Finite Group Databasebeta

← #61699

Subgroups — #61699

Order 256.

927 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic, complemented
2C2×#553C211yesmaximal, characteristic, complemented
3#3339C214yesmaximal, complemented
4#3337C214yesmaximal, complemented
5#3301C214yesmaximal, complemented
6#3297C214yesmaximal, complemented
7#3232C212yesmaximal, complemented
8#3188C212yesmaximal, complemented
9#3187C211yesmaximal, characteristic, complemented
10#3186C211yesmaximal, characteristic, complemented
11#3105C212yesmaximal, complemented
12#3091C212yesmaximal, complemented
13C2×#484C211yesmaximal, characteristic, complemented
14C2×#481C211yesmaximal, characteristic, complemented
15#2972C211yesmaximal, characteristic, complemented
16#553—22no—
17#553—22no—
18#553—22no—
19#553C2214yes—
20#553C2212yescomplemented
21#553C2214yescomplemented
22#552C2212yescomplemented
23#551—22no—
24#548C2212yescomplemented
25#546C2212yescomplemented
26#539—21no—
27#538—21no—
28#534—21no—
29#529C2212yes—
30C2×#126—21no—
31C2×#125C2211yescharacteristic, complemented
32#525C2212yes—
33C2×#123—21no—
34C2×C22⋊Q8C2211yescharacteristic
35C2×#121—21no—
36C2×#121C2211yescharacteristic
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic, complemented
39#518—21no—
40C2×C4×D4—21no—
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic, complemented
43C2×#117C2211yescharacteristic, complemented
44C22×C4⋊C4C2211yescharacteristic
45#499C2214yescomplemented
46#488C2214yescomplemented
47#485C2214yescomplemented
48#484C2214yescomplemented
49#484C2212yescomplemented
50#484C2212yescomplemented
51#483C2212yescomplemented
52#482C2214yescomplemented
53#482C2214yescomplemented
54#481C2212yes—
55#481C2212yes—
56#481C2212yescomplemented
57#481C2214yescomplemented
58#480C2212yes—
59#480C2212yescomplemented
60#479C2214yescomplemented
61#478C2214yescomplemented
62#478C2214yescomplemented
63#466C2214yescomplemented
64#466C2214yescomplemented
65#465C2214yescomplemented
66#465C2214yescomplemented
67#463C2214yescomplemented
68#450C2212yescomplemented
69#449C2214yescomplemented
70#448C2212yescomplemented
71#447C2212yes—
72#447C2212yescomplemented
73C4×SD16C2214yescomplemented
74C4×D8C2214yescomplemented
75C8×D4C2214yescomplemented
76C8×D4C2214yescomplemented
77#431C2212yescomplemented
78#427C2214yescomplemented
79C2×#107C2212yescomplemented
80C2×#106C2212yescomplemented
81#416C2214yescomplemented
82C2×#103C2212yescomplemented
83C2×#102C2212yescomplemented
84C2×C22⋊C8C2211yescharacteristic
85C2×C22⋊C8C2211yescharacteristic
86C2×C22⋊C8C2212yescomplemented
87C2×#41C2312yes—
88C22×Q8C2311yescharacteristic
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C2×SD16C2314yes—
100C2×D8C2314yes—
101C22×C8C2312yes—
102C22×C8C2312yes—
103C22×C8C2314yes—
104#126—24no—
105#126—24no—
106#126—22no—
107#126—22no—
108#126—22no—
109#125—22no—
110#125—22no—
111#125C2314yes—
112#125C2312yes—
113#124—22no—
114#124C2312yes—
115#123—22no—
116#123—22no—
117#123—22no—
118#123—22no—
119#123—24no—
120#123—24no—
121#123—22no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124C22⋊Q8—22no—
125C22⋊Q8C2312yes—
126C22⋊Q8C2312yes—
127C22⋊Q8C2312yes—
128#121—24no—
129#121—24no—
130#121—22no—
131#121—22no—
132#121—21no—
133#121—21no—
134#121—21no—
135#121—21no—
136#121—22no—
137#121—22no—
138#121C2312yes—
139#121C2312yes—
140#121C2312yes—
141#121C2312yes—
142#121C2314yes—
143#121C2312yes—
144#120—22no—
145#120—22no—
146C4×Q8C2312yes—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4D412yes—
156C4×D4D412yes—
157C4×D4C2314yes—
158C4×D4C2312yes—
159C4×D4C2312yes—
160C4×D4C2312yes—
161#117—22no—
162#117—22no—
163#117—22no—
164#117C2314yes—
165#117C2312yes—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4D411yescharacteristic
173C2×C4⋊C4C2311yescharacteristic
174C2×C4⋊C4C2311yescharacteristic
175C2×C4⋊C4C2312yes—
176C2×C4⋊C4C2312yes—
177C2×C4⋊C4C2311yescharacteristic
178C2×C4⋊C4C2311yescharacteristic
179C2×C4⋊C4C2311yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4D411yescharacteristic
188C2×C22⋊C4D411yescharacteristic
189C2×C42—21no—
190C2×C42C2311yescharacteristic
