Finite Group Databasebeta

← #498144

Subgroups — #498144

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2187C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C22×#357C211yesmaximal, characteristic
4#23024C211yesmaximal, characteristic, complemented
5C22⋊C4×D8C211yesmaximal, characteristic
6#13238C211yesmaximal, characteristic, complemented
7#13149C211yesmaximal, characteristic, complemented
8#13130C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12919C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic, complemented
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C22×#107C2211yescharacteristic
25#2360—21no—
26D4×C22⋊C4—21no—
27#2187C418yescomplemented
28#2175—24no—
29C2×#366C2211yescharacteristic, complemented
30C2×#366C2211yescharacteristic, complemented
31C2×#358C2211yescharacteristic, complemented
32C2×#358C2211yescharacteristic, complemented
33C2×#357—24no—
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic, complemented
42C2×#346C2211yescharacteristic, complemented
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic, complemented
46#1942C2211yescharacteristic, complemented
47#1928—21no—
48#1879—21no—
49#1879—21no—
50#1872—21no—
51#1863—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×Q16C2311yescharacteristic
64C22×D8—22no—
65C22×D8—22no—
66C22×D8—21no—
67C22×D8—22no—
68C22×D8—41no—
69C22×D8—41no—
70C22×D8C2×C414yes—
71C22×D8C2311yescharacteristic
72C22×D8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×C22⋊Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C22×C4⋊C4—21no—
79C22×C22⋊C4—41no—
80C2×SD32—22no—
81C2×SD32—22no—
82C2×SD32—41no—
83C2×SD32—41no—
84C2×SD32C2×C414yescomplemented
85C2×SD32C2×C414yescomplemented
86C22×C16—22no—
87C22×C16—22no—
88C22×C16—41no—
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92C22×C16C2311yescharacteristic
93#467—22no—
94#467C2×C414yescomplemented
95C4×Q16—22no—
96C4×Q16—41no—
97C4×D8—22no—
98C4×D8—22no—
99C4×D8—41no—
100C4×D8—41no—
101C2×#107—22no—
102C2×#107—21no—
103C2×#107—41no—
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—21no—
120#390—21no—
121#390—21no—
122#389—21no—
123#389—41no—
124#387—41no—
125#386—41no—
126#386—41no—
127#385—41no—
128#380—21no—
129#379—41no—
130C4×C4⋊C4—21no—
131C4×C22⋊C4—41no—
132C4×C22⋊C4D411yescharacteristic
133#366—22no—
134#366—41no—
135#358—22no—
136#358—41no—
137#357—22no—
138#357—22no—
139#357—22no—
140#357C2×C414yescomplemented
141#357C2×C414yescomplemented
142#357D414yes—
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×Q8—21no—
153C22×D4—24no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—41no—
159C22×D4—42no—
160C22×D4—81no—
161C22×D4—81no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16C22×C412yes—
172C2×D8—28no—
173C2×D8—24no—
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—44no—
183C2×D8—44no—
184C2×D8—44no—
185C2×D8—42no—
186C2×D8—42no—
187C2×D8C22×C412yes—
188C2×D8C22×C412yes—
189C2×D8C2×D412yes—
190C2×D8#4112yes—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8C22×C412yes—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8#4112yes—
200C22×C8C2411yescharacteristic
201C22⋊Q8—24no—
202C22⋊Q8—42no—
203C4×Q8—42no—
204C4×Q8—81no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—81no—
208C4×D4—81no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—81no—
220C2×C22⋊C4—41no—
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—84no—
229C2×C22⋊C4D811yescharacteristic
230C2×C22⋊C4D811yescharacteristic
231C2×C42—41no—
232C2×C42—41no—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235SD32—28no—
236SD32—44no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—44no—
247C2×C16C22×C412yes—
248C2×C16C22×C412yes—
249C2×C16C2×D412yes—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—42no—
259#107C22×C412yes—
260#107C2×D412yes—
261#103—24no—
262#103—42no—
263#103—81no—
264#102—42no—
265#102—42no—
266#102—81no—
267#102—81no—
268C22⋊C8—22no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95D811yescharacteristic
279#95D811yescharacteristic
280C24—41no—
281C24—41no—
282C24—44no—
283C24—82no—
284C24—82no—
285C24—82no—
286C24—82no—
287C24—161no—
288C24—161no—
289C24C2×D811yescharacteristic
