Finite Group Databasebeta

← #61731

Subgroups — #61731

Order 256.

908 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic, complemented
2C2×#555C211yesmaximal, characteristic, complemented
3#3354C212yesmaximal, complemented
4#3348C214yesmaximal, complemented
5#3340C212yesmaximal, complemented
6#3329C212yesmaximal, complemented
7#3321C214yesmaximal, complemented
8#3303C212yesmaximal, complemented
9#3238C211yesmaximal, characteristic, complemented
10#3229C211yesmaximal, characteristic, complemented
11#3194C212yesmaximal, complemented
12#3187C212yesmaximal, complemented
13#3110C211yesmaximal, characteristic, complemented
14#3105C211yesmaximal, characteristic, complemented
15#3095C212yesmaximal, complemented
16#3091C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18#556C2211yescharacteristic, complemented
19#555—21no—
20#555—22no—
21#555—21no—
22#555C2212yes—
23#555C2212yes—
24#555C2214yes—
25#554—21no—
26#552C2212yescomplemented
27#551—22no—
28D4×Q8C2211yescharacteristic, complemented
29#548C2212yescomplemented
30#547C2211yescharacteristic, complemented
31#547C2211yescharacteristic, complemented
32#546—21no—
33#542—21no—
34#539—22no—
35#533C2211yescharacteristic
36C2×C4⋊Q8C2211yescharacteristic
37#529C2211yescharacteristic
38C2×#124—22no—
39#525C2212yes—
40C2×C22⋊Q8C2212yes—
41C2×#121—21no—
42C2×#121C2212yes—
43#520—21no—
44C2×C4×D4C2211yescharacteristic
45C2×C4×D4C2211yescharacteristic
46C2×#117—21no—
47C2×#117C2212yes—
48#501C2212yescomplemented
49#498C2212yescomplemented
50#489C2212yescomplemented
51#487C2212yescomplemented
52#485C2214yescomplemented
53#485C2214yescomplemented
54#485C2214yescomplemented
55#484C2212yes—
56#484C2211yescharacteristic
57#483C2212yes—
58#483C2212yes—
59#483C2212yes—
60#482C2214yescomplemented
61#482C2214yescomplemented
62#481C2211yescharacteristic
63#481C2214yescomplemented
64#480C2212yes—
65#478C2212yescomplemented
66#477C2212yescomplemented
67#476C2212yescomplemented
68#475C2214yescomplemented
69#474C2212yescomplemented
70#469C2214yescomplemented
71#468C2214yescomplemented
72#465C2214yescomplemented
73#465C2214yescomplemented
74#463C2212yescomplemented
75#461C2212yescomplemented
76#450C2211yescharacteristic
77#450C2212yes—
78#449C2214yescomplemented
79#448C2214yescomplemented
80#447C2211yescharacteristic
81#440C2214yescomplemented
82C4×SD16C2212yescomplemented
83C4×D8C2212yescomplemented
84#436C2212yescomplemented
85#435C2214yescomplemented
86C8×D4C2212yescomplemented
87#432C2211yescharacteristic
88#431C2211yescharacteristic
89#428C2214yescomplemented
90#427C2214yescomplemented
91#419C2214yescomplemented
92#416C2214yescomplemented
93#407C2212yes—
94C2×C22⋊C8C2212yes—
95C2×#41—21no—
96C2×#41C2312yes—
97C22×Q8—21no—
98C22×Q8C2312yes—
99C22×D4—22no—
100C22×D4D411yescharacteristic
101C22×D4C2311yescharacteristic
102C23×C4—21no—
103C23×C4C2312yes—
104C2×SD16C2312yes—
105C2×SD16C2314yes—
106C2×D8C2312yes—
107C2×M16C2314yes—
108C22×C8C2314yes—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2312yes—
114C4⋊Q8C2312yes—
115C4⋊Q8C2311yescharacteristic
116#126—22no—
117#126—22no—
118#126—21no—
119#125—22no—
120#125—21no—
121#125C2311yescharacteristic
122#124—21no—
123#124—24no—
124#124—24no—
125#124—24no—
126#124—24no—
127#124—22no—
128#124C2311yescharacteristic
129#123—22no—
130#123—22no—
131#123—21no—
132#123—22no—
133#123—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—21no—
137C22⋊Q8—22no—
138C22⋊Q8—21no—
139C22⋊Q8—22no—
140C22⋊Q8C2311yescharacteristic
141C22⋊Q8C2312yes—
142C22⋊Q8C2312yes—
143C22⋊Q8C2312yes—
144C22⋊Q8C2312yes—
145#121—22no—
146#121—22no—
147#121—24no—
148#121—22no—
149#121—21no—
150#121—22no—
151#121—22no—
152#121C2314yes—
153#121C2314yes—
154#121C2311yescharacteristic
155#121C2312yes—
156#120—22no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8C2311yescharacteristic
160C4×Q8C2312yes—
161C4×D4—21no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—21no—
165C4×D4—22no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4D411yescharacteristic
169C4×D4D412yes—
170C4×D4D411yescharacteristic
171C4×D4C2312yes—
172C4×D4C2312yes—
173C4×D4C2311yescharacteristic
174C4×D4C2311yescharacteristic
175C4×D4C2311yescharacteristic
176C4×D4C2312yes—
177#117—22no—
178#117—22no—
179#117—21no—
180#117—22no—
181#117—24no—
182#117—22no—
183#117—22no—
184#117—21no—
185#117C2314yes—
186#117C2314yes—
187#117C2312yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4D411yescharacteristic
192C2×C4⋊C4C2311yescharacteristic
