Finite Group Databasebeta

← #365657

Subgroups — #365657

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3#14159C211yesmaximal, characteristic
4#14149C211yesmaximal, characteristic
5#13966C211yesmaximal, characteristic
6#11287C214yesmaximal
7#10918C214yesmaximal
8#10854C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11#2368—22no—
12#2360—22no—
13#2345—22no—
14#2345—21no—
15Q8×C22⋊C4—21no—
16#2329—22no—
17#2315C2212yes—
18C4×C22⋊Q8—22no—
19C2×#385C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22#2273C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1996C418yescomplemented
28C2×#342C2212yes—
29C2×#342C2214yes—
30C2×#342C2212yes—
31C2×#342C2214yes—
32#1446—21no—
33#1445—21no—
34#1442—22no—
35#1441—22no—
36#1436—22no—
37#1436—22no—
38#1434—22no—
39C22⋊#95C2212yes—
40C22⋊#95C2214yes—
41C22⋊#95C2214yes—
42C23×Q8D411yescharacteristic
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C414yes—
52C2×C22⋊Q8Q814yes—
53C2×C4×Q8—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2312yes—
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2312yes—
64C22×C22⋊C4C2312yes—
65C22×C22⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42D412yes—
69#391—24no—
70#391—22no—
71#389—24no—
72#389—22no—
73#387—24no—
74#387—24no—
75#387—24no—
76#385—24no—
77#385—24no—
78#385—28no—
79#380—22no—
80#380—22no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—22no—
86#380C2×C414yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—24no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4C2×C414yes—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—24no—
104C2×#95—22no—
105C2×#95—24no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—24no—
115C2×#95—22no—
116C2×#95—24no—
117C2×#95D412yes—
118C2×#95D412yes—
119#342—24no—
120#342—24no—
121#342—28no—
122#342C2×C4116yes—
123C25C2411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—82no—
128C22×Q8C4⋊C414yescomplemented
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—44no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D414yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q812yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C4×Q8—84no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C4⋊C414yescomplemented
181C2×C4⋊C4C4⋊C418yescomplemented
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22×C414yes—
211C2×C22⋊C4C22×C414yes—
212C2×C22⋊C4C22×C414yes—
213C2×C22⋊C4C2×Q814yes—
214C2×C22⋊C4#4114yes—
215C2×C22⋊C4#4114yes—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—42no—
222C2×C42—82no—
223C2×C42—82no—
224C2×C42—82no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—48no—
236#95—48no—
237#95—44no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—42no—
243#95—44no—
244#95—82no—
245#95—84no—
246#95—84no—
247C24—42no—
248C24—42no—
249C24C23×C411yescharacteristic
250C24C22×D411yescharacteristic
251C24C22×D411yescharacteristic
252C24C22×D411yescharacteristic
253C24C22×Q811yescharacteristic
254C24C2×#4111yescharacteristic
255C24C2×#4111yescharacteristic
256C24C2×#4112yes—
257C24C2×#4112yes—
258C242−1+412yes—
259C242−1+412yes—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—44no—
263C2×Q8—82no—
264C2×Q8—84no—
265C2×Q8—84no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4C2×C4⋊C414yes—
330C22×C4C2×C4⋊C414yes—
331C22×C4C2×C4⋊C412yes—
332C22×C4C2×C4⋊C412yes—
333C22×C4C4×D412yes—
334C22×C4C4×D412yes—
335C22×C4C4×D412yes—
336C22×C4C4×D412yes—
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4#12312yes—
346C22×C4#12312yes—
347C22×C4#12414yes—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—44no—
353C4⋊C4—82no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—88no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—28no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—44no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C42—44no—
380C42—82no—
381C42—84no—
382C42—84no—
383C42—84no—
384C42—84no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—44no—
394C23—41no—
395C23—41no—
396C23—44no—
397C23—41no—
398C23—41no—
399C23—44no—
400C23—41no—
401C23—41no—
402C23—44no—
403C23C22×C4⋊C411yescharacteristic
404C23C2×C4×D411yescharacteristic
405C23C2×C4×D411yescharacteristic
406C23C4×#4112yes—
407C23#51912yes—
408C23C2×#12011yescharacteristic
409C23C2×C22⋊Q811yescharacteristic
410C23C2×C22⋊Q811yescharacteristic
411C23C2×#12311yescharacteristic
412C23C2×#12411yescharacteristic
413C23#52912yes—
414C23#53211yescharacteristic
415C23#53611yescharacteristic
416C23#53611yescharacteristic
417C23#54112yes—
418C23#54712yes—
419C23#54812yes—
420C23#54812yes—
421C23#54812yes—
422C23#55212yes—
423C23#55612yes—
424C23#55712yes—
425C23#55712yes—
426Q8—48no—
427Q8—84no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—88no—
494C2×C4#38512yes—
495C2×C4#38512yes—
496C2×C4#38512yes—
497C2×C4#38512yes—
498C22—22no—
499C22—24no—
500C22—22no—
501C22—24no—
502C22—22no—
503C22—22no—
504C22—24no—
505C22—24no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22C2×#38511yescharacteristic
527C22C4×#12411yescharacteristic
528C22#232711yescharacteristic
529C22#235112yes—
530C22#236212yes—
531C22#242112yes—
532C22#243211yescharacteristic
533C22#243611yescharacteristic
534C22#245312yes—
535C22#256511yescharacteristic
536C22#256811yescharacteristic
537C22#302512yes—
538C22#302612yes—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—44no—
546C4—84no—
547C4—84no—
548C4—84no—
549C4—82no—
550C4—84no—
551C4—84no—
552C2—21no—
553C2—22no—
554C2—22no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—22no—
559C2—22no—
560C2#1397811yescharacteristic
561C2#2088811yescharacteristic
562C2#2088911yescharacteristic
563C2#2566512yes—
564C2#2594912yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1996C4semidirect
128. C22×Q8C4⋊C4semidirect
180. C2×C4⋊C4C4⋊C4semidirect
181. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342122
C2×#19962
#141592
#141492
#139662
#112872
#109182
#108542
C4×#3422