Finite Group Databasebeta

← #193670

Subgroups — #193670

Order 512.

1163 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3168C211yesmaximal, characteristic
2C2×#2994C211yesmaximal, characteristic, complemented
3#21848C211yesmaximal, characteristic, complemented
4C2×#1483C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8951C211yesmaximal, characteristic, complemented
8#8950C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8946C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14C2×#1277C211yesmaximal, characteristic, complemented
15#7848C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3168—24no—
19#3168—24no—
20C2×#499C411yescharacteristic
21C2×#495C411yescharacteristic
22C2×#488C411yescharacteristic
23C2×#488C411yescharacteristic
24C2×#467C411yescharacteristic
25C2×#466C411yescharacteristic
26C2×#465C411yescharacteristic
27C2×#465C411yescharacteristic
28#2994—24no—
29#2994—24no—
30C2×#446C2211yescharacteristic
31C2×C8×Q8C2211yescharacteristic
32C2×C4×Q16C2211yescharacteristic
33C2×C4×SD16C2211yescharacteristic
34C2×C4×SD16C2211yescharacteristic
35C2×C4×D8C2211yescharacteristic
36C2×#436C2211yescharacteristic
37C2×#435—21no—
38C2×C8×D4—21no—
39C2×C8×D4C2211yescharacteristic
40C2×#433—21no—
41C2×#431C2211yescharacteristic
42C2×#424—21no—
43C22×C4⋊C8C2211yescharacteristic
44C22×C4⋊C8C2211yescharacteristic
45C2×C4×M16C2211yescharacteristic
46C22×C4×C8C2211yescharacteristic
47#1984C2211yescharacteristic
48#1984C2211yescharacteristic
49#1954C2211yescharacteristic
50#1954C2211yescharacteristic
51#1869—21no—
52#1869—21no—
53#1798—21no—
54#1798—21no—
55#1483—24no—
56#1483—24no—
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71#1273C2211yescharacteristic
72#1273C2211yescharacteristic
73C22×Q16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×SD16C2×C411yescharacteristic
76C22×D8C2×C411yescharacteristic
77C22×M16—21no—
78C22×M16—21no—
79C23×C8—21no—
80C23×C8—21no—
81C23×C8—21no—
82C23×C8C2×C411yescharacteristic
83C23×C8C2×C411yescharacteristic
84#529—22no—
85#529—22no—
86#529—22no—
87#529—22no—
88#529—22no—
89#529—22no—
90#529D414yes—
91#529D414yes—
92C2×#126—21no—
93C2×#126—21no—
94C2×#125C2×C411yescharacteristic
95C2×#125C2×C411yescharacteristic
96C2×#124C2×C411yescharacteristic
97C2×#124C2×C411yescharacteristic
98C2×#123—21no—
99C2×#123—21no—
100C2×C22⋊Q8C2×C411yescharacteristic
101C2×C22⋊Q8C2×C411yescharacteristic
102C2×#121C2×C411yescharacteristic
103C2×#121C2×C411yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×#117—21no—
111C2×#117—21no—
112C22×C42C2311yescharacteristic
113#499—24no—
114#495—24no—
115#488—24no—
116#488—24no—
117#467—24no—
118#466—24no—
119#465—24no—
120#465—24no—
121#446—22no—
122#446—22no—
123C8×Q8—22no—
124C8×Q8—22no—
125C4×Q16—22no—
126C4×Q16—22no—
127C4×SD16—22no—
128C4×SD16—22no—
129C4×SD16—22no—
130C4×SD16—22no—
131C4×D8—22no—
132C4×D8—22no—
133#436—22no—
134#436—22no—
135#435—28no—
136C8×D4—28no—
137C8×D4—22no—
138C8×D4—22no—
139#433—28no—
140#431—22no—
141#431—22no—
142C2×#107C2×C411yescharacteristic
143C2×#107C2×C411yescharacteristic
144C2×#106C2×C411yescharacteristic
145C2×#106C2×C411yescharacteristic
146#424—28no—
147C2×C4⋊C8—21no—
148C2×C4⋊C8—21no—
149C2×C4⋊C8—21no—
150C2×C4⋊C8—22no—
151C2×C4⋊C8—22no—
152C2×C4⋊C8—22no—
153C2×C4⋊C8—21no—
154C2×C4⋊C8D411yescharacteristic
155C2×C4⋊C8D411yescharacteristic
156C2×C4⋊C8D414yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×#103C2×C411yescharacteristic
162C2×#103C2×C411yescharacteristic
163C2×#103C2×C411yescharacteristic
