Finite Group Databasebeta

← #116298

Subgroups — #116298

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1996C211yesmaximal, characteristic
2C2×#1520C211yesmaximal, characteristic
3C22×#328C211yesmaximal, characteristic
4#13908C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#380—21no—
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15C22×#95C2211yescharacteristic
16#1996—22no—
17#1996—22no—
18C2×#342—21no—
19C2×#342—21no—
20#1520—22no—
21#1520—22no—
22C2×#328—22no—
23C2×#328—21no—
24C2×#328—21no—
25C2×#328—22no—
26C2×#328C414yescomplemented
27#1444—21no—
28#1442—21no—
29#1436—22no—
30C22⋊#95—21no—
31#1294—22no—
32#1289—22no—
33#1282—22no—
34C23×Q8Q811yescharacteristic
35C24×C4—21no—
36C24×C4C2311yescharacteristic
37C23×C8—21no—
38C23×C8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8D412yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4Q811yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8C2×C412yes—
73C2×C22⋊C8D412yes—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—42no—
82#380—42no—
83#380D412yes—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95C2×C412yes—
97C2×#95D412yes—
98C2×#95D412yes—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#342—24no—
102#342—24no—
103#328—28no—
104#328—24no—
105#328—24no—
106C25C2×D411yescharacteristic
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—82no—
113C22×Q8C4⋊C412yescomplemented
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—44no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—82no—
130C23×C4—82no—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×Q811yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4#4111yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—41no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—82no—
152C22×C8—82no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C22⋊Q8—41no—
163C22⋊Q8—41no—
164C22⋊Q8—41no—
165C22⋊Q8—41no—
166C22⋊Q8—41no—
167C22⋊Q8—41no—
168C22⋊Q8—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C4⋊C412yescomplemented
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4SD1612yes—
219C2×C22⋊C4Q1612yes—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—82no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—44no—
239#95—44no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—82no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C24—42no—
247C24—42no—
248C24C4×D411yescharacteristic
249C24#12011yescharacteristic
250C24#12011yescharacteristic
251C24#12111yescharacteristic
252C24#12111yescharacteristic
253C24C22⋊Q811yescharacteristic
254C24#12411yescharacteristic
255C24C2×SD1611yescharacteristic
256C24C2×Q1611yescharacteristic
257C24#13511yescharacteristic
258C24#13511yescharacteristic
259C24AGL(1,ℤ/8)11yescharacteristic
260C24#13711yescharacteristic
261C24#13711yescharacteristic
262C24#13711yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C4⋊C411yescharacteristic
341C22×C4#11711yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4C22⋊Q811yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4C22⋊Q811yescharacteristic
356C22×C4C22⋊Q811yescharacteristic
357C22×C4#12311yescharacteristic
358C22×C4#12311yescharacteristic
359C22×C4#12311yescharacteristic
360C22×C4#12311yescharacteristic
361C22×C4#12311yescharacteristic
362C22×C4#12411yescharacteristic
363C22×C4#12411yescharacteristic
364C22×C4#12611yescharacteristic
365C22×C4#12611yescharacteristic
366C22×C4#12711yescharacteristic
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—82no—
370C2×C8—82no—
371C2×C8—82no—
372C2×C8—82no—
373C2×C8—82no—
374C2×C8—82no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—88no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—81no—
403C22⋊C4—81no—
404C22⋊C4—81no—
405C22⋊C4—81no—
406C22⋊C4—81no—
407C22⋊C4—81no—
408C42—82no—
409C42—82no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23#38511yescharacteristic
433C23#38611yescharacteristic
434C23#38811yescharacteristic
435C23#39211yescharacteristic
436C23#39311yescharacteristic
437C23#39411yescharacteristic
438C23#39611yescharacteristic
439C23C4×SD1611yescharacteristic
440C23C4×Q1611yescharacteristic
441C23#44011yescharacteristic
442C23#44111yescharacteristic
443C23#44811yescharacteristic
444C23#44911yescharacteristic
445C23#45011yescharacteristic
446C23#45111yescharacteristic
447C23#45111yescharacteristic
448C23#45211yescharacteristic
449C23#45211yescharacteristic
450C23#45211yescharacteristic
451C23#46111yescharacteristic
452C23#46211yescharacteristic
453C23#46311yescharacteristic
454C23#46411yescharacteristic
455C23#46511yescharacteristic
456C23#46711yescharacteristic
457C23#46811yescharacteristic
458C23#47011yescharacteristic
459C23#47511yescharacteristic
460C23#47711yescharacteristic
461C23#47911yescharacteristic
462C23#47911yescharacteristic
463C23#48711yescharacteristic
464C23#48811yescharacteristic
465C23#48911yescharacteristic
466C23#49111yescharacteristic
467Q8—82no—
468Q8—82no—
469Q8—82no—
470Q8—82no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—162no—
542C2×C4—162no—
543C2×C4#38011yescharacteristic
544C2×C4#38511yescharacteristic
545C2×C4#38811yescharacteristic
546C2×C4#39011yescharacteristic
547C2×C4#39411yescharacteristic
548C2×C4#39611yescharacteristic
549C2×C4#39711yescharacteristic
550C2×C4#39911yescharacteristic
551C8—82no—
552C8—82no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#143911yescharacteristic
594C22#186311yescharacteristic
595C22#186511yescharacteristic
596C22#187711yescharacteristic
597C22#187811yescharacteristic
598C22#193311yescharacteristic
599C22#193511yescharacteristic
600C22#200011yescharacteristic
601C22#200111yescharacteristic
602C22#202411yescharacteristic
603C22#202611yescharacteristic
604C22#203112yes—
605C22#203212yes—
606C22#204211yescharacteristic
607C22#204311yescharacteristic
608C22#206611yescharacteristic
609C22#206811yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—162no—
625C4—162no—
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2#1037512yes—
639C2#1037712yes—
640C2#1106711yescharacteristic
641C2#1108111yescharacteristic
642C2#1108911yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C2×#328C4semidirect
113. C22×Q8C4⋊C4semidirect
189. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19962
C2×#15202
C22×#3282
#139082
C2×#12942
C2×#12892
C2×#12822