Finite Group Databasebeta

← #18648

Subgroups — #18648

Order 256.

467 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic
2#3248C211yesmaximal, characteristic, complemented
3#3160C211yesmaximal, characteristic, complemented
4#3110C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1654C211yesmaximal, characteristic, complemented
7#1652C211yesmaximal, characteristic, complemented
8#1651C211yesmaximal, characteristic, complemented
9#1650C211yesmaximal, characteristic, complemented
10#1649C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1647C211yesmaximal, characteristic, complemented
13#1552C211yesmaximal, characteristic, complemented
14#1494C211yesmaximal, characteristic, complemented
15C2×D4⋊C8C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554—21no—
18#552—21no—
19#551—22no—
20D4×Q8C2211yescharacteristic
21#548—21no—
22#547C2211yescharacteristic
23#547C2211yescharacteristic
24#546—21no—
25#542—21no—
26#539—22no—
27#533C2211yescharacteristic
28#533C2211yescharacteristic
29#529C2211yescharacteristic
30#525—21no—
31C2×C22⋊Q8—21no—
32#520—21no—
33C2×C4×D4C2211yescharacteristic
34#500C2211yescharacteristic, complemented
35#500C2211yescharacteristic, complemented
36#499—22no—
37#498C2211yescharacteristic, complemented
38#498C2211yescharacteristic, complemented
39#484—21no—
40#479—21no—
41#475C2211yescharacteristic, complemented
42#474C2211yescharacteristic
43#474C2211yescharacteristic
44#474C2211yescharacteristic, complemented
45#467—21no—
46#465—21no—
47#464—21no—
48#462C2211yescharacteristic, complemented
49#461C2211yescharacteristic
50#461C2211yescharacteristic
51#461C2211yescharacteristic, complemented
52#450—21no—
53#440—21no—
54#440—21no—
55#432C2211yescharacteristic
56#427—22no—
57C2×C4⋊C8C2211yescharacteristic
58C2×C4×C8C2211yescharacteristic
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, 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×#41—21no—
73C22×Q8—21no—
74C22×D4D411yescharacteristic
75C23×C4—21no—
76C2×Q16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#126—22no—
88#126—22no—
89#126—21no—
90#125—22no—
91#125—21no—
92#125D411yescharacteristic
93#125D412yes—
94#124—22no—
95#124—21no—
96#124D411yescharacteristic
97#123—22no—
98#123—22no—
99#123—21no—
100#123—22no—
101#123—22no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C22⋊Q8—22no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8D411yescharacteristic
115#121—22no—
116#121—21no—
117#121—21no—
118#121D411yescharacteristic
119#120—22no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8C2311yescharacteristic
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4D411yescharacteristic
134C4×D4D411yescharacteristic
135C4×D4C2311yescharacteristic
136C4×D4C2311yescharacteristic
137C4×D4C2311yescharacteristic
138#117—22no—
139#117—21no—
140#117—21no—
141#117—21no—
142#117D412yes—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4D411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C42C2311yescharacteristic
150#107—22no—
151#107—22no—
152#107—21no—
153#107—21no—
154#107—21no—
155#106—22no—
156#106—22no—
157#106—21no—
158C4⋊C8—21no—
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163#103—21no—
164#103—21no—
165#103—21no—
166#103—21no—
167#103—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171C22⋊C8—21no—
172#97—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—21no—
177#41—42no—
178#41—42no—
179#41—81no—
180#41—81no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8C2×D411yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4C2×D411yescharacteristic
196C2×D4C2×D411yescharacteristic
197C2×D4C2×D411yescharacteristic
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212Q16—81no—
213SD16—42no—
214SD16—42no—
215SD16—81no—
216C2×C8—24no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—81no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4C2×D411yescharacteristic
244C4⋊C4C2×D411yescharacteristic
245C4⋊C4C2×D411yescharacteristic
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D412yes—
248C4⋊C4C2×D412yes—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—21no—
255C22⋊C4—22no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—41no—
265C42—22no—
266C42—22no—
267C42—21no—
268C42—41no—
269C42C2×D411yescharacteristic
270C42C2×D411yescharacteristic
271C42C2×D411yescharacteristic
272C42C2411yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—42no—
277C23—42no—
278C23—81no—
279C23#12011yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—82no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—42no—
297D4—44no—
298D4—42no—
299D4—82no—
300D4#13512yes—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—44no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#12011yescharacteristic
328C2×C4#12011yescharacteristic
329C2×C4#12011yescharacteristic
330C2×C4C22×D411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C22×D411yescharacteristic
333C8—22no—
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—41no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22#45612yes—
352C22C2×#12011yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—81no—
365C4—81no—
366C4C2×#13511yescharacteristic
367C4#57511yescharacteristic
368C4#57611yescharacteristic
369C4#57711yescharacteristic
370C2—22no—
371C2—42no—
372C2—81no—
373C2—81no—
374C2#300211yescharacteristic
375C2#301211yescharacteristic
376C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3520C2
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. #3248C2semidirect
3. #3160C2semidirect
4. #3110C2semidirect
5. #1654C2semidirect
6. #1654C2semidirect
7. #1652C2semidirect
8. #1651C2semidirect
9. #1650C2semidirect
10. #1649C2semidirect
11. #1647C2semidirect
12. #1647C2semidirect
13. #1552C2semidirect
14. #1494C2semidirect
15. C2×D4⋊C8C2semidirect
34. #500C22semidirect
35. #500C22semidirect
37. #498C22semidirect
38. #498C22semidirect
41. #475C22semidirect
44. #474C22semidirect
48. #462C22semidirect
51. #461C22semidirect
59. #332C22semidirect
60. #332C22semidirect
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
#35202
#32482
#31602
#31102
#16542
#16522
#16512
#16502
#16492
#16472
#15522
#14942
C2×D4⋊C82