Finite Group Databasebeta

← #24253

Subgroups — #24253

Order 256.

712 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#547C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#452C212yesmaximal, complemented
4C2×#440C211yesmaximal, characteristic, complemented
5#2536C211yesmaximal, characteristic, complemented
6#2036C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×Q16C2212yes—
14C22×SD16C2211yescharacteristic
15#547—24no—
16#547—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#123—22no—
19C2×C22⋊Q8C2212yes—
20C2×C22⋊Q8C2212yes—
21C2×#121—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4—21no—
26#462—24no—
27#452—28no—
28#440—24no—
29C2×#107C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2212yes—
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×#97C2211yescharacteristic
40#398—21no—
41#394—21no—
42#388—22no—
43#382C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—24no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8C2311yescharacteristic
53C22×Q8C2312yes—
54C22×D4—22no—
55C22×D4D411yescharacteristic
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58C23×C4—22no—
59C23×C4C2312yes—
60C2×Q16—24no—
61C2×Q16—24no—
62C2×Q16—42no—
63C2×SD16—22no—
64C2×SD16—22no—
65C2×SD16D414yescomplemented
66C22×C8C2312yes—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8D414yes—
71#123—216no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—42no—
76#121—28no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4D414yes—
83C2×C4⋊C4—28no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—22no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D412yes—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100#107—22no—
101C4⋊C8—22no—
102#103—24no—
103#103—24no—
104#103—22no—
105#103—42no—
106#102—22no—
107#102—24no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110#97—22no—
111#95—22no—
112#95—21no—
113#95—21no—
114C24—21no—
115C24C2×D412yes—
116#41—48no—
117#41—48no—
118#41—48no—
119#41—48no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8C2×D414yes—
131C2×Q8C2×D412yes—
132C2×D4—28no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—42no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4C2×D412yes—
141C2×D4C2×D412yes—
142C2×D4#4112yes—
143C22×C4—28no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4C2×D412yes—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2411yescharacteristic
163Q16—48no—
164SD16—28no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—21no—
173C2×C8C2×D412yes—
174C4⋊C4—28no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4C2×D412yes—
186C22⋊C4—28no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C42—42no—
192C42C2×D412yes—
193C42C2×D412yes—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23C22×D411yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—24no—
210Q8—24no—
211Q8—42no—
212Q8—41no—
213Q8—42no—
214Q8—41no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—44no—
219D4—416no—
220D4#12114yes—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—48no—
230C2×C4—416no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4#12012yes—
246C2×C4#12012yes—
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4H(5,𝔽2)11yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C8—22no—
254C8—41no—
255C8—42no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—44no—
260C22—44no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22C2×#12011yescharacteristic
269C22D4211yescharacteristic
270C22#54611yescharacteristic
271C22#57512yes—
272C22#57812yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—44no—
277C4—42no—
278C4—42no—
279C4—81no—
280C4—82no—
281C4C2×#12111yescharacteristic
282C4#53511yescharacteristic
283C4D4211yescharacteristic
284C4#54611yescharacteristic
285C2—44no—
286C2—82no—
287C2#240811yescharacteristic
288C2#300511yescharacteristic
289C2#301111yescharacteristic
290C2#326912yes—
291C2#331512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
3. C2×#452C2semidirect
4. C2×#440C2semidirect
5. #2536C2semidirect
6. #2036C2semidirect
7. #2001C2semidirect
8. #1960C2semidirect
9. #1955C2semidirect
10. #1804C2semidirect
11. #1788C2semidirect
65. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4622
C2×#4522
C2×#4402
#25362
#20362
#20012
#19602
#19552
#18042
#17882