Finite Group Databasebeta

← #242598

Subgroups — #242598

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21409C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5C2×#1569C211yesmaximal, characteristic
6C2×#1559C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10600C211yesmaximal, characteristic, complemented
9#10597C211yesmaximal, characteristic, complemented
10#10596C211yesmaximal, characteristic, complemented
11#10593C211yesmaximal, characteristic, complemented
12#10592C211yesmaximal, characteristic, complemented
13#10589C211yesmaximal, characteristic, complemented
14#10588C211yesmaximal, characteristic, complemented
15#8041C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423—21no—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1989—21no—
27#1989—21no—
28#1988C2211yescharacteristic
29#1988C2211yescharacteristic
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1940—21no—
39#1940—21no—
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1894—21no—
45#1894—21no—
46#1853—21no—
47#1852—21no—
48#1776—21no—
49#1776—21no—
50#1763—21no—
51#1763—21no—
52#1736—21no—
53#1736—21no—
54#1728—21no—
55#1569—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1559—24no—
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×M16—21no—
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73#533—22no—
74#533—22no—
75#533—22no—
76#533Q814yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×#125—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×#117—21no—
85C2×#117—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#432—22no—
91#432D414yes—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96#423—22no—
97#423—22no—
98#423—22no—
99#423—22no—
100#423—44no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×C22⋊C8—21no—
118C4×M16—22no—
119C4×M16—22no—
120C2×#97—21no—
121C2×#97D411yescharacteristic
122C2×#97D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#382—21no—
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C2×#95—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#335—22no—
160#335—22no—
161#334—22no—
162#334—22no—
163#333—22no—
164#333—22no—
165#332—22no—
166#332—22no—
167C22×Q8—21no—
168C22×Q8—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—41no—
179C2×M16—41no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—42no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8C2×Q812yes—
203C4⋊Q8C2×Q812yes—
204#125—24no—
205#125—42no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—42no—
213C4×Q8—42no—
214#117—24no—
215#117—42no—
216#117—82no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257#107—42no—
258#107—42no—
259#106—42no—
260#106—42no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—42no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8C2×D412yes—
278C4⋊C8C2×D412yes—
279#103—42no—
280#103—42no—
281#103—42no—
282#103—42no—
283C22⋊C8—24no—
284#97—24no—
285#97—22no—
286#97—22no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—41no—
296#95—41no—
297C24#12111yescharacteristic
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—81no—
307C2×Q8—81no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4C22⋊Q812yes—
338C22×C4C22⋊Q812yes—
339C22×C4C22⋊Q812yes—
340C22×C4#12612yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344M16—44no—
345M16—44no—
346M16—84no—
347M16—84no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8AGL(1,ℤ/8)12yes—
377C2×C8#13712yes—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4#13712yes—
407C4⋊C4#13712yes—
408C22⋊C4—44no—
409C22⋊C4—82no—
410C42—42no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—44no—
417C42—82no—
418C42#12112yes—
419C42#12112yes—
420C42C22⋊Q812yes—
421C42C22⋊Q812yes—
422C42C22⋊Q812yes—
423C42#12612yes—
424C42C22×D411yescharacteristic
425C42C22×Q811yescharacteristic
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23#39612yes—
435C23C2×#12111yescharacteristic
436Q8—42no—
437Q8—42no—
438Q8—82no—
439Q8—82no—
440Q8—81no—
441Q8—81no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4#39612yes—
487C2×C4#39612yes—
488C2×C4#39612yes—
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×#12611yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4C2×#13711yescharacteristic
498C2×C4C2×#13711yescharacteristic
499C2×C4#57511yescharacteristic
500C2×C4#57511yescharacteristic
501C2×C4#57511yescharacteristic
502C2×C4#57511yescharacteristic
503C8—22no—
504C8—22no—
505C8—42no—
506C8—42no—
507C8—41no—
508C8—41no—
509C8—81no—
510C8—81no—
511C8—81no—
512C8—81no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22C2×#39611yescharacteristic
521C22#303611yescharacteristic
522C22#303711yescharacteristic
523C22#305611yescharacteristic
524C22#305611yescharacteristic
525C22#306811yescharacteristic
526C22#306911yescharacteristic
527C4—21no—
528C4—21no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—82no—
535C4—82no—
536C4—81no—
537C4—81no—
538C4—161no—
539C4—161no—
540C4#303911yescharacteristic
541C4#304011yescharacteristic
542C4#305511yescharacteristic
543C4#305611yescharacteristic
544C4#307811yescharacteristic
545C4#307811yescharacteristic
546C4#307911yescharacteristic
547C4#308011yescharacteristic
548C2—42no—
549C2#1930511yescharacteristic
550C2#1930711yescharacteristic
551C2#1930911yescharacteristic
552C2#1931111yescharacteristic
553C2#2360411yescharacteristic
554C2#2361711yescharacteristic
555C2#2361911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10601C2semidirect
8. #10600C2semidirect
9. #10597C2semidirect
10. #10596C2semidirect
11. #10593C2semidirect
12. #10592C2semidirect
13. #10589C2semidirect
14. #10588C2semidirect

Maximal subgroups

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

subgroupindex
#214092
#213652
#213122
#211462
C2×#15692
C2×#15592
#106012
#106002
#105972
#105962
#105932
#105922
#105892
#105882
#80412