Finite Group Databasebeta

← #61898

Subgroups — #61898

Order 256.

922 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3522C211yesmaximal, characteristic, complemented
2C2×#559C211yesmaximal, characteristic, complemented
3#3403C214yesmaximal, complemented
4#3399C212yesmaximal, complemented
5#3396C212yesmaximal, complemented
6#3375C212yesmaximal, complemented
7#3370C214yesmaximal, complemented
8#3361C212yesmaximal, complemented
9#3217C211yesmaximal, characteristic, complemented
10#3213C212yesmaximal, complemented
11#3211C212yesmaximal, complemented
12#3205C211yesmaximal, characteristic, complemented
13#3114C212yesmaximal, complemented
14#3112C211yesmaximal, characteristic, complemented
15#3108C211yesmaximal, characteristic, complemented
16#3100C212yesmaximal, complemented
17#2984C211yesmaximal, characteristic, complemented
18#559—21no—
19#559—21no—
20#559—22no—
21#559C2212yes—
22#559C2214yes—
23#559C2212yes—
24#559C2211yescharacteristic, complemented
25#559C2212yescomplemented
26Q82C2211yescharacteristic, complemented
27#554C2211yescharacteristic, complemented
28#554C2212yescomplemented
29#550C2211yescharacteristic, complemented
30#548—21no—
31#547—21no—
32#547—22no—
33#542—22no—
34#542—21no—
35#532C2212yes—
36#532C2211yescharacteristic
37C2×#127C2211yescharacteristic
38#529C2211yescharacteristic
39#529C2212yes—
40C2×#124—21no—
41C2×#124C2212yes—
42C2×#123—22no—
43#520—21no—
44C2×C4×Q8C2211yescharacteristic
45C2×C4×Q8C2211yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2212yes—
48#496C2212yes—
49#496C2214yescomplemented
50#495C2214yescomplemented
51#493C2212yes—
52#490C2212yes—
53#489C2214yescomplemented
54#488C2214yescomplemented
55#486C2212yes—
56#483C2211yescharacteristic
57#480C2211yescharacteristic
58#480C2212yes—
59#479C2212yescomplemented
60#478C2214yescomplemented
61#477C2212yescomplemented
62#474C2214yescomplemented
63#464C2212yescomplemented
64#464C2214yescomplemented
65#463C2214yescomplemented
66#463C2214yescomplemented
67#463C2214yescomplemented
68#463C2214yescomplemented
69#460C2212yes—
70#460C2212yescomplemented
71#459C2212yes—
72#459C2214yes—
73#459C2214yescomplemented
74#451C2211yescharacteristic
75#450C2212yes—
76#448C2211yescharacteristic
77#448C2212yes—
78#447C2212yes—
79#447C2212yes—
80#447C2212yes—
81#446C2214yescomplemented
82#446C2212yescomplemented
83C8×Q8C2212yescomplemented
84#442C2214yescomplemented
85#440C2214yescomplemented
86#440C2214yescomplemented
87C4×Q16C2212yescomplemented
88C4×SD16C2212yescomplemented
89C4×D8C2214yescomplemented
90C4×D8C2214yescomplemented
91#432C2211yescharacteristic
92#432C2212yes—
93#431C2212yes—
94#431C2211yescharacteristic
95C2×#41—21no—
96C2×#41—22no—
97C2×#41—21no—
98C22×Q8D411yescharacteristic
99C22×Q8C2311yescharacteristic
100C22×D4—21no—
101C22×D4—21no—
102C22×D4C2312yes—
103C23×C4—22no—
104C2×Q16C2312yes—
105C2×SD16C2312yes—
106C2×SD16C2314yes—
107C2×SD16C2314yes—
108C2×D8C2314yes—
109C2×D8C2314yes—
110C2×D8C2314yes—
111C4⋊Q8—21no—
112C4⋊Q8—22no—
113C4⋊Q8C2312yes—
114C4⋊Q8C2311yescharacteristic
115#127—21no—
116#127—21no—
117#127C2312yes—
118#127C2312yes—
119#127C2312yes—
120#127C2311yescharacteristic
121#126—22no—
122#126—21no—
123#125C2312yes—
124#125C2311yescharacteristic
125#124—22no—
126#124—22no—
127#124—24no—
128#124—21no—
129#124—22no—
130#124—22no—
131#124—22no—
132#124—22no—
133#124—21no—
134#124C2314yes—
135#124C2314yes—
136#124C2312yes—
137#124C2311yescharacteristic
138#123—22no—
139#123—21no—
140#123—24no—
141#123—24no—
142#123—24no—
143#123—24no—
144#123—22no—
145#123—22no—
146#123—22no—
147C22⋊Q8—21no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8C2311yescharacteristic
153#121—21no—
154#121—22no—
155#121—21no—
156#121—22no—
157#121C2311yescharacteristic
158#121C2312yes—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×Q8D411yescharacteristic
163C4×Q8D412yes—
164C4×Q8D411yescharacteristic
165C4×Q8C2312yes—
166C4×Q8C2312yes—
167C4×Q8C2311yescharacteristic
168C4×Q8C2312yes—
169C4×Q8C2311yescharacteristic
170C4×Q8C2311yescharacteristic
171C4×Q8C2312yes—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—24no—
177C4×D4—21no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4C2314yes—
184C4×D4C2314yes—
185C4×D4C2311yescharacteristic
186C4×D4C2312yes—
187C4×D4C2312yes—
188C4×D4C2312yes—
189C4×D4C2312yes—
190#117—22no—
191#117—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4D411yescharacteristic
196C2×C4⋊C4D412yes—
197C2×C4⋊C4C2311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—21no—
