Finite Group Databasebeta

← #355405

Subgroups — #355405

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3C2×#1768C211yesmaximal, characteristic
4#13797C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic
6#10908C211yesmaximal, characteristic
7#10907C211yesmaximal, characteristic
8#10907C211yesmaximal, characteristic
9#10907C211yesmaximal, characteristic
10#10863C211yesmaximal, characteristic
11#10862C211yesmaximal, characteristic
12#10850C211yesmaximal, characteristic
13#10849C211yesmaximal, characteristic
14#10821C211yesmaximal, characteristic
15#10820C211yesmaximal, characteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1949C418yescomplemented
32#1947C418yescomplemented
33#1768C418yescomplemented
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#1436—21no—
49#1436—21no—
50#1436—21no—
51#1436—21no—
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58#1434—21no—
59#1434—21no—
60#1434—21no—
61#1434—21no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
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—41no—
84C2×#107—41no—
85C2×#107—41no—
86C2×#107—41no—
87C2×#107—41no—
88C2×#107—41no—
89C2×#107C2×C414yes—
90C2×#107C2×C414yes—
91C2×#107C2×C414yes—
92C2×#106—41no—
93C2×#106—41no—
94C2×#106C2×C414yes—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8C2×C414yes—
98C2×C4⋊C8C2×C414yes—
99C2×C4⋊C8C2×C414yes—
100C2×C4⋊C8C2×C414yes—
101C2×#97—41no—
102C2×#97—41no—
103C2×#97C2×C414yes—
104C2×C4×C8—41no—
105C2×C4×C8—41no—
106C2×C4×C8C2×C414yes—
107#384—22no—
108#384C2×C414yes—
109#382—22no—
110#382C2×C414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—41no—
114C4×C4⋊C4—41no—
115C4×C4⋊C4—41no—
116C4×C4⋊C4—41no—
117C4×C4⋊C4C2×C414yes—
118C4×C4⋊C4C2×C414yes—
119#376C2×C414yes—
120#376Q814yes—
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×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340—41no—
144#340—41no—
145#340—41no—
146#340—41no—
147#340—41no—
148#340—41no—
149#340—41no—
150#340—41no—
151#340C2×C414yes—
152#340C2×C414yes—
153#340C2×C414yes—
154#340C2×C414yes—
155#336C2×C414yes—
156#336C2×C414yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
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×C8C22×C412yes—
195C22×C8C22×C412yes—
196C22×C8C22×C412yes—
197C22×C8C22×C412yes—
198C22×C8C22×C412yes—
199C22×C8C22×C412yes—
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—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—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—81no—
243C2×C4⋊C4—81no—
244C2×C4⋊C4—81no—
245C2×C4⋊C4—81no—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
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—24no—
263C2×C42—24no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—81no—
267C2×C42—81no—
268C2×C42—81no—
269C2×C42—81no—
270C2×C42—81no—
271C2×C42—81no—
272C2×C42—81no—
273C2×C42—81no—
274C2×C42C22×C412yes—
275C2×C42C22×C412yes—
276C2×C42C22×C412yes—
277C2×C42C22×C412yes—
278C2×C42C22×C412yes—
279C2×C42C2×Q812yes—
280C2×C42C2×Q812yes—
281C2×C42C2×Q812yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284#107—44no—
285#107—44no—
286#107—44no—
287#106—44no—
288C4⋊C8—28no—
289C4⋊C8C42116yescomplemented
290#97—44no—
291C4×C8—44no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—42no—
297#95—42no—
298#95—81no—
299#95—81no—
300#95—81no—
301#95—81no—
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#95C4⋊C414yes—
311#95C4⋊C414yes—
312#95C4⋊C414yes—
313#95C4⋊C414yes—
314C24C22×D411yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
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—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
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×C4—41no—
403C22×C4C2×C4⋊C412yes—
404C22×C4C2×C4⋊C412yes—
405C22×C4C2×C4⋊C412yes—
406C22×C4C2×C4⋊C412yes—
407C22×C4C2×C4⋊C412yes—
408C22×C4C2×C4⋊C412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C4×D412yes—
412C22×C4C4×D412yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C23×C411yescharacteristic
415C22×C4C23×C411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
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—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8C2×C4214yes—
454C2×C8C2×C4214yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
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—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—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—84no—
