Finite Group Databasebeta

← #428465

Subgroups — #428465

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#2072C211yesmaximal, characteristic
4C2×#2043C212yesmaximal
5C2×#1863C212yesmaximal
6#13910C211yesmaximal, characteristic
7#11346C211yesmaximal, characteristic
8#11328C212yesmaximal
9#11067C212yesmaximal
10#10932C211yesmaximal, characteristic
11#10932C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2446—24no—
20#2446—24no—
21C2×#400—21no—
22C2×#398—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387—21no—
30C2×#387—21no—
31C2×#384—21no—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#2095—24no—
36#2072—24no—
37#2043—28no—
38#1863—28no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46#1444—21no—
47#1444—21no—
48C4⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—42no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103D418yes—
95#400—28no—
96#398—28no—
97#391—22no—
98#391—22no—
99#391—22no—
100#391—22no—
101#391D414yes—
102#391D414yes—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—24no—
107#389—42no—
108#389—42no—
109#387—28no—
110#387—28no—
111#384—22no—
112#384—22no—
113#384—44no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129#340—24no—
130#340—24no—
131#340—24no—
132#336—24no—
133#336—24no—
134#336—22no—
135#336—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C2×D412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×D414yes—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D414yes—
179C4⋊Q8—48no—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C4×Q8—48no—
183C4×Q8—44no—
184C4×Q8—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4Q1614yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4#4114yes—
230C2×C4⋊C4#4114yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248#107—48no—
249#103—216no—
250#103—48no—
251#103—48no—
252#103—48no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—22no—
263#95—44no—
264#95—44no—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8#12714yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
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—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4#12012yes—
358C22×C4#12012yes—
359C22×C4C2×Q1612yes—
360C22×C4C2×Q1612yes—
361C22×C4C2×Q1612yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+412yes—
371C22×C42−1+412yes—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—88no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—88no—
409C4⋊C4#12318yes—
410C42—24no—
411C42—24no—
412C42—48no—
413C42—48no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—42no—
419C42—42no—
420C23C2×#12011yescharacteristic
421C23#52511yescharacteristic
422C23#53611yescharacteristic
423C23D4212yes—
424C23#54812yes—
425C23C22×Q1611yescharacteristic
426C23C2×AGL(1,ℤ/8)11yescharacteristic
427C23C2×#13711yescharacteristic
428C23#57511yescharacteristic
429C23#57712yes—
430C23#57812yes—
431Q8—28no—
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—82no—
438Q8—82no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#45114yes—
505C2×C4C2×#12312yes—
506C2×C4C2×#12711yescharacteristic
507C2×C4#53211yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#53612yes—
511C2×C4#53912yes—
512C2×C4#54212yes—
513C2×C4D4212yes—
514C2×C4#54712yes—
515C2×C4#54712yes—
516C2×C4#54712yes—
517C2×C4#54712yes—
518C2×C4#54712yes—
519C2×C4#55312yes—
520C2×C4#56312yes—
521C8—44no—
522C8—82no—
523C22#240611yescharacteristic
524C22#243211yescharacteristic
525C22#243311yescharacteristic
526C22#244312yes—
527C22#267412yes—
528C22C2×#45111yescharacteristic
529C22#300611yescharacteristic
530C22#301011yescharacteristic
531C22#301111yescharacteristic
532C22#310411yescharacteristic
533C22#311011yescharacteristic
534C22#312012yes—
535C22#323011yescharacteristic
536C22#323611yescharacteristic
537C22#324212yes—
538C22#326512yes—
539C22#327112yes—
540C22#327612yes—
541C22D4×Q1612yes—
542C22#336312yes—
543C22#336712yes—
544C22#337212yes—
545C22#337712yes—
546C4—24no—
547C4—42no—
548C4—44no—
549C4—48no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—82no—
555C4—81no—
556C4—82no—
557C4—162no—
558C4#242212yes—
559C4#250112yes—
560C4#268012yes—
561C4#271812yes—
562C2#1538411yescharacteristic
563C2#2406411yescharacteristic
564C2#2407111yescharacteristic
565C2#2565111yescharacteristic
566C2#2566111yescharacteristic
567C2#2576011yescharacteristic
568C2#2576811yescharacteristic
569C2#2617112yes—
570C2#2618212yes—
571C2#3081612yes—
572C2#3083412yes—
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
C2×#24462
C2×#20952
C2×#20722
C2×#20432
C2×#18632
#139102
#113462
#113282
#110672
#109322