Finite Group Databasebeta

← #18878

Subgroups — #18878

Order 256.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic, complemented
2#3228C211yesmaximal, characteristic, complemented
3#3212C211yesmaximal, characteristic, complemented
4#3100C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1652C211yesmaximal, characteristic, complemented
7#1649C211yesmaximal, characteristic, complemented
8#1647C211yesmaximal, characteristic, complemented
9#1628C211yesmaximal, characteristic, complemented
10#1627C211yesmaximal, characteristic, complemented
11#1625C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1544C211yesmaximal, characteristic, complemented
14#1482C211yesmaximal, characteristic, complemented
15#1478C211yesmaximal, 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#532C2211yescharacteristic
29#529C2211yescharacteristic
30#525—21no—
31C2×C22⋊Q8—21no—
32#520—21no—
33C2×C4×D4C2211yescharacteristic
34#500C2211yescharacteristic, complemented
35#498C2211yescharacteristic, complemented
36#495—21no—
37#488—21no—
38#486C2211yescharacteristic, complemented
39#486C2211yescharacteristic, complemented
40#482—22no—
41#480—21no—
42#478—22no—
43#474C2211yescharacteristic
44#474C2211yescharacteristic
45#462C2211yescharacteristic, complemented
46#461C2211yescharacteristic, complemented
47#460C2211yescharacteristic, complemented
48#459C2211yescharacteristic
49#459C2211yescharacteristic
50#459C2211yescharacteristic, complemented
51#452—21no—
52#449—21no—
53#447—21no—
54C4×D8—21no—
55C4×D8—21no—
56#431C2211yescharacteristic
57#431C2211yescharacteristic
58#431C2211yescharacteristic
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—
73C2×#41—21no—
74C22×Q8—21no—
75C22×D4D411yescharacteristic
76C23×C4—21no—
77C2×Q16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C4⋊Q8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127C2311yescharacteristic
88#126—22no—
89#126—22no—
90#126—21no—
91#125—22no—
92#125D411yescharacteristic
93#125D412yes—
94#124—22no—
95#124—21no—
96#124—21no—
97#124D411yescharacteristic
98#123—22no—
99#123—22no—
100#123—21no—
101#123—22no—
102#123—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C22⋊Q8—21no—
111C22⋊Q8—22no—
112C22⋊Q8—21no—
113C22⋊Q8D411yescharacteristic
114#121—22no—
115#121—21no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121D411yescharacteristic
120#120—22no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8C2311yescharacteristic
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4D411yescharacteristic
135C4×D4D411yescharacteristic
136C4×D4C2311yescharacteristic
137C4×D4C2311yescharacteristic
138C4×D4C2311yescharacteristic
139#117—22no—
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—21no—
151#107—21no—
152#107—22no—
153#106—22no—
154C4⋊C8—21no—
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159#103—21no—
160#103—21no—
161#102—21no—
162#102—21no—
163#102—21no—
164#102—21no—
165#102—21no—
166#102—21no—
167#102—22no—
168#102—22no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—21no—
177#41—42no—
178#41—42no—
179#41—81no—
180#41—81no—
181#41—81no—
182#41—81no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—81no—
188C2×Q8—81no—
189C2×Q8C2×D411yescharacteristic
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4C2×D411yescharacteristic
201C2×D4C2×D411yescharacteristic
202C2×D4C2×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218Q16—81no—
219SD16—81no—
220SD16—81no—
221D8—42no—
222D8—42no—
223D8—81no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—22no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4C2×D411yescharacteristic
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D412yes—
248C4⋊C4C2×D411yescharacteristic
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D412yes—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—21no—
257C22⋊C4—22no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—22no—
269C42—22no—
270C42—21no—
271C42C2×D411yescharacteristic
272C42C2×D411yescharacteristic
273C42C2×D411yescharacteristic
274C42C2411yescharacteristic
275C23—21no—
276C23—21no—
277C23—21no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—81no—
282C23#12011yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—82no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—44no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—82no—
306D4—82no—
307D4#13512yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4#12011yescharacteristic
335C2×C4#12011yescharacteristic
336C2×C4#12011yescharacteristic
337C2×C4C22×D411yescharacteristic
338C2×C4C22×D411yescharacteristic
339C2×C4C22×D411yescharacteristic
340C8—41no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—41no—
345C8—41no—
346C22—21no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C2×#12011yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4C2×#13511yescharacteristic
376C4#57511yescharacteristic
377C4#57611yescharacteristic
378C4#57611yescharacteristic
379C2—41no—
380C2—42no—
381C2—81no—
382C2—81no—
383C2—161no—
384C2#300211yescharacteristic
385C2#300811yescharacteristic
386C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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
1. #3520C2semidirect
2. #3228C2semidirect
3. #3212C2semidirect
4. #3100C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1649C2semidirect
8. #1647C2semidirect
9. #1628C2semidirect
10. #1627C2semidirect
11. #1625C2semidirect
12. #1625C2semidirect
13. #1544C2semidirect
14. #1482C2semidirect
15. #1478C2semidirect
34. #500C22semidirect
35. #498C22semidirect
38. #486C22semidirect
39. #486C22semidirect
45. #462C22semidirect
46. #461C22semidirect
47. #460C22semidirect
50. #459C22semidirect
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
#32282
#32122
#31002
#16542
#16522
#16492
#16472
#16282
#16272
#16252
#15442
#14822
#14782