Finite Group Databasebeta

← #627803

Subgroups — #627803

Order 512.

2394 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3#15552C214yesmaximal, complemented
4#15548C214yesmaximal, complemented
5#15388C214yesmaximal, complemented
6#15382C214yesmaximal, complemented
7#15019C214yesmaximal
8#14702C211yesmaximal, characteristic
9#14686C211yesmaximal, characteristic
10#14347C211yesmaximal, characteristic
11#14281C211yesmaximal, characteristic
12#14047C214yesmaximal
13C2×#1436C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17#2627—28no—
18C2×#400C2212yes—
19C2×#398C2214yes—
20C2×#395C2212yes—
21#2313—28no—
22C2×#391C2212yes—
23C2×#387C2214yes—
24C2×#387C2214yes—
25C2×#387C2214yes—
26C2×#385C2212yes—
27C2×#384C2212yes—
28C2×#381C2212yes—
29C2×#381C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2214yes—
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C22×#95C2211yescharacteristic
36C22×#95C2212yes—
37C22×#95C2212yes—
38#1446C2212yes—
39#1445C2214yes—
40#1445C2214yes—
41#1445C2214yes—
42#1445C2212yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2212yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4D418yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C42—24no—
96C22×C42—44no—
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101#400—28no—
102#398—216no—
103#395—28no—
104#391—44no—
105#387—216no—
106#387—216no—
107#387—48no—
108#385—28no—
109#384—44no—
110#381—28no—
111#381—28no—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—216no—
116C4×C22⋊C4—28no—
117C2×#95—28no—
118C2×#95—24no—
119C2×#95—44no—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2411yescharacteristic
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—42no—
154C22×Q8—42no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D414yes—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4#4118yes—
182C23×C4#4114yes—
183C23×C4#4114yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C22⋊Q8—416no—
197C2×C4⋊C4—232no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D418yes—
227C2×C4⋊C4C2×D418yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—28no—
243C2×C42—416no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42—44no—
248#95—28no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24C22×D412yes—
282C24C2×#4112yes—
283C24C2×#4112yes—
284C24C2×#4114yes—
285C242−1+414yes—
286C24C2511yescharacteristic
287C2×Q8—48no—
288C2×Q8—44no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8—84no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—216no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4#123116yes—
381C22×C4C22×D414yes—
382C22×C4C22×D414yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4114yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4114yes—
391C22×C4H(5,𝔽2)14yes—
392C22×C42−1+414yes—
393C4⋊C4—216no—
394C4⋊C4—216no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—416no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—88no—
402C4⋊C4—88no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C42—28no—
408C42—48no—
409C42—44no—
410C42—88no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23C2×#12314yes—
428C23#53214yes—
429C23#53614yes—
430C23#55218yes—
431C23#55618yes—
432C23C23×D411yescharacteristic
433C23C22×#4111yescharacteristic
434C23C22×#4111yescharacteristic
435C23C22×#4112yes—
436C23C2×H(5,𝔽2)12yes—
437C23C2×2−1+412yes—
438C23C2×2−1+412yes—
439C23#58512yes—
440C23#58512yes—
441Q8—88no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—416no—
465C2×C4—432no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—416no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—88no—
515C2×C4—816no—
516C2×C4C2×#12314yes—
517C2×C4C2×#12314yes—
518C2×C4#53918yes—
519C22—24no—
520C22—28no—
521C22—28no—
522C22—24no—
523C22—28no—
524C22—28no—
525C22—28no—
526C22—28no—
527C22#249918yes—
528C22C22×#12311yescharacteristic
529C22C2×#53211yescharacteristic
530C22C2×#53611yescharacteristic
531C22C2×#53912yes—
532C22#345512yes—
533C22C2×#55212yes—
534C22C2×#55612yes—
535C22#349011yescharacteristic
536C22#349111yescharacteristic
537C22#349512yes—
538C22#349612yes—
539C22#349712yes—
540C22#350512yes—
541C22#350512yes—
542C22#350712yes—
543C22#350911yescharacteristic
544C22#351011yescharacteristic
545C22#353512yes—
546C22#353812yes—
547C22#356312yes—
548C22#356412yes—
549C4—24no—
550C4—24no—
551C4—28no—
552C4—44no—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4—84no—
559C4—84no—
560C4—84no—
561C2—28no—
562C2C2×#249911yescharacteristic
563C2#3500612yes—
564C2#3536011yescharacteristic
565C2#3567311yescharacteristic
566C2#3674212yes—
567C2#3788012yes—
568C2#3788612yes—
569C2#3950812yes—
570C2#3951212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15552C2semidirect
4. #15548C2semidirect
5. #15388C2semidirect
6. #15382C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#23132
#155522
#155482
#153882
#153822
#150192
#147022
#146862
#143472
#142812
#140472
C2×#14362