Finite Group Databasebeta

← #415433

Subgroups — #415433

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic
4C2×#1776C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6#13909C211yesmaximal, characteristic
7#11379C211yesmaximal, characteristic
8#11378C211yesmaximal, characteristic
9#11378C211yesmaximal, characteristic
10#11378C211yesmaximal, characteristic
11#11372C211yesmaximal, characteristic
12#11371C211yesmaximal, characteristic
13#11325C211yesmaximal, characteristic
14#11324C211yesmaximal, characteristic
15#10835C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1993—24no—
30#1988—24no—
31#1949—24no—
32#1776—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49#1446—22no—
50#1441—22no—
51#1436—22no—
52C4⋊#95—22no—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C23×Q8—21no—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8Q814yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—41no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—41no—
97C2×#106Q814yescomplemented
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8D414yes—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97D414yes—
104#384—22no—
105#384—41no—
106#382—22no—
107#382—22no—
108#382—22no—
109#382—41no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D412yes—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#336—22no—
141#336—22no—
142#336—41no—
143#336—41no—
144C22×Q8—24no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—82no—
148C22×Q8—82no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
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×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×Q812yes—
189C22×C8#4112yes—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192C4⋊Q8—44no—
193C4⋊Q8—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
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—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
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—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C2×Q812yes—
246C2×C4⋊C4C2×Q812yes—
247C2×C4⋊C4C2×Q812yes—
248C2×C4⋊C4C2×Q812yes—
249C2×C4⋊C4#4112yes—
250C2×C4⋊C4#4112yes—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
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—21no—
263C2×C42—44no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—82no—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×Q812yes—
270C2×C42#4112yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273#107—28no—
274#107—44no—
275#106—28no—
276#106—44no—
277C4⋊C8—28no—
278#97—28no—
279#95—24no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—44no—
286#95—82no—
287#95—82no—
288#95—82no—
289#95—82no—
290#95—82no—
291C24C22×D411yescharacteristic
292C2×Q8—48no—
293C2×Q8—88no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×Q8—84no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4#12312yes—
357C22×C4#12312yes—
358C22×C4#12712yes—
359C22×C4#12712yes—
360C22×C4#13714yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8C22⋊Q814yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—48no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—48no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—88no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—84no—
447C42—84no—
448C42C22⋊Q814yes—
449C42C22⋊Q814yes—
450C42#12314yes—
451C42#12314yes—
452C23C2×C22⋊Q811yescharacteristic
453C23C2×#12311yescharacteristic
454C23C2×#12711yescharacteristic
455C23#54611yescharacteristic
456C23#54711yescharacteristic
457C23#54811yescharacteristic
458C23D4×Q811yescharacteristic
459C23C2×#13712yes—
460C23#57512yes—
461C23#57612yes—
462C23#57712yes—
463Q8—816no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#39914yes—
527C2×C4#39914yes—
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4#52911yescharacteristic
535C2×C4#52911yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#53311yescharacteristic
538C2×C4#53611yescharacteristic
539C2×C4#53711yescharacteristic
540C2×C4#54111yescharacteristic
541C2×C4#54111yescharacteristic
542C2×C4#54211yescharacteristic
543C2×C4#54211yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4#54711yescharacteristic
546C2×C4#54811yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4#55611yescharacteristic
549C2×C4#55611yescharacteristic
550C2×C4#55711yescharacteristic
551C2×C4#55711yescharacteristic
552C2×C4#55711yescharacteristic
553C2×C4#55711yescharacteristic
554C2×C4Q8211yescharacteristic
555C2×C4#55911yescharacteristic
556C8—24no—
557C8—42no—
558C8—42no—
559C8—42no—
560C22C2×#39911yescharacteristic
561C22#244611yescharacteristic
562C22#244711yescharacteristic
563C22#245111yescharacteristic
564C22#245311yescharacteristic
565C22#254011yescharacteristic
566C22#254211yescharacteristic
567C22#307812yes—
568C22#308112yes—
569C22#309512yes—
570C22#309612yes—
571C22#315211yescharacteristic
572C22#315311yescharacteristic
573C22#315411yescharacteristic
574C22#315511yescharacteristic
575C22#331612yes—
576C22#331912yes—
577C22#334612yes—
578C22#334812yes—
579C22#336912yes—
580C22#337012yes—
581C22#338512yes—
582C22#338812yes—
583C4—24no—
584C4—24no—
585C4—44no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—161no—
594C4—161no—
595C4#244611yescharacteristic
596C4#244711yescharacteristic
597C4#245111yescharacteristic
598C4#245311yescharacteristic
599C4#258111yescharacteristic
600C4#258211yescharacteristic
601C4#262511yescharacteristic
602C4#285511yescharacteristic
603C2#1519311yescharacteristic
604C2#2373211yescharacteristic
605C2#2373511yescharacteristic
606C2#2629312yes—
607C2#2631512yes—
608C2#2645312yes—
609C2#2651912yes—
610C2#2842211yescharacteristic
611C2#2843211yescharacteristic
612C2#2854311yescharacteristic
613C2#2854911yescharacteristic
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

Complements

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

NH ≅ G/Ntype
97. C2×#106Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19932
C2×#19882
C2×#19492
C2×#17762
#141612
#139092
#113792
#113782
#113722
#113712
#113252
#113242
#108352