Finite Group Databasebeta

← #24183

Subgroups — #24183

Order 256.

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

normal subgroups only

This group has 501 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#3507C211yesmaximal, characteristic
2#3078C211yesmaximal, characteristic, complemented
3#3037C211yesmaximal, characteristic
4#2327C211yesmaximal, characteristic, complemented
5#2038C212yesmaximal, complemented
6#2005C212yesmaximal, complemented
7#1995C211yesmaximal, characteristic, complemented
8#1963C211yesmaximal, characteristic
9#1882C212yesmaximal, complemented
10#1860C212yesmaximal, complemented
11#1801C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13#559—21no—
14#557—21no—
15#556—21no—
16#556—22no—
17#554—21no—
18#552—22no—
19#539—22no—
20#536C2212yes—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#529—22no—
25#520C2212yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28#479—22no—
29#477—21no—
30#475—21no—
31#464—22no—
32#462—21no—
33#460—21no—
34#428C2212yes—
35#423C2211yescharacteristic
36#423C2211yescharacteristic
37C2×#104C2212yes—
38C2×#104C2212yes—
39#419C2212yes—
40#417C2212yes—
41C2×#101C2212yes—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#387—22no—
45#387—22no—
46#382C2211yescharacteristic
47#376C2211yescharacteristic
48#343C2212yes—
49#339C2212yes—
50#339C2212yes—
51C2×#41C2312yes—
52C22×Q8C2311yescharacteristic
53C22×Q8C2311yescharacteristic
54#137—24no—
55#137—42no—
56C2×Q16—22no—
57C2×SD16—22no—
58C2×M16C2312yes—
59C2×M16C2312yes—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66#127D411yescharacteristic
67#126—24no—
68#126—24no—
69#125—22no—
70#125—24no—
71#125D412yes—
72#124—22no—
73#124—22no—
74#124D412yes—
75#123—24no—
76#123—24no—
77C22⋊Q8—24no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82#121—21no—
83#121—21no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—24no—
99C4×D4—21no—
100C4×D4—21no—
101#117—24no—
102#117—22no—
103#117C2312yes—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4D412yes—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C42—21no—
111C2×C42D412yes—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#107—22no—
116#106—22no—
117C4⋊C8—21no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121#104—24no—
122#104—42no—
123#103—22no—
124#103—22no—
125#103—22no—
126#102—22no—
127#101—42no—
128#95—21no—
129#95—21no—
130#95—22no—
131#95—22no—
132#95—22no—
133#41—22no—
134#41—22no—
135#41—22no—
136#41—22no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D412yes—
151C2×Q8#4112yes—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4C2×D412yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167Q16—44no—
168SD16—44no—
169M16—22no—
170M16—22no—
171M16—22no—
172M16—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—24no—
176C4⋊C4—28no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4C2×D412yes—
194C4⋊C4C2×D412yes—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C42—24no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42—41no—
210C42C2×D411yescharacteristic
211C42C2×D411yescharacteristic
212C42C2×D411yescharacteristic
213C42C2×D411yescharacteristic
214C42#4112yes—
215C23—22no—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—22no—
221Q8—42no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—81no—
229D4—22no—
230D4—22no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—84no—
259C2×C4#12011yescharacteristic
260C2×C4#12011yescharacteristic
261C2×C4#12011yescharacteristic
262C2×C4#12011yescharacteristic
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C8—42no—
270C8—42no—
271C22—21no—
272C22—21no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22C2×#12011yescharacteristic
277C22#52511yescharacteristic
278C22#53711yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—42no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4#54611yescharacteristic
293C4#54711yescharacteristic
294C4#54811yescharacteristic
295C4#55011yescharacteristic
296C2—21no—
297C2—21no—
298C2—82no—
299C2#240711yescharacteristic
300C2#302011yescharacteristic
301C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
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. #3078C2semidirect
4. #2327C2semidirect
5. #2038C2semidirect
6. #2005C2semidirect
7. #1995C2semidirect
9. #1882C2semidirect
10. #1860C2semidirect
11. #1801C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#30782
#30372
#23272
#20382
#20052
#19952
#19632
#18822
#18602
#18012