498C42—84no—
499C42—84no—
500C42—84no—
501C42C2×C4214yes—
502C42#11714yes—
503C42C4×Q814yes—
504C42C4×Q814yes—
505C23C22×C4⋊C411yescharacteristic
506C23C2×C4×D411yescharacteristic
507C23C2×C4×D411yescharacteristic
508C23#53511yescharacteristic
509C23#53611yescharacteristic
510C23#53711yescharacteristic
511C23#53711yescharacteristic
512C23#57611yescharacteristic
513C23#57611yescharacteristic
514C23#57611yescharacteristic
515C23#57711yescharacteristic
516C23#57711yescharacteristic
517C23#57811yescharacteristic
518C23#57811yescharacteristic
519C23#57811yescharacteristic
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
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—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—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
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—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
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—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
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×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4C4×C4⋊C414yes—
617C2×C4C4×C4⋊C414yes—
618C2×C4C22×C4211yescharacteristic
619C2×C4C2×#11711yescharacteristic
620C2×C4C2×C4×Q811yescharacteristic
621C2×C4C2×C4×Q811yescharacteristic
622C2×C4#51811yescharacteristic
623C2×C4#51811yescharacteristic
624C2×C4#51811yescharacteristic
625C2×C4#51911yescharacteristic
626C2×C4#51911yescharacteristic
627C2×C4#51911yescharacteristic
628C2×C4#52011yescharacteristic
629C2×C4#52011yescharacteristic
630C2×C4#52011yescharacteristic
631C2×C4#52011yescharacteristic
632C2×C4#52011yescharacteristic
633C2×C4#52011yescharacteristic
634C2×C4#54011yescharacteristic
635C2×C4#54011yescharacteristic
636C2×C4#54111yescharacteristic
637C2×C4#54111yescharacteristic
638C2×C4#54211yescharacteristic
639C2×C4#54211yescharacteristic
640C2×C4#54211yescharacteristic
641C2×C4#54211yescharacteristic
642C2×C4#54411yescharacteristic
643C2×C4#54411yescharacteristic
644C2×C4#54411yescharacteristic
645C2×C4#54411yescharacteristic
646C8—24no—
647C8—24no—
648C8—42no—
649C8—42no—
650C22C2×C4×C4⋊C411yescharacteristic
651C22#231611yescharacteristic
652C22#231711yescharacteristic
653C22#232311yescharacteristic
654C22#232311yescharacteristic
655C22#232411yescharacteristic
656C22#232411yescharacteristic
657C22#291211yescharacteristic
658C22#291311yescharacteristic
659C22#291311yescharacteristic
660C22#291311yescharacteristic
661C22#294611yescharacteristic
662C22#294611yescharacteristic
663C22#294711yescharacteristic
664C22#294711yescharacteristic
665C22#294811yescharacteristic
666C22#295011yescharacteristic
667C22#295111yescharacteristic
668C22#295111yescharacteristic
669C22#322011yescharacteristic
670C22#322311yescharacteristic
671C22#322411yescharacteristic
672C22#322511yescharacteristic
673C22#324111yescharacteristic
674C22#324211yescharacteristic
675C22#324311yescharacteristic
676C22#324311yescharacteristic
677C22#325711yescharacteristic
678C22#325711yescharacteristic
679C22#325811yescharacteristic
680C22#325811yescharacteristic
681C22#325911yescharacteristic
682C22#326111yescharacteristic
683C22#326211yescharacteristic
684C22#326211yescharacteristic
685C4—24no—
686C4—24no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#227511yescharacteristic
700C4#227611yescharacteristic
701C4#231911yescharacteristic
702C4#232011yescharacteristic
703C4#233011yescharacteristic
704C4#233011yescharacteristic
705C4#233111yescharacteristic
706C4#233111yescharacteristic
707C2#1378311yescharacteristic
708C2#2045311yescharacteristic
709C2#2045511yescharacteristic
710C2#2196611yescharacteristic
711C2#2196811yescharacteristic
712C2#2199111yescharacteristic
713C2#2199211yescharacteristic
714C2#2212411yescharacteristic
715C2#2212511yescharacteristic
716C2#2212611yescharacteristic
717C2#2212711yescharacteristic
718C2#2215711yescharacteristic
719C2#2215811yescharacteristic
720C2#2216311yescharacteristic
721C2#2216411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×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
31. #1949C4semidirect
32. #1947C4semidirect
33. #1768C4semidirect
289. C4⋊C8C42semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#19472
C2×#17682
#137972
#137792
#109082
#109072
#108632
#108622
#108502
#108492
#108212
#108202