Finite Group Databasebeta

← #18931

Subgroups — #18931

Order 256.

493 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3543C211yesmaximal, characteristic
2#3252C211yesmaximal, characteristic, complemented
3#3233C211yesmaximal, characteristic, complemented
4#3208C211yesmaximal, characteristic, complemented
5#1642C211yesmaximal, characteristic, complemented
6#1641C211yesmaximal, characteristic, complemented
7#1640C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1631C211yesmaximal, characteristic, complemented
10#1631C211yesmaximal, characteristic, complemented
11#1630C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, 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#533C2211yescharacteristic, complemented
31C2×#127C2211yescharacteristic
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#499—21no—
35#492C2211yescharacteristic
36#492C2211yescharacteristic, complemented
37#488—21no—
38#486C2211yescharacteristic
39#486C2211yescharacteristic, complemented
40#485—21no—
41#482—21no—
42#479—21no—
43#477C2211yescharacteristic, complemented
44#477C2211yescharacteristic, complemented
45#476C2211yescharacteristic, complemented
46#476C2211yescharacteristic, complemented
47#475C2211yescharacteristic, complemented
48#474C2211yescharacteristic, complemented
49#470—21no—
50#468—21no—
51#464—21no—
52#462C2211yescharacteristic, complemented
53#461C2211yescharacteristic, complemented
54#460—22no—
55#450—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—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4D412yes—
79C2×Q16—21no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—21no—
83C2×M16—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127—21no—
88#127—21no—
89#127—21no—
90#127D412yes—
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#126—21no—
94#126—21no—
95#125—21no—
96#125D411yescharacteristic
97#125D411yescharacteristic
98#124—21no—
99#124—21no—
100#124D411yescharacteristic
101#124D411yescharacteristic
102#123—22no—
103#123—22no—
104#123—22no—
105#123—22no—
106#123—21no—
107#123—21no—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C22⋊Q8D411yescharacteristic
111C22⋊Q8D411yescharacteristic
112#121—22no—
113#121—22no—
114#121—22no—
115#121—22no—
116#121—22no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121—21no—
124#121D411yescharacteristic
125#121D411yescharacteristic
126#120—22no—
127#120—22no—
128#120—22no—
129#120—22no—
130#120—22no—
131#120—22no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8C2311yescharacteristic
135C4×Q8C2311yescharacteristic
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4C2311yescharacteristic
143C4×D4C2311yescharacteristic
144#117—21no—
145#117—21no—
146#117—21no—
147C2×C22⋊C4—22no—
148C2×C42C2311yescharacteristic
149#107—21no—
150#107—21no—
151#107—21no—
152#106—21no—
153#106—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#103—21no—
165#103—21no—
166#102—22no—
167#102—22no—
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—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D411yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—22no—
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—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C2×D4—82no—
210C2×D4—84no—
211C2×D4C2×D411yescharacteristic
212C2×D4C2×D411yescharacteristic
213C2×D4C2×D412yes—
214C2×D4C2×D412yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223Q16—81no—
224SD16—81no—
225SD16—82no—
226SD16—82no—
227M16—81no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4C2×D411yescharacteristic
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D411yescharacteristic
251C4⋊C4C2×D411yescharacteristic
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
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—21no—
267C22⋊C4—21no—
268C22⋊C4—41no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42C2×D411yescharacteristic
273C42C2×D411yescharacteristic
274C42C2×D411yescharacteristic
275C42C2411yescharacteristic
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—44no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23#12011yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—41no—
293Q8—41no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—82no—
311D4—82no—
312D4—82no—
313D4—82no—
314D4—82no—
315D4—82no—
316D4—82no—
317D4—82no—
318D4—82no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4#12011yescharacteristic
340C2×C4#12011yescharacteristic
341C2×C4#12011yescharacteristic
342C2×C4C22×D411yescharacteristic
343C2×C4C22×D411yescharacteristic
344C2×C4C22×D411yescharacteristic
345C8—41no—
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—41no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—42no—
356C22—41no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22—164no—
367C22—164no—
368C22C2×#12011yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4#57611yescharacteristic
382C4#57711yescharacteristic
383C4#57711yescharacteristic
384C4#57711yescharacteristic
385C2—41no—
386C2—82no—
387C2—81no—
388C2—81no—
389C2—82no—
390C2#300911yescharacteristic
391C2#301211yescharacteristic
392C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3543C2
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. #3252C2semidirect
3. #3233C2semidirect
4. #3208C2semidirect
5. #1642C2semidirect
6. #1641C2semidirect
7. #1640C2semidirect
8. #1639C2semidirect
9. #1631C2semidirect
10. #1631C2semidirect
11. #1630C2semidirect
12. #1629C2semidirect
13. #1545C2semidirect
14. #1495C2semidirect
15. #1482C2semidirect
30. #533C22semidirect
36. #492C22semidirect
39. #486C22semidirect
43. #477C22semidirect
44. #477C22semidirect
45. #476C22semidirect
46. #476C22semidirect
47. #475C22semidirect
48. #474C22semidirect
52. #462C22semidirect
53. #461C22semidirect
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
#32522
#32332
#32082
#16422
#16412
#16402
#16392
#16312
#16302
#16292
#15452
#14952
#14822