Finite Group Databasebeta

← #498148

Subgroups — #498148

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2184C211yesmaximal, characteristic
2C2×#2177C211yesmaximal, characteristic, complemented
3C22×#357C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic, complemented
5C22⋊C4×D8C211yesmaximal, characteristic, complemented
6#13239C211yesmaximal, characteristic, complemented
7#13149C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic, complemented
24C22×#107C2211yescharacteristic
25#2361—21no—
26D4×C22⋊C4—21no—
27#2184C418yescomplemented
28#2177—24no—
29C2×#367C2211yescharacteristic, complemented
30C2×#367C2211yescharacteristic, complemented
31C2×#357—24no—
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic, complemented
39C2×#357C2211yescharacteristic, complemented
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic, complemented
42C2×#346C2211yescharacteristic, complemented
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic, complemented
46#1942C2211yescharacteristic, complemented
47#1925—21no—
48#1879—21no—
49#1879—21no—
50#1874—21no—
51#1862—21no—
52#1808C2211yescharacteristic
53#1786—21no—
54C4×#107C2211yescharacteristic, complemented
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61#1374C2211yescharacteristic, complemented
62C23×D4—21no—
63C22×D8—22no—
64C22×D8—22no—
65C22×D8—21no—
66C22×D8—22no—
67C22×D8—41no—
68C22×D8—41no—
69C22×D8C2×C414yescomplemented
70C22×D8C2311yescharacteristic
71C22×D8C2311yescharacteristic
72C22×D8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×#121—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—41no—
79C22×C4⋊C4—21no—
80C22×C22⋊C4—41no—
81C2×D16—22no—
82C2×D16—22no—
83C2×D16—41no—
84C2×D16—41no—
85C2×D16C2×C414yescomplemented
86C2×D16C2×C414yescomplemented
87C22×C16—22no—
88C22×C16—22no—
89C22×C16—41no—
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92C22×C16C2311yescharacteristic
93C22×C16C2311yescharacteristic
94#466—22no—
95#466C2×C414yescomplemented
96C4×D8—22no—
97C4×D8—22no—
98C4×D8—22no—
99C4×D8—41no—
100C4×D8—41no—
101C4×D8—41no—
102C2×#107—22no—
103C2×#107—21no—
104C2×#107—41no—
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×C22⋊C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#390—21no—
121#390—21no—
122#390—21no—
123#389—21no—
124#388—41no—
125#386—41no—
126#386—41no—
127#386—41no—
128#385—41no—
129#380—21no—
130#379—41no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4D411yescharacteristic
133#367—22no—
134#367—41no—
135#357—22no—
136#357—22no—
137#357—22no—
138#357—22no—
139#357—41no—
140#357C2×C414yescomplemented
141#357C2×C414yescomplemented
142#357D414yescomplemented
143C22⋊C16C2×C414yescomplemented
144#346—22no—
145#346—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C25—41no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4—81no—
162C22×D4—81no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4C2×D411yescharacteristic
168C2×D8—28no—
169C2×D8—24no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—44no—
184C2×D8—44no—
185C2×D8—44no—
186C2×D8—42no—
187C2×D8—42no—
188C2×D8C22×C412yes—
189C2×D8C22×C412yes—
190C2×D8C22×C412yes—
191C2×D8C2×D412yes—
192C2×D8#4112yes—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8C22×C412yes—
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8#4112yes—
202C22×C8C2411yescharacteristic
203#121—24no—
204#121—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—81no—
209C4×D4—81no—
210C4×D4—81no—
211C4×D4—84no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—81no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—84no—
233C2×C22⋊C4D811yescharacteristic
234C2×C22⋊C4D811yescharacteristic
235C2×C42—41no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238D16—28no—
239D16—44no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—44no—
250C2×C16C22×C412yes—
251C2×C16C22×C412yes—
252C2×C16C2×D412yes—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—42no—
262#107C22×C412yes—
263#107C2×D412yes—
264#102—24no—
265#102—42no—
266#102—42no—
267#102—42no—
268#102—81no—
269#102—81no—
270#102—81no—
271C22⋊C8—22no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95D811yescharacteristic
280#95D811yescharacteristic
281C24—41no—
282C24—41no—
