Finite Group Databasebeta

← #61737

Subgroups — #61737

Order 256.

1006 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic, complemented
3#3353C212yesmaximal, complemented
4#3343C214yesmaximal, complemented
5#3334C212yesmaximal, complemented
6#3330C212yesmaximal, complemented
7#3308C214yesmaximal, complemented
8#3298C212yesmaximal, complemented
9#3239C211yesmaximal, characteristic, complemented
10#3230C211yesmaximal, characteristic, complemented
11#3193C212yesmaximal, complemented
12#3188C212yesmaximal, complemented
13#3110C211yesmaximal, characteristic, complemented
14#3105C211yesmaximal, characteristic, complemented
15#3092C212yesmaximal, complemented
16C2×#484C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18C22×#41—21no—
19#556C2211yescharacteristic, complemented
20#554—21no—
21#552C2212yescomplemented
22#551—22no—
23D4×Q8C2211yescharacteristic, complemented
24#548C2212yescomplemented
25#547—21no—
26#547—22no—
27#547—21no—
28#547C2212yes—
29#547C2212yes—
30#547C2214yes—
31#547C2211yescharacteristic, complemented
32#547C2211yescharacteristic, complemented
33#546—21no—
34#542—21no—
35#539—22no—
36#533C2211yescharacteristic
37C2×C4⋊Q8C2211yescharacteristic
38#529C2211yescharacteristic
39#525C2212yes—
40C2×#123—22no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8C2212yes—
43C2×#121C2212yes—
44#520—21no—
45C2×C4×D4C2211yescharacteristic
46C2×C4×D4C2211yescharacteristic
47C22×C4⋊C4C2212yes—
48#501C2212yescomplemented
49#500C2212yescomplemented
50#489C2212yescomplemented
51#487C2212yescomplemented
52#484C2211yescharacteristic
53#484C2212yes—
54#484C2212yes—
55#484C2212yes—
56#484C2212yes—
57#484C2214yescomplemented
58#484C2214yescomplemented
59#483C2214yescomplemented
60#482C2214yescomplemented
61#481C2211yescharacteristic
62#481C2212yes—
63#480C2214yescomplemented
64#480C2214yescomplemented
65#478C2212yescomplemented
66#477C2214yescomplemented
67#475C2212yescomplemented
68#474C2212yescomplemented
69#474C2212yescomplemented
70#468C2214yescomplemented
71#468C2214yescomplemented
72#466C2214yescomplemented
73#465C2214yescomplemented
74#463C2212yescomplemented
75#461C2212yescomplemented
76#450C2211yescharacteristic
77#450C2212yes—
78#449C2214yescomplemented
79#448C2214yescomplemented
80#447C2211yescharacteristic
81#440C2214yescomplemented
82C4×SD16C2212yescomplemented
83C4×D8C2212yescomplemented
84#436C2212yescomplemented
85#435C2214yescomplemented
86C8×D4C2212yescomplemented
87#432C2211yescharacteristic
88#431C2211yescharacteristic
89#428C2214yescomplemented
90C2×#107C2214yescomplemented
91#417C2214yescomplemented
92C2×#103C2214yescomplemented
93#407C2212yes—
94C2×C22⋊C8C2212yes—
95C2×#41—21no—
96C2×#41—24no—
97C2×#41—22no—
98C2×#41—22no—
99C2×#41—48no—
100C2×#41C2312yes—
101C22×Q8—21no—
102C22×Q8C2312yes—
103C22×D4—22no—
104C22×D4D411yescharacteristic
105C22×D4C2311yescharacteristic
106C23×C4—22no—
107C23×C4—21no—
108C23×C4C2312yes—
109C2×SD16C2312yes—
110C2×SD16C2314yes—
111C2×D8C2312yes—
112C2×M16C2314yes—
113C22×C8C2314yes—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C2311yescharacteristic
118C4⋊Q8C2312yes—
119C4⋊Q8C2312yes—
120C4⋊Q8C2311yescharacteristic
121#126—22no—
122#126—22no—
123#126—21no—
124#125—22no—
125#125—21no—
126#125C2311yescharacteristic
127#124—21no—
128#124—22no—
129#124C2311yescharacteristic
130#123—22no—
131#123—22no—
132#123—21no—
133#123—24no—
134#123—24no—
135#123—24no—
136#123—24no—
137#123—22no—
138#123—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—21no—
142C22⋊Q8—22no—
143C22⋊Q8—24no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—21no—
147C22⋊Q8—22no—
148C22⋊Q8C2311yescharacteristic
149C22⋊Q8C2312yes—
150C22⋊Q8C2312yes—
151C22⋊Q8C2312yes—
152C22⋊Q8C2312yes—
153#121—22no—
154#121—21no—
155#121—22no—
156#121—22no—
157#121C2314yes—
158#121C2314yes—
159#121C2311yescharacteristic
160#121C2312yes—
161#120—22no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8C2311yescharacteristic
165C4×Q8C2312yes—
166C4×D4—21no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4D411yescharacteristic
174C4×D4D412yes—
175C4×D4D411yescharacteristic
176C4×D4C2312yes—
177C4×D4C2312yes—
178C4×D4C2311yescharacteristic
179C4×D4C2311yescharacteristic
180C4×D4C2311yescharacteristic
181C4×D4C2312yes—
182#117—21no—
183#117—22no—
184#117—21no—
185#117C2312yes—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4D411yescharacteristic
193C2×C4⋊C4C2311yescharacteristic
194C2×C4⋊C4C2312yes—
195C2×C4⋊C4C2314yes—
196C2×C4⋊C4C2314yes—
197C2×C4⋊C4C2312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4D412yes—
