Finite Group Databasebeta

← #498138

Subgroups — #498138

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2184C211yesmaximal, characteristic
2C2×C4×D16C211yesmaximal, characteristic, complemented
3C2×#2140C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic, complemented
5C22⋊C4×D8C211yesmaximal, characteristic, complemented
6#13239C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13130C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C22×M32C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic, complemented
24C2×#427C2211yescharacteristic
25#2363—21no—
26D4×C22⋊C4—21no—
27#2184C418yescomplemented
28C4×D16—24no—
29C2×#367C2211yescharacteristic, complemented
30C2×#366C2211yescharacteristic, complemented
31#2140—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×C4×C16C2211yescharacteristic, complemented
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic, complemented
46#1942C2211yescharacteristic, complemented
47#1926—21no—
48#1879—21no—
49#1879—21no—
50#1875—21no—
51#1864—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×#117—21no—
79C22×C22⋊C4—41no—
80C2×D16—22no—
81C2×D16—22no—
82C2×D16—41no—
83C2×D16—41no—
84C2×D16C2×C414yes—
85C2×D16C2×C414yescomplemented
86C2×M32—22no—
87C2×M32—22no—
88C2×M32—41no—
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92C22×C16C2311yescharacteristic
93#466—22no—
94#466C2×C414yescomplemented
95C4×D8—22no—
96C4×D8—22no—
97C4×D8—22no—
98C4×D8—41no—
99C4×D8—41no—
100C4×D8—41no—
101#427—22no—
102#427—41no—
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#106—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#390—21no—
120#390—21no—
121#390—21no—
122#390—41no—
123#389—21no—
124#388—41no—
125#386—41no—
126#386—41no—
127#386—41no—
128#380—21no—
129#379—41no—
130C4×C4⋊C4—21no—
131C4×C22⋊C4—41no—
132C4×C22⋊C4D411yescharacteristic
133#367—41no—
134#366—22no—
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—41no—
145C4×C16—22no—
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×C4C2×D411yescharacteristic
167C2×D8—28no—
168C2×D8—24no—
169C2×D8—21no—
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—44no—
183C2×D8—44no—
184C2×D8—44no—
185C2×D8—42no—
186C2×D8—42no—
187C2×D8C22×C412yes—
188C2×D8C22×C412yes—
189C2×D8C22×C412yes—
190C2×D8C2×D412yes—
191C2×D8#4112yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C22×C412yes—
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8#4112yes—
201C22×C8C2411yescharacteristic
202#121—24no—
203#121—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—81no—
208C4×D4—81no—
209C4×D4—81no—
210#117—42no—
211#117—81no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
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—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—84no—
231C2×C22⋊C4D811yescharacteristic
232C2×C22⋊C4D811yescharacteristic
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237D16—28no—
238D16—44no—
239M32—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×C16C22×C412yes—
250C2×C16C22×C412yes—
251C2×C16C2×D412yes—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107C22×C412yes—
261#107C2×D412yes—
262#106—42no—
263#102—24no—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—81no—
268#102—81no—
269#102—81no—
270C22⋊C8—22no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95D811yescharacteristic
279#95D811yescharacteristic
280C24—41no—
281C24—41no—
282C24—41no—
283C24—44no—
284C24—82no—
285C24—82no—
286C24—82no—
287C24—82no—
288C24—161no—
289C24—161no—
290C24C2×D811yescharacteristic
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—48no—
297C2×D4—44no—
298C2×D4—41no—
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—44no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—161no—
318C2×D4—161no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—161no—
335C22×C4—161no—
336C22×C4—161no—
337C22×C4C4×D412yes—
338C22×C4C2×D811yescharacteristic
339C22×C4C2×D811yescharacteristic
340C22×C4C2×D811yescharacteristic
341C22×C4C2×D811yescharacteristic
342C22×C4C2×D811yescharacteristic
343C22×C4#13512yes—
344C22×C4C22×D411yescharacteristic
345D8—22no—
346D8—22no—
347D8—22no—
348D8—22no—
349D8—22no—
350D8—22no—
351D8—22no—
352D8—48no—
353D8—41no—
354D8—41no—
355D8—41no—
356D8—41no—
357D8C4×D414yes—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8C4×D412yes—
371C2×C8C23×C411yescharacteristic
372C2×C8C22×D411yescharacteristic
373C2×C8C2×#4111yescharacteristic
374C2×C8C2×#4111yescharacteristic
375C2×C8H(5,𝔽2)11yescharacteristic
376C2×C8H(5,𝔽2)11yescharacteristic
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—161no—
388C22⋊C4—22no—
389C22⋊C4—41no—
390C22⋊C4—44no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—161no—
400C22⋊C4—161no—
401C22⋊C4—161no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—82no—
407C42—161no—
408C16—22no—
409C16—22no—
410C16—22no—
411C16—41no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—88no—
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—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—164no—
442C23—164no—
443C23—164no—
444C23—164no—
445C23—164no—
446C23C4×D812yes—
447C23C22×D811yescharacteristic
448D4—24no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—88no—
457D4—81no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—164no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
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—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
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—164no—
495C2×C4—164no—
496C2×C4—164no—
497C2×C4C4×D812yes—
498C2×C4#50912yes—
499C2×C4C2×C4×D411yescharacteristic
500C2×C4D4211yescharacteristic
501C2×C4#54611yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4#57611yescharacteristic
504C2×C4#57711yescharacteristic
505C8—22no—
506C8—41no—
507C8—41no—
508C8C2×C4×D411yescharacteristic
509C8#51811yescharacteristic
510C8#54611yescharacteristic
511C8#55111yescharacteristic
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—44no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—1616no—
533C22—161no—
534C22—161no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22—161no—
541C22C2×C4×D811yescharacteristic
542C22D4×D811yescharacteristic
543C22#329411yescharacteristic
544C22C2×#50911yescharacteristic
545C22#341411yescharacteristic
546C22#341511yescharacteristic
547C22#341511yescharacteristic
548C4—22no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—82no—
554C4—82no—
555C4—161no—
556C4—161no—
557C4—161no—
558C4#235811yescharacteristic
559C4#294611yescharacteristic
560C4#328211yescharacteristic
561C4#329711yescharacteristic
562C2—42no—
563C2—84no—
564C2—162no—
565C2—162no—
566C2#2287011yescharacteristic
567C2C4×#50911yescharacteristic
568C2#3345611yescharacteristic
569C2#3383011yescharacteristic
570C2#3383411yescharacteristic
571C2#3388911yescharacteristic
572C2#3389211yescharacteristic
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×C4×D16C2semidirect
3. C2×#2140C2semidirect
4. #23027C2semidirect
5. C22⋊C4×D8C2semidirect
6. #13239C2semidirect
7. #13147C2semidirect
8. #13130C2semidirect
9. #13061C2semidirect
10. #13012C2semidirect
11. #13012C2semidirect
12. #12937C2semidirect
13. #12921C2semidirect
14. #12637C2semidirect
15. #12612C2semidirect
23. C2×C4×D8C22semidirect
27. #2184C4semidirect
29. C2×#367C22semidirect
30. C2×#366C22semidirect
38. C2×#357C22semidirect
39. C2×#357C22semidirect
41. C2×#346C22semidirect
42. C2×C4×C16C22semidirect
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
94. #466C2×C4semidirect
140. #357C2×C4semidirect
141. #357C2×C4semidirect
142. #357D4semidirect
143. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21842
C2×C4×D162
C2×#21402
#230272
C22⋊C4×D82
#132392
#131472
#131302
#130612
#130122
#129372
#129212
#126372
#126122