Finite Group Databasebeta

← #27290

Subgroups — #27290

Order 256.

437 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3137C211yesmaximal, characteristic, complemented
2#3083C211yesmaximal, characteristic, complemented
3#3009C211yesmaximal, characteristic
4#2897C211yesmaximal, characteristic
5#2078C211yesmaximal, characteristic, complemented
6#2046C211yesmaximal, characteristic, complemented
7#2038C211yesmaximal, characteristic, complemented
8#2027C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1938C211yesmaximal, characteristic, complemented
11#1931C211yesmaximal, characteristic, complemented
12#1908C211yesmaximal, characteristic, complemented
13#1882C211yesmaximal, characteristic, complemented
14#1792C211yesmaximal, characteristic
15#1765C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×#131C2211yescharacteristic
20#544C2211yescharacteristic, complemented
21#536C2211yescharacteristic
22#520C2211yescharacteristic
23#491—21no—
24#489—21no—
25#488—21no—
26#487—21no—
27#479—21no—
28#478—21no—
29#477—21no—
30#474—21no—
31#452—21no—
32#452—21no—
33#450—21no—
34#450—21no—
35#428C2211yescharacteristic, complemented
36C2×#107C2211yescharacteristic, complemented
37#424C2211yescharacteristic, complemented
38C2×#104C2211yescharacteristic
39C2×#104C2211yescharacteristic
40#419C2211yescharacteristic, complemented
41#418—21no—
42#418C2211yescharacteristic
43#418C2211yescharacteristic
44#417—21no—
45#417C2211yescharacteristic
46#417C2211yescharacteristic
47#417C2211yescharacteristic
48#416—21no—
49#416C2211yescharacteristic
50#416C2211yescharacteristic
51C2×#103C2211yescharacteristic, complemented
52C2×#102—21no—
53C2×#102C2211yescharacteristic
54C2×#101C2211yescharacteristic, complemented
55C2×#100C2211yescharacteristic
56#408C2211yescharacteristic
57#405C2211yescharacteristic, complemented
58C2×#97C2211yescharacteristic, complemented
59#391C2211yescharacteristic, complemented
60#390C2211yescharacteristic
61#343C2211yescharacteristic, complemented
62#343C2211yescharacteristic, complemented
63#341C2211yescharacteristic, complemented
64#341C2211yescharacteristic, complemented
65#339C2211yescharacteristic, complemented
66#339C2211yescharacteristic, complemented
67H(5,𝔽2)—22no—
68H(5,𝔽2)—22no—
69H(5,𝔽2)—22no—
70H(5,𝔽2)—41no—
71C2×#41—21no—
72C2×#41—21no—
73C2×#41C2311yescharacteristic
74C2×#41C2311yescharacteristic
75C22×Q8C2311yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—41no—
79C22×D4C2311yescharacteristic
80#137—22no—
81#137—22no—
82C2×Q16—21no—
83C2×SD16—21no—
84C2×SD16—22no—
85C2×SD16—21no—
86C2×SD16—21no—
87C2×SD16—41no—
88#131—22no—
89#131—41no—
90C2×M16—21no—
91C2×M16C2311yescharacteristic
92C2×M16C2311yescharacteristic
93C2×M16C2311yescharacteristic
94C2×M16C2311yescharacteristic
95C22×C8—21no—
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C4⋊Q8—21no—
99C4⋊Q8—21no—
100C4⋊Q8—21no—
101#125—21no—
102#125—21no—
103#124—21no—
104#123—41no—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107C22⋊Q8—41no—
108C4×Q8—21no—
109C4×D4—21no—
110C4×D4—41no—
111#117—21no—
112#117C2311yescharacteristic
113#117C2311yescharacteristic
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—41no—
118C2×C42C2311yescharacteristic
119#107—21no—
120#107—21no—
121#107—21no—
122#106—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125#104—22no—
126#104—41no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103—21no—
133#103—21no—
134#103D411yescharacteristic
135#103D411yescharacteristic
136#103D411yescharacteristic
137#103D411yescharacteristic
138#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#102D411yescharacteristic
148#102D411yescharacteristic
149#102D411yescharacteristic
150#102D411yescharacteristic
151#101—41no—
152#100—22no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155#97—21no—
156#97—41no—
157C4×C8—21no—
158#95—41no—
159C24—21no—
160C24—41no—
161#41—21no—
162#41—21no—
163#41—24no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—42no—
169#41—42no—
170#41—42no—
171#41#4112yes—
172#41#4112yes—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—41no—
180C2×Q8C2×D411yescharacteristic
181C2×Q8C2×D411yescharacteristic
182C2×D4—21no—
183C2×D4—24no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—42no—
197C2×D4C2×D411yescharacteristic
198C2×D4C2×D411yescharacteristic
199C2×D4#4112yes—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4C2411yescharacteristic
209Q16—42no—
210SD16—42no—
211SD16—42no—
212SD16—42no—
213M16—21no—
214M16—21no—
215M16—21no—
216M16—21no—
217M16—21no—
218M16—21no—
219M16—21no—
220M16—21no—
221M16—42no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—42no—
231C2×C8C2×D411yescharacteristic
232C2×C8C2×D411yescharacteristic
233C2×C8C2×D411yescharacteristic
234C2×C8C2×D411yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4C2×D411yescharacteristic
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D411yescharacteristic
248C4⋊C4C2×D411yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—82no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—44no—
264C23—41no—
265C23—41no—
266C23—81no—
267C23—81no—
268C23—161no—
269C23—161no—
270C23H(5,𝔽2)11yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—41no—
278Q8—161no—
279Q8#12412yes—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—22no—
284D4—21no—
285D4—22no—
286D4—21no—
287D4—44no—
288D4—44no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4#12412yes—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4C22×D411yescharacteristic
317C2×C4C22×D411yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C8—21no—
323C8—21no—
324C8—21no—
325C8—21no—
326C8—41no—
327C8—41no—
328C8—41no—
329C8—41no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—82no—
340C22—81no—
341C22—81no—
342C22—161no—
343C22—164no—
344C22#53811yescharacteristic
345C22D4211yescharacteristic
346C22#55111yescharacteristic
347C4—22no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4#52511yescharacteristic
358C4C2×#12411yescharacteristic
359C4#54611yescharacteristic
360C4#54611yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—81no—
365C2—82no—
366C2#247211yescharacteristic
367C2#328811yescharacteristic
368C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×H(5,𝔽2)C22
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. #3137C2semidirect
2. #3083C2semidirect
5. #2078C2semidirect
6. #2046C2semidirect
7. #2038C2semidirect
8. #2027C2semidirect
9. #1973C2semidirect
10. #1938C2semidirect
11. #1931C2semidirect
12. #1908C2semidirect
13. #1882C2semidirect
15. #1765C2semidirect
20. #544C22semidirect
35. #428C22semidirect
36. C2×#107C22semidirect
37. #424C22semidirect
40. #419C22semidirect
51. C2×#103C22semidirect
54. C2×#101C22semidirect
57. #405C22semidirect
58. C2×#97C22semidirect
59. #391C22semidirect
61. #343C22semidirect
62. #343C22semidirect
63. #341C22semidirect
64. #341C22semidirect
65. #339C22semidirect
66. #339C22semidirect

Maximal subgroups

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

subgroupindex
#31372
#30832
#30092
#28972
#20782
#20462
#20382
#20272
#19732
#19382
#19312
#19082
#18822
#17922
#17652