Finite Group Databasebeta

← #25833

Subgroups — #25833

Order 256.

467 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 143 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#3504C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic
4#2660C211yesmaximal, characteristic, complemented
5#2061C212yesmaximal, complemented
6#2060C212yesmaximal, complemented
7#1998C211yesmaximal, characteristic, complemented
8#1995C211yesmaximal, characteristic, complemented
9#1860C212yesmaximal
10#1859C212yesmaximal
11#1759C211yesmaximal, characteristic
12#557—21no—
13#557—21no—
14#554—21no—
15#554—21no—
16#554—22no—
17#554—22no—
18#544—21no—
19#544—21no—
20#544C2211yescharacteristic
21#544C2211yescharacteristic
22#533—21no—
23#533C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×#125—21no—
26C2×#125C2211yescharacteristic
27#520C2211yescharacteristic
28#520C2211yescharacteristic
29C4×#41C2211yescharacteristic
30#501—21no—
31#501—21no—
32#500—21no—
33#498—21no—
34#477—21no—
35#476—21no—
36#475—21no—
37#474—21no—
38#428C2212yes—
39#428C2212yes—
40#428C2212yes—
41#423C2211yescharacteristic
42C2×#104C2212yes—
43#417C2212yes—
44C4×M16C2211yescharacteristic
45#398—22no—
46#395—22no—
47#384C2212yes—
48#382C2212yes—
49#376C2211yescharacteristic
50#343C2212yes—
51#343C2212yes—
52#339C2212yes—
53#339C2212yes—
54#339C2212yes—
55C2×#41C2311yescharacteristic
56C22×Q8—21no—
57C2×M16C2312yes—
58C2×M16C2312yes—
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
70C4⋊Q8D411yescharacteristic
71#125—22no—
72#125—22no—
73#125—22no—
74#125—22no—
75#125—21no—
76#125—21no—
77#125—21no—
78#125—21no—
79#125—21no—
80#125—21no—
81#125—21no—
82#125—21no—
83#125D411yescharacteristic
84#125D411yescharacteristic
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8D411yescharacteristic
102C4×Q8D411yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4D411yescharacteristic
110C4×D4D411yescharacteristic
111#117—21no—
112#117—21no—
113#117—21no—
114#117C2311yescharacteristic
115#117C2311yescharacteristic
116#117C2311yescharacteristic
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4D412yes—
124C2×C4⋊C4D412yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C42—21no—
129C2×C42C2312yes—
130C2×C42C2311yescharacteristic
131#107—22no—
132#107—22no—
133#107—22no—
134#106—22no—
135#106—22no—
136#106—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139#104Q818yescomplemented
140#103—22no—
141#102—22no—
142#97—21no—
143C4×C8—21no—
144#95—21no—
145#95—21no—
146#95—24no—
147#41—22no—
148#41C2×Q814yes—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—82no—
156C2×Q8C2×D411yescharacteristic
157C2×D4—21no—
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4C2×D412yes—
167C22×C4C2×D412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171M16—22no—
172M16—24no—
173M16C2×Q814yes—
174C2×C8—22no—
175C2×C8—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—84no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—22no—
226C42C2×D411yescharacteristic
227C42C2×D411yescharacteristic
228C42C2×D411yescharacteristic
229C42C2×D411yescharacteristic
230C42C2×Q814yes—
231C23—21no—
232C23C22×D411yescharacteristic
233Q8—21no—
234Q8—81no—
235Q8—81no—
236Q8—82no—
237Q8—82no—
238Q8—82no—
239Q8—82no—
240Q8C4⋊Q812yes—
241D4—24no—
242D4—21no—
243D4C4⋊Q812yes—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4#12011yescharacteristic
270C2×C4#12011yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C2×C4C22×Q812yes—
274C2×C42−1+412yes—
275C8—24no—
276C8—42no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22C2×#12011yescharacteristic
283C22D4×Q812yes—
284C4—22no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—44no—
290C4—42no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—82no—
299C4C2×C4⋊Q811yescharacteristic
300C4#53611yescharacteristic
301C4D4×Q812yes—
302C2—22no—
303C2—42no—
304C2#243411yescharacteristic
305C2#302111yescharacteristic
306C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
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. #3165C2semidirect
4. #2660C2semidirect
5. #2061C2semidirect
6. #2060C2semidirect
7. #1998C2semidirect
8. #1995C2semidirect
139. #104Q8semidirect

Maximal subgroups

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

subgroupindex
#35042
#31652
#30792
#26602
#20612
#20602
#19982
#19952
#18602
#18592
#17592