Finite Group Databasebeta

← #25876

Subgroups — #25876

Order 256.

611 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 131 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#3499C211yesmaximal, characteristic
2#3141C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic, complemented
4#2862C211yesmaximal, characteristic, complemented
5#2038C214yesmaximal, complemented
6#1966C211yesmaximal, characteristic, complemented
7#1962C211yesmaximal, characteristic, complemented
8#1793C214yesmaximal, complemented
9C4×#104C211yesmaximal, characteristic, complemented
10C2×2−1+4C2212yes—
11D4×Q8—21no—
12D4×Q8—21no—
13#547—21no—
14#547—21no—
15#547—22no—
16#547—22no—
17#536—22no—
18#536C2212yes—
19#532—21no—
20#532C2211yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C4×#41C2211yescharacteristic
25#489—22no—
26#487—21no—
27#486—21no—
28#477—21no—
29#476—21no—
30#475—21no—
31#474—21no—
32#428C2212yes—
33#423C2211yescharacteristic
34C2×#104C2212yes—
35C2×#104C2214yes—
36#417C2212yes—
37#417C2214yes—
38C2×#101C2214yes—
39C4×M16C2211yescharacteristic
40#391—22no—
41#391C2214yes—
42#382—22no—
43C43C2211yescharacteristic
44#339C2212yes—
452−1+4—28no—
462−1+4—44no—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—24no—
50C2×#41C2312yes—
51C2×#41C2311yescharacteristic
52C22×Q8—22no—
53C22×Q8—22no—
54C22×Q8—21no—
55C22×Q8C2312yes—
56C2×M16C2312yes—
57C2×M16C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8D411yescharacteristic
70#127D411yescharacteristic
71#124—22no—
72#124D412yes—
73#123—24no—
74#123—24no—
75#123—24no—
76C22⋊Q8—24no—
77C22⋊Q8—24no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80#121—22no—
81#121—22no—
82#121—21no—
83#121—21no—
84C4×Q8D411yescharacteristic
85C4×Q8D411yescharacteristic
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4D411yescharacteristic
93C4×D4D411yescharacteristic
94#117—21no—
95#117C2311yescharacteristic
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4D412yes—
100C2×C4⋊C4D412yes—
101C2×C4⋊C4C2312yes—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2312yes—
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111#104—24no—
112#104—44no—
113#103—22no—
114#103—24no—
115#102—22no—
116#102—24no—
117#101—44no—
118#97—21no—
119C4×C8—21no—
120#95—28no—
121#41—22no—
122#41—24no—
123#41—24no—
124#41—48no—
125#41—416no—
126#41—44no—
127#41—44no—
128#41#4114yes—
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—24no—
133C2×Q8—24no—
134C2×Q8—48no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8C2×D412yes—
141C2×Q8C2×D411yescharacteristic
142C2×D4—21no—
143C2×D4—24no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C2×D412yes—
148C2×D4C2×D411yescharacteristic
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4C2×D414yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161M16—22no—
162M16—22no—
163M16—24no—
164C2×C8—22no—
165C2×C8—22no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4C2×D411yescharacteristic
181C4⋊C4C2×D411yescharacteristic
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C42—24no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—22no—
194C42—22no—
195C42—42no—
196C42C2×D411yescharacteristic
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C23—21no—
201C23—22no—
202C23C22×D411yescharacteristic
203Q8—21no—
204Q8—24no—
205Q8—48no—
206Q8—42no—
207Q8—42no—
208Q8—44no—
209Q8—82no—
210Q8—82no—
211Q8#12412yes—
212D4—24no—
213D4—21no—
214D4—24no—
215D4—48no—
216D4—44no—
217D4—42no—
218D4—42no—
219D4#12412yes—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—28no—
230C2×C4—48no—
231C2×C4—42no—
232C2×C4—48no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4#12011yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C2×C4C2×#4112yes—
247C2×C4H(5,𝔽2)12yes—
248C8—42no—
249C8—42no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22C2×#12011yescharacteristic
259C22#54612yes—
260C4—22no—
261C4—21no—
262C4—21no—
263C4—44no—
264C4—44no—
265C4—42no—
266C4—41no—
267C4—41no—
268C4—82no—
269C4#45412yes—
270C4#45612yes—
271C4C2×#12411yescharacteristic
272C4#53511yescharacteristic
273C4#54612yes—
274C2—22no—
275C2—42no—
276C2—82no—
277C2#243511yescharacteristic
278C2C2×#45411yescharacteristic
279C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3499C2
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. #3141C2semidirect
3. #3079C2semidirect
4. #2862C2semidirect
5. #2038C2semidirect
6. #1966C2semidirect
7. #1962C2semidirect
8. #1793C2semidirect
9. C4×#104C2semidirect

Maximal subgroups

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

subgroupindex
#34992
#31412
#30792
#28622
#20382
#19662
#19622
#17932
C4×#1042