Finite Group Databasebeta

← #26852

Subgroups — #26852

Order 256.

349 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3137C211yesmaximal, characteristic
2#3097C211yesmaximal, characteristic
3#3082C211yesmaximal, characteristic, complemented
4#2947C211yesmaximal, characteristic
5#2064C211yesmaximal, characteristic, complemented
6#2061C211yesmaximal, characteristic, complemented
7#2052C211yesmaximal, characteristic
8#2005C211yesmaximal, characteristic
9#1960C211yesmaximal, characteristic, complemented
10#1938C211yesmaximal, characteristic
11#1930C211yesmaximal, characteristic, complemented
12#1930C211yesmaximal, characteristic, complemented
13#1886C211yesmaximal, characteristic, complemented
14#1860C211yesmaximal, characteristic
15#1773C211yesmaximal, characteristic, complemented
16C2×#137C2211yescharacteristic
17#544C2211yescharacteristic
18#536C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#520C2211yescharacteristic
21#519C2211yescharacteristic
22#491—21no—
23#489—21no—
24#488—21no—
25#487—21no—
26#485—21no—
27#485—21no—
28#484—21no—
29#483—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#419C2211yescharacteristic
47#419C2211yescharacteristic
48#417C2211yescharacteristic
49#416C2211yescharacteristic
50C2×#103C2211yescharacteristic
51C2×#103C2211yescharacteristic
52C2×#103C2211yescharacteristic
53C2×#103C2211yescharacteristic
54C2×#101C2211yescharacteristic
55C2×#101C2211yescharacteristic
56#408C2211yescharacteristic
57#405C2211yescharacteristic
58#405C2211yescharacteristic
59C2×#97C2211yescharacteristic
60#382C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
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—
82C4⋊Q8—21no—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85#125—21no—
86#125—21no—
87#124—21no—
88#123—41no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—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×C42C2311yescharacteristic
109#108—22no—
110#107—21no—
111#107—21no—
112#107—21no—
113#107D411yescharacteristic
114#107D411yescharacteristic
115#106—21no—
116#106—21no—
117#106—21no—
118#106D411yescharacteristic
119#106D411yescharacteristic
120C4⋊C8—21no—
121C4⋊C8—21no—
122#104—22no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103—21no—
137#103—21no—
138#103D411yescharacteristic
139#103D411yescharacteristic
140#102—21no—
141#102D411yescharacteristic
142#102D411yescharacteristic
143#101—22no—
144#101—22no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147#97—21no—
148#97—21no—
149#97—22no—
150C4×C8—21no—
151C4×C8—21no—
152#95—41no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—41no—
169C2×Q8C2×D411yescharacteristic
170C2×Q8#4112yes—
171C2×D4—21no—
172C2×D4C2×D411yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4C2411yescharacteristic
181Q16—24no—
182SD16—24no—
183M16—21no—
184M16—21no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16#4112yes—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—44no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
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—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42#4112yes—
236C23—21no—
237C23H(5,𝔽2)11yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—21no—
241Q8—22no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—161no—
249D4—21no—
250D4—21no—
251D4—41no—
252D4—41no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—161no—
276C2×C4C22×D411yescharacteristic
277C2×C4C22×D411yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C42−1+411yescharacteristic
282C8—22no—
283C8—21no—
284C8—21no—
285C8—21no—
286C8—21no—
287C8—41no—
288C8—41no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22#54211yescharacteristic
296C22D4211yescharacteristic
297C22#55111yescharacteristic
298C4—42no—
299C4—41no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4#52511yescharacteristic
311C4#53211yescharacteristic
312C4#54711yescharacteristic
313C4#54811yescharacteristic
314C2—21no—
315C2—21no—
316C2—81no—
317C2#246111yescharacteristic
318C2#328911yescharacteristic
319C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
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
5. #2064C2semidirect
6. #2061C2semidirect
9. #1960C2semidirect
11. #1930C2semidirect
12. #1930C2semidirect
13. #1886C2semidirect
15. #1773C2semidirect

Maximal subgroups

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

subgroupindex
#31372
#30972
#30822
#29472
#20642
#20612
#20522
#20052
#19602
#19382
#19302
#18862
#18602
#17732