Finite Group Databasebeta

← #62143

Subgroups — #62143

Order 256.

915 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic, complemented
2#3481C211yesmaximal, characteristic, complemented
3#3380C212yesmaximal, complemented
4#3361C212yesmaximal, complemented
5#3357C212yesmaximal, complemented
6#3355C214yesmaximal, complemented
7#3325C212yesmaximal, complemented
8#3322C214yesmaximal, complemented
9#3253C211yesmaximal, characteristic, complemented
10#3251C212yesmaximal, complemented
11#3233C212yesmaximal, complemented
12#3226C211yesmaximal, characteristic, complemented
13#3106C211yesmaximal, characteristic, complemented
14#3100C212yesmaximal, complemented
15#3079C211yesmaximal, characteristic, complemented
16#3077C212yesmaximal, complemented
17#2991C211yesmaximal, characteristic, complemented
18#559C2211yescharacteristic
19#556—21no—
20#555C2212yes—
21#554C2211yescharacteristic
22#554C2211yescharacteristic
23#554C2212yescomplemented
24#553—21no—
25#553C2212yes—
26#553C2214yescomplemented
27#551—22no—
28#548—22no—
29#547—21no—
30#546—21no—
31#546C2212yes—
32D42C2211yescharacteristic
33#542—22no—
34#537—21no—
35#533C2212yescomplemented
36#532C2211yescharacteristic
37#532C2212yescomplemented
38#532C2211yescharacteristic, complemented
39C2×C4⋊Q8C2211yescharacteristic, complemented
40#529—21no—
41#529C2212yes—
42#525—22no—
43#520—21no—
44C4×#41C2211yescharacteristic
45C4×#41C2211yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2212yes—
48#500C2212yescomplemented
49#499C2214yescomplemented
50#488C2214yescomplemented
51#486C2212yes—
52#485C2214yescomplemented
53#482C2214yes—
54#482C2212yescomplemented
55#480C2214yescomplemented
56#480C2214yescomplemented
57#478C2214yes—
58#478C2214yescomplemented
59#478C2214yescomplemented
60#477C2212yescomplemented
61#477C2211yescharacteristic, complemented
62#476C2211yescharacteristic
63#476C2212yes—
64#476C2212yescomplemented
65#476C2212yescomplemented
66#475C2212yescomplemented
67#474C2212yes—
68#474C2212yescomplemented
69#474C2212yescomplemented
70#469C2214yes—
71#469C2214yescomplemented
72#468C2214yescomplemented
73#463C2214yescomplemented
74#460C2211yescharacteristic, complemented
75#459C2211yescharacteristic
76#459C2212yes—
77#459C2212yescomplemented
78#449C2212yes—
79#447C2214yes—
80C8×Q8C2212yescomplemented
81C4×SD16C2212yescomplemented
82C4×D8C2214yes—
83C4×D8C2212yes—
84C4×D8C2214yescomplemented
85#436C2212yes—
86#436C2214yescomplemented
87#431C2212yescomplemented
88#431C2211yescharacteristic, complemented
89#428C2214yescomplemented
90#428C2212yescomplemented
91#423C2212yescomplemented
92#423C2211yescharacteristic, complemented
93#418C2214yes—
94#417C2212yes—
95C2×#41—21no—
96C2×#41—21no—
97C2×#41—22no—
98C2×#41D411yescharacteristic
99C2×#41C2311yescharacteristic
100C22×Q8—21no—
101C22×D4—21no—
102C22×D4C2312yes—
103C23×C4—22no—
104C2×SD16C2312yes—
105C2×D8C2312yes—
106C2×D8C2314yes—
107C2×M16C2312yes—
108C2×M16C2314yes—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8C2312yes—
112C4⋊Q8C2312yes—
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115#127C2311yescharacteristic
116#127C2311yescharacteristic
117#126—22no—
118#126—24no—
119#126—22no—
120#125—21no—
121#125C2312yes—
122#125C2314yes—
123#124—21no—
124#124—24no—
125#124—21no—
126#124—21no—
127#124—22no—
128#124C2312yes—
129#123—22no—
130#123—22no—
131#123—22no—
132#123—22no—
133#123—24no—
134C22⋊Q8—21no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—24no—
140C22⋊Q8—22no—
141C22⋊Q8C2312yes—
142#121—22no—
143#121—22no—
144#121—24no—
145#121—21no—
146#121—22no—
147#121—21no—
148#121—21no—
149#121—22no—
150#121C2312yes—
151#121C2312yes—
152#121C2312yes—
153#121C2314yes—
154#120—22no—
155#120—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8D411yescharacteristic
159C4×Q8C2311yescharacteristic
160C4×Q8C2311yescharacteristic
161C4×Q8C2311yescharacteristic
162C4×D4—21no—
163C4×D4—22no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—22no—
168C4×D4—24no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—21no—
174C4×D4D411yescharacteristic
175C4×D4D412yes—
176C4×D4C2311yescharacteristic
177C4×D4C2312yes—
178C4×D4C2312yes—
179C4×D4C2312yes—
180C4×D4C2311yescharacteristic
181C4×D4C2314yes—
182C4×D4C2312yes—
183C4×D4C2311yescharacteristic
184C4×D4C2312yes—
185#117—22no—
186#117—21no—
187#117—22no—
188#117D411yescharacteristic
189#117D412yes—
190#117C2311yescharacteristic
191#117C2314yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C2312yes—
196C2×C4⋊C4C2312yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—21no—
200C2×C42C2312yes—
201C2×C42C2311yescharacteristic
202#107C2314yes—
203#107C2314yes—
204#107C2314yes—
205#107C2312yes—
206#106C2312yes—
