Finite Group Databasebeta

← #524578

Subgroups — #524578

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2#14714C211yesmaximal, characteristic
3#14430C214yesmaximal
4#14378C214yesmaximal
5#14266C212yesmaximal
6#14161C211yesmaximal, characteristic
7#14133C212yesmaximal
8#14041C212yesmaximal
9#13981C212yesmaximal
10#13936C212yesmaximal
11#13918C212yesmaximal
12#13911C212yesmaximal
13#13910C211yesmaximal, characteristic
14C4⋊C42C211yesmaximal, characteristic
15#13783C212yesmaximal
16C4×#389C211yesmaximal, characteristic
17#13755C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19#2369C4116yescomplemented
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#391C2214yes—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#387C2214yes—
30C2×#384C2211yescharacteristic
31C2×#384C2212yes—
32C2×#382C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×#376C2212yes—
43C2×C43C2211yescharacteristic
44#1446C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1436C2212yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78#1434C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C23×Q8C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8C2×C418yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119#391—44no—
120#391—44no—
121#391C2×C4132yes—
122#389—24no—
123#389—28no—
124#389—44no—
125#389C2×C4116yes—
126#389C2×C4116yes—
127#389Q818yescomplemented
128#387—48no—
129#384—24no—
130#384—44no—
131#382—48no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—44no—
135C4×C4⋊C4—44no—
136C4×C4⋊C4—44no—
137C4×C4⋊C4C2×C4116yes—
138C4×C4⋊C4C2×C4116yes—
139C4×C4⋊C4C2×C418yes—
140C4×C4⋊C4C2×C418yes—
141#376—44no—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C43—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8C22×C418yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4⋊Q8—232no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C4116yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C2×Q814yes—
252C2×C4⋊C4C2×Q818yes—
253C2×C4⋊C4C2×Q814yes—
254C2×C4⋊C4#4114yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C22×C418yes—
288C2×C42C22×C418yes—
289C2×C42C22×C4116yes—
290C2×C42C22×C414yes—
291C2×C42C2×Q814yes—
292C2×C42#4114yes—
293C2×C42#4114yes—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—42no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C4116yes—
323#95C22×C4116yes—
324#95C22×C418yes—
325#95C22×C418yes—
326#95C2×Q818yes—
327C24C2511yescharacteristic
328C2×Q8—216no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8—84no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4C23×C414yes—
426C22×C4C23×C414yes—
427C22×C4C23×C414yes—
428C22×C4C23×C418yes—
429C22×C4C23×C412yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C4⋊C4C4×Q8116yes—
472C42—216no—
473C42—216no—
474C42—28no—
475C42—416no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—48no—
481C42—44no—
482C42—48no—
483C42—88no—
484C42#117116yes—
485C23C24×C411yescharacteristic
486C23C23×Q811yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×2−1+411yescharacteristic
495C23C2×2−1+411yescharacteristic
496C23C2×2−1+411yescharacteristic
497C23C2×2−1+411yescharacteristic
498C23#58511yescharacteristic
499C23#58511yescharacteristic
500Q8—816no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4C2×#11714yes—
605C2×C4C2×#11714yes—
606C2×C4C2×#11714yes—
607C2×C4C2×C4×Q818yes—
608C2×C4C2×C4×Q814yes—
609C2×C4#51914yes—
610C2×C4#52014yes—
611C2×C4#52014yes—
612C2×C4#54114yes—
613C2×C4#54114yes—
614C2×C4#54214yes—
615C2×C4#54214yes—
616C2×C4#55414yes—
617C2×C4#55614yes—
618C2×C4#55714yes—
619C2×C4#55914yes—
620C22C22×#11711yescharacteristic
621C22C22×C4×Q811yescharacteristic
622C22C2×#51911yescharacteristic
623C22C2×#52011yescharacteristic
624C22C2×#52011yescharacteristic
625C22C4×H(5,𝔽2)11yescharacteristic
626C22C4×H(5,𝔽2)11yescharacteristic
627C22C2×#54111yescharacteristic
628C22C2×#54111yescharacteristic
629C22C2×#54211yescharacteristic
630C22C2×#54211yescharacteristic
631C22C2×#55411yescharacteristic
632C22C2×#55611yescharacteristic
633C22C2×#55711yescharacteristic
634C22C2×#55911yescharacteristic
635C22#349411yescharacteristic
636C22#349411yescharacteristic
637C22#350111yescharacteristic
638C22#350211yescharacteristic
639C22#350311yescharacteristic
640C22#350311yescharacteristic
641C22#350811yescharacteristic
642C22#351011yescharacteristic
643C22#351511yescharacteristic
644C22#351811yescharacteristic
645C22#351911yescharacteristic
646C22#352211yescharacteristic
647C22#354411yescharacteristic
648C22#354411yescharacteristic
649C22#354711yescharacteristic
650C22#354711yescharacteristic
651C22#354711yescharacteristic
652C22#354811yescharacteristic
653C22#354911yescharacteristic
654C22#355511yescharacteristic
655C4—28no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—48no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4#232018yes—
671C4#235518yes—
672C2C2×#232011yescharacteristic
673C2C2×#235511yescharacteristic
674C2#3425511yescharacteristic
675C2#3433211yescharacteristic
676C2#3433511yescharacteristic
677C2#3448411yescharacteristic
678C2#3448511yescharacteristic
679C2#3915711yescharacteristic
680C2#3915811yescharacteristic
681C2#3920111yescharacteristic
682C2#3920311yescharacteristic
683C2#3960711yescharacteristic
684C2#3960811yescharacteristic
685C2#3964211yescharacteristic
686C2#3964311yescharacteristic
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
19. #2369C4semidirect
127. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#147142
#144302
#143782
#142662
#141612
#141332
#140412
#139812
#139362
#139182
#139112
#139102
C4⋊C422
#137832
C4×#3892
#137552