Finite Group Databasebeta

← #17597

Subgroups — #17597

Order 256.

530 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#2318C211yesmaximal, characteristic, complemented
4#1521C211yesmaximal, characteristic, complemented
5#1519C211yesmaximal, characteristic, complemented
6#1517C211yesmaximal, characteristic, complemented
7#1516C211yesmaximal, characteristic, complemented
8#1514C211yesmaximal, characteristic, complemented
9#1513C211yesmaximal, characteristic, complemented
10#1512C211yesmaximal, characteristic, complemented
11#1510C211yesmaximal, characteristic, complemented
12#1507C211yesmaximal, characteristic, complemented
13#1504C211yesmaximal, characteristic, complemented
14#1501C211yesmaximal, characteristic, complemented
15C2×#327C211yesmaximal, characteristic, complemented
16#555C411yescharacteristic, complemented
17#554C411yescharacteristic, complemented
18#553—21no—
19#553C411yescharacteristic, complemented
20#553C411yescharacteristic, complemented
21#550C411yescharacteristic, complemented
22#546—21no—
23#546—21no—
24#546C411yescharacteristic, complemented
25#546C411yescharacteristic, complemented
26D42—21no—
27D42C411yescharacteristic, complemented
28#540—21no—
29#538—21no—
30#538—21no—
31#532C2211yescharacteristic
32#529C2211yescharacteristic
33#525C2211yescharacteristic
34#525C2211yescharacteristic
35C2×#121C2211yescharacteristic
36C2×#121C2211yescharacteristic
37#518—21no—
38C2×C4×D4C2211yescharacteristic
39#436—21no—
40#435—21no—
41#435—21no—
42C8×D4—21no—
43#432C2211yescharacteristic, complemented
44#431C2211yescharacteristic, complemented
45#407C2211yescharacteristic, complemented
46#407C2211yescharacteristic, complemented
47C2×C22⋊C8C2211yescharacteristic, complemented
48C2×C22⋊C8C2211yescharacteristic, complemented
49#388—21no—
50#387—21no—
51#386—21no—
52#385—21no—
53#383C2211yescharacteristic, complemented
54#382C2211yescharacteristic, complemented
55#381C2211yescharacteristic, complemented
56#380C2211yescharacteristic, complemented
57C4×C22⋊C4C2211yescharacteristic, complemented
58C2×#95C2211yescharacteristic, complemented
59#328C2211yescharacteristic, complemented
60#328C2211yescharacteristic, complemented
61#328C2211yescharacteristic, complemented
62#328C2211yescharacteristic, complemented
63#327C2211yescharacteristic
64#327C2211yescharacteristic
65#327C2211yescharacteristic
66#327C2211yescharacteristic
67#327C2211yescharacteristic, complemented
68#327C2211yescharacteristic, complemented
69#327C2211yescharacteristic, complemented
70#327C2211yescharacteristic, complemented
71#327C2211yescharacteristic, complemented
72#327C2211yescharacteristic, complemented
73#327C2211yescharacteristic, complemented
74#327C2211yescharacteristic, complemented
75C2×#41—21no—
76C2×#41C2×C411yescharacteristic
77C2×#41C2×C411yescharacteristic, complemented
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4C2×C411yescharacteristic
82C22×D4C2×C411yescharacteristic, complemented
83C22×D4D411yescharacteristic
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C2×M16—21no—
88C2×M16—21no—
89C22×C8—21no—
90C22×C8—21no—
91C4⋊Q8C2×C411yescharacteristic, complemented
92#127—21no—
93#127C2×C411yescharacteristic, complemented
94#126—21no—
95#126—21no—
96#126—21no—
97#125—21no—
98#125C2×C411yescharacteristic, complemented
99#124—21no—
100#124—21no—
101#124—21no—
102#124C2×C411yescharacteristic, complemented
103#123—21no—
104#123—21no—
105#123—21no—
106#123—21no—
107#123—21no—
108#123—21no—
