C8—22no—
223C8—42no—
224C8—41no—
225C8—42no—
226C22—21no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22C2×#12111yescharacteristic
232C22#55211yescharacteristic
233C22#55611yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—81no—
242C4—82no—
243C4#47312yes—
244C4#52511yescharacteristic
245C4#53611yescharacteristic
246C4#54711yescharacteristic
247C4#54811yescharacteristic
248C2—21no—
249C2—21no—
250C2—81no—
251C2#241511yescharacteristic
252C2C2×#47311yescharacteristic
253C2#306811yescharacteristic
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×#477C2semidirect
7. #1997C2semidirect
8. #1952C2semidirect
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
#30792
C2×#4772
C4×#1372
#29252
#20782
#20522
#19972
#19522
#19502
#19302
#18012