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—48no—
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—44no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—82no—
314C2×D4—82no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—161no—
332C22×C4—161no—
333C22×C4—161no—
334C22×C4C4×D412yes—
335C22×C4C2×D811yescharacteristic
336C22×C4C2×D811yescharacteristic
337C22×C4C2×D811yescharacteristic
338C22×C4C2×D811yescharacteristic
339C22×C4C2×D811yescharacteristic
340C22×C4#13512yes—
341C22×C4C22×D411yescharacteristic
342Q16—22no—
343Q16—22no—
344Q16—41no—
345Q16—41no—
346D8—22no—
347D8—22no—
348D8—22no—
349D8—22no—
350D8—22no—
351D8—48no—
352D8—41no—
353D8—41no—
354D8C4×D414yes—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8C4×D412yes—
368C2×C8C23×C411yescharacteristic
369C2×C8C22×D411yescharacteristic
370C2×C8C2×#4111yescharacteristic
371C2×C8C2×#4111yescharacteristic
372C2×C8H(5,𝔽2)11yescharacteristic
373C2×C8H(5,𝔽2)11yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—44no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—161no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—44no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—161no—
397C22⋊C4—161no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—82no—
403C42—161no—
404C16—22no—
405C16—22no—
406C16—22no—
407C16—41no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—88no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—88no—
426C23—81no—
427C23—81no—
428C23—164no—
429C23—164no—
430C23—164no—
431C23—164no—
432C23—164no—
433C23C4×D812yes—
434C23#50912yes—
435C23#51012yes—
436C23C22×D811yescharacteristic
437Q8—42no—
438Q8—42no—
439Q8—81no—
440Q8—81no—
441D4—24no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—88no—
448D4—81no—
449D4—81no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—164no—
487C2×C4—164no—
488C2×C4—164no—
489C2×C4C4×D812yes—
490C2×C4C2×C4×D411yescharacteristic
491C2×C4D4211yescharacteristic
492C2×C4#54611yescharacteristic
493C2×C4C2×#13511yescharacteristic
494C2×C4#57611yescharacteristic
495C2×C4#57711yescharacteristic
496C8—22no—
497C8—41no—
498C8—41no—
499C8C2×C4×D411yescharacteristic
500C8#51811yescharacteristic
501C8#54611yescharacteristic
502C8#55111yescharacteristic
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—1616no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22#217514yescomplemented
525C22C2×C4×D811yescharacteristic
526C22D4×D811yescharacteristic
527C22#329411yescharacteristic
528C22C2×#50911yescharacteristic
529C22C2×#51011yescharacteristic
530C22#341511yescharacteristic
531C22#341611yescharacteristic
532C4—22no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—161no—
541C4—161no—
542C4—161no—
543C4#235811yescharacteristic
544C4#294611yescharacteristic
545C4#328211yescharacteristic
546C4#329711yescharacteristic
547C2—24no—
548C2—84no—
549C2—162no—
550C2#2287011yescharacteristic
551C2C2×#217511yescharacteristic
552C2#3345411yescharacteristic
553C2#3383011yescharacteristic
554C2#3383211yescharacteristic
555C2#3387711yescharacteristic
556C2#3388111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
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×#2175C2semidirect
4. #23024C2semidirect
6. #13238C2semidirect
7. #13149C2semidirect
8. #13130C2semidirect
9. #13060C2semidirect
10. #13014C2semidirect
11. #13013C2semidirect
12. #12935C2semidirect
13. #12919C2semidirect
14. #12637C2semidirect
15. #12611C2semidirect
21. C2×C4×Q16C22semidirect
27. #2187C4semidirect
29. C2×#366C22semidirect
30. C2×#366C22semidirect
31. C2×#358C22semidirect
32. C2×#358C22semidirect
41. C2×#346C22semidirect
42. C2×#346C22semidirect
45. #1969C22semidirect
46. #1942C22semidirect
54. C4×#107C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1374C22semidirect
61. #1374C22semidirect
84. C2×SD32C2×C4semidirect
85. C2×SD32C2×C4semidirect
94. #467C2×C4semidirect
140. #357C2×C4semidirect
141. #357C2×C4semidirect
143. C22⋊C16C2×C4semidirect
524. C22#2175semidirect

Maximal subgroups

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

subgroupindex
C2×#21872
C2×#21752
C22×#3572
#230242
C22⋊C4×D82
#132382
#131492
#131302
#130602
#130142
#130132
#129352
#129192
#126372
#126112