283C24—41no—
284C24—44no—
285C24—82no—
286C24—82no—
287C24—82no—
288C24—82no—
289C24—161no—
290C24—161no—
291C24C2×D811yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—44no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—161no—
338C22×C4—161no—
339C22×C4—161no—
340C22×C4—161no—
341C22×C4C4×D412yes—
342C22×C4C2×D811yescharacteristic
343C22×C4C2×D811yescharacteristic
344C22×C4C2×D811yescharacteristic
345C22×C4C2×D811yescharacteristic
346C22×C4C2×D811yescharacteristic
347C22×C4#13512yes—
348C22×C4C22×D411yescharacteristic
349D8—22no—
350D8—22no—
351D8—22no—
352D8—22no—
353D8—22no—
354D8—22no—
355D8—22no—
356D8—48no—
357D8—41no—
358D8—41no—
359D8—41no—
360D8—41no—
361D8C4×D414yescomplemented
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8C4×D412yes—
375C2×C8C23×C411yescharacteristic
376C2×C8C22×D411yescharacteristic
377C2×C8C2×#4111yescharacteristic
378C2×C8C2×#4111yescharacteristic
379C2×C8H(5,𝔽2)11yescharacteristic
380C2×C8H(5,𝔽2)11yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—161no—
404C22⋊C4—161no—
405C22⋊C4—161no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—82no—
411C16—22no—
412C16—22no—
413C16—22no—
414C16—41no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—88no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—88no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—164no—
443C23—164no—
444C23—164no—
445C23—164no—
446C23—164no—
447C23C4×D812yes—
448C23#50912yes—
449C23#50912yes—
450C23C22×D811yescharacteristic
451D4—24no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—88no—
460D4—88no—
461D4—81no—
462D4—81no—
463D4—81no—
464D4—81no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—164no—
499C2×C4—164no—
500C2×C4—164no—
501C2×C4—164no—
502C2×C4C4×D812yes—
503C2×C4C2×C4×D411yescharacteristic
504C2×C4D4211yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4C2×#13511yescharacteristic
507C2×C4#57611yescharacteristic
508C2×C4#57711yescharacteristic
509C8—22no—
510C8—41no—
511C8—41no—
512C8C2×C4×D411yescharacteristic
513C8#51811yescharacteristic
514C8#54611yescharacteristic
515C8#55111yescharacteristic
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—1616no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22—161no—
544C22#217714yescomplemented
545C22C2×C4×D811yescharacteristic
546C22D4×D811yescharacteristic
547C22#329411yescharacteristic
548C22C2×#50911yescharacteristic
549C22C2×#50911yescharacteristic
550C22#341511yescharacteristic
551C22#341611yescharacteristic
552C4—22no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—82no—
558C4—82no—
559C4—161no—
560C4—161no—
561C4—161no—
562C4#235811yescharacteristic
563C4#294611yescharacteristic
564C4#328211yescharacteristic
565C4#329711yescharacteristic
566C2—24no—
567C2—84no—
568C2—162no—
569C2—162no—
570C2#2287011yescharacteristic
571C2C2×#217711yescharacteristic
572C2#3345411yescharacteristic
573C2#3383011yescharacteristic
574C2#3383111yescharacteristic
575C2#3387311yescharacteristic
576C2#3388511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2184C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2177C2semidirect
3. C22×#357C2semidirect
4. #23025C2semidirect
5. C22⋊C4×D8C2semidirect
6. #13239C2semidirect
7. #13149C2semidirect
8. #13131C2semidirect
9. #13059C2semidirect
10. #13012C2semidirect
11. #13012C2semidirect
12. #12937C2semidirect
13. #12920C2semidirect
14. #12637C2semidirect
15. #12611C2semidirect
23. C2×C4×D8C22semidirect
27. #2184C4semidirect
29. C2×#367C22semidirect
30. C2×#367C22semidirect
38. C2×#357C22semidirect
39. C2×#357C22semidirect
41. C2×#346C22semidirect
42. C2×#346C22semidirect
45. #1970C22semidirect
46. #1942C22semidirect
54. C4×#107C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1374C22semidirect
61. #1374C22semidirect
69. C22×D8C2×C4semidirect
85. C2×D16C2×C4semidirect
86. C2×D16C2×C4semidirect
95. #466C2×C4semidirect
140. #357C2×C4semidirect
141. #357C2×C4semidirect
142. #357D4semidirect
143. C22⋊C16C2×C4semidirect
361. D8C4×D4semidirect
544. C22#2177semidirect

Maximal subgroups

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

subgroupindex
C2×#21842
C2×#21772
C22×#3572
#230252
C22⋊C4×D82
#132392
#131492
#131312
#130592
#130122
#129372
#129202
#126372
#126112