191#107C2314yes—
192#107C2314yes—
193#107C2314yes—
194#106C2314yes—
195#106C2314yes—
196#106C2314yes—
197C4⋊C8C2314yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#102C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208C22⋊C8C2312yes—
209C22⋊C8C2312yes—
210C22⋊C8C2314yes—
211C4×C8C2314yes—
212C24—21no—
213C24—21no—
214C24C2×D411yescharacteristic
215C24C2×D411yescharacteristic
216#41—44no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×Q8C2412yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—24no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—82no—
235C2×D4C2×D412yes—
236C2×D4C2×D412yes—
237C2×D4C2412yes—
238C2×D4C2412yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D411yescharacteristic
260C22×C4C2×D412yes—
261C22×C4C2411yescharacteristic
262C22×C4C2411yescharacteristic
263C22×C4C2411yescharacteristic
264C22×C4C2412yes—
265SD16—44no—
266D8—44no—
267C2×C8—42no—
268C2×C8—44no—
269C2×C8—42no—
270C2×C8C2414yes—
271C2×C8C2414yes—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4C2×D412yes—
288C4⋊C4C2×D412yes—
289C4⋊C4C2412yes—
290C4⋊C4C2412yes—
291C4⋊C4C2412yes—
292C4⋊C4C2412yes—
293C4⋊C4C2412yes—
294C22⋊C4—22no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—24no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—24no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4C2×D412yes—
310C22⋊C4C2×D412yes—
311C22⋊C4C2×D412yes—
312C22⋊C4C2×D412yes—
313C42—24no—
314C42—22no—
315C42—22no—
316C42C2×D412yes—
317C42C2412yes—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23#13514yes—
332C23C22×D412yes—
333C23C22×D411yescharacteristic
334C23C22×D411yescharacteristic
335C23C22×D411yescharacteristic
336Q8—24no—
337Q8—81no—
338D4—24no—
339D4—24no—
340D4—24no—
341D4—24no—
342D4—42no—
343D4—44no—
344D4—44no—
345D4—44no—
346D4—44no—
347D4—81no—
348D4—82no—
349D4—84no—
350D4—82no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4C22×D412yes—
380C2×C4C22×D412yes—
381C2×C4C22×D411yescharacteristic
382C2×C4C22×D412yes—
383C2×C4C22×D411yescharacteristic
384C2×C4C22×D411yescharacteristic
385C2×C4C2511yescharacteristic
386C8—24no—
387C8—24no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—84no—
406C22—162no—
407C22C2×#13512yes—
408C22C2×#13512yes—
409C22C2×#13512yes—
410C22C23×D411yescharacteristic
411C4—22no—
412C4—21no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4#58512yes—
424C2—22no—
425C2—22no—
426C2—22no—
427C2—42no—
428C2—82no—
429C2—82no—
430C2#348511yescharacteristic
431C2C22×#13511yescharacteristic
432C2#358511yescharacteristic
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. #3505C2semidirect
2. C2×#553C2semidirect
3. #3339C2semidirect
4. #3337C2semidirect
5. #3301C2semidirect
6. #3297C2semidirect
7. #3232C2semidirect
8. #3188C2semidirect
9. #3187C2semidirect
10. #3186C2semidirect
11. #3105C2semidirect
12. #3091C2semidirect
13. C2×#484C2semidirect
14. C2×#481C2semidirect
15. #2972C2semidirect
20. #553C22semidirect
21. #553C22semidirect
22. #552C22semidirect
24. #548C22semidirect
25. #546C22semidirect
31. C2×#125C22semidirect
38. C2×#121C22semidirect
42. C2×C4×D4C22semidirect
43. C2×#117C22semidirect
45. #499C22semidirect
46. #488C22semidirect
47. #485C22semidirect
48. #484C22semidirect
49. #484C22semidirect
50. #484C22semidirect
51. #483C22semidirect
52. #482C22semidirect
53. #482C22semidirect
56. #481C22semidirect
57. #481C22semidirect
59. #480C22semidirect
60. #479C22semidirect
61. #478C22semidirect
62. #478C22semidirect
63. #466C22semidirect
64. #466C22semidirect
65. #465C22semidirect
66. #465C22semidirect
67. #463C22semidirect
68. #450C22semidirect
69. #449C22semidirect
70. #448C22semidirect
72. #447C22semidirect
73. C4×SD16C22semidirect
74. C4×D8C22semidirect
75. C8×D4C22semidirect
76. C8×D4C22semidirect
77. #431C22semidirect
78. #427C22semidirect
79. C2×#107C22semidirect
80. C2×#106C22semidirect
81. #416C22semidirect
82. C2×#103C22semidirect
83. C2×#102C22semidirect
86. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5532
#33392
#33372
#33012
#32972
#32322
#31882
#31872
#31862
#31052
#30912
C2×#4842
C2×#4812
#29722