Finite Group Databasebeta

← #62590

Subgroups — #62590

Order 256.

895 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3515C211yesmaximal, characteristic, complemented
2#3508C211yesmaximal, characteristic, complemented
3#3365C212yesmaximal, complemented
4#3361C212yesmaximal, complemented
5#3333C212yesmaximal, complemented
6#3332C212yesmaximal, complemented
7#3323C214yesmaximal, complemented
8#3322C214yesmaximal, complemented
9#3237C211yesmaximal, characteristic, complemented
10#3234C212yesmaximal, complemented
11#3219C212yesmaximal, complemented
12#3217C211yesmaximal, characteristic, complemented
13#3172C211yesmaximal, characteristic, complemented
14#3171C212yesmaximal, complemented
15#3142C212yesmaximal, complemented
16#3138C211yesmaximal, characteristic, complemented
17#2971C211yesmaximal, characteristic, complemented
18#559—21no—
19#559C2212yes—
20#557C2211yescharacteristic, complemented
21#554C2211yescharacteristic, complemented
22#554C2211yescharacteristic, complemented
23#553C2212yes—
24#552—22no—
25#552—21no—
26#552C2212yes—
27#550—21no—
28#550C2212yes—
29#547—21no—
30#547C2211yescharacteristic, complemented
31#546—22no—
32#546C2214yes—
33#541—21no—
34#540—22no—
35#539—22no—
36#535—21no—
37#533C2211yescharacteristic
38#533C2211yescharacteristic
39#532C2212yes—
40#532C2212yes—
41C2×#127C2211yescharacteristic
42#529C2211yescharacteristic
43#529C2212yescomplemented
44C2×#125C2212yescomplemented
45#520—21no—
46#518—21no—
47C4×#41C2211yescharacteristic
48#501C2212yescomplemented
49#501C2211yescharacteristic, complemented
50#497C2211yescharacteristic, complemented
51#496C2212yes—
52#491C2211yescharacteristic, complemented
53#491C2212yescomplemented
54#490C2212yes—
55#490C2212yes—
56#490C2212yes—
57#489C2211yescharacteristic, complemented
58#488C2214yescomplemented
59#488C2214yescomplemented
60#486C2212yes—
61#486C2212yes—
62#485C2214yes—
63#481C2214yescomplemented
64#480C2214yescomplemented
65#478C2214yescomplemented
66#478C2214yescomplemented
67#477C2212yescomplemented
68#475C2212yescomplemented
69#470C2212yes—
70#469C2214yescomplemented
71#468C2212yes—
72#468C2214yescomplemented
73#464C2214yescomplemented
74#463C2214yescomplemented
75#460C2212yes—
76#459C2212yes—
77#452C2214yescomplemented
78#450C2212yes—
79#449C2214yescomplemented
80#447C2212yes—
81C8×Q8C2212yescomplemented
82C4×Q16C2212yescomplemented
83C4×SD16C2212yescomplemented
84C4×SD16C2214yescomplemented
85C4×D8C2214yescomplemented
86#436C2214yes—
87#436C2212yescomplemented
88#432C2211yescharacteristic
89#432C2212yescomplemented
90#418C2214yes—
91#417C2214yescomplemented
92#417C2214yescomplemented
93C4×M16C2211yescharacteristic
94C4×M16C2212yescomplemented
95C2×#41—21no—
96C2×#41—22no—
97C2×#41D411yescharacteristic
98C2×#41C2314yes—
99C22×D4—21no—
100C22×D4—21no—
101C22×D4C2312yes—
102C2×Q16C2312yes—
103C2×SD16C2312yes—
104C2×SD16C2314yes—
105C2×D8C2314yes—
106C2×M16C2312yes—
107C2×M16C2314yes—
108C4⋊Q8—21no—
109C4⋊Q8C2311yescharacteristic
110C4⋊Q8C2311yescharacteristic
111C4⋊Q8C2311yescharacteristic
112C4⋊Q8C2311yescharacteristic
113#127—22no—
114#127—21no—
115#127—21no—
116#127C2312yes—
117#127C2312yes—
118#126—22no—
119#126—22no—
120#126—21no—
121#126—22no—
122#125—22no—
123#125—22no—
124#125—21no—
125#125—22no—
126#125—22no—
127#125—21no—
128#125C2311yescharacteristic
129#125C2312yes—
130#125C2312yes—
131#125C2312yes—
132#124—22no—
133#124—22no—
134#124C2311yescharacteristic
135#124C2314yes—
136#123—22no—
137#123—21no—
138#123—22no—
139#123—24no—
140#123—24no—
141#123—22no—
142#123—22no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8—22no—
148C22⋊Q8—21no—
149C22⋊Q8—22no—
150C22⋊Q8C2311yescharacteristic
151C22⋊Q8C2312yes—
152C22⋊Q8C2312yes—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—22no—
157#121—24no—
158#121—24no—
159#121C2311yescharacteristic
160#121C2312yes—
161#121C2314yes—
162#120—24no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8D411yescharacteristic
168C4×Q8C2311yescharacteristic
169C4×Q8C2312yes—
170C4×Q8C2312yes—
171C4×Q8C2311yescharacteristic
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—21no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4D411yescharacteristic
182C4×D4D412yes—
183C4×D4C2311yescharacteristic
184C4×D4C2312yes—
185C4×D4C2314yes—
186C4×D4C2312yes—
187C4×D4C2311yescharacteristic
188#117—21no—
189#117—22no—
190#117—21no—
191#117—22no—
192#117D411yescharacteristic
193#117D412yes—
194#117C2312yes—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2312yes—
