Finite Group Databasebeta

← #27676

Subgroups — #27676

Order 256.

352 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3093C211yesmaximal, characteristic
2#3092C211yesmaximal, characteristic
3#2930C211yesmaximal, characteristic
4C2×#428C211yesmaximal, characteristic, complemented
5#2090C211yesmaximal, characteristic, complemented
6#2089C211yesmaximal, characteristic, complemented
7#2079C211yesmaximal, characteristic, complemented
8#2078C211yesmaximal, characteristic, complemented
9#1952C211yesmaximal, characteristic, complemented
10#1951C211yesmaximal, characteristic, complemented
11#1822C211yesmaximal, characteristic, complemented
12#1821C211yesmaximal, characteristic, complemented
13#1816C211yesmaximal, characteristic, complemented
14#1815C211yesmaximal, characteristic, complemented
15#1811C211yesmaximal, characteristic, complemented
16C2×#131C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#525C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#485—21no—
22#485—21no—
23#484—21no—
24#483—21no—
25#482—21no—
26#482—21no—
27#481—21no—
28#480—21no—
29#436—21no—
30#436—21no—
31#435—21no—
32#435—21no—
33C2×#108C2211yescharacteristic
34C2×#108C2211yescharacteristic
35#428—21no—
36#428—21no—
37#428—21no—
38#428—21no—
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#428C2211yescharacteristic
42#428C2211yescharacteristic
43C2×#107C2211yescharacteristic
44C2×#107C2211yescharacteristic
45C2×#106C2211yescharacteristic
46C2×#106C2211yescharacteristic
47#424C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#408C2211yescharacteristic
53#407C2211yescharacteristic
54#407C2211yescharacteristic
55#407C2211yescharacteristic
56#405C2211yescharacteristic
57#343C2211yescharacteristic
58#343C2211yescharacteristic
59#343C2211yescharacteristic
60#343C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#343C2211yescharacteristic
65#341C2211yescharacteristic
66#341C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70#131—22no—
71#131—22no—
72C2×M16—21no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16D411yescharacteristic
76C2×M16D411yescharacteristic
77C2×M16D411yescharacteristic
78C2×M16D411yescharacteristic
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C2×M16C2311yescharacteristic
83C2×M16C2311yescharacteristic
84C2×M16C2311yescharacteristic
85C22×C8—21no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88#123—21no—
89C22⋊Q8D411yescharacteristic
90C22⋊Q8D411yescharacteristic
91#121D411yescharacteristic
92#121D411yescharacteristic
93#120—21no—
94C4×D4—21no—
95C4×D4—21no—
96#117—21no—
97#117—21no—
98#117—21no—
99#117—21no—
100#117C2311yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C42—21no—
112#108—22no—
113#108—22no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107—21no—
118#107—21no—
119#107—21no—
120#107—21no—
121#107—21no—
122#106—21no—
123#106—21no—
124#106—21no—
125#106—21no—
126#106—21no—
127#106—21no—
128#106—21no—
129#106—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#102—21no—
137#102—21no—
138#102—21no—
139#102—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C22⋊C8—21no—
144C22⋊C8—21no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148#97—21no—
149C4×C8—21no—
150C24#4111yescharacteristic
151#41—21no—
152#41—21no—
153#41—21no—
154#41—21no—
155C2×Q8C2×D411yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4C2×D411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—82no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4#4111yescharacteristic
177C22×C4C2411yescharacteristic
178M16—22no—
179M16—22no—
180M16—22no—
181M16—22no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—21no—
192M16—21no—
193M16—21no—
194M16—42no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—42no—
204C2×C8C2×D411yescharacteristic
205C2×C8C2×D411yescharacteristic
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C2×C8#4112yes—
209C2×C8#4112yes—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4C2×D411yescharacteristic
223C4⋊C4C2×D411yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C42—21no—
232C42—41no—
233C42—41no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#12111yescharacteristic
240C23#12111yescharacteristic
241C23C2×#4111yescharacteristic
242Q8—21no—
243Q8—21no—
244D4—21no—
245D4—21no—
246D4—41no—
247D4—41no—
248D4—81no—
249D4—81no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4#12111yescharacteristic
276C2×C4#12111yescharacteristic
277C2×C4C22×D411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C42−1+411yescharacteristic
282C2×C42−1+411yescharacteristic
283C8—21no—
284C8—21no—
285C8—41no—
286C8—41no—
287C8—41no—
288C8—41no—
289C8—41no—
290C8—41no—
291C8#12312yes—
292C8#12312yes—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22C2×#12111yescharacteristic
307C22#55211yescharacteristic
308C22#55211yescharacteristic
309C4—21no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4C2×#12311yescharacteristic
319C4#53611yescharacteristic
320C4#54711yescharacteristic
321C4#54711yescharacteristic
322C2—21no—
323C2—21no—
324C2—41no—
325C2—41no—
326C2—81no—
327C2#248611yescharacteristic
328C2#306811yescharacteristic
329C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#428C2semidirect
5. #2090C2semidirect
6. #2089C2semidirect
7. #2079C2semidirect
8. #2078C2semidirect
9. #1952C2semidirect
10. #1951C2semidirect
11. #1822C2semidirect
12. #1821C2semidirect
13. #1816C2semidirect
14. #1815C2semidirect
15. #1811C2semidirect

Maximal subgroups

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

subgroupindex
#30932
#30922
#29302
C2×#4282
#20902
#20892
#20792
#20782
#19522
#19512
#18222
#18212
#18162
#18152
#18112