Finite Group Databasebeta

← #25605

Subgroups — #25605

Order 256.

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

normal subgroups only

This group has 659 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×#529C211yesmaximal, characteristic, complemented
2C2×#496C211yesmaximal, characteristic, complemented
3C2×#490C211yesmaximal, characteristic, complemented
4#2856C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1992C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15#529—24no—
16#529—24no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121—21no—
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2212yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30C22×C42C2211yescharacteristic
31#496—24no—
32#490—24no—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×#97C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#390—21no—
41#390—21no—
42#390C2212yes—
43#384C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×SD16—24no—
59C2×SD16—24no—
60C2×SD16—42no—
61C2×D8—24no—
62C2×D8—24no—
63C2×D8—42no—
64C22×C8C2312yes—
65C22×C8C2312yes—
66#127—22no—
67#127—22no—
68#126—28no—
69#125—22no—
70#125D414yescomplemented
71#124—22no—
72#124D414yescomplemented
73#123—28no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76#121—22no—
77#121—22no—
78#121—42no—
79#121—42no—
80#121—44no—
81#121—44no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×D4—28no—
85C4×D4—22no—
86C4×D4—22no—
87#117—28no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
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×C22⋊C4—22no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42D411yescharacteristic
109C2×C42D411yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#103—42no—
113#102—24no—
114#102—24no—
115#102—42no—
116C22⋊C8—24no—
117#97—22no—
118#97—22no—
119C24—21no—
120C24—22no—
121C24C2×D411yescharacteristic
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8C2×D412yes—
126C2×D4—24no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4C2×D412yes—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—48no—
172D8—48no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8#4114yes—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D412yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—24no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C42—24no—
200C42—22no—
201C42—22no—
202C42—21no—
203C42—21no—
204C42C2×D412yes—
205C42C2×D412yes—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23AGL(1,ℤ/8)12yes—
227C23C22×D411yescharacteristic
228Q8—24no—
229Q8—42no—
230D4—24no—
231D4—48no—
232D4—42no—
233D4—42no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—84no—
238D4—84no—
239D4—88no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4#12012yes—
270C2×C4#12012yes—
271C2×C4AGL(1,ℤ/8)12yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4C2×#4112yes—
275C2×C4H(5,𝔽2)12yes—
276C8—24no—
277C8—42no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22C2×#12011yescharacteristic
300C22#53211yescharacteristic
301C22#53511yescharacteristic
302C22C2×AGL(1,ℤ/8)11yescharacteristic
303C22C2×AGL(1,ℤ/8)11yescharacteristic
304C22#57511yescharacteristic
305C22#57511yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—42no—
313C4—42no—
314C4—82no—
315C4#54612yes—
316C4#55012yes—
317C2—42no—
318C2—82no—
319C2—162no—
320C2#243111yescharacteristic
321C2#300611yescharacteristic
322C2#300611yescharacteristic
323C2#317111yescharacteristic
324C2#318011yescharacteristic
325C2#321711yescharacteristic
326C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#529C2semidirect
2. C2×#496C2semidirect
3. C2×#490C2semidirect
4. #2856C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1992C2semidirect
8. #1959C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
11. #1749C2semidirect
70. #125D4semidirect
72. #124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4962
C2×#4902
#28562
#20002
#19992
#19922
#19592
#18802
#18792
#17492