Finite Group Databasebeta

← #24014

Subgroups — #24014

Order 256.

638 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 135 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×#536C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#483C211yesmaximal, characteristic, complemented
4#2311C211yesmaximal, characteristic
5#2036C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1873C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1787C212yesmaximal, complemented
11#1780C211yesmaximal, characteristic
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14#536—24no—
15#536—24no—
16C2×C4⋊Q8—21no—
17C2×#124—21no—
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#484—24no—
28#483—24no—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39#391—21no—
40#390—21no—
41#386—21no—
42#386—21no—
43#380C2212yes—
44#380C2212yes—
45#340C2212yes—
46#340C2212yes—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—24no—
50C2×#41—42no—
51C22×Q8—22no—
52C22×Q8—22no—
53C22×Q8—21no—
54C22×Q8—24no—
55C22×Q8—42no—
56C22×Q8D411yescharacteristic
57C22×Q8C2311yescharacteristic
58C22×Q8C2311yescharacteristic
59C22×D4D411yescharacteristic
60C22×D4D411yescharacteristic
61C22×D4D411yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C22×C8C2312yes—
68C22×C8C2312yes—
69C4⋊Q8—28no—
70#124—28no—
71#123—28no—
72#123—28no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8D414yes—
80C22⋊Q8D414yes—
81#117—22no—
82#117—22no—
83#117—42no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C42—21no—
104C2×C42—21no—
105#107—24no—
106#106—24no—
107#103—24no—
108#103—24no—
109#103—42no—
110#103—42no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115#95—22no—
116#95—22no—
117C24C2×D411yescharacteristic
118C24C2×D411yescharacteristic
119C24C2×D411yescharacteristic
120#41—48no—
121#41—48no—
122C2×Q8—24no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—48no—
133C2×Q8—48no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8C2×D412yes—
137C2×Q8C2×D412yes—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—82no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4C2×D411yescharacteristic
172C22×C4#4114yes—
173C22×C4C2411yescharacteristic
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4C2×D412yes—
192C4⋊C4C2×D412yes—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C42—24no—
202C42—42no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23#12012yes—
217C23#12312yes—
218C23C22×D411yescharacteristic
219Q8—24no—
220Q8—24no—
221Q8—44no—
222Q8—48no—
223Q8—42no—
224Q8—42no—
225D4—44no—
226D4—44no—
227D4—44no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—48no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4#12012yes—
260C2×C4#12312yes—
261C2×C4C22×D411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4C2×#4112yes—
264C2×C4H(5,𝔽2)12yes—
265C8—42no—
266C8—42no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22C2×#12011yescharacteristic
283C22C2×#12311yescharacteristic
284C22#53411yescharacteristic
285C22#57712yes—
286C22#57812yes—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—42no—
294C4—82no—
295C4—82no—
296C4#54612yes—
297C4#54612yes—
298C2—42no—
299C2—42no—
300C2—81no—
301C2#240511yescharacteristic
302C2#301011yescharacteristic
303C2#301111yescharacteristic
304C2#309712yes—
305C2#320112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
3. C2×#483C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #1873C2semidirect
8. #1873C2semidirect
9. #1854C2semidirect
10. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4842
C2×#4832
#23112
#20362
#20352
#18732
#18542
#17872
#17802