164C2×#103C2×C411yescharacteristic
165C2×#102C2×C411yescharacteristic
166C2×#102C2×C411yescharacteristic
167C2×#102C2×C411yescharacteristic
168C2×#102C2×C411yescharacteristic
169C2×C22⋊C8—21no—
170C2×C22⋊C8—21no—
171C2×C22⋊C8—21no—
172C2×C22⋊C8—21no—
173C2×C22⋊C8—21no—
174C2×C22⋊C8—21no—
175C2×C22⋊C8D411yescharacteristic
176C2×C22⋊C8D411yescharacteristic
177C4×M16—22no—
178C4×M16—22no—
179C2×#97—21no—
180C2×#97—21no—
181C2×C4×C8—21no—
182C2×C4×C8—22no—
183C2×C4×C8—22no—
184C2×C4×C8C2×C411yescharacteristic
185C2×C4×C8C2×C411yescharacteristic
186C2×C4×C8C2311yescharacteristic
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2311yescharacteristic
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196Q8⋊C8—22no—
197Q8⋊C8—22no—
198Q8⋊C8—22no—
199Q8⋊C8—22no—
200D4⋊C8—22no—
201D4⋊C8—22no—
202D4⋊C8—22no—
203D4⋊C8—22no—
204C22×Q8C22×C411yescharacteristic
205C22×Q8C22×C411yescharacteristic
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C22×C411yescharacteristic
209C22×D4C22×C411yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4#4111yescharacteristic
215C2×Q16—22no—
216C2×Q16—22no—
217C2×Q16—41no—
218C2×SD16—22no—
219C2×SD16—22no—
220C2×SD16—22no—
221C2×SD16—22no—
222C2×SD16—41no—
223C2×SD16—41no—
224C2×D8—22no—
225C2×D8—22no—
226C2×D8—41no—
227C2×M16—24no—
228C2×M16—24no—
229C2×M16—42no—
230C2×M16—42no—
231C2×M16—81no—
232C2×M16—81no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—41no—
246C22×C8—41no—
247C22×C8—42no—
248C22×C8—81no—
249C22×C8—81no—
250C22×C8—81no—
251C22×C8—81no—
252C22×C8—81no—
253C22×C8—81no—
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C22×C411yescharacteristic
257C22×C8C22×C411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262#126—24no—
263#126—24no—
264#126—42no—
265#126—42no—
266#125—22no—
267#125—22no—
268#125—22no—
269#125C22⋊C414yes—
270#124—22no—
271#124—22no—
272#124—22no—
273#124C22⋊C414yes—
274#123—24no—
275#123—24no—
276#123—42no—
277#123—42no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8C22⋊C414yes—
282#121—22no—
283#121—22no—
284#121—22no—
285#121C22⋊C414yes—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8C2×D412yes—
293C4×Q8C2×D412yes—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4C2×D412yes—
305C4×D4C2×D412yes—
306#117—24no—
307#117—24no—
308#117—42no—
309#117—42no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C22×C411yescharacteristic
319C2×C4⋊C4C22×C411yescharacteristic
320C2×C4⋊C4C22×C411yescharacteristic
321C2×C4⋊C4C22×C411yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42C22×C411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42#4112yes—
340C2×C42#4111yescharacteristic
341C2×C42C2411yescharacteristic
342#107—22no—
343#107—22no—
344#106—22no—
345#106—22no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—42no—
358C4⋊C8C2×D412yes—
359C4⋊C8C2×D412yes—
360#103—22no—
361#103—22no—
362#103—22no—
363#103—22no—
364#102—22no—
365#102—22no—
366#102—22no—
367#102—22no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—81no—
373C22⋊C8—81no—
374C22⋊C8—81no—
375C22⋊C8—81no—
376#97—24no—
377#97—42no—
378C4×C8—24no—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8—21no—
382C4×C8—21no—
383C4×C8—21no—
384C4×C8—21no—
385C4×C8—21no—
386C4×C8—21no—
387C4×C8—21no—
388C4×C8—21no—
389C4×C8—21no—
390C4×C8—21no—
391C4×C8—21no—
392C4×C8—21no—
393C24—21no—
394C24—21no—
395C24C4×D411yescharacteristic
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8C2×C22⋊C412yes—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4C2×C22⋊C412yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4C2×C22⋊C412yes—