202C2×C42C2311yescharacteristic
203#107C2312yes—
204#107C2314yes—
205#107C2314yes—
206#107C2314yes—
207#107C2314yes—
208#106C2312yes—
209#106C2314yes—
210C4⋊C8C2312yes—
211C4⋊C8C2312yes—
212#103C2312yes—
213#103C2314yes—
214#103C2314yes—
215#103C2314yes—
216#103C2314yes—
217#102C2312yes—
218#102C2314yes—
219#102C2312yes—
220#102C2312yes—
221#102C2314yes—
222C22⋊C8C2311yescharacteristic
223C22⋊C8C2312yes—
224C22⋊C8C2314yes—
225C22⋊C8C2311yescharacteristic
226C22⋊C8C2314yes—
227#97C2312yes—
228C4×C8C2312yes—
229C24—21no—
230C24C2×D412yes—
231#41—44no—
232#41—416no—
233#41—48no—
234#41—44no—
235#41—48no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8C2411yescharacteristic
243C2×Q8C2412yes—
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—42no—
252C2×D4—48no—
253C2×D4C2×D412yes—
254C2×D4C2×D411yescharacteristic
255C2×D4C2×D411yescharacteristic
256C2×D4C2411yescharacteristic
257C2×D4C2412yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D411yescharacteristic
273C22×C4C2×D412yes—
274C22×C4C2×D412yes—
275C22×C4C2×D412yes—
276C22×C4C2411yescharacteristic
277C22×C4C2412yes—
278C22×C4C2412yes—
279SD16—44no—
280SD16—42no—
281D8—42no—
282M16—44no—
283C2×C8—44no—
284C2×C8C2412yes—
285C2×C8C2412yes—
286C2×C8C2414yes—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4C2×D411yescharacteristic
307C4⋊C4C2×D412yes—
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2411yescharacteristic
310C4⋊C4C2411yescharacteristic
311C4⋊C4C2412yes—
312C4⋊C4C2412yes—
313C4⋊C4C2414yes—
314C22⋊C4—21no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—22no—
328C22⋊C4—21no—
329C22⋊C4C2×D412yes—
330C22⋊C4C2×D412yes—
331C22⋊C4C2×D412yes—
332C22⋊C4C2×D412yes—
333C42—22no—
334C42—22no—
335C42—21no—
336C42C2×D411yescharacteristic
337C42C2×D411yescharacteristic
338C42C2411yescharacteristic
339C42C2411yescharacteristic
340C23—24no—
341C23—24no—
342C23—21no—
343C23—22no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23C22×D412yes—
349C23C22×D412yes—
350C23C22×D411yescharacteristic
351Q8—22no—
352Q8—24no—
353Q8—44no—
354Q8—44no—
355Q8—42no—
356Q8—82no—
357D4—24no—
358D4—22no—
359D4—22no—
360D4—22no—
361D4—22no—
362D4—42no—
363D4—48no—
364D4—48no—
365D4—44no—
366D4—42no—
367D4—82no—
368D4AGL(1,ℤ/8)14yes—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D412yes—
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C2×C4C22×D411yescharacteristic
399C2×C4C22×D412yes—
400C2×C4C22×D411yescharacteristic
401C2×C4C2511yescharacteristic
402C8—22no—
403C8—24no—
404C8—22no—
405C22—21no—
406C22—22no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—48no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—84no—
420C22C2×AGL(1,ℤ/8)14yes—
421C22#57812yes—
422C22C23×D411yescharacteristic
423C4—21no—
424C4—22no—
425C4—22no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—41no—
431C4—41no—
432C4—42no—
433C4—44no—
434C4—42no—
435C4—41no—
436C4C2×AGL(1,ℤ/8)12yes—
437C4#58512yes—
438C2—22no—
439C2—24no—
440C2—42no—
441C2—81no—
442C2—82no—
443C2#348511yescharacteristic
444C2C22×AGL(1,ℤ/8)11yescharacteristic
445C2C2×#57811yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3520C2semidirect
2. C2×#547C2semidirect
3. #3353C2semidirect
4. #3343C2semidirect
5. #3334C2semidirect
6. #3330C2semidirect
7. #3308C2semidirect
8. #3298C2semidirect
9. #3239C2semidirect
10. #3230C2semidirect
11. #3193C2semidirect
12. #3188C2semidirect
13. #3110C2semidirect
14. #3105C2semidirect
15. #3092C2semidirect
16. C2×#484C2semidirect
17. #2974C2semidirect
19. #556C22semidirect
21. #552C22semidirect
23. D4×Q8C22semidirect
24. #548C22semidirect
31. #547C22semidirect
32. #547C22semidirect
48. #501C22semidirect
49. #500C22semidirect
50. #489C22semidirect
51. #487C22semidirect
57. #484C22semidirect
58. #484C22semidirect
59. #483C22semidirect
60. #482C22semidirect
63. #480C22semidirect
64. #480C22semidirect
65. #478C22semidirect
66. #477C22semidirect
67. #475C22semidirect
68. #474C22semidirect
69. #474C22semidirect
70. #468C22semidirect
71. #468C22semidirect
72. #466C22semidirect
73. #465C22semidirect
74. #463C22semidirect
75. #461C22semidirect
78. #449C22semidirect
79. #448C22semidirect
81. #440C22semidirect
82. C4×SD16C22semidirect
83. C4×D8C22semidirect
84. #436C22semidirect
85. #435C22semidirect
86. C8×D4C22semidirect
89. #428C22semidirect
90. C2×#107C22semidirect
91. #417C22semidirect
92. C2×#103C22semidirect

Maximal subgroups

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

subgroupindex
#35202
C2×#5472
#33532
#33432
#33342
#33302
#33082
#32982
#32392
#32302
#31932
#31882
#31102
#31052
#30922
C2×#4842
#29742