Finite Group Databasebeta

← #553201

Subgroups — #553201

Order 512.

3366 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#2320C211yesmaximal, characteristic
4#14664C212yesmaximal
5#14474C214yesmaximal
6#14420C214yesmaximal
7#14346C212yesmaximal
8#14289C212yesmaximal
9#14283C211yesmaximal, characteristic
10#14160C211yesmaximal, characteristic
11#14054C212yesmaximal
12#14053C212yesmaximal
13#13942C212yesmaximal
14#13931C212yesmaximal
15#13931C212yesmaximal
16#13926C211yesmaximal, characteristic
17#13850C211yesmaximal, characteristic
18C2×#557—28no—
19C2×#557—28no—
20C2×#557C4116yescomplemented
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2211yescharacteristic
25C22×#125C2212yes—
26C22×#125C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32#2355—28no—
33#2320—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#389C2212yes—
39C2×#387C2214yes—
40C2×#387C2214yes—
41C2×#384C2211yescharacteristic
42C2×#383C2212yes—
43C2×#382C2212yes—
44C2×#382C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×#376C2211yescharacteristic
50#1446C2214yes—
51#1446C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2214yes—
55#1445C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1441C2211yescharacteristic
59#1441C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2211yescharacteristic
63#1441C2211yescharacteristic
64#1441C2211yescharacteristic
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2214yes—
83#1434C2214yes—
84#1434C2214yes—
85#1434C2214yes—
86C4×#95C2212yes—
87C4×#95C2212yes—
88C23×Q8C2312yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91#557—2128no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8C2×C4116yes—
97C2×C4⋊Q8C2×C418yes—
98C2×#125—24no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—28no—
104C2×#125—24no—
105C2×#125—28no—
106C2×#125C2×C4116yes—
107C2×#125C2×C418yes—
108C2×#125C2×C4116yes—
109C2×#125C2×C418yes—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—28no—
116C2×C4×Q8—24no—
117C2×C4×Q8—28no—
118C2×C4×Q8C2×C418yes—
119C2×C4×Q8C2×C4116yes—
120C2×C4×Q8C2×C418yes—
121C2×C4×Q8C2×C418yes—
122C2×C4×Q8C2×C418yes—
123C2×C4×Q8D418yes—
124C2×C4×Q8D418yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2314yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C42C2312yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151C22×C42C2312yes—
152C22×C42C2314yes—
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155C22×C42C2312yes—
156#391—28no—
157#391—28no—
158#391—44no—
159#391—48no—
160#389—28no—
161#387—216no—
162#387—216no—
163#384—24no—
164#383—28no—
165#382—28no—
166#382—28no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—28no—
171#376—24no—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—41no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—41no—
217C22×Q8C22×C418yes—
218C22×Q8C22×C414yes—
219C22×Q8C22×C414yes—
220C22×Q8C2×D414yes—
221C22×Q8C2×D414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—264no—
240C4⋊Q8—232no—
241#125—264no—
242#125—232no—
243#125—264no—
244#125—232no—
245C4×Q8—232no—
246C4×Q8—232no—
247C4×Q8—264no—
248C4×Q8—232no—
249C4×Q8C22⋊C4164yes—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C22×C414yes—
293C2×C4⋊C4C22×C418yes—
294C2×C4⋊C4C22×C414yes—
295C2×C4⋊C4C22×C418yes—
296C2×C4⋊C4C22×C414yes—
297C2×C4⋊C4C22×C418yes—
298C2×C4⋊C4C22×C414yes—
299C2×C4⋊C4C22×C418yes—
300C2×C4⋊C4C22×C414yes—
301C2×C4⋊C4C22×C418yes—
302C2×C4⋊C4C22×C418yes—
