Finite Group Databasebeta

← #418243

Subgroups — #418243

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4C2×#1863C212yesmaximal
5#15128C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11124C212yesmaximal
8#11067C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#487—28no—
16C2×#487—24no—
17C22×#107C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2446—24no—
24#2446—24no—
25C2×#400—21no—
26C2×#398—21no—
27C2×#395—21no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#387—21no—
34C2×#387—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#2057—24no—
37#1863—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8D414yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#487—232no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103D418yes—
98C2×C4×C8—24no—
99C2×C4×C8—22no—
100#400—28no—
101#398—28no—
102#395—44no—
103#391—22no—
104#391—22no—
105#391—22no—
106#391—22no—
107#391—22no—
108#391—22no—
109#389—22no—
110#389—24no—
111#389—42no—
112#389D414yes—
113#387—28no—
114#387—28no—
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—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336—24no—
136#336—22no—
137C22×Q8—21no—
138C22×Q8—21no—
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×Q8—22no—
150C22×Q8—44no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—82no—
155C22×Q8C2×D412yes—
156C22×Q8C2×D414yes—
157C22×Q8C2×D412yes—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C22×C8#4114yes—
185C4⋊Q8—28no—
186C4⋊Q8—28no—
187C4⋊Q8—48no—
188C4⋊Q8—44no—
189C4⋊Q8—48no—
190C4⋊Q8—44no—
191C4×Q8—48no—
192C4×Q8—44no—
193C4×Q8—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
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—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4Q1614yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4114yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248#107—48no—
249#103—216no—
250#103—216no—
251#103—216no—
252#103—48no—
253C4×C8—28no—
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—
265#95—42no—
266#95—42no—
267#95—82no—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—48no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
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—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8#12014yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4C2×Q1612yes—
367C22×C4C2×Q1612yes—
368C22×C4C2×Q1612yes—
369C22×C4#13512yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4#13712yes—
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C22×C42−1+412yes—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—88no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—48no—
420C42—41no—
421C42—41no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23C2×#12011yescharacteristic
426C23#53211yescharacteristic
427C23#53611yescharacteristic
428C23D4212yes—
429C23#54812yes—
430C23C22×Q1611yescharacteristic
431C23C2×#13511yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23C2×#13711yescharacteristic
434C23#57712yes—
435C23#57812yes—
436Q8—28no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—816no—
443Q8—82no—
444Q8—82no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
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—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#44914yes—
510C2×C4#45114yes—
511C2×C4C2×#12011yescharacteristic
512C2×C4#52512yes—
513C2×C4#53211yescharacteristic
514C2×C4#53411yescharacteristic
515C2×C4#53612yes—
516C2×C4#53611yescharacteristic
517C2×C4#54212yes—
518C2×C4#54212yes—
519C2×C4D4212yes—
520C2×C4#54712yes—
521C2×C4#54712yes—
522C2×C4#54712yes—
523C2×C4#54712yes—
524C2×C4#54812yes—
525C2×C4#55112yes—
526C2×C4#56312yes—
527C8—44no—
528C8—44no—
529C22#240311yescharacteristic
530C22#243211yescharacteristic
531C22#243211yescharacteristic
532C22#246112yes—
533C22#267412yes—
534C22C2×#45111yescharacteristic
535C22C2×#44911yescharacteristic
536C22#301011yescharacteristic
537C22#301111yescharacteristic
538C22#317011yescharacteristic
539C22#317812yes—
540C22#318211yescharacteristic
541C22#323011yescharacteristic
542C22#323111yescharacteristic
543C22#324212yes—
544C22#326512yes—
545C22#327012yes—
546C22#328212yes—
547C22D4×Q1612yes—
548C22#336312yes—
549C22#336412yes—
550C22#336712yes—
551C22#337212yes—
552C4—24no—
553C4—24no—
554C4—42no—
555C4—42no—
556C4—44no—
557C4—81no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—81no—
562C4—82no—
563C4—84no—
564C4#240612yes—
565C4#246112yes—
566C4#267412yes—
567C4#271712yes—
568C2#1526211yescharacteristic
569C2#2385611yescharacteristic
570C2#2385811yescharacteristic
571C2#2565011yescharacteristic
572C2#2565111yescharacteristic
573C2#2567411yescharacteristic
574C2#2568611yescharacteristic
575C2#2682212yes—
576C2#2683612yes—
577C2#3080612yes—
578C2#3081612yes—
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×#4872
C2×#24462
C2×#20572
C2×#18632
#151282
#111902
#111242
#110672
#110102
#108802
#108742