Finite Group Databasebeta

← #26853

Subgroups — #26853

Order 256.

361 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#3136C211yesmaximal, characteristic
2#3096C211yesmaximal, characteristic
3#3082C211yesmaximal, characteristic, complemented
4#2947C211yesmaximal, characteristic, complemented
5#2063C211yesmaximal, characteristic, complemented
6#2060C211yesmaximal, characteristic, complemented
7#2052C211yesmaximal, characteristic, complemented
8#2004C211yesmaximal, characteristic
9#1959C211yesmaximal, characteristic, complemented
10#1938C211yesmaximal, characteristic, complemented
11#1931C211yesmaximal, characteristic, complemented
12#1930C211yesmaximal, characteristic, complemented
13#1888C211yesmaximal, characteristic, complemented
14#1859C211yesmaximal, characteristic, complemented
15#1773C211yesmaximal, characteristic, complemented
16C2×#137C2211yescharacteristic
17#544C2211yescharacteristic
18#535C2211yescharacteristic
19C2×#124C2211yescharacteristic
20#520C2211yescharacteristic
21#519C2211yescharacteristic
22#490—21no—
23#489—21no—
24#488—21no—
25#486—21no—
26#482—21no—
27#482—21no—
28#481—21no—
29#480—21no—
30#479—21no—
31#479—21no—
32#477—21no—
33#475—21no—
34#441—21no—
35#440—21no—
36C4×Q16—21no—
37C4×SD16—21no—
38C2×#108C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#428C2211yescharacteristic
42#427C2211yescharacteristic
43#424C2211yescharacteristic
44C2×#104C2211yescharacteristic
45#419C2211yescharacteristic
46#418C2211yescharacteristic
47#418C2211yescharacteristic
48#417C2211yescharacteristic
49#416C2211yescharacteristic
50C2×#103C2211yescharacteristic
51C2×#103C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#102C2211yescharacteristic
54C2×#101C2211yescharacteristic
55C2×#100C2211yescharacteristic
56#408C2211yescharacteristic
57#405C2211yescharacteristic
58#405C2211yescharacteristic
59C2×#97C2211yescharacteristic
60#384C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4C2311yescharacteristic
70#137—22no—
71#137—22no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×M16C2311yescharacteristic
75C2×M16C2311yescharacteristic
76C2×M16C2311yescharacteristic
77C2×M16C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82#127—21no—
83#125—21no—
84#125—21no—
85#124—21no—
86#124—22no—
87#124—22no—
88C22⋊Q8—41no—
89#121—21no—
90#121—21no—
91#120—41no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×D4—21no—
98C4×D4—41no—
99#117—21no—
100#117—41no—
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103#117C2311yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—41no—
109C2×C42C2311yescharacteristic
110#108—22no—
111#107—21no—
112#107—21no—
113#107—21no—
114#107D411yescharacteristic
115#107D411yescharacteristic
116#106—21no—
117#106—21no—
118#106—21no—
119#106D411yescharacteristic
120#106D411yescharacteristic
121C4⋊C8—21no—
122C4⋊C8—21no—
123#104—22no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103D411yescharacteristic
132#103D411yescharacteristic
133#102—21no—
134#102—21no—
135#102—21no—
136#102—21no—
137#102—21no—
138#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102D411yescharacteristic
143#102D411yescharacteristic
144#101—22no—
145#100—22no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148#97—21no—
149#97—21no—
150#97—22no—
151C4×C8—21no—
152C4×C8—21no—
153C24—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8C2×D411yescharacteristic
165C2×Q8#4112yes—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4C2×D411yescharacteristic
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4C2411yescharacteristic
183Q16—24no—
184SD16—24no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—21no—
192M16#4112yes—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C2×C8C2×D411yescharacteristic
204C2×C8C2×D411yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—82no—
216C4⋊C4C2×D411yescharacteristic
217C4⋊C4C2×D411yescharacteristic
218C4⋊C4C2×D411yescharacteristic
219C4⋊C4C2×D411yescharacteristic
220C4⋊C4C2×D411yescharacteristic
221C4⋊C4C2×D411yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—44no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42#4112yes—
237C23—21no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—81no—
242C23—81no—
243C23H(5,𝔽2)11yescharacteristic
244Q8—22no—
245Q8—21no—
246Q8—21no—
247Q8—22no—
248Q8—41no—
249Q8—41no—
250Q8—161no—
251D4—21no—
252D4—21no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—161no—
260D4—161no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
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—81no—
279C2×C4—81no—
280C2×C4—161no—
281C2×C4C22×D411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C42−1+411yescharacteristic
287C8—22no—
288C8—21no—
289C8—21no—
290C8—21no—
291C8—21no—
292C8—41no—
293C8—41no—
294C8—41no—
295C22—21no—
296C22—21no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—161no—
305C22#54211yescharacteristic
306C22D4211yescharacteristic
307C22#55111yescharacteristic
308C4—42no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4#52511yescharacteristic
320C4#53211yescharacteristic
321C4#54711yescharacteristic
322C4#54811yescharacteristic
323C2—21no—
324C2—21no—
325C2—81no—
326C2—161no—
327C2#246111yescharacteristic
328C2#328811yescharacteristic
329C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#535C22
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
3. #3082C2semidirect
4. #2947C2semidirect
5. #2063C2semidirect
6. #2060C2semidirect
7. #2052C2semidirect
9. #1959C2semidirect
10. #1938C2semidirect
11. #1931C2semidirect
12. #1930C2semidirect
13. #1888C2semidirect
14. #1859C2semidirect
15. #1773C2semidirect

Maximal subgroups

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

subgroupindex
#31362
#30962
#30822
#29472
#20632
#20602
#20522
#20042
#19592
#19382
#19312
#19302
#18882
#18592
#17732