207#106C2314yes—
208#106C2314yes—
209C4⋊C8C2312yes—
210C4⋊C8C2311yescharacteristic
211C4⋊C8C2312yes—
212C4⋊C8C2314yes—
213C4⋊C8C2311yescharacteristic
214#103C2314yes—
215#103C2312yes—
216#102C2314yes—
217#102C2312yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221#102C2312yes—
222#102C2312yes—
223#102C2314yes—
224C22⋊C8C2314yes—
225C22⋊C8C2312yes—
226C4×C8C2312yes—
227C4×C8C2314yes—
228C24—22no—
229#41—22no—
230#41—42no—
231#41—41no—
232#41—44no—
233#41—82no—
234#41—84no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—21no—
238C2×Q8—22no—
239C2×Q8—82no—
240C2×Q8C2×D411yescharacteristic
241C2×Q8C2411yescharacteristic
242C2×D4—24no—
243C2×D4—21no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4C2×D411yescharacteristic
256C2×D4C2×D412yes—
257C2×D4C2412yes—
258C2×D4C2412yes—
259C2×D4C2411yescharacteristic
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—84no—
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D411yescharacteristic
273C22×C4C2×D412yes—
274C22×C4C2×D412yes—
275C22×C4C2411yescharacteristic
276C22×C4C2412yes—
277SD16—42no—
278D8—24no—
279D8—44no—
280M16—42no—
281M16—44no—
282C2×C8C2412yes—
283C2×C8C2412yes—
284C2×C8C2414yes—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4C2×D411yescharacteristic
299C4⋊C4C2×D411yescharacteristic
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D412yes—
302C4⋊C4C2412yes—
303C4⋊C4C2414yes—
304C4⋊C4C2411yescharacteristic
305C4⋊C4C2411yescharacteristic
306C4⋊C4C2412yes—
307C22⋊C4—22no—
308C22⋊C4—21no—
309C22⋊C4—24no—
310C22⋊C4—21no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—24no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—22no—
323C22⋊C4C2×D412yes—
324C22⋊C4C2×D412yes—
325C22⋊C4C2×D412yes—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—24no—
330C42C2×D411yescharacteristic
331C42C2×D411yescharacteristic
332C42C2×D412yes—
333C42C2×D412yes—
334C42C2412yes—
335C42C2411yescharacteristic
336C42C2411yescharacteristic
337C23—21no—
338C23—22no—
339C23—22no—
340C23—44no—
341C23—44no—
342C23—82no—
343C23C22×D412yes—
344C23C22×D411yescharacteristic
345Q8—22no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8—44no—
350Q8—82no—
351Q8—81no—
352D4—22no—
353D4—22no—
354D4—22no—
355D4—21no—
356D4—22no—
357D4—22no—
358D4—42no—
359D4—44no—
360D4—42no—
361D4—42no—
362D4—44no—
363D4—44no—
364D4—42no—
365D4—42no—
366D4—81no—
367D4—82no—
368D4—84no—
369D42−1+412yes—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C22×D412yes—
395C2×C4C22×D411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D412yes—
398C2×C4C22×D411yescharacteristic
399C2×C4C22×D412yes—
400C2×C4C22×D411yescharacteristic
401C2×C4C22×D411yescharacteristic
402C2×C4C2511yescharacteristic
403C8—22no—
404C8—24no—
405C8—22no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—88no—
422C22C23×D411yescharacteristic
423C4—21no—
424C4—22no—
425C4—21no—
426C4—22no—
427C4—22no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—42no—
432C4—41no—
433C4—42no—
434C4—44no—
435C4#57512yes—
436C4#57612yes—
437C4C2×2−1+411yescharacteristic
438C4#58511yescharacteristic
439C2—41no—
440C2—42no—
441C2—42no—
442C2—82no—
443C2—82no—
444C2#348911yescharacteristic
445C2C2×#57511yescharacteristic
446C2C2×#57611yescharacteristic
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. #3511C2semidirect
2. #3481C2semidirect
3. #3380C2semidirect
4. #3361C2semidirect
5. #3357C2semidirect
6. #3355C2semidirect
7. #3325C2semidirect
8. #3322C2semidirect
9. #3253C2semidirect
10. #3251C2semidirect
11. #3233C2semidirect
12. #3226C2semidirect
13. #3106C2semidirect
14. #3100C2semidirect
15. #3079C2semidirect
16. #3077C2semidirect
17. #2991C2semidirect
23. #554C22semidirect
26. #553C22semidirect
35. #533C22semidirect
37. #532C22semidirect
38. #532C22semidirect
39. C2×C4⋊Q8C22semidirect
48. #500C22semidirect
49. #499C22semidirect
50. #488C22semidirect
52. #485C22semidirect
54. #482C22semidirect
55. #480C22semidirect
56. #480C22semidirect
58. #478C22semidirect
59. #478C22semidirect
60. #477C22semidirect
61. #477C22semidirect
64. #476C22semidirect
65. #476C22semidirect
66. #475C22semidirect
68. #474C22semidirect
69. #474C22semidirect
71. #469C22semidirect
72. #468C22semidirect
73. #463C22semidirect
74. #460C22semidirect
77. #459C22semidirect
80. C8×Q8C22semidirect
81. C4×SD16C22semidirect
84. C4×D8C22semidirect
86. #436C22semidirect
87. #431C22semidirect
88. #431C22semidirect
89. #428C22semidirect
90. #428C22semidirect
91. #423C22semidirect
92. #423C22semidirect

Maximal subgroups

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

subgroupindex
#35112
#34812
#33802
#33612
#33572
#33552
#33252
#33222
#32532
#32512
#32332
#32262
#31062
#31002
#30792
#30772
#29912