Finite Group Databasebeta

← #4397245

Subgroups — #4397245

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#17063C212yesmaximal, complemented
4#16889C212yesmaximal, complemented
5#15681C214yesmaximal, complemented
6#15591C214yesmaximal, complemented
7#15470C214yesmaximal
8#15385C214yesmaximal
9#15242C212yesmaximal, complemented
10#15209C212yesmaximal, complemented
11#14660C211yesmaximal, characteristic
12#14342C211yesmaximal, characteristic
13#14292C211yesmaximal, characteristic
14#13904C211yesmaximal, characteristic
15#13780C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2608—28no—
19#2502—28no—
20C2×#400C2214yes—
21C2×#398C2214yes—
22C2×#396C2212yes—
23C2×#396C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2214yes—
34C2×#381C2211yescharacteristic
35C2×#381C2211yescharacteristic
36C2×#381C2214yes—
37C2×#380C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2212yes—
41#1446C2212yes—
42#1445C2214yes—
43#1445C2212yes—
44#1445C2212yes—
45#1445C2214yes—
46#1445C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2212yes—
62#1440C2214yes—
63#1440C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#400—48no—
113#398—216no—
114#396—28no—
115#396—28no—
116#391—24no—
117#391—24no—
118#389—28no—
119#389—28no—
120#387—28no—
121#387—28no—
122#385—216no—
123#384—24no—
124#384—24no—
125#382—48no—
126#381—24no—
127#381—24no—
128#381—48no—
129#380—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4D418yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C25C2411yescharacteristic
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—42no—
167C22×Q8—42no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C2×D414yes—
176C23×C4#4114yes—
177C23×C4#4114yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C22⋊Q8—416no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
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—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—416no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×D418yes—
249C2×C42—28no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D414yes—
259C2×C42C2×D414yes—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95C2×D418yes—
290C24—22no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24C22×D412yes—
296C24C2×#4112yes—
297C24C2×#4112yes—
298C24C2511yescharacteristic
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—84no—
303C2×Q8—84no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
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—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—88no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4#124116yes—
378C22×C4C22×D414yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D414yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C42−1+418yes—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—88no—
402C22⋊C4—216no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C42—44no—
410C42—44no—
411C42—416no—
412C42—82no—
413C42—82no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23C2×#12414yes—
430C23C23×D411yescharacteristic
431C23C22×#4111yescharacteristic
432C23C22×#4111yescharacteristic
433C23C2×H(5,𝔽2)12yes—
434C23C2×2−1+414yes—
435C23#58512yes—
436C23#58512yes—
437C23#58512yes—
438Q8—88no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—416no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
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—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
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—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—816no—
512C2×C4C2×#12414yes—
513C2×C4C2×#12414yes—
514C2×C4#53618yes—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—48no—
523C22C22×#12411yescharacteristic
524C22C2×#53612yes—
525C22#348511yescharacteristic
526C22#348511yescharacteristic
527C22#348812yes—
528C22#349112yes—
529C22#349112yes—
530C22#350612yes—
531C22#350811yescharacteristic
532C22#350912yes—
533C22#350912yes—
534C22#351011yescharacteristic
535C22#354114yes—
536C22#355314yes—
537C22#355914yes—
538C22#356514yes—
539C4—24no—
540C4—24no—
541C4—28no—
542C4—44no—
543C4—44no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—84no—
553C2—44no—
554C2#3532011yescharacteristic
555C2#3545511yescharacteristic
556C2#3555611yescharacteristic
557C2#4140812yes—
558C2#4268212yes—
559C2#4685512yes—
560C2#4722412yes—
561C2#4887714yes—
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. #17063C2semidirect
4. #16889C2semidirect
5. #15681C2semidirect
6. #15591C2semidirect
9. #15242C2semidirect
10. #15209C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
C2×#25022
#170632
#168892
#156812
#155912
#154702
#153852
#152422
#152092
#146602
#143422
#142922
#139042
#137802