Finite Group Databasebeta

← #414938

Subgroups — #414938

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic
2C2×#1940C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11257C212yesmaximal
8#11077C212yesmaximal
9#11077C212yesmaximal
10#11024C212yesmaximal
11#10843C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#441—24no—
16C2×#441—24no—
17C2×#441—24no—
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#97C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1940—24no—
32#1805—24no—
33#1803—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—21no—
40#1441—21no—
41#1441—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—21no—
46#1434—21no—
47C23×Q8C2311yescharacteristic
48C23×Q8C2311yescharacteristic
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16Q814yes—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8D414yes—
70C2×C4×Q8D414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#441—232no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—42no—
93C2×C4⋊C8—42no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97D414yes—
109#391—24no—
110#391—24no—
111#391—42no—
112#391—42no—
113#384—22no—
114#384—22no—
115#382—22no—
116#382—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—24no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8C2×D412yes—
159C22×Q8C2×D412yes—
160C22×Q8C2×Q812yes—
161C22×Q8C2×Q812yes—
162C22×Q8#4112yes—
163C22×Q8#4112yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C2×Q16—28no—
176C2×Q16—28no—
177C2×Q16—28no—
178C2×Q16—28no—
179C2×Q16—28no—
180C2×Q16—28no—
181C2×Q16—28no—
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—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×Q812yes—
202C22×C8#4112yes—
203C4×Q8—28no—
204C4×Q8—28no—
205C4×Q8—28no—
206C4×Q8—28no—
207C4×Q8—44no—
208C4×Q8—44no—
209C4×Q8—44no—
210C4×Q8—44no—
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—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266#106—28no—
267C4⋊C8—48no—
268C4⋊C8—44no—
269#103—28no—
270#103—28no—
271#103—48no—
272#103—48no—
273#97—28no—
274#95—22no—
275#95—22no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95—82no—
285#95—82no—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8C22⋊Q814yes—
342C2×Q8C22⋊Q814yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
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—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
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—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4#13714yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C42−1+411yescharacteristic
419Q16—232no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8C22⋊Q814yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—48no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—48no—
467C42—48no—
468C42#12014yes—
469C42#12314yes—
470C42#12314yes—
471C42#12414yes—
472C23C2×#12111yescharacteristic
473C23C2×#12111yescharacteristic
474C23C2×#12111yescharacteristic
475C23#54611yescharacteristic
476C23#54611yescharacteristic
477C23D4×Q811yescharacteristic
478C23#55011yescharacteristic
479C23C2×#13712yes—
480C23#57512yes—
481C23#57712yes—
482C23#57812yes—
483Q8—24no—
484Q8—24no—
485Q8—24no—
486Q8—24no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#39414yes—
560C2×C4#39414yes—
561C2×C4C2×#12011yescharacteristic
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4C2×C22⋊Q811yescharacteristic
564C2×C4C2×C22⋊Q811yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4C2×#12311yescharacteristic
567C2×C4#52511yescharacteristic
568C2×C4#52511yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4C2×#12411yescharacteristic
571C2×C4#53511yescharacteristic
572C2×C4#53511yescharacteristic
573C2×C4#53511yescharacteristic
574C2×C4#53711yescharacteristic
575C2×C4#53711yescharacteristic
576C2×C4#53711yescharacteristic
577C2×C4#55211yescharacteristic
578C2×C4#55211yescharacteristic
579C2×C4#55311yescharacteristic
580C2×C4#55311yescharacteristic
581C2×C4#55311yescharacteristic
582C2×C4#55311yescharacteristic
583C2×C4#55411yescharacteristic
584C2×C4#55411yescharacteristic
585C2×C4#55411yescharacteristic
586C2×C4#55511yescharacteristic
587C2×C4#55511yescharacteristic
588C2×C4#55611yescharacteristic
589C8—24no—
590C8—42no—
591C8—82no—
592C22C2×#39411yescharacteristic
593C22#241411yescharacteristic
594C22#241411yescharacteristic
595C22#241711yescharacteristic
596C22#241711yescharacteristic
597C22#253511yescharacteristic
598C22#253811yescharacteristic
599C22#303712yes—
600C22#303911yescharacteristic
601C22#304011yescharacteristic
602C22#304311yescharacteristic
603C22#304411yescharacteristic
604C22#304612yes—
605C22#305612yes—
606C22#306312yes—
607C22#331212yes—
608C22#331312yes—
609C22#331612yes—
610C22#331712yes—
611C22#338712yes—
612C22#339012yes—
613C22#339812yes—
614C22#340112yes—
615C4—24no—
616C4—24no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—162no—
624C4—162no—
625C4#248511yescharacteristic
626C4#248511yescharacteristic
627C4#248711yescharacteristic
628C4#248711yescharacteristic
629C4#256211yescharacteristic
630C4#256311yescharacteristic
631C4#261511yescharacteristic
632C4#284811yescharacteristic
633C2#1518111yescharacteristic
634C2#2354511yescharacteristic
635C2#2354711yescharacteristic
636C2#2464112yes—
637C2#2464811yescharacteristic
638C2#2464911yescharacteristic
639C2#2482612yes—
640C2#2483111yescharacteristic
641C2#2483211yescharacteristic
642C2#2815312yes—
643C2#2827812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4412
C2×#19402
C2×#18052
C2×#18032
#139422
#139262
#112572
#110772
#110242
#108432