Finite Group Databasebeta

← #26196

Subgroups — #26196

Order 256.

371 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3097C211yesmaximal, characteristic, complemented
2#3046C211yesmaximal, characteristic
3#2951C211yesmaximal, characteristic
4#2913C211yesmaximal, characteristic, complemented
5#2095C211yesmaximal, characteristic, complemented
6#2089C211yesmaximal, characteristic, complemented
7#2078C211yesmaximal, characteristic, complemented
8#2038C211yesmaximal, characteristic, complemented
9#1930C211yesmaximal, characteristic, complemented
10#1922C211yesmaximal, characteristic, complemented
11#1908C211yesmaximal, characteristic, complemented
12#1882C211yesmaximal, characteristic, complemented
13#1876C211yesmaximal, characteristic
14#1859C211yesmaximal, characteristic, complemented
15#1859C211yesmaximal, characteristic, complemented
16C2×#137C2211yescharacteristic
17C2×#135C2211yescharacteristic
18C2×#131C2211yescharacteristic
19#536C2211yescharacteristic
20#520C2211yescharacteristic
21#520C2211yescharacteristic
22#520C2211yescharacteristic
23#485—21no—
24#485—21no—
25#484—21no—
26#483—21no—
27#464—21no—
28#463—21no—
29#462—21no—
30#461—21no—
31#440—21no—
32#440—21no—
33C4×Q16—21no—
34C4×D8—21no—
35#428—21no—
36#428—21no—
37#428C2211yescharacteristic
38#428C2211yescharacteristic
39#428C2211yescharacteristic
40#427—21no—
41#427C2211yescharacteristic
42C2×#107—21no—
43C2×#107C2211yescharacteristic
44C2×#107C2211yescharacteristic
45#424C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#419C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52C2×#103C2211yescharacteristic
53C2×#103C2211yescharacteristic
54C2×#101C2211yescharacteristic
55#408C2211yescharacteristic
56#405C2211yescharacteristic
57#391C2211yescharacteristic
58#384C2211yescharacteristic
59#343C2211yescharacteristic
60#343C2211yescharacteristic
61#343C2211yescharacteristic
62#341C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70#137—22no—
71#137—22no—
72#135—22no—
73#135—22no—
74C2×Q16—21no—
75C2×Q16D411yescharacteristic
76C2×SD16—21no—
77C2×SD16D411yescharacteristic
78C2×SD16D411yescharacteristic
79C2×D8D411yescharacteristic
80#131—22no—
81#131—22no—
82C2×M16—21no—
83C2×M16C2311yescharacteristic
84C2×M16C2311yescharacteristic
85C2×M16C2311yescharacteristic
86C2×M16C2311yescharacteristic
87C22×C8—21no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C4⋊Q8—21no—
91#124—21no—
92#123—41no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104#117—21no—
105#117—21no—
106#117—21no—
107#117C2311yescharacteristic
108#117C2311yescharacteristic
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—21no—
118#107—21no—
119#107—21no—
120#107—21no—
121#107—21no—
122#107—21no—
123#107—21no—
124#107D411yescharacteristic
125#107D411yescharacteristic
126#106—21no—
127#106—21no—
128#106—21no—
129#106D411yescharacteristic
130#106D411yescharacteristic
131C4⋊C8—21no—
132C4⋊C8—21no—
133#104—22no—
134#104—22no—
135#103—21no—
136#103—21no—
137#103—21no—
138#103—21no—
139#103—21no—
140#103—21no—
141#103—21no—
142#103—21no—
143#103—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#101—22no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150#97—21no—
151C4×C8—21no—
152#95—41no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41#4112yes—
160#41#4112yes—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8C2×D411yescharacteristic
168C2×Q8C2×D411yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×D411yescharacteristic
172C2×D4C2×D411yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4C2411yescharacteristic
184Q16—22no—
185Q16—42no—
186SD16—22no—
187SD16—22no—
188SD16—42no—
189D8—22no—
190M16—24no—
191M16—21no—
192M16—21no—
193M16—21no—
194M16—21no—
195M16—21no—
196M16—21no—
197M16—21no—
198M16#4112yes—
199C2×C8—24no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8C2×D411yescharacteristic
209C2×C8C2×D411yescharacteristic
210C2×C8C2×D411yescharacteristic
211C2×C8C2×D411yescharacteristic
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C4⋊C4C2×D411yescharacteristic
225C4⋊C4C2×D411yescharacteristic
226C4⋊C4C2×D411yescharacteristic
227C4⋊C4C2×D411yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C42—41no—
242C23—21no—
243C23—21no—
244C23H(5,𝔽2)11yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—21no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8#12312yes—
253D4—22no—
254D4—22no—
255D4—21no—
256D4—21no—
257D4—21no—
258D4—41no—
259D4—41no—
260D4#12312yes—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4—161no—
286C2×C4C22×D411yescharacteristic
287C2×C4C22×D411yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C42−1+411yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—22no—
296C8—21no—
297C8—41no—
298C8—41no—
299C8—41no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#53911yescharacteristic
309C22D4211yescharacteristic
310C22#55311yescharacteristic
311C4—22no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4C2×#12311yescharacteristic
323C4#52511yescharacteristic
324C4#54711yescharacteristic
325C4#54811yescharacteristic
326C2—21no—
327C2—21no—
328C2—42no—
329C2—81no—
330C2#244311yescharacteristic
331C2#328811yescharacteristic
332C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3097C2semidirect
4. #2913C2semidirect
5. #2095C2semidirect
6. #2089C2semidirect
7. #2078C2semidirect
8. #2038C2semidirect
9. #1930C2semidirect
10. #1922C2semidirect
11. #1908C2semidirect
12. #1882C2semidirect
14. #1859C2semidirect
15. #1859C2semidirect

Maximal subgroups

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

subgroupindex
#30972
#30462
#29512
#29132
#20952
#20892
#20782
#20382
#19302
#19222
#19082
#18822
#18762
#18592