Finite Group Databasebeta

← #384456

Subgroups — #384456

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1987C211yesmaximal, characteristic
2#14053C211yesmaximal, characteristic
3#13921C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5#11316C211yesmaximal, characteristic
6#11315C211yesmaximal, characteristic
7#11292C211yesmaximal, characteristic
8#11257C211yesmaximal, characteristic
9#11231C211yesmaximal, characteristic
10#10908C211yesmaximal, characteristic
11#10880C211yesmaximal, characteristic
12#10876C211yesmaximal, characteristic
13#10820C211yesmaximal, characteristic
14#10807C211yesmaximal, characteristic
15#10806C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#391—21no—
21C2×#389C2211yescharacteristic
22C2×#383—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1987C418yescomplemented
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45#1446—21no—
46#1443—21no—
47#1443—21no—
48#1441—21no—
49#1436—21no—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58#1434—21no—
59C4×#95C2211yescharacteristic
60C23×Q8—21no—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—41no—
69C2×C4⋊Q8—41no—
70C2×C4⋊Q8C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#106—22no—
87C2×#106—41no—
88C2×#106—41no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8C2×C414yes—
92C2×C4×C8—22no—
93C2×C4×C8—41no—
94C2×C4×C8—41no—
95#391—44no—
96#389—22no—
97#389—41no—
98#383—44no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—41no—
102C4×C4⋊C4C2×C414yes—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—41no—
132#340—41no—
133#340—41no—
134#340—41no—
135#340—41no—
136#340—41no—
137#340C2×C414yes—
138#340C2×C414yes—
139#340C2×C414yes—
140#340C2×C414yes—
141#336—22no—
142#336—22no—
143#336—41no—
144#336—41no—
145C22×Q8—24no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—81no—
151C22×Q8—81no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8C22×C412yes—
193C22×C8C22×C412yes—
194C22×C8#4112yes—
195C4⋊Q8—28no—
196C4⋊Q8—44no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—81no—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4#4112yes—
251C2×C4⋊C4#4112yes—
252C2×C4⋊C4#4112yes—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—81no—
275C2×C42C22×C412yes—
276C2×C42#4112yes—
277#106—44no—
278C4⋊C8—44no—
279C4×C8—44no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—41no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—81no—
298#95—81no—
299#95—81no—
300#95—81no—
301#95—81no—
302#95SD1614yes—
303C24C22×D411yescharacteristic
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4C2×SD1612yes—
405C22×C4C2×SD1612yes—
406C22×C4C2×SD1612yes—
407C22×C4#13512yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C42−1+411yescharacteristic
419C22×C42−1+411yescharacteristic
420C22×C42−1+411yescharacteristic
421C22×C42−1+411yescharacteristic
422C22×C42−1+411yescharacteristic
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—48no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—84no—
484C4⋊C4#11714yes—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—48no—
498C42—84no—
499C23C2×C4×D411yescharacteristic
500C23#53611yescharacteristic
501C23#53711yescharacteristic
502C23#54611yescharacteristic
503C23#54611yescharacteristic
504C23#54711yescharacteristic
505C23#54811yescharacteristic
506C23C22×SD1611yescharacteristic
507C23C2×#13511yescharacteristic
508C23C2×AGL(1,ℤ/8)11yescharacteristic
509C23#57511yescharacteristic
510C23#57611yescharacteristic
511C23#57711yescharacteristic
512C23#57811yescharacteristic
513C23#57811yescharacteristic
514Q8—816no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4C4×SD1614yes—
613C2×C4C2×#11711yescharacteristic
614C2×C4C4×#4111yescharacteristic
615C2×C4#51811yescharacteristic
616C2×C4#51911yescharacteristic
617C2×C4#51911yescharacteristic
618C2×C4#52011yescharacteristic
619C2×C4#53811yescharacteristic
620C2×C4#53811yescharacteristic
621C2×C4#54111yescharacteristic
622C2×C4#54111yescharacteristic
623C2×C4#54111yescharacteristic
624C2×C4#54211yescharacteristic
625C2×C4#54211yescharacteristic
626C2×C4#54211yescharacteristic
627C2×C4#54211yescharacteristic
628C2×C4#54211yescharacteristic
629C2×C4#55111yescharacteristic
630C2×C4#55111yescharacteristic
631C2×C4#55211yescharacteristic
632C2×C4#55211yescharacteristic
633C2×C4#55211yescharacteristic
634C2×C4#55611yescharacteristic
635C2×C4#55611yescharacteristic
636C2×C4#55611yescharacteristic
637C2×C4#56311yescharacteristic
638C2×C4#56311yescharacteristic
639C2×C4#56311yescharacteristic
640C2×C4#56311yescharacteristic
641C8—42no—
642C8—42no—
643C8—42no—
644C8—81no—
645C22#232311yescharacteristic
646C22#234211yescharacteristic
647C22#236211yescharacteristic
648C22#269811yescharacteristic
649C22#270211yescharacteristic
650C22#271511yescharacteristic
651C22#271611yescharacteristic
652C22C2×C4×SD1611yescharacteristic
653C22C4×AGL(1,ℤ/8)11yescharacteristic
654C22#294611yescharacteristic
655C22#295011yescharacteristic
656C22#322711yescharacteristic
657C22#323311yescharacteristic
658C22#324211yescharacteristic
659C22#324311yescharacteristic
660C22#324611yescharacteristic
661C22#324711yescharacteristic
662C22#325811yescharacteristic
663C22#326211yescharacteristic
664C22#329811yescharacteristic
665C22#329811yescharacteristic
666C22#330011yescharacteristic
667C22#330011yescharacteristic
668C22#331111yescharacteristic
669C22#331811yescharacteristic
670C22#332211yescharacteristic
671C22#332411yescharacteristic
672C22#333511yescharacteristic
673C22#334111yescharacteristic
674C22#334311yescharacteristic
675C22#334911yescharacteristic
676C22#336011yescharacteristic
677C22#336211yescharacteristic
678C22#336711yescharacteristic
679C22#337711yescharacteristic
680C4—24no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—81no—
691C4—161no—
692C4—161no—
693C4—161no—
694C4#232011yescharacteristic
695C4#233711yescharacteristic
696C4#235611yescharacteristic
697C4#238111yescharacteristic
698C4#275811yescharacteristic
699C4#275911yescharacteristic
700C4#276211yescharacteristic
701C4#276411yescharacteristic
702C2#1438011yescharacteristic
703C2#2211111yescharacteristic
704C2#2212511yescharacteristic
705C2#2257111yescharacteristic
706C2#2258011yescharacteristic
707C2#2306111yescharacteristic
708C2#2307311yescharacteristic
709C2#3168811yescharacteristic
710C2#3169511yescharacteristic
711C2#3180411yescharacteristic
712C2#3181111yescharacteristic
713C2#3219811yescharacteristic
714C2#3220611yescharacteristic
715C2#3223011yescharacteristic
716C2#3223811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1987C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19872
#140532
#139212
#113842
#113162
#113152
#112922
#112572
#112312
#109082
#108802
#108762
#108202
#108072
#108062