Finite Group Databasebeta

← #18930

Subgroups — #18930

Order 256.

510 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3543C211yesmaximal, characteristic, complemented
2#3251C211yesmaximal, characteristic, complemented
3#3212C211yesmaximal, characteristic, complemented
4#3205C211yesmaximal, characteristic, complemented
5#1642C211yesmaximal, characteristic, complemented
6#1641C211yesmaximal, characteristic, complemented
7#1640C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1621C211yesmaximal, characteristic, complemented
10#1621C211yesmaximal, characteristic, complemented
11#1620C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1545C211yesmaximal, characteristic, complemented
14#1495C211yesmaximal, characteristic, complemented
15#1482C211yesmaximal, characteristic, complemented
16#560—22no—
17#559C2211yescharacteristic
18#550C2211yescharacteristic
19#550C2211yescharacteristic
20#546—21no—
21#546—21no—
22D42—21no—
23D42—21no—
24D42C2211yescharacteristic
25#540—21no—
26#540—21no—
27#535—21no—
28#535—21no—
29#534—22no—
30#532C2211yescharacteristic, complemented
31C2×#127C2211yescharacteristic
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#495—21no—
35#493C2211yescharacteristic
36#493C2211yescharacteristic, complemented
37#488—21no—
38#486C2211yescharacteristic
39#486C2211yescharacteristic, complemented
40#478—21no—
41#475C2211yescharacteristic, complemented
42#474C2211yescharacteristic, complemented
43#469—21no—
44#468—21no—
45#463—21no—
46#462C2211yescharacteristic, complemented
47#461C2211yescharacteristic, complemented
48#460C2211yescharacteristic, complemented
49#460C2211yescharacteristic, complemented
50#459—22no—
51#459C2211yescharacteristic, complemented
52#459C2211yescharacteristic, complemented
53#452—21no—
54#449—21no—
55#447—22no—
56#431C2211yescharacteristic, complemented
57#431C2211yescharacteristic, complemented
58#423C2211yescharacteristic, complemented
59#331C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62#331C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65Q8⋊C8C2211yescharacteristic, complemented
66Q8⋊C8C2211yescharacteristic, complemented
67D4⋊C8C2211yescharacteristic, complemented
68D4⋊C8C2211yescharacteristic, complemented
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41—21no—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4D412yes—
80C2×Q16—21no—
81C2×SD16—21no—
82C2×SD16—21no—
83C2×SD16—21no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8—21no—
87C2×D8—21no—
88C2×M16—21no—
89C4⋊Q8C2311yescharacteristic
90#127—21no—
91#127—21no—
92#127—21no—
93#127—21no—
94#127D412yes—
95#127C2311yescharacteristic
96#127C2311yescharacteristic
97#127C2311yescharacteristic
98#126—21no—
99#126—21no—
100#125D411yescharacteristic
101#125D411yescharacteristic
102#124—21no—
103#124—21no—
104#124—21no—
105#124D411yescharacteristic
106#124D411yescharacteristic
107#123—22no—
108#123—22no—
109#123—22no—
110#123—22no—
111#123—21no—
112#123—21no—
113C22⋊Q8D411yescharacteristic
114C22⋊Q8D411yescharacteristic
115#121—22no—
116#121—22no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—21no—
127#121—21no—
128#121—21no—
129#121D411yescharacteristic
130#121D411yescharacteristic
131#120—22no—
132#120—22no—
133#120—22no—
134#120—22no—
135#120—22no—
136#120—22no—
137C4×Q8C2311yescharacteristic
138C4×Q8C2311yescharacteristic
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2311yescharacteristic
148C4×D4C2311yescharacteristic
149#117—21no—
150#117—21no—
151C2×C22⋊C4—22no—
152C2×C42C2311yescharacteristic
153#107—21no—
154#106—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161#103—21no—
162#103—21no—
163#103—21no—
164#102—22no—
165#102—22no—
166#102—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171C22⋊C8—21no—
172C22⋊C8—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—22no—
177C24—22no—
178#41—81no—
179#41—81no—
180#41—81no—
181#41—81no—
182#41—81no—
183#41—81no—
184C2×Q8—21no—
185C2×Q8C2×D411yescharacteristic
186C2×Q8C2×D411yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4—81no—
212C2×D4—81no—
213C2×D4—81no—
214C2×D4—82no—
215C2×D4—84no—
216C2×D4C2×D411yescharacteristic
217C2×D4C2×D411yescharacteristic
218C2×D4C2×D412yes—
219C2×D4C2×D412yes—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229Q16—81no—
230SD16—81no—
231SD16—81no—
232SD16—81no—
233D8—81no—
234D8—81no—
235D8—82no—
236D8—82no—
237M16—81no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4C2×D411yescharacteristic
255C4⋊C4C2×D411yescharacteristic
256C4⋊C4C2×D411yescharacteristic
257C4⋊C4C2×D411yescharacteristic
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42C2×D411yescharacteristic
279C42C2×D411yescharacteristic
280C42C2×D411yescharacteristic
281C42C2411yescharacteristic
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23#12011yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—41no—
299Q8—41no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—42no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—82no—
322D4—82no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#12011yescharacteristic
352C2×C4#12011yescharacteristic
353C2×C4#12011yescharacteristic
354C2×C4C22×D411yescharacteristic
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D411yescharacteristic
357C8—41no—
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—41no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—42no—
368C22—41no—
369C22—42no—
370C22—41no—
371C22—42no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—164no—
382C22—164no—
383C22C2×#12011yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4#57611yescharacteristic
396C4#57611yescharacteristic
397C4#57611yescharacteristic
398C4#57711yescharacteristic
399C2—41no—
400C2—82no—
401C2—81no—
402C2—81no—
403C2—82no—
404C2—161no—
405C2#300811yescharacteristic
406C2#301211yescharacteristic
407C2#301911yescharacteristic
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. #3543C2semidirect
2. #3251C2semidirect
3. #3212C2semidirect
4. #3205C2semidirect
5. #1642C2semidirect
6. #1641C2semidirect
7. #1640C2semidirect
8. #1639C2semidirect
9. #1621C2semidirect
10. #1621C2semidirect
11. #1620C2semidirect
12. #1619C2semidirect
13. #1545C2semidirect
14. #1495C2semidirect
15. #1482C2semidirect
30. #532C22semidirect
36. #493C22semidirect
39. #486C22semidirect
41. #475C22semidirect
42. #474C22semidirect
46. #462C22semidirect
47. #461C22semidirect
48. #460C22semidirect
49. #460C22semidirect
51. #459C22semidirect
52. #459C22semidirect
56. #431C22semidirect
57. #431C22semidirect
58. #423C22semidirect
61. #331C22semidirect
62. #331C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
65. Q8⋊C8C22semidirect
66. Q8⋊C8C22semidirect
67. D4⋊C8C22semidirect
68. D4⋊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
#35432
#32512
#32122
#32052
#16422
#16412
#16402
#16392
#16212
#16202
#16192
#15452
#14952
#14822