Finite Group Databasebeta

← #517181

Subgroups — #517181

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic
3C2×#2330C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#14712C211yesmaximal, characteristic
6#14709C211yesmaximal, characteristic
7#14707C211yesmaximal, characteristic
8#14354C211yesmaximal, characteristic
9#14277C211yesmaximal, characteristic
10#14276C211yesmaximal, characteristic
11#14237C214yesmaximal
12#14236C214yesmaximal
13#14214C214yesmaximal
14#14211C214yesmaximal
15#13779C211yesmaximal, characteristic
16#13779C211yesmaximal, characteristic
17C4×#382C211yesmaximal, characteristic
18C4×#384C211yesmaximal, characteristic
19C4×#376C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22#2660C4116yescomplemented
23C2×#398C2214yes—
24C2×#395C2214yes—
25#2331—28no—
26#2330—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2214yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2214yes—
56C2×#376C2211yescharacteristic
57C2×#376C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C2×#376C2211yescharacteristic
60C2×C43C2211yescharacteristic
61C2×C43C2211yescharacteristic
62C2×C43C2211yescharacteristic
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2214yes—
88#1434C2214yes—
89C4×#95C2212yes—
90C4×#95C2212yes—
91C4×#95C2212yes—
92C4×#95C2214yes—
93C4×#95C2212yes—
94C23×Q8C2312yes—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8C2×C418yes—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125C2×C418yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#398C2×C4132yes—
140#395C2×C4132yes—
141#391—28no—
142#391—28no—
143#391—48no—
144#389—24no—
145#389—24no—
146#389—24no—
147#389—24no—
148#389—28no—
149#389—28no—
150#389—48no—
151#387—28no—
152#387—28no—
153#387—28no—
154#387—28no—
155#384—24no—
156#384—24no—
157#384—24no—
158#384C2×C418yes—
159#384C2×C418yes—
160#382—24no—
161#382—24no—
162#382—24no—
163#382C2×C418yes—
164#382C2×C418yes—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—48no—
170#376—24no—
171#376C2×C418yes—
172#376D418yes—
173#376Q818yescomplemented
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2312yes—
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C2×#95C2312yes—
205C2×#95C2312yes—
206C43—24no—
207C43—24no—
208C43Q818yescomplemented
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8C22×C418yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C4⋊Q8—232no—
234#125—232no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C22×C414yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C418yes—
286C2×C4⋊C4C22×C418yes—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C414yes—
289C2×C4⋊C4C22×C418yes—
290C2×C4⋊C4C22×C4116yes—
291C2×C4⋊C4C22×C4116yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C22×C414yes—
323C2×C42C22×C414yes—
324C2×C42C22×C414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×Q814yes—
329C2×C42C2×Q818yes—
330C2×C42C2×Q814yes—
331C2×C42C2×Q818yes—
332C2×C42C2×Q814yes—
333C2×C42C2×Q814yes—
334C2×C42C2×Q814yes—
335C2×C42C2×Q814yes—
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42#4114yes—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95C22×C418yes—
362#95C22×C418yes—
363#95C22×C4116yes—
364#95C22×C4116yes—
365#95C2×D414yes—
366#95C2×D414yes—
367#95C2×D414yes—
368#95C2×D414yes—
369#95C2×Q818yes—
370#95C2×Q818yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—416no—
374C2×Q8—88no—
375C2×Q8—84no—
376C2×Q8—84no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4C23×C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C23×C414yes—
454C22×C4C23×C414yes—
455C22×C4C23×C412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q814yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q812yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C42−1+414yes—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—416no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—88no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42—216no—
507C42—48no—
508C42—48no—
509C42—48no—
510C42—48no—
511C42—48no—
512C42—84no—
513C42—84no—
514C42—84no—
515C42—84no—
516C42C4×D4116yes—
517C42C4×Q8116yes—
518C42C4×Q8116yes—
519C42C4⋊Q8116yes—
520C23C24×C411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C23×Q811yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×2−1+412yes—
529C23#58512yes—
530C23#58512yes—
531Q8—816no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4C2×C4×D414yes—
624C2×C4C2×C4×D414yes—
625C2×C4C2×C4×D414yes—
626C2×C4C2×C4×Q818yes—
627C2×C4C2×C4×Q818yes—
628C2×C4C2×C4×Q814yes—
629C2×C4C2×C4×Q814yes—
630C2×C4C2×C4⋊Q814yes—
631C2×C4C2×C4⋊Q814yes—
632C2×C4C2×C4⋊Q814yes—
633C2×C4#53214yes—
634C2×C4#53314yes—
635C2×C4#53314yes—
636C2×C4#54418yes—
637C22C22×C4×D411yescharacteristic
638C22C22×C4×Q811yescharacteristic
639C22C22×C4×Q811yescharacteristic
640C22C4×H(5,𝔽2)12yes—
641C22C4×2−1+412yes—
642C22C22×C4⋊Q811yescharacteristic
643C22C2×#53211yescharacteristic
644C22C2×#53311yescharacteristic
645C22C2×#53311yescharacteristic
646C22C2×#54412yes—
647C22#345812yes—
648C22#346012yes—
649C22#346112yes—
650C22#349412yes—
651C22#349412yes—
652C22#349512yes—
653C22#349512yes—
654C22#349811yescharacteristic
655C22#349911yescharacteristic
656C22#350011yescharacteristic
657C22#350011yescharacteristic
658C22#350312yes—
659C22#350412yes—
660C4—24no—
661C4—28no—
662C4—24no—
663C4—28no—
664C4—24no—
665C4—24no—
666C4—28no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4C4×C4⋊Q818yes—
679C2C2×C4×C4⋊Q811yescharacteristic
680C2C4×#54412yes—
681C2#3427212yes—
682C2#3427511yescharacteristic
683C2#3427611yescharacteristic
684C2#3527511yescharacteristic
685C2#3527611yescharacteristic
686C2#3528212yes—
687C2#3530412yes—
688C2#3531811yescharacteristic
689C2#3533011yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #2660C4semidirect
173. #376Q8semidirect
208. C43Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#23312
C2×#23302
#147142
#147122
#147092
#147072
#143542
#142772
#142762
#142372
#142362
#142142
#142112
#137792
C4×#3822
C4×#3842
C4×#3762