Finite Group Databasebeta

← #18801

Subgroups — #18801

Order 256.

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

normal subgroups only

This group has 468 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
1Q8×#41C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic
4C2×#462C211yesmaximal, characteristic, complemented
5#1649C212yesmaximal, complemented
6#1648C212yesmaximal, complemented
7#1645C212yesmaximal, complemented
8#1644C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1487C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic
12C22×Q16—21no—
13Q82C2211yescharacteristic
14#557—21no—
15#557C2211yescharacteristic
16#554—22no—
17#554—21no—
18#554—21no—
19#554C2211yescharacteristic
20D4×Q8—21no—
21D4×Q8C2211yescharacteristic
22#533—22no—
23#533—21no—
24#533—21no—
25#533C2211yescharacteristic
26C2×C4⋊Q8C2211yescharacteristic
27C4×#41—21no—
28C4×#41C2211yescharacteristic
29C2×C4×Q8—21no—
30C2×C4×Q8C2211yescharacteristic
31#500C2212yes—
32#487C2212yes—
33#484—22no—
34#477—21no—
35#477—22no—
36#476—21no—
37#475C2212yes—
38#474C2212yes—
39#462—21no—
40#462—21no—
41#462C2212yes—
42#462C2212yes—
43#431C2211yescharacteristic
44#428—21no—
45#423C2211yescharacteristic
46C2×C4⋊C8C2211yescharacteristic
47#417—21no—
48C2×#103—21no—
49#333C2212yes—
50#332C2212yes—
51Q8⋊C8C2212yes—
52Q8⋊C8C2212yes—
53Q8⋊C8C2212yes—
54D4⋊C8C2212yes—
55C2×#41D411yescharacteristic
56C22×Q8—21no—
57C22×Q8—21no—
58C22×Q8D411yescharacteristic
59C2×Q16—22no—
60C2×Q16—22no—
61C2×Q16—44no—
62C2×M16—21no—
63C22×C8—21no—
64C4⋊Q8—22no—
65C4⋊Q8—21no—
66C4⋊Q8—22no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8C2312yes—
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2311yescharacteristic
76#125—22no—
77#125—22no—
78#125—22no—
79#125—21no—
80#125—21no—
81#125—21no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—21no—
88C22⋊Q8—22no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—22no—
92C22⋊Q8D412yes—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8D411yescharacteristic
104C4×Q8D411yescharacteristic
105C4×Q8D411yescharacteristic
106C4×Q8D412yes—
107C4×Q8C2311yescharacteristic
108C4×Q8C2311yescharacteristic
109C4×Q8C2311yescharacteristic
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4D411yescharacteristic
117C4×D4C2311yescharacteristic
118#117—22no—
119#117—21no—
120#117—21no—
121#117D411yescharacteristic
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4D411yescharacteristic
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#107—24no—
134#106—22no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8C2312yes—
139C4⋊C8C2312yes—
140#103—22no—
141#103—24no—
142#103—22no—
143#103—22no—
144#102—22no—
145C22⋊C8—21no—
146C4×C8C2312yes—
147#41—22no—
148#41Q1614yescomplemented
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—41no—
161C2×Q8C2×D411yescharacteristic
162C2×Q8C2×D411yescharacteristic
163C2×Q8C2×D411yescharacteristic
164C2×Q8C2×D412yes—
165C2×D4—21no—
166C2×D4C2×D411yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178Q16—44no—
179Q16—44no—
180M16—42no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—42no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C4⋊C4C2×D411yescharacteristic
213C4⋊C4C2×D412yes—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C42—22no—
221C42—22no—
222C42—21no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42C2×D411yescharacteristic
231C42C2×D411yescharacteristic
232C42C2×D411yescharacteristic
233C42C2411yescharacteristic
234C23—21no—
235C23#12011yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—21no—
239Q8—21no—
240Q8—42no—
241Q8—42no—
242Q8—44no—
243Q8—41no—
244Q8—41no—
245Q8—42no—
246Q8—42no—
247Q8—82no—
248Q8—82no—
249Q8C2×Q1612yescomplemented
250Q8AGL(1,ℤ/8)12yes—
251D4—24no—
252D4—21no—
253D4C2×Q1612yescomplemented
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#12011yescharacteristic
277C2×C4#12011yescharacteristic
278C2×C4#12011yescharacteristic
279C2×C4C2×Q1612yes—
280C2×C4#13712yes—
281C2×C4C22×D411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D411yescharacteristic
284C8—42no—
285C8—42no—
286C8—42no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22#45112yes—
293C22C2×#12011yescharacteristic
294C4—22no—
295C4—21no—
296C4—22no—
297C4—21no—
298C4—21no—
299C4—41no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4#45112yes—
308C4C22×Q1611yescharacteristic
309C4C2×AGL(1,ℤ/8)11yescharacteristic
310C4C2×#13711yescharacteristic
311C4#57511yescharacteristic
312C2—22no—
313C2—42no—
314C2C2×#45111yescharacteristic
315C2#300611yescharacteristic
316C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3230C2semidirect
4. C2×#462C2semidirect
5. #1649C2semidirect
6. #1648C2semidirect
7. #1645C2semidirect
8. #1644C2semidirect
9. #1550C2semidirect
10. #1487C2semidirect
148. #41Q16semidirect
249. Q8C2×Q16semidirect
253. D4C2×Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×#412
#32302
#30792
C2×#4622
#16492
#16482
#16452
#16442
#15502
#14872
#14772