Finite Group Databasebeta

← #375894

Subgroups — #375894

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic, complemented
2C2×#2025C211yesmaximal, characteristic
3C2×#1864C212yesmaximal
4#13785C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11049C211yesmaximal, characteristic
8#10931C212yesmaximal
9#10908C212yesmaximal, complemented
10#10884C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2330—28no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389—22no—
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#2025C418yescomplemented
28#1864—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1437—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—22no—
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#125—24no—
47C2×#125—24no—
48C2×#125—22no—
49C2×#125C2×C414yescomplemented
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102C2×C418yes—
79#390—22no—
80#390—22no—
81#390C2×C414yes—
82#390Q814yes—
83#389—216no—
84#389—24no—
85#389D418yes—
86#387—216no—
87#384—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—42no—
91#376—24no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340C2×C418yescomplemented
104#336—24no—
105#336—22no—
106#336—24no—
107#336C2×C414yescomplemented
108C25—21no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4C22×C412yes—
117C22×D4C2×Q812yes—
118C22×D4#4112yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—82no—
128C23×C4—82no—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8C22×C414yes—
148#125—216no—
149#125—216no—
150C4×D4—48no—
151C4×D4—48no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×Q812yes—
183C2×C4⋊C4#4112yes—
184C2×C4⋊C4#4114yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—28no—
202C2×C42—42no—
203C2×C42—82no—
204C2×C42—82no—
205C2×C42—82no—
206C2×C42C22×C412yes—
207C2×C42C2×D414yes—
208C2×C42C2×Q812yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211#106—48no—
212#102—216no—
213#102—48no—
214#95—28no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—42no—
219#95—82no—
220#95—82no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24C22×D411yescharacteristic
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4C4×Q814yes—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—88no—
300C22×C4—88no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4C4×D414yes—
304C22×C4#12012yes—
305C22×C4#12012yes—
306C22×C4C22⋊Q814yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C22×Q811yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C42−1+412yes—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—232no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—88no—
346C4⋊C4—88no—
347C4⋊C4C4×D418yes—
348C22⋊C4—48no—
349C22⋊C4—88no—
350C22⋊C4—88no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—88no—
358C42—84no—
359C23—24no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—44no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23C2×C4×D411yescharacteristic
383C23C2×#12011yescharacteristic
384C23C2×C22⋊Q811yescharacteristic
385C23#52511yescharacteristic
386C23#52511yescharacteristic
387C23#53511yescharacteristic
388C23#53611yescharacteristic
389C23#57612yes—
390C23#57712yes—
391C23#57712yes—
392C23#57712yes—
393D4—28no—
394D4—44no—
395D4—44no—
396D4—44no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—28no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—816no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4C2×C4×D412yes—
452C2×C4C2×C4×Q811yescharacteristic
453C2×C4C4×#4111yescharacteristic
454C2×C4#52012yes—
455C2×C4#52512yes—
456C2×C4#52911yescharacteristic
457C2×C4#53311yescharacteristic
458C2×C4#54212yes—
459C2×C4#54612yes—
460C2×C4#54612yes—
461C2×C4#54812yes—
462C2×C4D4×Q812yes—
463C2×C4#55212yes—
464C2×C4#55412yes—
465C2×C4#55612yes—
466C2×C4#55912yes—
467C8—44no—
468C8—82no—
469C22—44no—
470C22—48no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—84no—
477C22—82no—
478C22—84no—
479C22—84no—
480C22C4×#12011yescharacteristic
481C22C4×C22⋊Q811yescharacteristic
482C22#232611yescharacteristic
483C22#240411yescharacteristic
484C22#240911yescharacteristic
485C22#258011yescharacteristic
486C22#259911yescharacteristic
487C22#294612yes—
488C22#295212yes—
489C22#301012yes—
490C22#301212yes—
491C22#308112yes—
492C22#308412yes—
493C22#311912yes—
494C22#311912yes—
495C22#312312yes—
496C22#312512yes—
497C22#322011yescharacteristic
498C22#322111yescharacteristic
499C22#322211yescharacteristic
500C22#322211yescharacteristic
501C22#324112yes—
502C22#324412yes—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—48no—
507C4—44no—
508C4—82no—
509C4—84no—
510C4—82no—
511C4—82no—
512C4#236012yes—
513C4#236712yes—
514C4#245812yes—
515C4#246512yes—
516C2—84no—
517C2#1417211yescharacteristic
518C2#2146412yes—
519C2#2161912yes—
520C2#2222711yescharacteristic
521C2#2223211yescharacteristic
522C2#2394111yescharacteristic
523C2#2394211yescharacteristic
524C2#2430912yes—
525C2#2926511yescharacteristic
526C2#2926911yescharacteristic
527C2#2940612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2330C2semidirect
5. #11385C2semidirect
6. #11323C2semidirect
9. #10908C2semidirect
10. #10884C2semidirect
11. #10796C2semidirect
27. #2025C4semidirect
49. C2×#125C2×C4semidirect
103. #340C2×C4semidirect
107. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#20252
C2×#18642
#137852
#113852
#113232
#110492
#109312
#109082
#108842
#107962