202C2×C42C2312yes—
203C2×C42C2311yescharacteristic
204#107C2312yes—
205#107C2314yes—
206#106C2312yes—
207C4⋊C8C2312yes—
208C4⋊C8C2314yes—
209C4⋊C8C2312yes—
210C4⋊C8C2314yes—
211#103C2312yes—
212#103C2312yes—
213#103C2312yes—
214#103C2314yes—
215#102C2312yes—
216#102C2314yes—
217#102C2314yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2311yescharacteristic
223C22⋊C8C2312yes—
224#97C2312yes—
225#97C2314yes—
226C4×C8C2314yes—
227C4×C8C2312yes—
228C24—22no—
229#41—44no—
230#41—84no—
231#41—82no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—21no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8C2×D411yescharacteristic
238C2×Q8C2×D411yescharacteristic
239C2×Q8C2×D412yes—
240C2×Q8C2412yes—
241C2×Q8C2411yescharacteristic
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—82no—
256C2×D4C2414yes—
257C2×D4C2411yescharacteristic
258C2×D4C2412yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—84no—
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D412yes—
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D412yes—
273C22×C4C2411yescharacteristic
274Q16—42no—
275SD16—42no—
276SD16—44no—
277SD16—44no—
278D8—44no—
279D8—44no—
280D8—44no—
281C2×C8C2412yes—
282C2×C8C2414yes—
283C2×C8C2412yes—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4C2×D411yescharacteristic
297C4⋊C4C2×D412yes—
298C4⋊C4C2×D411yescharacteristic
299C4⋊C4C2×D412yes—
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D412yes—
302C4⋊C4C2×D412yes—
303C4⋊C4C2411yescharacteristic
304C4⋊C4C2412yes—
305C4⋊C4C2411yescharacteristic
306C4⋊C4C2412yes—
307C22⋊C4—21no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—22no—
315C22⋊C4—21no—
316C22⋊C4—22no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—21no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—21no—
329C42C2×D412yes—
330C42C2×D411yescharacteristic
331C42C2×D412yes—
332C42C2×D411yescharacteristic
333C42C2412yes—
334C42C2411yescharacteristic
335C42C2412yes—
336C42C2411yescharacteristic
337C23—22no—
338C23—21no—
339C23—24no—
340C23—44no—
341C23—44no—
342C23—44no—
343C23C22×D411yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—22no—
347Q8—22no—
348Q8—22no—
349Q8—42no—
350Q8—42no—
351Q8—82no—
352Q8—81no—
353D4—24no—
354D4—22no—
355D4—24no—
356D4—24no—
357D4—44no—
358D4—44no—
359D4—44no—
360D4—42no—
361D4—44no—
362D4—44no—
363D4—42no—
364D4—82no—
365D4—82no—
366D4—82no—
367D4—81no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D412yes—
392C2×C4C22×D412yes—
393C2×C4C22×D412yes—
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C2511yescharacteristic
398C8—22no—
399C8—22no—
400C8—24no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—42no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—44no—
409C22—41no—
410C22—42no—
411C22—44no—
412C22—88no—
413C22#57612yes—
414C22C23×D411yescharacteristic
415C4—21no—
416C4—21no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—22no—
422C4—21no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—42no—
429C4#57512yes—
430C4#58512yes—
431C2—22no—
432C2—82no—
433C2—81no—
434C2—84no—
435C2#348711yescharacteristic
436C2C2×#57511yescharacteristic
437C2C2×#57611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3522C2semidirect
2. C2×#559C2semidirect
3. #3403C2semidirect
4. #3399C2semidirect
5. #3396C2semidirect
6. #3375C2semidirect
7. #3370C2semidirect
8. #3361C2semidirect
9. #3217C2semidirect
10. #3213C2semidirect
11. #3211C2semidirect
12. #3205C2semidirect
13. #3114C2semidirect
14. #3112C2semidirect
15. #3108C2semidirect
16. #3100C2semidirect
17. #2984C2semidirect
24. #559C22semidirect
25. #559C22semidirect
26. Q82C22semidirect
27. #554C22semidirect
28. #554C22semidirect
29. #550C22semidirect
49. #496C22semidirect
50. #495C22semidirect
53. #489C22semidirect
54. #488C22semidirect
59. #479C22semidirect
60. #478C22semidirect
61. #477C22semidirect
62. #474C22semidirect
63. #464C22semidirect
64. #464C22semidirect
65. #463C22semidirect
66. #463C22semidirect
67. #463C22semidirect
68. #463C22semidirect
70. #460C22semidirect
73. #459C22semidirect
81. #446C22semidirect
82. #446C22semidirect
83. C8×Q8C22semidirect
84. #442C22semidirect
85. #440C22semidirect
86. #440C22semidirect
87. C4×Q16C22semidirect
88. C4×SD16C22semidirect
89. C4×D8C22semidirect
90. C4×D8C22semidirect

Maximal subgroups

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

subgroupindex
#35222
C2×#5592
#34032
#33992
#33962
#33752
#33702
#33612
#32172
#32132
#32112
#32052
#31142
#31122
#31082
#31002
#29842