Finite Group Databasebeta

← #62144

Subgroups — #62144

Order 256.

887 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509C211yesmaximal, characteristic, complemented
2#3481C211yesmaximal, characteristic
3#3381C212yesmaximal, complemented
4#3365C212yesmaximal, complemented
5#3356C212yesmaximal, complemented
6#3351C214yesmaximal, complemented
7#3324C212yesmaximal, complemented
8#3323C214yesmaximal, complemented
9#3254C211yesmaximal, characteristic, complemented
10#3252C212yesmaximal, complemented
11#3233C212yesmaximal, complemented
12#3227C211yesmaximal, characteristic, complemented
13#3107C211yesmaximal, characteristic, complemented
14#3101C212yesmaximal, complemented
15#3079C211yesmaximal, characteristic, complemented
16#3077C212yesmaximal, complemented
17#2991C211yesmaximal, characteristic, complemented
18#559C2211yescharacteristic
19#557—21no—
20#557C2212yescomplemented
21#556—21no—
22#556C2212yes—
23#554C2211yescharacteristic
24#554C2211yescharacteristic
25#553—21no—
26#553C2212yes—
27#552—22no—
28#552C2214yescomplemented
29#551—22no—
30#546—21no—
31#546C2212yes—
32D42C2211yescharacteristic
33#541—22no—
34#536—21no—
35#533C2212yescomplemented
36#533C2211yescharacteristic, complemented
37#533C2212yescomplemented
38#532C2211yescharacteristic
39C2×C4⋊Q8C2211yescharacteristic, complemented
40#529—21no—
41#529C2212yes—
42#525—22no—
43#519—21no—
44C4×#41C2211yescharacteristic
45C4×#41C2211yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2212yes—
48#499C2214yescomplemented
49#498C2212yescomplemented
50#488C2214yescomplemented
51#486C2212yes—
52#485C2212yescomplemented
53#485C2214yescomplemented
54#483C2214yescomplemented
55#482C2214yes—
56#481C2214yescomplemented
57#479C2214yescomplemented
58#478C2214yes—
59#478C2214yescomplemented
60#477C2212yescomplemented
61#477C2212yescomplemented
62#476C2212yes—
63#476C2212yescomplemented
64#476C2212yescomplemented
65#475C2212yescomplemented
66#475C2211yescharacteristic, complemented
67#474C2211yescharacteristic
68#474C2212yes—
69#474C2212yescomplemented
70#470C2214yescomplemented
71#468C2214yes—
72#468C2214yescomplemented
73#464C2214yescomplemented
74#462C2211yescharacteristic, complemented
75#461C2211yescharacteristic
76#461C2212yescomplemented
77#460C2212yes—
78#452C2212yes—
79#450C2214yes—
80C8×Q8C2212yescomplemented
81C4×Q16C2212yescomplemented
82C4×SD16C2214yes—
83C4×SD16C2212yes—
84C4×SD16C2214yescomplemented
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×#41D411yescharacteristic
97C2×#41C2311yescharacteristic
98C22×Q8—21no—
99C22×D4—21no—
100C22×D4C2312yes—
101C23×C4—22no—
102C2×Q16C2312yes—
103C2×SD16C2312yes—
104C2×SD16C2314yes—
105C2×M16C2312yes—
106C2×M16C2314yes—
107C4⋊Q8—22no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8C2311yescharacteristic
111C4⋊Q8C2312yes—
112C4⋊Q8C2312yes—
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115#127C2311yescharacteristic
116#126—22no—
117#126—24no—
118#126—24no—
119#126—22no—
120#125—21no—
121#125—24no—
122#125—24no—
123#125—21no—
124#125—22no—
125#125C2312yes—
126#125C2314yes—
127#124—21no—
128#124—22no—
129#124—21no—
130#124C2312yes—
131#123—24no—
132#123—22no—
133#123—22no—
134#123—22no—
135#123—22no—
136C22⋊Q8—21no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—21no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8C2312yes—
144C22⋊Q8C2312yes—
145C22⋊Q8C2314yes—
146#121—22no—
147#121—21no—
148#121—21no—
149#121—22no—
150#121C2312yes—
151#121C2312yes—
152#120—22no—
153#120—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—21no—
160C4×Q8D411yescharacteristic
161C4×Q8C2311yescharacteristic
162C4×Q8C2311yescharacteristic
163C4×Q8C2314yes—
164C4×Q8C2312yes—
165C4×Q8C2311yescharacteristic
166C4×D4—21no—
167C4×D4—22no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4D411yescharacteristic
176C4×D4D412yes—
177C4×D4C2311yescharacteristic
178C4×D4C2312yes—
179C4×D4C2312yes—
180C4×D4C2312yes—
181C4×D4C2311yescharacteristic
182C4×D4C2311yescharacteristic
183C4×D4C2312yes—
184#117—21no—
185#117—22no—
186#117—24no—
187#117—21no—
188#117—22no—
189#117D411yescharacteristic
190#117D412yes—
191#117C2311yescharacteristic
192#117C2314yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2312yes—
196C2×C4⋊C4C2312yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—21no—
200C2×C42C2312yes—
201C2×C42C2311yescharacteristic
202#107C2312yes—
203#107C2314yes—
