Finite Group Databasebeta

← #115895

Subgroups — #115895

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1767C211yesmaximal, characteristic, complemented
2C22×#329C212yesmaximal
3#14708C211yesmaximal, characteristic
4#10693C211yesmaximal, characteristic, complemented
5C2×#1295C212yesmaximal, complemented
6C22×#124C2211yescharacteristic
7C22×C4⋊C8—21no—
8C22×#97C2212yes—
9C22×#97C2212yes—
10C2×#391C411yescharacteristic
11C2×#391C411yescharacteristic
12C2×#390C411yescharacteristic
13C2×#390C411yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C43C2211yescharacteristic
16#1767—24no—
17C2×#336—21no—
18C2×#336—21no—
19C2×#336—21no—
20C2×#329—24no—
21C2×#329—24no—
22C2×#329—24no—
23#1437—21no—
24#1437—21no—
25#1437—21no—
26#1437—21no—
27#1295—24no—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C23×C8—22no—
31C23×C8—22no—
32C2×#124—22no—
33C2×#124—22no—
34C2×#124—21no—
35C2×#124—21no—
36C2×#124—22no—
37C2×#124—22no—
38C2×#124D412yes—
39C2×#124Q812yes—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—24no—
56C2×C4⋊C8—24no—
57C2×C4⋊C8—24no—
58C2×#97—22no—
59C2×#97—22no—
60C2×#97—22no—
61C2×#97—22no—
62C2×#97—22no—
63C2×#97—22no—
64C2×#97—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×#97—22no—
68C2×#97D414yes—
69C2×#97D414yes—
70#391—42no—
71#391—42no—
72#390—42no—
73#390—42no—
74#382—22no—
75#382—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C43—22no—
83C43—22no—
84#336—24no—
85#336—24no—
86#336—24no—
87#329—216no—
88C25—21no—
89C22×Q8—21no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8C4⋊C412yes—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—41no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4—41no—
104C22×D4C4⋊C412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C22×C8—24no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8—42no—
129#124—24no—
130#124—24no—
131#124—24no—
132#124—24no—
133#124—42no—
134#124—42no—
135#124—42no—
136#124—42no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×Q811yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4112yes—
191C2×C42#4112yes—
192C2×C42#4111yescharacteristic
193C4⋊C8—216no—
194#97—24no—
195#97—24no—
196#97—24no—
197#97—24no—
198#97—42no—
199#97—42no—
200#97—42no—
201#97—42no—
202#95—42no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—41no—
208#95—41no—
209C24—22no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24C2×C22⋊C411yescharacteristic
225C2×Q8—24no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×D4—24no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—82no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4—82no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4#10012yes—
301C22×C4#10012yes—
302C22×C4#10412yes—
303C22×C4#10412yes—
304C22×C4#10412yes—
305C22×C4#10412yes—
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4#11711yescharacteristic
308C22×C4C4×D412yes—
309C22×C4C4×D412yes—
310C22×C4#12011yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4C22⋊Q811yescharacteristic
316C22×C4C22⋊Q811yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12411yescharacteristic
319C22×C4#12711yescharacteristic
320C2×C8—28no—
321C2×C8—28no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C42—24no—
347C42—24no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42—44no—
352C42—42no—
353C42—42no—
354C42#12012yes—
355C42#12112yes—
356C42C22⋊Q812yes—
357C42C22⋊Q812yes—
358C42#12312yes—
359C42#12312yes—
360C42#12412yes—
361C42#12612yes—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23#38011yescharacteristic
398C23#38611yescharacteristic
399C23#39011yescharacteristic
400C23C2×#10011yescharacteristic
401C23C2×#10011yescharacteristic
402C23#41311yescharacteristic
403C23#41311yescharacteristic
404C23C2×#10411yescharacteristic
405C23C2×#10411yescharacteristic
406C23C2×#10411yescharacteristic
407C23C2×#10411yescharacteristic
408C23#42111yescharacteristic
409C23#42111yescharacteristic
410C23#42111yescharacteristic
411C23#42111yescharacteristic
412Q8—84no—
413Q8—84no—
414D4—84no—
415D4—84no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4#32914yescomplemented
493C2×C4#38512yes—
494C2×C4#38812yes—
495C2×C4#39211yescharacteristic
496C2×C4#39311yescharacteristic
497C2×C4#39412yes—
498C2×C4#39612yes—
499C2×C4#39711yescharacteristic
500C2×C4#39911yescharacteristic
501C8—48no—
502C8—48no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—161no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22#143911yescharacteristic
519C22C2×#32911yescharacteristic
520C22#152411yescharacteristic
521C22#152411yescharacteristic
522C22#152511yescharacteristic
523C22#152711yescharacteristic
524C22#152711yescharacteristic
525C22#152911yescharacteristic
526C22#152911yescharacteristic
527C22#153011yescharacteristic
528C22#153011yescharacteristic
529C22#153211yescharacteristic
530C22#153211yescharacteristic
531C22#153311yescharacteristic
532C22#153511yescharacteristic
533C22#153511yescharacteristic
534C22#153611yescharacteristic
535C22#180011yescharacteristic
536C22#180011yescharacteristic
537C22#180611yescharacteristic
538C22#180611yescharacteristic
539C22#180611yescharacteristic
540C22#180611yescharacteristic
541C22#187111yescharacteristic
542C22#187111yescharacteristic
543C22#189611yescharacteristic
544C22#189711yescharacteristic
545C22#190011yescharacteristic
546C22#190011yescharacteristic
547C22#196211yescharacteristic
548C22#196311yescharacteristic
549C22#196511yescharacteristic
550C22#196711yescharacteristic
551C22#198011yescharacteristic
552C22#198011yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—24no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—161no—
567C4—161no—
568C2—162no—
569C2#857911yescharacteristic
570C2#858111yescharacteristic
571C2#858111yescharacteristic
572C2#858311yescharacteristic
573C2#911911yescharacteristic
574C2#912011yescharacteristic
575C2#912311yescharacteristic
576C2#912411yescharacteristic
577C2#968511yescharacteristic
578C2#968711yescharacteristic
579C2#968811yescharacteristic
580C2#968911yescharacteristic
581C2#1104411yescharacteristic
582C2#1105311yescharacteristic
583C2#1107511yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1767C2semidirect
4. #10693C2semidirect
5. C2×#1295C2semidirect
492. C2×C4#329semidirect

Maximal subgroups

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

subgroupindex
C2×#17672
C22×#3292
#147082
#106932
C2×#12952