109#123—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C22⋊Q8C2×C411yescharacteristic
113C22⋊Q8C2×C411yescharacteristic, complemented
114C22⋊Q8C2311yescharacteristic
115#121—21no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—21no—
127#121—21no—
128#121C2×C411yescharacteristic
129#121C2×C411yescharacteristic
130#121C2×C411yescharacteristic
131#121C2×C411yescharacteristic, complemented
132#121C2×C411yescharacteristic, complemented
133#121C2×C411yescharacteristic, complemented
134#121C2311yescharacteristic
135#121C2311yescharacteristic
136#121C2311yescharacteristic
137#120—21no—
138#120—21no—
139#120—21no—
140#120—21no—
141#120—21no—
142#120—21no—
143C4×Q8C2×C411yescharacteristic, complemented
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4C2×C411yescharacteristic
155C4×D4C2×C411yescharacteristic
156C4×D4C2×C411yescharacteristic
157C4×D4C2×C411yescharacteristic
158C4×D4C2×C411yescharacteristic, complemented
159C4×D4C2×C411yescharacteristic, complemented
160C4×D4C2×C411yescharacteristic, complemented
161C4×D4D411yescharacteristic
162C4×D4D411yescharacteristic
163C4×D4D411yescharacteristic
164C4×D4D411yescharacteristic
165#117—21no—
166#117—21no—
167#117C2×C411yescharacteristic
168#117C2×C411yescharacteristic, complemented
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C2×C411yescharacteristic
172C2×C4⋊C4C2×C411yescharacteristic, complemented
173C2×C4⋊C4D411yescharacteristic
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4D411yescharacteristic
182C2×C22⋊C4D411yescharacteristic
183C2×C42—21no—
184C2×C42C2311yescharacteristic
185C4⋊C8—21no—
186C4⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8C2311yescharacteristic
192C22⋊C8C2311yescharacteristic
193C22⋊C8C2311yescharacteristic
194C22⋊C8C2311yescharacteristic
195#97—21no—
196C4×C8—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2311yescharacteristic
203#95C2311yescharacteristic
204#95C2311yescharacteristic
205#95C2311yescharacteristic
206C24—21no—
207C24—21no—
208C24C2×D411yescharacteristic
209C24C2×D411yescharacteristic
210#41—81no—
211#41—81no—
212#41—81no—
213#41—81no—
214C2×Q8—21no—
215C2×Q8C22×C411yescharacteristic
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—42no—
232C2×D4—44no—
233C2×D4—81no—
234C2×D4—81no—
235C2×D4—81no—
236C2×D4—81no—
237C2×D4—81no—
238C2×D4—81no—
239C2×D4C22⋊C411yescharacteristic
240C2×D4C22⋊C411yescharacteristic
241C2×D4C22⋊C411yescharacteristic
242C2×D4C22⋊C411yescharacteristic
243C2×D4C22×C411yescharacteristic
244C2×D4C22×C411yescharacteristic
245C2×D4C22×C411yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4C22×C411yescharacteristic
263C22×C4C22×C411yescharacteristic
264C22×C4C22×C411yescharacteristic
265C22×C4C22×C411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D411yescharacteristic
268C22×C4C2×D411yescharacteristic
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D411yescharacteristic
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D411yescharacteristic
273C22×C4C2×D411yescharacteristic
274C22×C4C2411yescharacteristic
275M16—81no—
276M16—81no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—81no—
282C2×C8—81no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4C22⋊C411yescharacteristic
295C4⋊C4C22⋊C411yescharacteristic
296C4⋊C4C22⋊C411yescharacteristic
297C4⋊C4C22⋊C411yescharacteristic
298C4⋊C4C22×C411yescharacteristic
299C4⋊C4C22×C411yescharacteristic
300C4⋊C4C22×C411yescharacteristic
301C4⋊C4C22×C411yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—42no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4C22⋊C411yescharacteristic