455C22×C4C4×D412yes—
456C22×C4C4×D411yescharacteristic
457C22×C4C4×D411yescharacteristic
458C22×C4C4×D411yescharacteristic
459C22×C4#12012yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4#12312yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4111yescharacteristic
466Q16—44no—
467SD16—44no—
468SD16—44no—
469D8—44no—
470M16—48no—
471M16—84no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—44no—
499C2×C8—44no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—84no—
515C2×C8—84no—
516C2×C8—84no—
517C2×C8#13112yes—
518C2×C8#13112yes—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4C2×C22⋊C412yes—
536C4⋊C4C2×C22⋊C412yes—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—42no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42C2×C22⋊C412yes—
554C42C4×D412yes—
555C42#12012yes—
556C42#12112yes—
557C42#12112yes—
558C42#12312yes—
559C42C22×D411yescharacteristic
560C42C22×D411yescharacteristic
561C42C22×D411yescharacteristic
562C42C2×#4111yescharacteristic
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23#38612yes—
582C23#44312yes—
583C23#44412yes—
584C23C2×C4×D411yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592Q8—41no—
593D4—22no—
594D4—22no—
595D4—48no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—84no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4#38612yes—
648C2×C4#38612yes—
649C2×C4#38612yes—
650C2×C4C22×C22⋊C411yescharacteristic
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×#12011yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×#12111yescharacteristic
655C2×C4C2×#12311yescharacteristic
656C2×C4C2×#13111yescharacteristic
657C2×C4C2×#13111yescharacteristic
658C2×C4#56811yescharacteristic
659C2×C4#56811yescharacteristic
660C2×C4#57611yescharacteristic
661C2×C4#57711yescharacteristic
662C2×C4#57711yescharacteristic
663C2×C4#57811yescharacteristic
664C8—22no—
665C8—22no—
666C8—42no—
667C8—42no—
668C8—41no—
669C8—41no—
670C8—41no—
671C8—41no—
672C8—42no—
673C8—42no—
674C8—41no—
675C8—41no—
676C8—81no—
677C8—81no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22#187614yes—
699C22C2×#38611yescharacteristic
700C22#293511yescharacteristic
701C22#293511yescharacteristic
702C22#295111yescharacteristic
703C22#295211yescharacteristic
704C22C2×#44311yescharacteristic
705C22C2×#44411yescharacteristic
706C4—22no—
707C4—21no—
708C4—21no—
709C4—22no—
710C4—22no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4#288011yescharacteristic
718C4#288011yescharacteristic
719C4#293111yescharacteristic
720C4#293211yescharacteristic
721C4#301211yescharacteristic
722C4#301311yescharacteristic
723C4#304511yescharacteristic
724C4#304611yescharacteristic
725C2—24no—
726C2—82no—
727C2—161no—
728C2#1808711yescharacteristic
729C2#1808811yescharacteristic
730C2#1809211yescharacteristic
731C2#1809311yescharacteristic
732C2#2090011yescharacteristic
733C2C2×#187611yescharacteristic
734C2#2093711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#466C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2994C2semidirect
3. #21848C2semidirect
4. C2×#1483C2semidirect
5. C2×#1483C2semidirect
6. #8952C2semidirect
7. #8951C2semidirect
8. #8950C2semidirect
9. #8949C2semidirect
10. #8948C2semidirect
11. #8947C2semidirect
12. #8946C2semidirect
13. #8945C2semidirect
14. C2×#1277C2semidirect
15. #7848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31682
C2×#29942
#218482
C2×#14832
#89522
#89512
#89502
#89492
#89482
#89472
#89462
#89452
C2×#12772
#78482