Finite Group Databasebeta

← #220157

Subgroups — #220157

Order 512.

917 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3172C211yesmaximal, characteristic
2#22531C211yesmaximal, characteristic
3#22509C211yesmaximal, characteristic, complemented
4C2×#1591C211yesmaximal, characteristic, complemented
5C2×#1579C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9655C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8950C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8946C211yesmaximal, characteristic, complemented
12#8597C211yesmaximal, characteristic, complemented
13#8315C211yesmaximal, characteristic
14#8314C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3172—24no—
19#3172—24no—
20C2×#501C411yescharacteristic
21C2×#497C411yescharacteristic
22C2×#491C411yescharacteristic
23C2×#489C411yescharacteristic
24C2×#470C411yescharacteristic
25C2×#470C411yescharacteristic
26C2×#468C411yescharacteristic
27C2×#468C411yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30C2×C8×Q8C2211yescharacteristic
31C2×C8×Q8C2211yescharacteristic
32C2×C4×Q16C2211yescharacteristic
33C2×C4×Q16C2211yescharacteristic
34C2×C4×SD16C2211yescharacteristic
35C2×C4×SD16C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37#1985—21no—
38#1984—21no—
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1955—21no—
42#1954—21no—
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1798—21no—
46#1798—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1750—21no—
50#1750—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1591—24no—
54#1579—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×C82C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×Q16C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×M16C2×C411yescharacteristic
76C22×M16C2×C411yescharacteristic
77#533—22no—
78#533—22no—
79#533—41no—
80#533—41no—
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×C4⋊Q8C2×C411yescharacteristic
83#529—22no—
84#529—22no—
85#529—41no—
86#529—41no—
87C2×#126—21no—
88C2×#125—21no—
89C2×#125C2×C411yescharacteristic
90C2×#124C2×C411yescharacteristic
91C2×#123—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2×C411yescharacteristic
94C2×C22⋊Q8C2×C411yescharacteristic
95C2×C22⋊Q8C2×C411yescharacteristic
96C2×#121C2×C411yescharacteristic
97C2×C4×Q8—21no—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4—21no—
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C22×C42C2311yescharacteristic
106#501—24no—
107#497—24no—
108#491—24no—
109#489—24no—
110#470—24no—
111#470—24no—
112#468—24no—
113#468—24no—
114#446—22no—
115#446—22no—
116#446—41no—
117#446—41no—
118C8×Q8—22no—
119C8×Q8—22no—
120C8×Q8—41no—
121C8×Q8—41no—
122C4×Q16—22no—
123C4×Q16—22no—
124C4×Q16D414yes—
125C4×Q16D414yes—
126C4×SD16—22no—
127C4×SD16—22no—
128C4×SD16—22no—
129C4×SD16—22no—
130C2×#107C2×C411yescharacteristic
131C2×#107C2×C411yescharacteristic
132C2×#106C2×C411yescharacteristic
133C2×#106C2×C411yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×C22⋊C8—21no—
151C2×C22⋊C8—21no—
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8D411yescharacteristic
158C2×C22⋊C8D411yescharacteristic
159C4×M16—22no—
160C4×M16—22no—
161C2×#97—21no—
162C2×#97—21no—
163C2×#97C2×C411yescharacteristic
164C2×#97C2×C411yescharacteristic
165C2×C4×C8—21no—
166C2×C4×C8—21no—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336D411yescharacteristic
181#336D411yescharacteristic
182#335—22no—
183#334—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—22no—
187Q8⋊C8—22no—
188Q8⋊C8—41no—
189Q8⋊C8—41no—
190D4⋊C8—41no—
191D4⋊C8—41no—
192#322—22no—
193C82—22no—
194C22×Q8—21no—
195C22×Q8C22×C411yescharacteristic
196C22×Q8C22×C411yescharacteristic
197C22×Q8C22×C411yescharacteristic
198C22×D4—21no—
199C22×D4C22×C411yescharacteristic
200C23×C4—21no—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4111yescharacteristic
204C2×Q16—22no—
205C2×Q16—22no—
206C2×Q16—22no—
207C2×Q16—22no—
208C2×Q16—41no—
209C2×Q16C22⋊C414yes—
210C2×SD16—22no—
211C2×SD16—22no—
212C2×SD16—22no—
213C2×SD16—22no—
214C2×SD16—22no—
215C2×SD16—41no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—22no—
221C2×M16—41no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—41no—
239C4⋊Q8—41no—
240#126—42no—
241#126—42no—
242#125—22no—
243#125—41no—
244#125—42no—
245#125—42no—
246#124—22no—
