Finite Group Databasebeta

← #368604

Subgroups — #368604

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1768C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic
6#11316C211yesmaximal, characteristic
7#11315C211yesmaximal, characteristic
8#11293C211yesmaximal, characteristic
9#11292C211yesmaximal, characteristic
10#10881C211yesmaximal, characteristic
11#10880C211yesmaximal, characteristic
12#10850C211yesmaximal, characteristic
13#10850C211yesmaximal, characteristic
14#10821C211yesmaximal, characteristic
15#10820C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1988—24no—
28#1987C418yescomplemented
29#1768—24no—
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×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48#1446—21no—
49#1446—21no—
50#1445—21no—
51#1445—21no—
52#1436—21no—
53#1436—21no—
54C4⋊#95—21no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61#1434—21no—
62C23×Q8—21no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8C2×C414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yes—
94C2×C4⋊C8D414yes—
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—41no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—41no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389Q814yescomplemented
105#382—22no—
106#382—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C414yes—
111#376—22no—
112#376—22no—
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×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—41no—
144#340—41no—
145#340—41no—
146#340—41no—
147#340C2×C414yes—
148#340C2×C414yes—
149#340C2×C414yes—
150#340C2×C414yes—
151#336—22no—
152#336—22no—
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—81no—
157C22×Q8—81no—
158C22×Q8—81no—
159C22×Q8—81no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
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×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C22×C412yes—
200C22×C8C22×C412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
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—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—24no—
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—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—81no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4—81no—
255C2×C4⋊C4—81no—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C2×Q812yes—
261C2×C4⋊C4C2×Q812yes—
262C2×C4⋊C4#4112yes—
263C2×C4⋊C4#4112yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—24no—
275C2×C42—42no—
276C2×C42—81no—
277C2×C42—81no—
278C2×C42—81no—
279C2×C42—81no—
280C2×C42C22×C412yes—
281C2×C42C2×D412yes—
282C2×C42C2×Q812yes—
283C2×C42#4112yes—
284C2×C42#4112yes—
285C2×C42#4112yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288C4⋊C8—28no—
289C4⋊C8—28no—
290#97—44no—
291C4×C8—44no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—81no—
303#95—81no—
304#95—81no—
305#95—81no—
306#95—81no—
307#95—81no—
308#95—81no—
309#95—81no—
310#95—81no—
311#95—81no—
312#95—81no—
313#95—81no—
314C24C22×D411yescharacteristic
315C2×Q8—48no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—84no—
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—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
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×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4C4×D412yes—
413C22×C4C4×D412yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4C22⋊Q812yes—
417C22×C4C22⋊Q812yes—
418C22×C4#12412yes—
419C22×C4#12412yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q811yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C42−1+411yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
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—41no—
484C4⋊C4—41no—
485C4⋊C4—48no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4C4×Q814yes—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—84no—
506C42—84no—
507C42#11714yes—
508C42C4×D414yes—
509C42#12314yes—
510C42#12614yes—
511C23C2×C4×D411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×C22⋊Q811yescharacteristic
514C23#52511yescharacteristic
515C23C2×#12411yescharacteristic
516C23#53611yescharacteristic
517C23#53711yescharacteristic
518C23#57611yescharacteristic
519C23#57611yescharacteristic
520C23#57711yescharacteristic
521C23#57711yescharacteristic
522C23#57711yescharacteristic
523C23#57711yescharacteristic
524C23#57811yescharacteristic
525C23#57811yescharacteristic
526Q8—816no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
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×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4#38814yes—
624C2×C4#38814yes—
625C2×C4C2×#11711yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×Q811yescharacteristic
628C2×C4C4×#4111yescharacteristic
629C2×C4#51911yescharacteristic
630C2×C4#52011yescharacteristic
631C2×C4C2×#12311yescharacteristic
632C2×C4C2×#12611yescharacteristic
633C2×C4#52911yescharacteristic
634C2×C4#53311yescharacteristic
635C2×C4#54111yescharacteristic
636C2×C4#54211yescharacteristic
637C2×C4#54611yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54811yescharacteristic
640C2×C4D4×Q811yescharacteristic
641C2×C4#55211yescharacteristic
642C2×C4#55211yescharacteristic
643C2×C4#55411yescharacteristic
644C2×C4#55611yescharacteristic
645C2×C4#55611yescharacteristic
646C2×C4#55611yescharacteristic
647C2×C4#55611yescharacteristic
648C2×C4#55611yescharacteristic
649C2×C4#55711yescharacteristic
650C2×C4#55711yescharacteristic
651C2×C4#55911yescharacteristic
652C2×C4#55911yescharacteristic
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22C2×#38811yescharacteristic
658C22C4×C22⋊Q811yescharacteristic
659C22#232311yescharacteristic
660C22#241511yescharacteristic
661C22#241711yescharacteristic
662C22#256811yescharacteristic
663C22#259211yescharacteristic
664C22#294611yescharacteristic
665C22#294611yescharacteristic
666C22#294911yescharacteristic
667C22#295011yescharacteristic
668C22#305711yescharacteristic
669C22#305811yescharacteristic
670C22#306311yescharacteristic
671C22#306311yescharacteristic
672C22#308111yescharacteristic
673C22#308211yescharacteristic
674C22#308311yescharacteristic
675C22#308411yescharacteristic
676C22#311811yescharacteristic
677C22#312111yescharacteristic
678C22#312411yescharacteristic
679C22#312511yescharacteristic
680C22#313611yescharacteristic
681C22#313611yescharacteristic
682C22#313711yescharacteristic
683C22#313711yescharacteristic
684C22#324111yescharacteristic
685C22#324211yescharacteristic
686C22#324311yescharacteristic
687C22#324411yescharacteristic
688C22#325711yescharacteristic
689C22#325811yescharacteristic
690C22#326211yescharacteristic
691C22#326311yescharacteristic
692C4—24no—
693C4—24no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—81no—
702C4—82no—
703C4—81no—
704C4—161no—
705C4—161no—
706C4#235511yescharacteristic
707C4#235611yescharacteristic
708C4#236511yescharacteristic
709C4#236711yescharacteristic
710C4#244611yescharacteristic
711C4#244711yescharacteristic
712C4#251111yescharacteristic
713C4#251411yescharacteristic
714C2#1402111yescharacteristic
715C2#2108811yescharacteristic
716C2#2109111yescharacteristic
717C2#2162311yescharacteristic
718C2#2162511yescharacteristic
719C2#2212311yescharacteristic
720C2#2212511yescharacteristic
721C2#2471211yescharacteristic
722C2#2471811yescharacteristic
723C2#2484711yescharacteristic
724C2#2485311yescharacteristic
725C2#2922811yescharacteristic
726C2#2922911yescharacteristic
727C2#2936211yescharacteristic
728C2#2936311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1987C4semidirect
104. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19872
C2×#17682
#142822
#137832
#113162
#113152
#112932
#112922
#108812
#108802
#108502
#108212
#108202