Finite Group Databasebeta

← #188680

Subgroups — #188680

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20815C211yesmaximal, characteristic
4C2×#1559C211yesmaximal, characteristic
5C2×#1554C211yesmaximal, characteristic
6#8594C211yesmaximal, characteristic, complemented
7#8593C211yesmaximal, characteristic, complemented
8#8591C211yesmaximal, characteristic, complemented
9#8590C211yesmaximal, characteristic, complemented
10#8590C211yesmaximal, characteristic, complemented
11#8589C211yesmaximal, characteristic, complemented
12#8587C211yesmaximal, characteristic, complemented
13#8586C211yesmaximal, characteristic, complemented
14#8043C211yesmaximal, characteristic
15#8040C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2360—21no—
22#2360—21no—
23#2360C412yes—
24#2360C412yes—
25#2355—21no—
26#2355C412yes—
27Q8×C4⋊C4—21no—
28Q8×C4⋊C4C412yes—
29#2330—21no—
30#2330—21no—
31#2317—21no—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C2×#384C2211yescharacteristic
37#2274—21no—
38C4×#117C2211yescharacteristic
39#1940—21no—
40#1940—21no—
41#1851—21no—
42#1851—21no—
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1798—21no—
52#1798—21no—
53#1768—21no—
54#1768—21no—
55#1559—24no—
56#1554—24no—
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×M16—21no—
74#533—22no—
75#533—22no—
76#533D414yes—
77#533Q814yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×#125—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8C2×C412yes—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8C2×C412yes—
87C2×C4×Q8C2×C412yes—
88C2×C4×Q8C2×C412yes—
89C2×#117—21no—
90C2×#117D411yescharacteristic
91C2×#117D411yescharacteristic
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2×C412yes—
95C22×C42C2311yescharacteristic
96#432—22no—
97#432—22no—
98#432—22no—
99#432—22no—
100#431—22no—
101#431—22no—
102#423—22no—
103#423—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127#391—21no—
128#391—21no—
129#391C2×C412yes—
130#391C2×C412yes—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389C2×C412yes—
142#389C2×C412yes—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#385—22no—
148#385—22no—
149#384—22no—
150#384—21no—
151#384—21no—
152#384C2×C412yes—
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#382—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4C2×C412yes—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4C2×C411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4C2×C411yescharacteristic
173C4×C22⋊C4C2×C411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175C4×C22⋊C4D411yescharacteristic
176#376D411yescharacteristic
177#376D411yescharacteristic
178C43C2311yescharacteristic
179C43C2311yescharacteristic
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#333—22no—
189#333—22no—
190#333—22no—
191#333—22no—
192#332—22no—
193#332—22no—
194#332—22no—
195#332—22no—
196C22×Q8—21no—
197C22×Q8C22×C412yes—
198C23×C4—22no—
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C2×M16—42no—
203C2×M16—81no—
204C2×M16—81no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8C2×D412yes—
218C4⋊Q8C2×D412yes—
219C4⋊Q8C2×Q812yes—
220C4⋊Q8C2×Q812yes—
221#125—24no—
222#125—24no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8C4⋊C418yes—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—81no—
231C4×Q8—81no—
232C4×Q8—81no—
233C4×Q8—81no—
234C4×Q8—82no—
235C4×Q8—82no—
236C4×Q8C4⋊C418yes—
237#117—24no—
238#117—24no—
239#117—22no—
240#117—22no—
241#117—41no—
242#117—41no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—81no—
262C2×C4⋊C4—81no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C22×C412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×Q812yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300C2×C42C2411yescharacteristic
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321C4⋊C8—42no—
322C4⋊C8—81no—
323C4⋊C8—81no—
324C4⋊C8—81no—
325C4⋊C8—81no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329#97—42no—
330#97—42no—
331C4×C8—42no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348C24C2×C22⋊C411yescharacteristic
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8C2×C4⋊C414yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—44no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C4⋊C412yes—
388C22×C4#11712yes—
389C22×C4#12112yes—
390C22×C4#12112yes—
391C22×C4C22⋊Q812yes—
392C22×C4C22⋊Q812yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396M16—84no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4C2×C4⋊C414yes—
452C22⋊C4—28no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—28no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—82no—
477C42—82no—
478C42—82no—
479C42—82no—
480C42C2×C4⋊C412yes—
481C42#11712yes—
482C42#12112yes—
483C42#12112yes—
484C42C22⋊Q812yes—
485C42C22⋊Q812yes—
486C42#13712yes—
487C42#13712yes—
488C42C22×D411yescharacteristic
489C42C22×Q811yescharacteristic
490C42C2×#4111yescharacteristic
491C42C2×#4111yescharacteristic
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23#38012yes—
499C23C22×C22⋊C411yescharacteristic
500Q8—44no—
501Q8—44no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—164no—
559C2×C4#38012yes—
560C2×C4#38012yes—
561C2×C4#38012yes—
562C2×C4#41912yes—
563C2×C4#41912yes—
564C2×C4C22×C4⋊C411yescharacteristic
565C2×C4C2×#11711yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×#13711yescharacteristic
571C2×C4C2×#13711yescharacteristic
572C2×C4#57511yescharacteristic
573C2×C4#57511yescharacteristic
574C2×C4#57611yescharacteristic
575C2×C4#57711yescharacteristic
576C2×C4#57711yescharacteristic
577C2×C4#57811yescharacteristic
578C8—81no—
579C8—81no—
580C8—81no—
581C8—81no—
582C8—81no—
583C8—81no—
584C8—81no—
585C8—81no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#38011yescharacteristic
594C22#288811yescharacteristic
595C22#288911yescharacteristic
596C22C2×#41911yescharacteristic
597C22#289611yescharacteristic
598C22#289711yescharacteristic
599C22#289811yescharacteristic
600C4—22no—
601C4—22no—
602C4—21no—
603C4—21no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—44no—
614C4—44no—
615C4—82no—
616C4—82no—
617C4#303711yescharacteristic
618C4#303711yescharacteristic
619C4#304511yescharacteristic
620C4#304611yescharacteristic
621C4#307811yescharacteristic
622C4#307811yescharacteristic
623C4#308311yescharacteristic
624C4#308411yescharacteristic
625C2—42no—
626C2#1776011yescharacteristic
627C2#1776111yescharacteristic
628C2#1776611yescharacteristic
629C2#1776711yescharacteristic
630C2#2064011yescharacteristic
631C2#2064711yescharacteristic
632C2#2065211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8594C2semidirect
7. #8593C2semidirect
8. #8591C2semidirect
9. #8590C2semidirect
10. #8590C2semidirect
11. #8589C2semidirect
12. #8587C2semidirect
13. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#342142
#219782
#208152
C2×#15592
C2×#15542
#85942
#85932
#85912
#85902
#85892
#85872
#85862
#80432
#80402