Finite Group Databasebeta

← #31187

Subgroups — #31187

Order 256.

334 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3097C211yesmaximal, characteristic
2#3084C211yesmaximal, characteristic
3#3082C211yesmaximal, characteristic, complemented
4#3044C211yesmaximal, characteristic
5#2090C211yesmaximal, characteristic, complemented
6#2064C211yesmaximal, characteristic, complemented
7#2052C211yesmaximal, characteristic
8#2047C211yesmaximal, characteristic, complemented
9#2038C211yesmaximal, characteristic
10#2028C211yesmaximal, characteristic, complemented
11#1971C211yesmaximal, characteristic, complemented
12#1946C211yesmaximal, characteristic, complemented
13#1938C211yesmaximal, characteristic
14#1882C211yesmaximal, characteristic
15#1857C211yesmaximal, characteristic, complemented
16C2×#137C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18#544C2211yescharacteristic
19#536C2211yescharacteristic
20#520C2211yescharacteristic
21#519C2211yescharacteristic
22#485—21no—
23#485—21no—
24#484—21no—
25#483—21no—
26#479—21no—
27#479—21no—
28#479—21no—
29#478—21no—
30#477—21no—
31#476—21no—
32#475—21no—
33#475—21no—
34#464—21no—
35#464—21no—
36#462—21no—
37#462—21no—
38C2×#108C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#428C2211yescharacteristic
42#427C2211yescharacteristic
43C2×#106C2211yescharacteristic
44#424C2211yescharacteristic
45#424C2211yescharacteristic
46#424C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#419C2211yescharacteristic
49#419C2211yescharacteristic
50#419C2211yescharacteristic
51#417C2211yescharacteristic
52#416C2211yescharacteristic
53C2×#103C2211yescharacteristic
54C2×#103C2211yescharacteristic
55C2×#101C2211yescharacteristic
56C2×#101C2211yescharacteristic
57#408C2211yescharacteristic
58C2×#97C2211yescharacteristic
59#391C2211yescharacteristic
60#391C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70#137—22no—
71#137—41no—
72C2×Q16—21no—
73C2×Q16—22no—
74C2×Q16—21no—
75C2×Q16—21no—
76C2×Q16—41no—
77C2×SD16—21no—
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87#125—21no—
88#125—21no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×D4—21no—
100C4×D4—41no—
101#117—21no—
102#117—41no—
103#117C2311yescharacteristic
104#117C2311yescharacteristic
105#117C2311yescharacteristic
106C2×C4⋊C4—21no—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#108—41no—
111#107—21no—
112#107—21no—
113#107—21no—
114#107D411yescharacteristic
115#107D411yescharacteristic
116#106—21no—
117#106—21no—
118#106—21no—
119#106—21no—
120#106—21no—
121#106D411yescharacteristic
122#106D411yescharacteristic
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129#104—41no—
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#103—21no—
139#103—21no—
140#103—21no—
141#103D411yescharacteristic
142#103D411yescharacteristic
143#102—21no—
144#102D411yescharacteristic
145#102D411yescharacteristic
146#101—22no—
147#101—41no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150#97—41no—
151#95—41no—
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—42no—
182Q16—42no—
183Q16—42no—
184SD16—42no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—21no—
192M16—21no—
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—41no—
216C4⋊C4—41no—
217C4⋊C4C2×D411yescharacteristic
218C4⋊C4C2×D411yescharacteristic
219C4⋊C4C2×D411yescharacteristic
220C4⋊C4C2×D411yescharacteristic
221C4⋊C4C2×D411yescharacteristic
222C4⋊C4C2×D411yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—21no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—41no—
235C42—41no—
236C42—41no—
237C23—21no—
238C23H(5,𝔽2)11yescharacteristic
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—22no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—161no—
250D4—21no—
251D4—21no—
252D4—41no—
253D4—41no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—41no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—161no—
277C2×C4C22×D411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4H(5,𝔽2)11yescharacteristic
281C2×C42−1+411yescharacteristic
282C2×C42−1+411yescharacteristic
283C8—21no—
284C8—21no—
285C8—21no—
286C8—21no—
287C8—41no—
288C8—41no—
289C8—41no—
290C8—41no—
291C22—21no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22#53911yescharacteristic
297C22D4211yescharacteristic
298C22#56211yescharacteristic
299C4—41no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4#53611yescharacteristic
312C4#53711yescharacteristic
313C4#54611yescharacteristic
314C4#54811yescharacteristic
315C2—21no—
316C2—21no—
317C2—81no—
318C2#268111yescharacteristic
319C2#328911yescharacteristic
320C2#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. #2090C2semidirect
6. #2064C2semidirect
8. #2047C2semidirect
10. #2028C2semidirect
11. #1971C2semidirect
12. #1946C2semidirect
15. #1857C2semidirect

Maximal subgroups

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

subgroupindex
#30972
#30842
#30822
#30442
#20902
#20642
#20522
#20472
#20382
#20282
#19712
#19462
#19382
#18822
#18572