199C2×C4⋊C4C2312yes—
200C2×C22⋊C4—24no—
201C2×C42C2312yes—
202C2×C42C2311yescharacteristic
203#107C2314yes—
204#107C2312yes—
205#106C2314yes—
206#106C2312yes—
207C4⋊C8C2312yes—
208C4⋊C8C2314yes—
209#103C2312yes—
210#103C2312yes—
211#103C2314yes—
212#103C2314yes—
213#103C2312yes—
214#102C2314yes—
215#102C2314yes—
216#102C2314yes—
217#102C2314yes—
218#102C2314yes—
219#102C2314yes—
220#102C2312yes—
221C22⋊C8C2312yes—
222C22⋊C8C2314yes—
223#97C2311yescharacteristic
224#97C2311yescharacteristic
225#97C2312yes—
226C4×C8C2312yes—
227C4×C8C2314yes—
228C24—22no—
229#41—42no—
230#41—44no—
231#41—44no—
232#41—44no—
233#41—84no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8C2×D411yescharacteristic
238C2×Q8C2412yes—
239C2×Q8C2411yescharacteristic
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—82no—
252C2×D4C2×D411yescharacteristic
253C2×D4C2×D412yes—
254C2×D4C2414yes—
255C2×D4C2411yescharacteristic
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D412yes—
264C22×C4C2×D412yes—
265C22×C4C2×D411yescharacteristic
266C22×C4C2412yes—
267C22×C4C2411yescharacteristic
268Q16—42no—
269SD16—44no—
270SD16—42no—
271D8—44no—
272M16—44no—
273M16—42no—
274C2×C8C2412yes—
275C2×C8C2412yes—
276C2×C8C2414yes—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4C2×D411yescharacteristic
299C4⋊C4C2×D411yescharacteristic
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2412yes—
304C4⋊C4C2411yescharacteristic
305C4⋊C4C2412yes—
306C4⋊C4C2412yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—21no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—22no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4C2×D412yes—
322C22⋊C4C2×D412yes—
323C22⋊C4C2×D412yes—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—21no—
328C42C2×D412yes—
329C42C2×D411yescharacteristic
330C42C2×D411yescharacteristic
331C42C2×D412yes—
332C42C2412yes—
333C42C2411yescharacteristic
334C42C2411yescharacteristic
335C23—21no—
336C23—24no—
337C23—44no—
338C23—82no—
339C23—82no—
340C23C22×D411yescharacteristic
341C23C22×D412yes—
342Q8—22no—
343Q8—22no—
344Q8—22no—
345Q8—22no—
346Q8—42no—
347Q8—42no—
348Q8—81no—
349D4—22no—
350D4—22no—
351D4—24no—
352D4—24no—
353D4—42no—
354D4—44no—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—82no—
359D4—82no—
360D4—81no—
361D4—84no—
362D4—84no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4C22×D411yescharacteristic
383C2×C4C22×D412yes—
384C2×C4C22×D411yescharacteristic
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D412yes—
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C2511yescharacteristic
391C8—22no—
392C8—24no—
393C8—22no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—44no—
402C22—44no—
403C22—41no—
404C22—41no—
405C22—44no—
406C22—164no—
407C22C23×D411yescharacteristic
408C4—21no—
409C4—22no—
410C4—21no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—41no—
421C4—42no—
422C4#57512yes—
423C4#58512yes—
424C2—42no—
425C2—41no—
426C2—81no—
427C2—84no—
428C2#350011yescharacteristic
429C2C2×#57511yescharacteristic
430C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3515C2semidirect
2. #3508C2semidirect
3. #3365C2semidirect
4. #3361C2semidirect
5. #3333C2semidirect
6. #3332C2semidirect
7. #3323C2semidirect
8. #3322C2semidirect
9. #3237C2semidirect
10. #3234C2semidirect
11. #3219C2semidirect
12. #3217C2semidirect
13. #3172C2semidirect
14. #3171C2semidirect
15. #3142C2semidirect
16. #3138C2semidirect
17. #2971C2semidirect
20. #557C22semidirect
21. #554C22semidirect
22. #554C22semidirect
30. #547C22semidirect
43. #529C22semidirect
44. C2×#125C22semidirect
48. #501C22semidirect
49. #501C22semidirect
50. #497C22semidirect
52. #491C22semidirect
53. #491C22semidirect
57. #489C22semidirect
58. #488C22semidirect
59. #488C22semidirect
63. #481C22semidirect
64. #480C22semidirect
65. #478C22semidirect
66. #478C22semidirect
67. #477C22semidirect
68. #475C22semidirect
70. #469C22semidirect
72. #468C22semidirect
73. #464C22semidirect
74. #463C22semidirect
77. #452C22semidirect
79. #449C22semidirect
81. C8×Q8C22semidirect
82. C4×Q16C22semidirect
83. C4×SD16C22semidirect
84. C4×SD16C22semidirect
85. C4×D8C22semidirect
87. #436C22semidirect
89. #432C22semidirect
91. #417C22semidirect
92. #417C22semidirect
94. C4×M16C22semidirect

Maximal subgroups

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

subgroupindex
#35152
#35082
#33652
#33612
#33332
#33322
#33232
#33222
#32372
#32342
#32192
#32172
#31722
#31712
#31422
#31382
#29712