303C2×C4⋊C4C22×C418yes—
304C2×C4⋊C4C22×C418yes—
305C2×C4⋊C4C22×C414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D418yes—
308C2×C4⋊C4C2×D418yes—
309C2×C4⋊C4C2×D414yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42C22×C418yes—
331C2×C42C22×C414yes—
332C2×C42C22×C418yes—
333C2×C42C22×C414yes—
334C2×C42C22×C418yes—
335C2×C42C22×C414yes—
336C2×C42C2×D414yes—
337C2×C42C2×D418yes—
338C2×C42C2×D414yes—
339C2×C42C2×D418yes—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—24no—
357#95—42no—
358#95—42no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—42no—
365#95—44no—
366#95—44no—
367#95—42no—
368#95—44no—
369#95—44no—
370#95—44no—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—28no—
374C2×Q8—216no—
375C2×Q8—216no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—216no—
379C2×Q8—28no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8C2×C22⋊C4116yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4C23×C412yes—
473C22×C4C23×C414yes—
474C22×C4C23×C412yes—
475C22×C4C23×C414yes—
476C22×C4C23×C412yes—
477C22×C4C23×C412yes—
478C22×C4C23×C414yes—
479C22×C4C23×C414yes—
480C22×C4C23×C412yes—
481C22×C4C23×C412yes—
482C22×C4C23×C412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+414yes—
495C22×C42−1+412yes—
496C22×C42−1+414yes—
497C22×C42−1+414yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—84no—
516C4⋊C4—88no—
517C4⋊C4C2×C22⋊C4132yes—
518C4⋊C4C2×C22⋊C4116yes—
519C42—216no—
520C42—28no—
521C42—216no—
522C42—28no—
523C42—48no—
524C42—48no—
525C42—88no—
526C42—84no—
527C42C2×C22⋊C4132yes—
528C42C2×C22⋊C4116yes—
529C23C24×C411yescharacteristic
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×2−1+412yes—
534C23C2×2−1+412yes—
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+412yes—
537C23#58511yescharacteristic
538C23#58511yescharacteristic
539C23#58512yes—
540Q8—232no—
541Q8—416no—
542Q8—432no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4C22×C22⋊C418yes—
641C2×C4C22×C22⋊C414yes—
642C2×C4C22×C22⋊C418yes—
643C2×C4C22×C22⋊C414yes—
644C2×C4C22×C22⋊C414yes—
645C2×C4#51918yes—
646C2×C4#52014yes—
647C2×C4#53614yes—
648C2×C4#53618yes—
649C2×C4#53614yes—
650C22C23×C22⋊C411yescharacteristic
651C22C2×#51912yes—
652C22C2×#52011yescharacteristic
653C22#342811yescharacteristic
654C22C4×2−1+412yes—
655C22C2×#53612yes—
656C22C2×#53611yescharacteristic
657C22C2×#53611yescharacteristic
658C22#348611yescharacteristic
659C22#348711yescharacteristic
660C22#348811yescharacteristic
661C22#348811yescharacteristic
662C22#348911yescharacteristic
663C22#348911yescharacteristic
664C22#349712yes—
665C22#355212yes—
666C22#355412yes—
667C22#355512yes—
668C22#355512yes—
669C22#355612yes—
670C22#355712yes—
671C22#356312yes—
672C22#356512yes—
673C4—24no—
674C4—28no—
675C4—24no—
676C4—24no—
677C4—28no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—84no—
687C4—84no—
688C4—82no—
689C4#231018yes—
690C2C2×#231011yescharacteristic
691C2#3417511yescharacteristic
692C2#3425711yescharacteristic
693C2#3448812yes—
694C2#3449012yes—
695C2#4280112yes—
696C2#4280412yes—
697C2#4281111yescharacteristic
698C2#4281411yescharacteristic
699C2#4288611yescharacteristic
700C2#4289111yescharacteristic
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
20. C2×#557C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#23552
C2×#23202
#146642
#144742
#144202
#143462
#142892
#142832
#141602
#140542
#140532
#139422
#139312
#139262
#138502