Finite Group Databasebeta

← #18807

Subgroups — #18807

Order 256.

461 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2#3231C211yesmaximal, characteristic, complemented
3#3110C211yesmaximal, characteristic, complemented
4#3107C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1652C211yesmaximal, characteristic, complemented
7#1650C211yesmaximal, characteristic, complemented
8#1647C211yesmaximal, characteristic, complemented
9#1646C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1644C211yesmaximal, characteristic, complemented
12#1643C211yesmaximal, characteristic, complemented
13#1554C211yesmaximal, characteristic, complemented
14#1488C211yesmaximal, characteristic, complemented
15#1478C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#553—21no—
20#552—21no—
21#552—21no—
22#551—22no—
23#547C2211yescharacteristic
24#546—21no—
25#533C2211yescharacteristic
26#533C2211yescharacteristic
27#529—22no—
28#529—21no—
29#525—21no—
30C4×#41—21no—
31C4×#41C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×#117—21no—
34#500C2211yescharacteristic, complemented
35#498C2211yescharacteristic, complemented
36#487C2211yescharacteristic, complemented
37#487C2211yescharacteristic, complemented
38#485—22no—
39#485—21no—
40#484—21no—
41#479—22no—
42#475C2211yescharacteristic, complemented
43#474C2211yescharacteristic
44#474C2211yescharacteristic
45#474C2211yescharacteristic, complemented
46#462C2211yescharacteristic, complemented
47#461C2211yescharacteristic
48#461C2211yescharacteristic
49#461C2211yescharacteristic, complemented
50#452—21no—
51#450—21no—
52#440—21no—
53#440—21no—
54C4×Q16—21no—
55C4×SD16—21no—
56#432C2211yescharacteristic
57#431C2211yescharacteristic
58#431C2211yescharacteristic
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#332C2211yescharacteristic, complemented
62#332C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41D411yescharacteristic
73C22×D4D411yescharacteristic
74C23×C4—21no—
75C2×Q16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#126—22no—
84#126—22no—
85#126—22no—
86#125—22no—
87#125—21no—
88#125—21no—
89#125—21no—
90#124—22no—
91#124—21no—
92#123—22no—
93#123—22no—
94#123—22no—
95#123—22no—
96C22⋊Q8—21no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8—21no—
100C22⋊Q8—21no—
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8D412yes—
104#121—22no—
105#121—21no—
106#121—21no—
107#121—21no—
108#120—22no—
109C4×Q8—21no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8D411yescharacteristic
114C4×Q8D412yes—
115C4×Q8C2311yescharacteristic
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4—21no—
122C4×D4—21no—
123C4×D4—21no—
124C4×D4D411yescharacteristic
125C4×D4D411yescharacteristic
126C4×D4D411yescharacteristic
127C4×D4C2311yescharacteristic
128C4×D4C2311yescharacteristic
129C4×D4C2311yescharacteristic
130#117—22no—
131#117—22no—
132#117—22no—
133#117—21no—
134#117—21no—
135#117—21no—
136#117—21no—
137#117—21no—
138#117—21no—
139#117D411yescharacteristic
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4D411yescharacteristic
143C2×C22⋊C4—22no—
144C2×C22⋊C4—21no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42C2311yescharacteristic
148#107—21no—
149#107—21no—
150#107—21no—
151#107—22no—
152#106—21no—
153#106—22no—
154C4⋊C8—21no—
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159#103—21no—
160#103—22no—
161#103—22no—
162#103—21no—
163#103—21no—
164#103—21no—
165#103—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172#97—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—21no—
177#41—41no—
178#41—41no—
179#41—42no—
180#41—42no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4C2×D411yescharacteristic
193C2×D4C2×D411yescharacteristic
194C2×D4C2×D411yescharacteristic
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210Q16—42no—
211SD16—42no—
212SD16—42no—
213SD16—42no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4C2×D411yescharacteristic
239C4⋊C4C2×D411yescharacteristic
240C4⋊C4C2×D411yescharacteristic
241C4⋊C4C2×D411yescharacteristic
242C4⋊C4C2×D412yes—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—21no—
250C22⋊C4—22no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—41no—
258C42—22no—
259C42—22no—
260C42—21no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—41no—
265C42C2×D411yescharacteristic
266C42C2×D411yescharacteristic
267C42C2×D411yescharacteristic
268C42C2411yescharacteristic
269C23—21no—
270C23—21no—
271C23—21no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23#12011yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—42no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—44no—
292D4—42no—
293D4—82no—
294D4#13512yes—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4#12011yescharacteristic
323C2×C4#12011yescharacteristic
324C2×C4#12011yescharacteristic
325C2×C4C22×D411yescharacteristic
326C2×C4C22×D411yescharacteristic
327C2×C4C22×D411yescharacteristic
328C8—41no—
329C8—41no—
330C8—41no—
331C8—41no—
332C8—41no—
333C8—41no—
334C22—21no—
335C22—22no—
336C22—21no—
337C22—21no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22C2×#12011yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4C2×#13511yescharacteristic
362C4#57511yescharacteristic
363C4#57511yescharacteristic
364C4#57511yescharacteristic
365C2—41no—
366C2—42no—
367C2—81no—
368C2—81no—
369C2#300211yescharacteristic
370C2#300511yescharacteristic
371C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3475C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3231C2semidirect
3. #3110C2semidirect
4. #3107C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1650C2semidirect
8. #1647C2semidirect
9. #1646C2semidirect
10. #1646C2semidirect
11. #1644C2semidirect
12. #1643C2semidirect
13. #1554C2semidirect
14. #1488C2semidirect
15. #1478C2semidirect
34. #500C22semidirect
35. #498C22semidirect
36. #487C22semidirect
37. #487C22semidirect
42. #475C22semidirect
45. #474C22semidirect
46. #462C22semidirect
49. #461C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #332C22semidirect
62. #332C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34752
#32312
#31102
#31072
#16542
#16522
#16502
#16472
#16462
#16442
#16432
#15542
#14882
#14782