193C2×C4⋊C4C2312yes—
194C2×C4⋊C4C2312yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4D412yes—
200C2×C42—22no—
201C2×C42C2311yescharacteristic
202#107C2314yes—
203#107C2314yes—
204#107C2312yes—
205#106C2312yes—
206#106C2314yes—
207#106C2314yes—
208#106C2314yes—
209C4⋊C8C2312yes—
210C4⋊C8C2312yes—
211#103C2314yes—
212#103C2314yes—
213#103C2312yes—
214#103C2314yes—
215#103C2314yes—
216#102C2312yes—
217#102C2314yes—
218#102C2312yes—
219#102C2312yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2312yes—
223C22⋊C8C2314yes—
224C22⋊C8C2311yescharacteristic
225C22⋊C8C2314yes—
226#97C2312yes—
227C4×C8C2312yes—
228C24—21no—
229C24C2×D412yes—
230#41—44no—
231#41—44no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—82no—
238C2×Q8C2411yescharacteristic
239C2×Q8C2412yes—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—84no—
249C2×D4C2×D411yescharacteristic
250C2×D4C2×D411yescharacteristic
251C2×D4C2×D412yes—
252C2×D4C2411yescharacteristic
253C2×D4C2412yes—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D412yes—
267C22×C4C2×D412yes—
268C22×C4C2×D412yes—
269C22×C4C2411yescharacteristic
270C22×C4C2412yes—
271C22×C4C2412yes—
272SD16—42no—
273SD16—44no—
274D8—42no—
275M16—44no—
276C2×C8—44no—
277C2×C8C2412yes—
278C2×C8C2412yes—
279C2×C8C2414yes—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
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—21no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4C2×D411yescharacteristic
298C4⋊C4C2×D412yes—
299C4⋊C4C2×D411yescharacteristic
300C4⋊C4C2411yescharacteristic
301C4⋊C4C2412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2414yes—
304C4⋊C4C2412yes—
305C22⋊C4—21no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—22no—
321C22⋊C4—21no—
322C22⋊C4C2×D412yes—
323C22⋊C4C2×D412yes—
324C22⋊C4C2×D412yes—
325C22⋊C4C2×D412yes—
326C42—24no—
327C42—24no—
328C42—22no—
329C42—22no—
330C42—21no—
331C42C2×D411yescharacteristic
332C42C2×D411yescharacteristic
333C42C2411yescharacteristic
334C42C2411yescharacteristic
335C23—24no—
336C23—21no—
337C23—22no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C22×D412yes—
344C23C22×D412yes—
345C23C22×D411yescharacteristic
346Q8—22no—
347Q8—24no—
348Q8—42no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352D4—22no—
353D4—22no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—42no—
358D4—42no—
359D4—44no—
360D4—42no—
361D4—84no—
362D4—84no—
363D4—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C22×D411yescharacteristic
394C2×C4C2511yescharacteristic
395C8—22no—
396C8—24no—
397C8—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22—84no—
413C22—84no—
414C22#57712yes—
415C22C23×D411yescharacteristic
416C4—21no—
417C4—22no—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—44no—
428C4—41no—
429C4#57512yes—
430C4#58512yes—
431C2—22no—
432C2—42no—
433C2—42no—
434C2—81no—
435C2—82no—
436C2#348511yescharacteristic
437C2C2×#57511yescharacteristic
438C2C2×#57711yescharacteristic
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. #3520C2semidirect
2. C2×#555C2semidirect
3. #3354C2semidirect
4. #3348C2semidirect
5. #3340C2semidirect
6. #3329C2semidirect
7. #3321C2semidirect
8. #3303C2semidirect
9. #3238C2semidirect
10. #3229C2semidirect
11. #3194C2semidirect
12. #3187C2semidirect
13. #3110C2semidirect
14. #3105C2semidirect
15. #3095C2semidirect
16. #3091C2semidirect
17. #2974C2semidirect
18. #556C22semidirect
26. #552C22semidirect
28. D4×Q8C22semidirect
29. #548C22semidirect
30. #547C22semidirect
31. #547C22semidirect
48. #501C22semidirect
49. #498C22semidirect
50. #489C22semidirect
51. #487C22semidirect
52. #485C22semidirect
53. #485C22semidirect
54. #485C22semidirect
60. #482C22semidirect
61. #482C22semidirect
63. #481C22semidirect
65. #478C22semidirect
66. #477C22semidirect
67. #476C22semidirect
68. #475C22semidirect
69. #474C22semidirect
70. #469C22semidirect
71. #468C22semidirect
72. #465C22semidirect
73. #465C22semidirect
74. #463C22semidirect
75. #461C22semidirect
78. #449C22semidirect
79. #448C22semidirect
81. #440C22semidirect
82. C4×SD16C22semidirect
83. C4×D8C22semidirect
84. #436C22semidirect
85. #435C22semidirect
86. C8×D4C22semidirect
89. #428C22semidirect
90. #427C22semidirect
91. #419C22semidirect
92. #416C22semidirect

Maximal subgroups

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

subgroupindex
#35202
C2×#5552
#33542
#33482
#33402
#33292
#33212
#33032
#32382
#32292
#31942
#31872
#31102
#31052
#30952
#30912
#29742