Finite Group Databasebeta

← #719158

Subgroups — #719158

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C212yesmaximal
2C2×#2399C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic
4#16524C214yesmaximal
5#15627C214yesmaximal
6#15626C214yesmaximal
7#15221C212yesmaximal
8#15202C212yesmaximal
9#14809C214yesmaximal
10#14276C211yesmaximal, characteristic
11#14017C212yesmaximal
12#13954C211yesmaximal, characteristic
13#13926C211yesmaximal, characteristic
14#13911C211yesmaximal, characteristic
15#13786C211yesmaximal, characteristic
16C22×C4×Q8C2212yes—
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19#2581—216no—
20#2399—28no—
21C2×#400C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#395C2212yes—
27#2351—28no—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×#376C2211yescharacteristic
48#1446C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77#1434C2214yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2312yes—
80C2×C4×Q8—28no—
81C2×C4×Q8—24no—
82C2×C4×Q8—28no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—24no—
86C2×C4×Q8—44no—
87C2×C4×Q8—44no—
88C2×C4×Q8Q818yescomplemented
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#400—216no—
118#398—28no—
119#398—28no—
120#395—28no—
121#395—216no—
122#395—28no—
123#391—28no—
124#391D418yes—
125#391Q818yescomplemented
126#389—28no—
127#389—216no—
128#389—28no—
129#389—28no—
130#389—44no—
131#387—216no—
132#387—216no—
133#384—24no—
134#382—28no—
135#382—24no—
136#382—28no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4D418yes—
142#376—24no—
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8C2×D414yes—
184C22×Q8C2×Q814yes—
185C22×Q8C2×Q814yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C4×Q8—232no—
202C4×Q8—432no—
203C4×Q8—432no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×Q814yes—
260C2×C4⋊C4C2×Q818yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42C2×D418yes—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42C2×Q818yes—
282C2×C42C2×Q814yes—
283C2×C42C2×Q814yes—
284C2×C42#4114yes—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—24no—
291#95—22no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C2×D414yes—
311#95C2×D414yes—
312#95C2×D4116yes—
313#95C2×Q8116yes—
314C24C2511yescharacteristic
315C2×Q8—28no—
316C2×Q8—28no—
317C2×Q8—28no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8C4⋊Q8116yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4C22×D414yes—
401C22×C4C22×D414yes—
402C22×C4C22×D414yes—
403C22×C4C22×D414yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×Q814yes—
410C22×C4C22×Q814yes—
411C22×C4C22×Q814yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q814yes—
415C22×C4C22×Q814yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q812yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+414yes—
423C22×C42−1+412yes—
424C4⋊C4—216no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C42—216no—
445C42—216no—
446C42—28no—
447C42—28no—
448C42—416no—
449C42—48no—
450C42—84no—
451C42C22⋊Q8116yes—
452C23C23×D411yescharacteristic
453C23C23×D411yescharacteristic
454C23C23×Q811yescharacteristic
455C23C23×Q811yescharacteristic
456C23C22×#4111yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×2−1+411yescharacteristic
459C23C2×2−1+412yes—
460C23#58512yes—
461C23#58512yes—
462C23#58512yes—
463Q8—232no—
464Q8—432no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4C2×C22⋊Q814yes—
551C2×C4C2×C22⋊Q814yes—
552C2×C4C2×C22⋊Q814yes—
553C2×C4C2×C4⋊Q818yes—
554C2×C4C2×C4⋊Q814yes—
555C2×C4#54414yes—
556C2×C4D4214yes—
557C2×C4#54714yes—
558C2×C4D4×Q814yes—
559C2×C4#55414yes—
560C22C22×C22⋊Q811yescharacteristic
561C22C22×C4⋊Q811yescharacteristic
562C22#344912yes—
563C22C2×#54411yescharacteristic
564C22C2×D4211yescharacteristic
565C22C2×#54711yescharacteristic
566C22C2×D4×Q811yescharacteristic
567C22C2×#55411yescharacteristic
568C22#348712yes—
569C22#348912yes—
570C22#348912yes—
571C22#350212yes—
572C22#350212yes—
573C22#350412yes—
574C22#350412yes—
575C22#350911yescharacteristic
576C22#351011yescharacteristic
577C22#351011yescharacteristic
578C22#351011yescharacteristic
579C22#354012yes—
580C22#354112yes—
581C22#355512yes—
582C22#355712yes—
583C4—24no—
584C4—24no—
585C4—28no—
586C4—44no—
587C4—44no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—84no—
595C4—82no—
596C4—84no—
597C4#244018yes—
598C2C2×#244011yescharacteristic
599C2#3533411yescharacteristic
600C2#3566211yescharacteristic
601C2#3595712yes—
602C2#3765912yes—
603C2#3959112yes—
604C2#4282112yes—
605C2#4526611yescharacteristic
606C2#4527611yescharacteristic
607C2#4542511yescharacteristic
608C2#4542911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
88. C2×C4×Q8Q8semidirect
125. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×#23992
C2×#23512
#165242
#156272
#156262
#152212
#152022
#148092
#142762
#140172
#139542
#139262
#139112
#137862