204#107C2314yes—
205#106C2314yes—
206#106C2314yes—
207#106C2312yes—
208#106C2314yes—
209C4⋊C8C2312yes—
210C4⋊C8C2311yescharacteristic
211C4⋊C8C2312yes—
212C4⋊C8C2314yes—
213C4⋊C8C2311yescharacteristic
214#103C2314yes—
215#103C2314yes—
216#103C2312yes—
217#103C2312yes—
218#103C2312yes—
219#102C2314yes—
220#102C2312yes—
221#102C2314yes—
222#102C2314yes—
223#102C2314yes—
224C22⋊C8C2314yes—
225C22⋊C8C2312yes—
226C4×C8C2312yes—
227C4×C8C2314yes—
228C24—22no—
229#41—22no—
230#41—42no—
231#41—41no—
232#41—82no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8—44no—
238C2×Q8C2×D411yescharacteristic
239C2×Q8C2411yescharacteristic
240C2×Q8C2412yes—
241C2×D4—24no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4C2×D411yescharacteristic
251C2×D4C2×D412yes—
252C2×D4C2412yes—
253C2×D4C2411yescharacteristic
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—84no—
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D412yes—
266C22×C4C2×D412yes—
267C22×C4C2411yescharacteristic
268C22×C4C2412yes—
269Q16—42no—
270SD16—24no—
271SD16—44no—
272M16—42no—
273M16—44no—
274C2×C8C2412yes—
275C2×C8C2412yes—
276C2×C8C2414yes—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4C2×D411yescharacteristic
297C4⋊C4C2×D412yes—
298C4⋊C4C2×D411yescharacteristic
299C4⋊C4C2×D412yes—
300C4⋊C4C2414yes—
301C4⋊C4C2412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2412yes—
304C4⋊C4C2411yescharacteristic
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—24no—
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—21no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4C2×D412yes—
320C22⋊C4C2×D412yes—
321C22⋊C4C2×D412yes—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—24no—
326C42—24no—
327C42—22no—
328C42C2×D411yescharacteristic
329C42C2×D411yescharacteristic
330C42C2×D412yes—
331C42C2×D412yes—
332C42C2412yes—
333C42C2411yescharacteristic
334C42C2411yescharacteristic
335C23—21no—
336C23—22no—
337C23—44no—
338C23—44no—
339C23—82no—
340C23C22×D412yes—
341C23C22×D411yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—21no—
345Q8—22no—
346Q8—21no—
347Q8—44no—
348Q8—44no—
349Q8—42no—
350Q8—81no—
351D4—22no—
352D4—22no—
353D4—21no—
354D4—22no—
355D4—42no—
356D4—44no—
357D4—42no—
358D4—44no—
359D4—42no—
360D4—81no—
361D4—82no—
362D42−1+412yes—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4C22×D412yes—
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C22×D412yes—
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C22×D411yescharacteristic
394C2×C4C2511yescharacteristic
395C8—22no—
396C8—24no—
397C8—22no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—21no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—88no—
411C22C23×D411yescharacteristic
412C4—21no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—41no—
419C4—42no—
420C4—41no—
421C4—44no—
422C4—41no—
423C4—42no—
424C4—42no—
425C4#57512yes—
426C4#57712yes—
427C4C2×2−1+411yescharacteristic
428C4#58511yescharacteristic
429C2—41no—
430C2—42no—
431C2—42no—
432C2—82no—
433C2#348911yescharacteristic
434C2C2×#57511yescharacteristic
435C2C2×#57711yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3481C2
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. #3509C2semidirect
3. #3381C2semidirect
4. #3365C2semidirect
5. #3356C2semidirect
6. #3351C2semidirect
7. #3324C2semidirect
8. #3323C2semidirect
9. #3254C2semidirect
10. #3252C2semidirect
11. #3233C2semidirect
12. #3227C2semidirect
13. #3107C2semidirect
14. #3101C2semidirect
15. #3079C2semidirect
16. #3077C2semidirect
17. #2991C2semidirect
20. #557C22semidirect
28. #552C22semidirect
35. #533C22semidirect
36. #533C22semidirect
37. #533C22semidirect
39. C2×C4⋊Q8C22semidirect
48. #499C22semidirect
49. #498C22semidirect
50. #488C22semidirect
52. #485C22semidirect
53. #485C22semidirect
54. #483C22semidirect
56. #481C22semidirect
57. #479C22semidirect
59. #478C22semidirect
60. #477C22semidirect
61. #477C22semidirect
63. #476C22semidirect
64. #476C22semidirect
65. #475C22semidirect
66. #475C22semidirect
69. #474C22semidirect
70. #470C22semidirect
72. #468C22semidirect
73. #464C22semidirect
74. #462C22semidirect
76. #461C22semidirect
80. C8×Q8C22semidirect
81. C4×Q16C22semidirect
84. C4×SD16C22semidirect
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
#35092
#34812
#33812
#33652
#33562
#33512
#33242
#33232
#32542
#32522
#32332
#32272
#31072
#31012
#30792
#30772
#29912