Finite Group Databasebeta

← #25697

Subgroups — #25697

Order 256.

515 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#3506C211yesmaximal, characteristic
2#3166C211yesmaximal, characteristic, complemented
3#3142C211yesmaximal, characteristic, complemented
4#2566C211yesmaximal, characteristic, complemented
5#2037C212yesmaximal, complemented
6#2037C212yesmaximal, complemented
7#1986C211yesmaximal, characteristic, complemented
8#1980C211yesmaximal, characteristic, complemented
9#1860C212yesmaximal, complemented
10#1859C212yesmaximal, complemented
11C4×#100C211yesmaximal, characteristic, complemented
12#559—22no—
13#556—21no—
14#555—21no—
15#553—22no—
16#550—21no—
17#547—21no—
18#538—22no—
19#535C2212yes—
20#533C2211yescharacteristic
21#532C2211yescharacteristic
22C2×#125C2211yescharacteristic
23#525—22no—
24#520C2212yes—
25C2×C4×D4C2211yescharacteristic
26#501—21no—
27#501—21no—
28#499—22no—
29#491—21no—
30#490—21no—
31#488—22no—
32#428C2212yes—
33#428C2212yes—
34C2×#104C2212yes—
35C2×#104C2212yes—
36#417C2212yes—
37#417C2212yes—
38C2×#100C2212yes—
39C4×M16C2211yescharacteristic
40C4×M16C2211yescharacteristic
41#396—22no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#385—22no—
45#384C2211yescharacteristic
46#382C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48#343C2212yes—
49#339C2212yes—
50#339C2212yes—
51C2×#41—22no—
52C2×#41C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4D412yes—
55C2×M16C2312yes—
56C2×M16C2312yes—
57C4⋊Q8D411yescharacteristic
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60#127—22no—
61#127D411yescharacteristic
62#126—24no—
63#125—21no—
64#125—21no—
65#125—21no—
66#125—21no—
67#125D412yes—
68#124—24no—
69#124—22no—
70#124D412yes—
71#123—24no—
72#123—24no—
73C22⋊Q8—22no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76#121—22no—
77#121—24no—
78#121—21no—
79#121—21no—
80#121—22no—
81#121—22no—
82#120—24no—
83C4×Q8—22no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×D4—24no—
87C4×D4—22no—
88C4×D4—24no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95#117—22no—
96#117—22no—
97#117C2312yes—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4D412yes—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#107—22no—
114#106—22no—
115#106—22no—
116#104—24no—
117#104—24no—
118#103—22no—
119#103—22no—
120#102—22no—
121#102—22no—
122#100—24no—
123#97—21no—
124#97—21no—
125C4×C8—21no—
126C4×C8—21no—
127#95—21no—
128#95—22no—
129C24C2×D412yes—
130#41—22no—
131#41—22no—
132#41—22no—
133#41—22no—
134#41—84no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8C2×D412yes—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4C2×D412yes—
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—82no—
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161M16—22no—
162M16—22no—
163M16—22no—
164M16#4114yescomplemented
165C2×C8—22no—
166C2×C8—22no—
167C4⋊C4—24no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—82no—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D412yes—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—44no—
197C22⋊C4—82no—
198C42—24no—
199C42—22no—
200C42—22no—
201C42—22no—
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2×D411yescharacteristic
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23C22×D411yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—81no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—42no—
223D4—42no—
224D4—81no—
225D4—84no—
226D4—84no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—82no—
248C2×C4—84no—
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4#12011yescharacteristic
252C2×C4#12011yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4112yes—
256C2×C42−1+412yes—
257C8—24no—
258C8—42no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22C2×#12011yescharacteristic
270C22#53211yescharacteristic
271C22#53611yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#54712yes—
285C4#54812yes—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—82no—
290C2#243211yescharacteristic
291C2#301811yescharacteristic
292C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3506C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3166C2semidirect
3. #3142C2semidirect
4. #2566C2semidirect
5. #2037C2semidirect
6. #2037C2semidirect
7. #1986C2semidirect
8. #1980C2semidirect
9. #1860C2semidirect
10. #1859C2semidirect
11. C4×#100C2semidirect
164. M16#41semidirect

Maximal subgroups

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

subgroupindex
#35062
#31662
#31422
#25662
#20372
#19862
#19802
#18602
#18592
C4×#1002