Finite Group Databasebeta

← #25985

Subgroups — #25985

Order 256.

421 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3084C211yesmaximal, characteristic, complemented
2#3083C211yesmaximal, characteristic, complemented
3#2898C211yesmaximal, characteristic
4#2898C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2064C211yesmaximal, characteristic, complemented
7#2064C211yesmaximal, characteristic, complemented
8#2028C211yesmaximal, characteristic, complemented
9#1866C211yesmaximal, characteristic, complemented
10#1866C211yesmaximal, characteristic, complemented
11#1860C211yesmaximal, characteristic, complemented
12#1860C211yesmaximal, characteristic, complemented
13#1816C211yesmaximal, characteristic, complemented
14#1815C211yesmaximal, characteristic, complemented
15#1793C211yesmaximal, characteristic
16C2×2−1+4C2211yescharacteristic
17C2×#131C2211yescharacteristic
18C2×#131C2211yescharacteristic
19#544C2211yescharacteristic
20#520C2211yescharacteristic
21#520C2211yescharacteristic
22#479—21no—
23#479—21no—
24#478—21no—
25#478—21no—
26#477—21no—
27#476—21no—
28#475—21no—
29#474—21no—
30C2×#108C2211yescharacteristic
31C2×#108C2211yescharacteristic
32#428C2211yescharacteristic
33#428C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36#424C2211yescharacteristic
37#424C2211yescharacteristic
38C2×#104C2211yescharacteristic
39C2×#104C2211yescharacteristic
40#419—21no—
41#419—21no—
42#419C2211yescharacteristic
43#419C2211yescharacteristic
44#417—21no—
45#417—21no—
46#417C2211yescharacteristic
47#417C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#416—21no—
51#416—21no—
52#416C2211yescharacteristic
53#416C2211yescharacteristic
54C2×#103—21no—
55C2×#103—21no—
56C2×#103C2211yescharacteristic
57C2×#103C2211yescharacteristic
58C2×#101C2211yescharacteristic
59#391C2211yescharacteristic
60#382C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
672−1+4—22no—
682−1+4—22no—
692−1+4—41no—
702−1+4Q814yescomplemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41—41no—
74C2×#41C2311yescharacteristic
75C2×#41C2311yescharacteristic
76C22×Q8—21no—
77C22×Q8—21no—
78C22×Q8C2311yescharacteristic
79#131—22no—
80#131—22no—
81#131—41no—
82#131—41no—
83C2×M16—21no—
84C2×M16—21no—
85C2×M16C2311yescharacteristic
86C2×M16C2311yescharacteristic
87C2×M16C2311yescharacteristic
88C2×M16C2311yescharacteristic
89C22×C8—21no—
90C22×C8—21no—
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95#125—21no—
96#125—21no—
97C22⋊Q8—41no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—41no—
103C4×D4—41no—
104#117—21no—
105#117—21no—
106#117C2311yescharacteristic
107#117C2311yescharacteristic
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#108—41no—
115#108—41no—
116#107—21no—
117#107—21no—
118#107—21no—
119#107—21no—
120#106—21no—
121#106—21no—
122#106—21no—
123#106—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128#104—22no—
129#104—22no—
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#103—21no—
142#103D411yescharacteristic
143#103D411yescharacteristic
144#103D411yescharacteristic
145#103D411yescharacteristic
146#102—21no—
147#102—21no—
148#102—21no—
149#102—21no—
150#102D411yescharacteristic
151#102D411yescharacteristic
152#102D411yescharacteristic
153#102D411yescharacteristic
154#101—41no—
155#95—41no—
156#95—41no—
157#41—21no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—24no—
162#41—24no—
163#41—44no—
164#41—42no—
165#41—42no—
166#41—44no—
167#41C2×Q812yes—
168#41C2×Q812yes—
169#41#4112yes—
170#41#4112yes—
171C2×Q8—21no—
172C2×Q8—24no—
173C2×Q8—24no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8C2×D411yescharacteristic
183C2×Q8C2×D411yescharacteristic
184C2×Q8C2×Q812yes—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4C2×D411yescharacteristic
189C2×D4C2×D411yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4C2411yescharacteristic
202M16—21no—
203M16—21no—
204M16—21no—
205M16—21no—
206M16—21no—
207M16—21no—
208M16—21no—
209M16—21no—
210M16—42no—
211M16—42no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8C2×D411yescharacteristic
223C2×C8C2×D411yescharacteristic
224C2×C8C2×D411yescharacteristic
225C2×C8C2×D411yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4C2×D411yescharacteristic
238C4⋊C4C2×D411yescharacteristic
239C4⋊C4C2×D411yescharacteristic
240C4⋊C4C2×D411yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—21no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—41no—
250C42—41no—
251C23—21no—
252C23—21no—
253C23H(5,𝔽2)11yescharacteristic
254Q8—21no—
255Q8—21no—
256Q8—22no—
257Q8—22no—
258Q8—44no—
259Q8—41no—
260Q8—41no—
261Q8—161no—
262Q8C22⋊Q812yes—
263Q8C22⋊Q812yes—
264D4—22no—
265D4—22no—
266D4—22no—
267D4—22no—
268D4—21no—
269D4—21no—
270D4—44no—
271D4C22⋊Q812yes—
272D4C22⋊Q812yes—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—161no—
298C2×C4—161no—
299C2×C4C22×D411yescharacteristic
300C2×C4C22×D411yescharacteristic
301C2×C4C22×Q811yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C8—21no—
306C8—21no—
307C8—21no—
308C8—21no—
309C8—41no—
310C8—41no—
311C8—41no—
312C8—41no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22#54311yescharacteristic
322C22D4211yescharacteristic
323C22#55111yescharacteristic
324C4—22no—
325C4—22no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4C2×C22⋊Q811yescharacteristic
336C4C2×C22⋊Q811yescharacteristic
337C4#54611yescharacteristic
338C4#54611yescharacteristic
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2#243711yescharacteristic
344C2#328911yescharacteristic
345C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×2−1+4C22
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. #3084C2semidirect
2. #3083C2semidirect
5. #2097C2semidirect
6. #2064C2semidirect
7. #2064C2semidirect
8. #2028C2semidirect
9. #1866C2semidirect
10. #1866C2semidirect
11. #1860C2semidirect
12. #1860C2semidirect
13. #1816C2semidirect
14. #1815C2semidirect
70. 2−1+4Q8semidirect

Maximal subgroups

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

subgroupindex
#30842
#30832
#28982
#20972
#20642
#20282
#18662
#18602
#18162
#18152
#17932