Finite Group Databasebeta

← #24251

Subgroups — #24251

Order 256.

728 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#448C212yesmaximal, complemented
4C2×#440C211yesmaximal, characteristic, complemented
5#2535C211yesmaximal, characteristic
6#2036C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1787C212yesmaximal, complemented
12C23×Q8C2212yes—
13C22×SD16C2212yes—
14C22×SD16C2211yescharacteristic
15D4×Q8—24no—
16D4×Q8—24no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—22no—
20C2×C22⋊Q8C2212yes—
21C2×#121C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#461—24no—
27#448—28no—
28#440—24no—
29C2×#107C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×#97C2211yescharacteristic
40#397—21no—
41#392—21no—
42#386—22no—
43#382C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46C22×Q8—24no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—24no—
50C22×Q8—24no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8C2312yes—
54C22×Q8C2311yescharacteristic
55C22×D4—22no—
56C22×D4D411yescharacteristic
57C22×D4C2311yescharacteristic
58C23×C4—22no—
59C23×C4—21no—
60C23×C4C2312yes—
61C2×SD16—22no—
62C2×SD16—22no—
63C2×SD16—24no—
64C2×SD16—24no—
65C2×SD16—42no—
66C2×SD16D414yescomplemented
67C22×C8C2312yes—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—28no—
71C4⋊Q8—22no—
72C4⋊Q8D414yescomplemented
73C22⋊Q8—216no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76#121—24no—
77#121—42no—
78C4×Q8—22no—
79C4×Q8D414yescomplemented
80C4×D4—28no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C2×C4⋊C4—21no—
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—22no—
103#103—24no—
104#103—24no—
105#103—24no—
106#103—42no—
107#102—22no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110#97—22no—
111#95—22no—
112#95—21no—
113#95—21no—
114C24—21no—
115C24C2×D412yes—
116C2×Q8—28no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—48no—
128C2×Q8—48no—
129C2×Q8—48no—
130C2×Q8—48no—
131C2×Q8C2×D414yes—
132C2×Q8C2×D412yes—
133C2×Q8C2×D412yes—
134C2×Q8#4112yes—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—82no—
145C2×D4C2×D412yes—
146C22×C4—21no—
147C22×C4—28no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—82no—
162C22×C4C2×D412yes—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166SD16—28no—
167SD16—48no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—21no—
176C2×C8C2×D412yes—
177C4⋊C4—28no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4C2×D412yes—
188C22⋊C4—28no—
189C22⋊C4—24no—
190C22⋊C4—44no—
191C22⋊C4—42no—
192C22⋊C4—82no—
193C42—24no—
194C42C2×D412yes—
195C42C2×D412yes—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23#13714yes—
208C23C22×D411yescharacteristic
209Q8—22no—
210Q8—22no—
211Q8—22no—
212Q8—24no—
213Q8—24no—
214Q8—416no—
215Q8—42no—
216Q8—42no—
217Q8#12114yes—
218D4—28no—
219D4—22no—
220D4—22no—
221D4—41no—
222D4—41no—
223D4—88no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—416no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—84no—
248C2×C4#12012yes—
249C2×C4#12012yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—22no—
257C8—41no—
258C8—42no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—81no—
269C22—81no—
270C22—82no—
271C22C2×#12011yescharacteristic
272C22D4211yescharacteristic
273C22#54611yescharacteristic
274C22C2×#13712yes—
275C22#57712yes—
276C4—24no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—44no—
281C4—42no—
282C4—81no—
283C4—82no—
284C4C2×#12111yescharacteristic
285C4#53511yescharacteristic
286C4D4211yescharacteristic
287C4#54611yescharacteristic
288C2—44no—
289C2—82no—
290C2#240811yescharacteristic
291C2#300411yescharacteristic
292C2#301011yescharacteristic
293C2#326612yes—
294C2#332012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
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
1. C2×D4×Q8C2semidirect
2. C2×#461C2semidirect
3. C2×#448C2semidirect
4. C2×#440C2semidirect
6. #2036C2semidirect
7. #2000C2semidirect
8. #1960C2semidirect
9. #1955C2semidirect
10. #1805C2semidirect
11. #1787C2semidirect
66. C2×SD16D4semidirect
72. C4⋊Q8D4semidirect
79. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4612
C2×#4482
C2×#4402
#25352
#20362
#20002
#19602
#19552
#18052
#17872