247#124—41no—
248#123—42no—
249#123—42no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—41no—
254C22⋊Q8—41no—
255C22⋊Q8—41no—
256C22⋊Q8—42no—
257C22⋊Q8—42no—
258#121—22no—
259#121—41no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8C2×D412yes—
271C4×Q8C2×D412yes—
272C4×Q8C2×D412yes—
273C4×Q8C2×D412yes—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—42no—
279C4×D4—42no—
280#117—42no—
281#117—42no—
282#117—42no—
283#117—42no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42C22×C411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42#4111yescharacteristic
316C2×C42C2411yescharacteristic
317#107—22no—
318#107C22⋊C414yes—
319#106—22no—
320#106—22no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332C4⋊C8—42no—
333#103—22no—
334#103—22no—
335#103—22no—
336#103—22no—
337#103C22⋊C414yes—
338#103C22⋊C414yes—
339#102—22no—
340#102—22no—
341C22⋊C8—22no—
342C22⋊C8—41no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—81no—
348C22⋊C8—81no—
349C22⋊C8—81no—
350#97—22no—
351#97—22no—
352#97—42no—
353#97—42no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—42no—
365C4×C8—42no—
366C4×C8C2×D412yes—
367C4×C8C2×D412yes—
368C24—21no—
369C24C4×D411yescharacteristic
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—81no—
389C2×Q8C2×C22⋊C412yes—
390C2×Q8C2×C22⋊C412yes—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—81no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435Q16—28no—
436Q16—44no—
437SD16—44no—
438SD16—44no—
439M16—44no—
440M16—44no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—22no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8C2×C22⋊C412yes—
478C2×C8C2×C22⋊C412yes—
479C2×C8#13112yes—
480C2×C8#13112yes—
481C2×C8#13112yes—
482C2×C8#13712yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C2×C22⋊C412yes—
504C4⋊C4C2×C22⋊C412yes—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C42—21no—
514C42—21no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—41no—
524C42—41no—
525C42C22×D411yescharacteristic
526C42C22×D411yescharacteristic
527C42H(5,𝔽2)11yescharacteristic
528C42H(5,𝔽2)11yescharacteristic
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C2×C4×D411yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—84no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—84no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4C22×C22⋊C411yescharacteristic
609C2×C4#51811yescharacteristic
610C2×C4D4211yescharacteristic
611C2×C4D4211yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54611yescharacteristic
614C2×C4C2×#13111yescharacteristic
615C2×C4C2×#13111yescharacteristic
616C2×C4C2×#13111yescharacteristic
617C2×C4#56811yescharacteristic
618C2×C4C2×#13711yescharacteristic
619C2×C4#57511yescharacteristic
620C2×C4#57711yescharacteristic
621C2×C4#57811yescharacteristic
622C8—22no—
623C8—22no—
624C8—22no—
625C8—42no—
626C8—41no—
627C8—41no—
628C8—41no—
629C8—41no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8—81no—
635C8#40814yes—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22D4×C22⋊C411yescharacteristic
651C22#293111yescharacteristic
652C22#293511yescharacteristic
653C22C4×#13711yescharacteristic
654C22#294711yescharacteristic
655C22#295511yescharacteristic
656C22#295611yescharacteristic
657C4—21no—
658C4—21no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—42no—
665C4—42no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4C2×#40811yescharacteristic
670C4#288011yescharacteristic
671C4#297911yescharacteristic
672C4#298111yescharacteristic
673C4#327011yescharacteristic
674C4#327111yescharacteristic
675C4#327711yescharacteristic
676C4#327811yescharacteristic
677C2—81no—
678C2—161no—
679C2#1796711yescharacteristic
680C2#1796911yescharacteristic
681C2#1807711yescharacteristic
682C2#1808111yescharacteristic
683C2#2240211yescharacteristic
684C2#2244511yescharacteristic
685C2#2245311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22509C2semidirect
4. C2×#1591C2semidirect
5. C2×#1579C2semidirect
6. #9660C2semidirect
7. #9655C2semidirect
8. #8952C2semidirect
9. #8950C2semidirect
10. #8948C2semidirect
11. #8946C2semidirect
12. #8597C2semidirect
14. #8314C2semidirect
15. #8146C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31722
#225312
#225092
C2×#15912
C2×#15792
#96602
#96552
#89522
#89502
#89482
#89462
#85972
#83152
#83142
#81462