Finite Group Databasebeta

← #368530

Subgroups — #368530

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1894C211yesmaximal, characteristic
2C2×#1873C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#14139C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10932C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#391C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1894C418yescomplemented
25#1873—24no—
26#1736—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1441—21no—
37#1441—21no—
38#1437—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42C22⋊#95—22no—
43C4×#95—21no—
44C23×Q8C2311yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—42no—
50C22×M16—42no—
51C22×M16C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8C2×C414yes—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—42no—
64C2×#117D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42C2312yes—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103C2×C418yescomplemented
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93#391—24no—
94#391—42no—
95#380—22no—
96#380C2×C414yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—42no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340C2×C418yescomplemented
118#336—42no—
119#336—42no—
120C25C2×D411yescharacteristic
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—44no—
144C23×C4—81no—
145C23×C4—81no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C2×M16—28no—
157C2×M16—48no—
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—48no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C22×C414yes—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182#117—28no—
183#117—44no—
184#117—48no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
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—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4#4114yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—81no—
229C2×C22⋊C4—81no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—82no—
241#103—48no—
242#103—44no—
243#103—44no—
244C22⋊C8—44no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—82no—
253#95—82no—
254#95—82no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24C4×D412yes—
264C24#12112yes—
265C24#12312yes—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—48no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4#11714yes—
356C22×C4C4×D414yes—
357C22×C4C4×D412yes—
358C22×C4#12112yes—
359C22×C4#12312yes—
360C22×C4#12414yes—
361C22×C4#12614yes—
362C22×C4#13714yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4111yescharacteristic
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373M16—416no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—48no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—48no—
410C4⋊C4#12314yes—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—84no—
415C42—48no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—88no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#38814yes—
444C23C2×C4×D411yescharacteristic
445C23C2×#12111yescharacteristic
446C23C2×#12311yescharacteristic
447C23#54612yes—
448C23#54612yes—
449C23C2×#13712yes—
450C23#57512yes—
451C23#57611yescharacteristic
452C23#57711yescharacteristic
453C23#57711yescharacteristic
454C23#57811yescharacteristic
455Q8—42no—
456Q8—44no—
457Q8—42no—
458Q8—81no—
459Q8—81no—
460Q8—82no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—816no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#38814yes—
523C2×C4C2×#11711yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C4×#4112yes—
526C2×C4#51812yes—
527C2×C4C2×#12311yescharacteristic
528C2×C4#52511yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12611yescharacteristic
531C2×C4#52912yes—
532C2×C4#53411yescharacteristic
533C2×C4#53511yescharacteristic
534C2×C4#53812yes—
535C2×C4#55112yes—
536C2×C4#55112yes—
537C2×C4#55312yes—
538C2×C4#55312yes—
539C2×C4#55312yes—
540C2×C4#55312yes—
541C8—44no—
542C8—81no—
543C8—81no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22C2×#38811yescharacteristic
556C22#236412yes—
557C22#241111yescharacteristic
558C22#241211yescharacteristic
559C22#244412yes—
560C22C4×#13712yes—
561C22#294611yescharacteristic
562C22#294711yescharacteristic
563C22#303911yescharacteristic
564C22#304011yescharacteristic
565C22#304511yescharacteristic
566C22#304611yescharacteristic
567C22#309512yes—
568C22#309811yescharacteristic
569C22#309911yescharacteristic
570C22#331112yes—
571C22#331212yes—
572C22#331412yes—
573C22#331712yes—
574C22#332212yes—
575C22#332712yes—
576C22#332912yes—
577C22#333112yes—
578C4—24no—
579C4—44no—
580C4—44no—
581C4—82no—
582C4—84no—
583C4—82no—
584C4—84no—
585C4—82no—
586C4—82no—
587C4C4×#12311yescharacteristic
588C4#232111yescharacteristic
589C4#235312yes—
590C4#250411yescharacteristic
591C4#251012yes—
592C4#259111yescharacteristic
593C2—44no—
594C2#1402011yescharacteristic
595C2#2106111yescharacteristic
596C2#2106211yescharacteristic
597C2#2314412yes—
598C2#2314512yes—
599C2#2449311yescharacteristic
600C2#2449811yescharacteristic
601C2#2454311yescharacteristic
602C2#2454811yescharacteristic
603C2#2622112yes—
604C2#2622312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11316C2semidirect
7. #11190C2semidirect
8. #10932C2semidirect
9. #10806C2semidirect
24. #1894C4semidirect
89. C2×#103C2×C4semidirect
117. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18942
C2×#18732
C2×#17362
#141392
#138012
#113162
#111902
#109322
#108062
#107302
#107152