Finite Group Databasebeta

← #366019

Subgroups — #366019

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1921C211yesmaximal, characteristic
2#13933C211yesmaximal, characteristic
3#13818C211yesmaximal, characteristic
4#11328C211yesmaximal, characteristic
5#11293C211yesmaximal, characteristic
6#11256C211yesmaximal, characteristic
7#11219C211yesmaximal, characteristic
8#10907C211yesmaximal, characteristic
9#10907C211yesmaximal, characteristic
10#10885C211yesmaximal, characteristic
11#10885C211yesmaximal, characteristic
12#10880C211yesmaximal, characteristic
13#10807C211yesmaximal, characteristic
14#10797C211yesmaximal, characteristic
15C4×#340C211yesmaximal, characteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1921C418yescomplemented
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×#340C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47#1442—21no—
48#1442—21no—
49#1436—21no—
50#1436—21no—
51C4⋊#95—21no—
52C4⋊#95—21no—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57#1434—21no—
58C4×#95—21no—
59C4×#95C2211yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—41no—
87C2×#107—41no—
88C2×#107C2×C414yes—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—41no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—41no—
98#389—41no—
99#389C2×C414yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—41no—
106C4×C4⋊C4—41no—
107C4×C4⋊C4C2×C414yes—
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×#95—21no—
123C2×#95—21no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—41no—
137#340—41no—
138#340C2×C414yes—
139#340C2×C414yes—
140#340C2×C414yes—
141#340Q814yes—
142#336—22no—
143#336—41no—
144#336—41no—
145#336—41no—
146#336C2×C414yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
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×C8C22×C412yes—
187C22×C8C22×C412yes—
188C22×C8C2×Q812yes—
189C22×C8#4112yes—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—21no—
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—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4—81no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4—81no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C4⋊C4C2×Q812yes—
244C2×C4⋊C4#4112yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C4⋊C4#4112yes—
248C2×C4⋊C4#4112yes—
249C2×C4⋊C4#4112yes—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
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—24no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—81no—
273C2×C42—81no—
274C2×C42—81no—
275C2×C42—81no—
276C2×C42—81no—
277C2×C42—81no—
278C2×C42C22×C412yes—
279C2×C42#4112yes—
280#107—28no—
281#107—44no—
282C4×C8—44no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—41no—
289#95—41no—
290#95—42no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—81no—
297#95—81no—
298#95—81no—
299#95—81no—
300#95—81no—
301#95—81no—
302#95Q1614yes—
303C24C22×D411yescharacteristic
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
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—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—41no—
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×C4C4×D412yes—
397C22×C4C4×D412yes—
398C22×C4C2×Q1612yes—
399C22×C4C2×Q1612yes—
400C22×C4C2×Q1612yes—
401C22×C4#13512yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C42−1+411yescharacteristic
415C22×C42−1+411yescharacteristic
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
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—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
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—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
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—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4#11714yes—
478C4⋊C4C4×Q814yes—
479C4⋊C4#12514yes—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—84no—
493C42—84no—
494C42—84no—
495C42—84no—
496C23C2×C4×D411yescharacteristic
497C23#52511yescharacteristic
498C23#52511yescharacteristic
499C23#54611yescharacteristic
500C23#54611yescharacteristic
501C23#54711yescharacteristic
502C23D4×Q811yescharacteristic
503C23C22×Q1611yescharacteristic
504C23C2×#13511yescharacteristic
505C23#57511yescharacteristic
506C23#57511yescharacteristic
507C23#57611yescharacteristic
508C23#57611yescharacteristic
509C23#57711yescharacteristic
510C23#57711yescharacteristic
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
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×C4C4×Q1614yes—
609C2×C4C2×#11711yescharacteristic
610C2×C4C2×C4×Q811yescharacteristic
611C2×C4C4×#4111yescharacteristic
612C2×C4C4×#4111yescharacteristic
613C2×C4#52011yescharacteristic
614C2×C4#52011yescharacteristic
615C2×C4C2×#12511yescharacteristic
616C2×C4#52911yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#52911yescharacteristic
619C2×C4#52911yescharacteristic
620C2×C4#53311yescharacteristic
621C2×C4#53811yescharacteristic
622C2×C4#53911yescharacteristic
623C2×C4#54111yescharacteristic
624C2×C4#54211yescharacteristic
625C2×C4#55111yescharacteristic
626C2×C4#55211yescharacteristic
627C2×C4#55211yescharacteristic
628C2×C4#55211yescharacteristic
629C2×C4#55211yescharacteristic
630C2×C4#55411yescharacteristic
631C2×C4#55411yescharacteristic
632C2×C4#55511yescharacteristic
633C2×C4#55511yescharacteristic
634C2×C4#55611yescharacteristic
635C2×C4#55711yescharacteristic
636C2×C4#55911yescharacteristic
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22#229011yescharacteristic
642C22#235911yescharacteristic
643C22#236011yescharacteristic
644C22#245911yescharacteristic
645C22#246211yescharacteristic
646C22#247311yescharacteristic
647C22#247411yescharacteristic
648C22C2×C4×Q1611yescharacteristic
649C22#294311yescharacteristic
650C22#294611yescharacteristic
651C22#294611yescharacteristic
652C22#310211yescharacteristic
653C22#310211yescharacteristic
654C22#310411yescharacteristic
655C22#310411yescharacteristic
656C22#311811yescharacteristic
657C22#311911yescharacteristic
658C22#312211yescharacteristic
659C22#312511yescharacteristic
660C22#329711yescharacteristic
661C22#329911yescharacteristic
662C22#329911yescharacteristic
663C22#330511yescharacteristic
664C22#330911yescharacteristic
665C22#331111yescharacteristic
666C22#332211yescharacteristic
667C22#332311yescharacteristic
668C22#333711yescharacteristic
669C22#333811yescharacteristic
670C22#334711yescharacteristic
671C22#335011yescharacteristic
672C22#338111yescharacteristic
673C22Q8×Q1611yescharacteristic
674C22#338811yescharacteristic
675C22#338911yescharacteristic
676C4—24no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—81no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—161no—
689C4—161no—
690C4C4×#12511yescharacteristic
691C4#233411yescharacteristic
692C4#234611yescharacteristic
693C4#235511yescharacteristic
694C4#250811yescharacteristic
695C4#252411yescharacteristic
696C4#257411yescharacteristic
697C4#257511yescharacteristic
698C2#1398511yescharacteristic
699C2#2111911yescharacteristic
700C2#2112611yescharacteristic
701C2#2292211yescharacteristic
702C2#2293311yescharacteristic
703C2#2296511yescharacteristic
704C2#2297211yescharacteristic
705C2#2672611yescharacteristic
706C2#2673011yescharacteristic
707C2#2686211yescharacteristic
708C2#2686611yescharacteristic
709C2#2730011yescharacteristic
710C2#2730411yescharacteristic
711C2#2733211yescharacteristic
712C2#2733611yescharacteristic
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
27. #1921C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19212
#139332
#138182
#113282
#112932
#112562
#112192
#109072
#108852
#108802
#108072
#107972
C4×#3402