Finite Group Databasebeta

← #29606

Subgroups — #29606

Order 256.

695 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 127 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, complemented
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4#2615C211yesmaximal, characteristic, complemented
5#2035C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1999C212yesmaximal, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1788C211yesmaximal, characteristic, complemented
11#1780C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C22×D8C2211yescharacteristic
16#537—24no—
17#537—24no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20C2×#121—21no—
21C2×#121—22no—
22C2×#121C2211yescharacteristic
23C2×#121C2212yes—
24C2×#121C2212yes—
25C22×C4⋊C4C2211yescharacteristic
26#481—24no—
27#449—24no—
28C2×#106C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37C2×C22⋊C8C2212yes—
38#393—21no—
39#392—22no—
40#392—21no—
41#380C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41—42no—
49C2×#41—42no—
50C2×#41—42no—
51C2×#41—42no—
52C2×#41D414yescomplemented
53C22×Q8D411yescharacteristic
54C22×Q8C2311yescharacteristic
55C22×D4—21no—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4—21no—
59C22×D4D412yes—
60C22×D4D411yescharacteristic
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67C2×SD16—22no—
68C2×SD16—42no—
69C2×D8—22no—
70C2×D8—42no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2312yes—
74C22⋊Q8—28no—
75C22⋊Q8—24no—
76C22⋊Q8—24no—
77#121—216no—
78#121—28no—
79#121—24no—
80#121—24no—
81#121—22no—
82#121—22no—
83#121—42no—
84#121—42no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4D414yescomplemented
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—21no—
98#106—24no—
99#103—22no—
100#103—42no—
101#102—22no—
102#102—24no—
103#102—42no—
104C22⋊C8—22no—
105C22⋊C8—22no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#95—22no—
109#95—21no—
110C24—21no—
111C24—21no—
112C24C2×D412yes—
113C24C2×D411yescharacteristic
114#41—44no—
115#41—44no—
116#41—44no—
117#41—44no—
118#41—48no—
119#41—48no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—42no—
125C2×Q8C2×D412yes—
126C2×D4—24no—
127C2×D4—24no—
128C2×D4—28no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—42no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—82no—
146C2×D4—82no—
147C2×D4—82no—
148C2×D4C2×D412yes—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—44no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4C2×D412yes—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2411yescharacteristic
171SD16—44no—
172D8—44no—
173C2×C8—22no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4C2×D414yes—
188C22⋊C4—28no—
189C22⋊C4—28no—
190C22⋊C4—44no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C22⋊C4—82no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23—42no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23#12012yes—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—44no—
220Q8—42no—
221D4—22no—
222D4—22no—
223D4—44no—
224D4—48no—
225D4—48no—
226D4—41no—
227D4—41no—
228D4—42no—
229D4—42no—
230D4—88no—
231D4—84no—
232D4—88no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—84no—
256C2×C4#12012yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4H(5,𝔽2)12yes—
260C2×C4H(5,𝔽2)12yes—
261C8—41no—
262C8—41no—
263C8—42no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22—82no—
281C22—81no—
282C22—82no—
283C22—82no—
284C22—81no—
285C22—81no—
286C22C2×#12011yescharacteristic
287C22#53412yes—
288C22#57611yescharacteristic
289C22#57712yes—
290C22#57711yescharacteristic
291C4—22no—
292C4—42no—
293C4—42no—
294C4—44no—
295C4—82no—
296C4—81no—
297C4#53411yescharacteristic
298C4#53411yescharacteristic
299C4#53512yes—
300C2—42no—
301C2—82no—
302C2—82no—
303C2—161no—
304C2#261311yescharacteristic
305C2#301011yescharacteristic
306C2#301211yescharacteristic
307C2#319812yes—
308C2#320412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#537C2semidirect
2. C2×#481C2semidirect
3. C2×#449C2semidirect
4. #2615C2semidirect
5. #2035C2semidirect
6. #2034C2semidirect
7. #2000C2semidirect
8. #1999C2semidirect
9. #1823C2semidirect
10. #1788C2semidirect
11. #1780C2semidirect
52. C2×#41D4semidirect
88. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4812
C2×#4492
#26152
#20352
#20342
#20002
#19992
#18232
#17882
#17802