Finite Group Databasebeta

← #385160

Subgroups — #385160

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C211yesmaximal, characteristic
2C2×#1854C212yesmaximal, complemented
3C2×#1807C212yesmaximal, complemented
4#13905C211yesmaximal, characteristic
5#13871C211yesmaximal, characteristic
6#11325C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1869C418yescomplemented
25#1854—28no—
26#1807—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33#1445—21no—
34#1440—21no—
35#1438—21no—
36#1438—21no—
37#1437—21no—
38#1437—21no—
39#1437—21no—
40#1436—21no—
41C4⋊#95C2212yes—
42C23×D4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8—24no—
46C23×C8—24no—
47C23×C8—41no—
48C23×C8—41no—
49C23×C8C2×C414yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4C2×C414yes—
57C2×C4×D4Q814yes—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117—41no—
61C2×#117—41no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—42no—
79C2×#107—42no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#106—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—41no—
89C2×C22⋊C8—41no—
90C2×C22⋊C8C2×C418yes—
91C2×C22⋊C8C2×C414yes—
92#382—41no—
93#382—41no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—42no—
98#380—42no—
99#380—42no—
100#380—41no—
101#380—41no—
102C4×C4⋊C4—41no—
103C4×C4⋊C4—41no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340D418yescomplemented
118#336C2×C418yes—
119C25C2×D411yescharacteristic
120C25C2×D411yescharacteristic
121C22×D4—24no—
122C22×D4—41no—
123C22×D4—41no—
124C22×D4—42no—
125C22×D4C4⋊C414yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q812yes—
153C23×C4C2×Q812yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—216no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—48no—
169C22×C8—44no—
170C22×C8C22×C414yes—
171C22×C8C22×C414yes—
172C22×C8C2×D414yes—
173C22×C8#4114yes—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178#117—44no—
179#117—48no—
180#117—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
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—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C4⋊C414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4C4⋊C414yes—
226C2×C22⋊C4C4⋊C414yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×Q812yes—
237#107—48no—
238#106—48no—
239#106—48no—
240C22⋊C8—216no—
241C22⋊C8—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—82no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24C2×C4⋊C412yes—
267C24C2×C4⋊C412yes—
268C24#13514yes—
269C24C22×D411yescharacteristic
270C2×D4—28no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—48no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
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—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C2×C4⋊C412yes—
356C22×C4C2×C4⋊C412yes—
357C22×C4C2×C4⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)12yes—
368C22×C42−1+412yes—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—416no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8C4×D418yes—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—48no—
404C22⋊C4—88no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—84no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—48no—
413C42—84no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23#42714yes—
456C23C22×C4⋊C411yescharacteristic
457C23#53411yescharacteristic
458C23#53411yescharacteristic
459C23D4212yes—
460C23#54712yes—
461C23C2×#13512yes—
462C23#57512yes—
463C23#57712yes—
464C23#57712yes—
465D4—416no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4C2×C4×D412yes—
527C2×C4#51811yescharacteristic
528C2×C4#51811yescharacteristic
529C2×C4#52012yes—
530C2×C4#53512yes—
531C2×C4#53712yes—
532C2×C4#54012yes—
533C2×C4#54212yes—
534C2×C4#54311yescharacteristic
535C2×C4#54311yescharacteristic
536C2×C4D4211yescharacteristic
537C2×C4#54612yes—
538C2×C4D4×Q812yes—
539C2×C4#55111yescharacteristic
540C2×C4#55412yes—
541C2×C4#56011yescharacteristic
542C2×C4#56212yes—
543C2×C4#56311yescharacteristic
544C8—28no—
545C8—44no—
546C22—28no—
547C22—24no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22#231511yescharacteristic
565C22D4×C4⋊C412yes—
566C22#266911yescharacteristic
567C22#267112yes—
568C22#271711yescharacteristic
569C22C2×#42711yescharacteristic
570C22#291111yescharacteristic
571C22#291212yes—
572C22#319112yes—
573C22#319112yes—
574C22#320011yescharacteristic
575C22#320011yescharacteristic
576C22#320412yes—
577C22#326712yes—
578C22#327112yes—
579C22#328212yes—
580C22#328212yes—
581C22#333712yes—
582C22#333712yes—
583C22#334812yes—
584C22#335012yes—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—84no—
590C4—84no—
591C4—162no—
592C4—162no—
593C4#232412yes—
594C4#235811yescharacteristic
595C4#237611yescharacteristic
596C4#268311yescharacteristic
597C4#269212yes—
598C4#274811yescharacteristic
599C2—44no—
600C2—82no—
601C2#1440111yescharacteristic
602C2#2193911yescharacteristic
603C2#2194611yescharacteristic
604C2#2265412yes—
605C2#2266212yes—
606C2#3050312yes—
607C2#3055812yes—
608C2#3056412yes—
609C2#3226212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1854C2semidirect
3. C2×#1807C2semidirect
6. #11325C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10908C2semidirect
24. #1869C4semidirect
117. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C2×#18542
C2×#18072
#139052
#138712
#113252
#110962
#110952
#109082