Finite Group Databasebeta

← #24011

Subgroups — #24011

Order 256.

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

normal subgroups only

This group has 670 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×#537C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#480C211yesmaximal, characteristic, complemented
4#2311C211yesmaximal, characteristic, complemented
5#2036C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1875C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic, complemented
9#1852C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1780C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14#537—24no—
15#537—24no—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121—21no—
20C2×#121—21no—
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#484—24no—
28#480—24no—
29C2×#107C2212yes—
30C2×#107C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
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—22no—
50C2×#41—22no—
51C2×#41—24no—
52C2×#41—24no—
53C2×#41—42no—
54C2×#41—42no—
55C22×Q8D411yescharacteristic
56C22×Q8C2311yescharacteristic
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4D411yescharacteristic
61C22×D4D411yescharacteristic
62C22×D4D411yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C22×C8C2312yes—
71C22×C8C2312yes—
72C22⋊Q8—28no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8D414yes—
77#121—28no—
78#121—28no—
79#121—28no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121D414yes—
84#117—42no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C42—21no—
105#107—24no—
106#107—24no—
107#103—24no—
108#103—42no—
109#102—24no—
110#102—42no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C24C2×D411yescharacteristic
120C24C2×D411yescharacteristic
121#41—48no—
122#41—48no—
123#41—48no—
124#41—48no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—41no—
131C2×Q8C2×D412yes—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—82no—
150C2×D4—82no—
151C2×D4C2×D412yes—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4#4114yes—
179C22×C4C2411yescharacteristic
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4C2×D412yes—
197C4⋊C4C2×D412yes—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—82no—
205C42—42no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—41no—
221C23—41no—
222C23—42no—
223C23—42no—
224C23#12012yes—
225C23#12312yes—
226C23C22×D411yescharacteristic
227Q8—24no—
228Q8—44no—
229Q8—42no—
230D4—24no—
231D4—44no—
232D4—44no—
233D4—44no—
234D4—48no—
235D4—42no—
236D4—84no—
237D4—84no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—48no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4#12012yes—
268C2×C4#12312yes—
269C2×C4C22×D411yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C2×#4112yes—
272C2×C4H(5,𝔽2)12yes—
273C8—42no—
274C8—42no—
275C22—24no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22C2×#12011yescharacteristic
296C22C2×#12311yescharacteristic
297C22#53411yescharacteristic
298C22#57612yes—
299C22#57812yes—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—82no—
307C4—82no—
308C4#54612yes—
309C4#54612yes—
310C2—42no—
311C2—42no—
312C2—81no—
313C2—82no—
314C2#240511yescharacteristic
315C2#300811yescharacteristic
316C2#301111yescharacteristic
317C2#309812yes—
318C2#320312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
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×#480C2semidirect
4. #2311C2semidirect
5. #2036C2semidirect
6. #2033C2semidirect
7. #1875C2semidirect
8. #1873C2semidirect
9. #1852C2semidirect
10. #1788C2semidirect
11. #1780C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4842
C2×#4802
#23112
#20362
#20332
#18752
#18732
#18522
#17882
#17802