Finite Group Databasebeta

← #25360

Subgroups — #25360

Order 256.

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

normal subgroups only

This group has 366 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#3141C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3#3037C211yesmaximal, characteristic
4#2935C211yesmaximal, characteristic
5#2078C212yesmaximal, complemented
6#2052C212yesmaximal, complemented
7#1997C211yesmaximal, characteristic, complemented
8#1950C211yesmaximal, characteristic, complemented
9#1908C212yesmaximal, complemented
10#1816C212yesmaximal, complemented
11C4×#101C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×#131C2212yes—
14#532C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17#489—22no—
18#487—21no—
19#486—21no—
20#477—21no—
21#477—22no—
22#477—21no—
23#464—22no—
24#462—21no—
25#460—21no—
26#436—21no—
27#435—22no—
28C8×D4—21no—
29C2×#108C2212yes—
30C2×#107C2212yes—
31#423C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33#417C2212yes—
34#417C2212yes—
35#417C2212yes—
36C2×#103C2212yes—
37C2×#101C2211yescharacteristic
38C2×#101C2211yescharacteristic
39#408C2212yes—
40C4×M16C2211yescharacteristic
41C4×M16C2211yescharacteristic
42#343C2212yes—
43#343C2211yescharacteristic
44#343C2212yes—
45#343C2211yescharacteristic
46#341C2212yes—
47C2×#41C2312yes—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50#137—24no—
51#137—42no—
52C2×Q16—22no—
53C2×SD16—22no—
54#131—24no—
55#131—42no—
56C2×M16—22no—
57C2×M16C2312yes—
58C2×M16C2311yescharacteristic
59C2×M16C2311yescharacteristic
60C2×M16C2311yescharacteristic
61C2×M16C2311yescharacteristic
62C22×C8—22no—
63C22×C8C2312yes—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8D411yescharacteristic
69#127D411yescharacteristic
70#124D412yes—
71#121—21no—
72#121—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×D4—21no—
78C4×D4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C42—21no—
84C2×C42C2311yescharacteristic
85#108—42no—
86#107—22no—
87#107—22no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8D411yescharacteristic
91C4⋊C8D412yes—
92C4⋊C8D411yescharacteristic
93#103—22no—
94#103—22no—
95#103—22no—
96#103—22no—
97#103—22no—
98#102—22no—
99#102—22no—
100#102—22no—
101#101—22no—
102#101—22no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105#97—21no—
106#97—21no—
107C4×C8—21no—
108C4×C8—21no—
109#41—22no—
110#41—22no—
111#41—22no—
112#41—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—82no—
121C2×Q8C2×D412yes—
122C2×Q8#4112yes—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4C2×D412yes—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4#4111yescharacteristic
132C22×C4#4111yescharacteristic
133C22×C4C2411yescharacteristic
134Q16—44no—
135SD16—44no—
136M16—21no—
137M16—21no—
138M16—21no—
139M16—21no—
140M16—22no—
141M16—21no—
142M16—22no—
143M16—21no—
144M16—21no—
145M16—44no—
146M16#4112yes—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—44no—
155C2×C8C2×D412yes—
156C2×C8C2×D412yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—41no—
170C42—41no—
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C23—22no—
176C23C2×#4111yescharacteristic
177Q8—22no—
178Q8—22no—
179Q8—22no—
180Q8—22no—
181Q8—41no—
182Q8—41no—
183Q8—82no—
184Q8—82no—
185D4—22no—
186D4—22no—
187D4—42no—
188D4—42no—
189D4—42no—
190D4—81no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4#12111yescharacteristic
204C2×C4#12111yescharacteristic
205C2×C4#12111yescharacteristic
206C2×C4#12111yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4H(5,𝔽2)11yescharacteristic
212C2×C42−1+411yescharacteristic
213C8—22no—
214C8—22no—
215C8—22no—
216C8—41no—
217C8—42no—
218C8—41no—
219C8—41no—
220C22—21no—
221C22—21no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22C2×#12111yescharacteristic
226C22#52911yescharacteristic
227C22#54211yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—42no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4#47312yes—
236C4#54611yescharacteristic
237C4#54611yescharacteristic
238C4#54711yescharacteristic
239C4#54811yescharacteristic
240C2—21no—
241C2—21no—
242C2—82no—
243C2#242711yescharacteristic
244C2C2×#47311yescharacteristic
245C2#306811yescharacteristic
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×#477C2semidirect
5. #2078C2semidirect
6. #2052C2semidirect
7. #1997C2semidirect
8. #1950C2semidirect
9. #1908C2semidirect
10. #1816C2semidirect
11. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
#31412
C2×#4772
#30372
#29352
#20782
#20522
#19972
#19502
#19082
#18162
C4×#1012