325C22⋊C4C22⋊C411yescharacteristic
326C22⋊C4C22⋊C411yescharacteristic
327C22⋊C4C22⋊C411yescharacteristic
328C22⋊C4C22⋊C411yescharacteristic
329C22⋊C4C22⋊C411yescharacteristic
330C22⋊C4C22⋊C411yescharacteristic
331C22⋊C4C22⋊C411yescharacteristic
332C42—21no—
333C42—21no—
334C42—21no—
335C42—81no—
336C42C22×C411yescharacteristic
337C42C22×C411yescharacteristic
338C42C2×D411yescharacteristic
339C42C2×D411yescharacteristic
340C23—21no—
341C23—21no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23C2×C22⋊C411yescharacteristic
358C23C2×C22⋊C411yescharacteristic
359C23C2×C22⋊C411yescharacteristic
360C23C2×C22⋊C411yescharacteristic
361C23C22×D411yescharacteristic
362Q8—41no—
363Q8—81no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—42no—
372D4—42no—
373D4—41no—
374D4—41no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—42no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—81no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—84no—
428C2×C4C2×C22⋊C411yescharacteristic
429C2×C4C2×C22⋊C411yescharacteristic
430C2×C4C2×C22⋊C411yescharacteristic
431C2×C4C2×C22⋊C411yescharacteristic
432C2×C4C2×C22⋊C411yescharacteristic
433C2×C4C2×C22⋊C411yescharacteristic
434C2×C4C2×C22⋊C411yescharacteristic
435C2×C4C2×C22⋊C411yescharacteristic
436C2×C4C23×C411yescharacteristic
437C2×C4C22×D411yescharacteristic
438C8—41no—
439C8—41no—
440C8—41no—
441C8—41no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—81no—
468C22—162no—
469C22—162no—
470C22#42112yes—
471C22C22×C22⋊C411yescharacteristic
472C22#57611yescharacteristic
473C22#57711yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C2—21no—
494C2—21no—
495C2—22no—
496C2—41no—
497C2—41no—
498C2—41no—
499C2—81no—
500C2—81no—
501C2—81no—
502C2#288311yescharacteristic
503C2#289711yescharacteristic
504C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3519C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2974C2semidirect
3. #2318C2semidirect
4. #1521C2semidirect
5. #1519C2semidirect
6. #1517C2semidirect
7. #1516C2semidirect
8. #1514C2semidirect
9. #1513C2semidirect
10. #1512C2semidirect
11. #1510C2semidirect
12. #1507C2semidirect
13. #1504C2semidirect
14. #1501C2semidirect
15. C2×#327C2semidirect
16. #555C4semidirect
17. #554C4semidirect
19. #553C4semidirect
20. #553C4semidirect
21. #550C4semidirect
24. #546C4semidirect
25. #546C4semidirect
27. D42C4semidirect
43. #432C22semidirect
44. #431C22semidirect
45. #407C22semidirect
46. #407C22semidirect
47. C2×C22⋊C8C22semidirect
48. C2×C22⋊C8C22semidirect
53. #383C22semidirect
54. #382C22semidirect
55. #381C22semidirect
56. #380C22semidirect
57. C4×C22⋊C4C22semidirect
58. C2×#95C22semidirect
59. #328C22semidirect
60. #328C22semidirect
61. #328C22semidirect
62. #328C22semidirect
67. #327C22semidirect
68. #327C22semidirect
69. #327C22semidirect
70. #327C22semidirect
71. #327C22semidirect
72. #327C22semidirect
73. #327C22semidirect
74. #327C22semidirect
77. C2×#41C2×C4semidirect
82. C22×D4C2×C4semidirect
91. C4⋊Q8C2×C4semidirect
93. #127C2×C4semidirect
98. #125C2×C4semidirect
102. #124C2×C4semidirect
113. C22⋊Q8C2×C4semidirect
131. #121C2×C4semidirect
132. #121C2×C4semidirect
133. #121C2×C4semidirect
143. C4×Q8C2×C4semidirect
158. C4×D4C2×C4semidirect
159. C4×D4C2×C4semidirect
160. C4×D4C2×C4semidirect
168. #117C2×C4semidirect
172. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35192
#29742
#23182
#15212
#15192
#15172
#15162
#15142
#15132
#15122
#15102
#15072
#15042
#15012
C2×#3272