Finite Group Databasebeta

← #189129

Subgroups — #189129

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2962C211yesmaximal, characteristic
2#21894C211yesmaximal, characteristic
3#21870C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5C2×#1569C211yesmaximal, characteristic
6#8657C211yesmaximal, characteristic, complemented
7#8655C211yesmaximal, characteristic, complemented
8#8653C211yesmaximal, characteristic, complemented
9#8651C211yesmaximal, characteristic, complemented
10#8649C211yesmaximal, characteristic, complemented
11#8647C211yesmaximal, characteristic, complemented
12#8645C211yesmaximal, characteristic, complemented
13#8643C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic
15#8044C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17#2962—24no—
18#2962C418yescomplemented
19C2×#446—21no—
20C2×#446—21no—
21C2×#446C2211yescharacteristic
22C2×#446C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#424—21no—
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#405—21no—
30C2×C4×M16C2211yescharacteristic
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#1993—21no—
34#1988—21no—
35#1981—21no—
36#1960—21no—
37#1894—21no—
38#1894—21no—
39#1873—21no—
40#1873—21no—
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1809—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1803C2211yescharacteristic
54#1803C2211yescharacteristic
55#1776—21no—
56#1776—21no—
57#1569—24no—
58C2×#335C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#334C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C23×C8—21no—
74#533—22no—
75#533—22no—
76#533C2×C414yes—
77#533Q814yes—
78C2×C4⋊Q8D411yescharacteristic
79C2×C4⋊Q8D411yescharacteristic
80C2×#125—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8D411yescharacteristic
83C2×C22⋊Q8D411yescharacteristic
84C2×C4×Q8—21no—
85C2×C4×Q8C2311yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×#117—21no—
88C2×#117—21no—
89C2×#117—21no—
90C22×C42C2311yescharacteristic
91#446—28no—
92#446—28no—
93#446—22no—
94#446—22no—
95#446C2×C414yes—
96#446C2×C414yes—
97C8×Q8—22no—
98C8×Q8—22no—
99C8×Q8C2×C414yes—
100C8×Q8C2×C414yes—
101#432—22no—
102#432—22no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#106—21no—
106C2×#106—21no—
107#424—28no—
108#423—22no—
109#423—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2×C414yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×C22⋊C8—21no—
133#405—28no—
134C4×M16C2×C414yes—
135C4×M16D414yes—
136C2×#97—21no—
137C2×#97—21no—
138C2×#97—21no—
139C2×#97D411yescharacteristic
140C2×#97D411yescharacteristic
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382—21no—
150#382—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#335—22no—
172#335—22no—
173#334—22no—
174#334—22no—
175C22×Q8—21no—
176C22×Q8C2×D411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—41no—
184C2×M16—41no—
185C2×M16—42no—
186C2×M16—81no—
187C2×M16—81no—
188C2×M16C22⋊C414yes—
189C2×M16C22⋊C414yes—
190C22×C8—24no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8C4⋊C414yes—
206C4⋊Q8C4⋊C414yes—
207#125—24no—
208#125—24no—
209C22⋊Q8—24no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—24no—
213C22⋊Q8C4⋊C414yes—
214C22⋊Q8C4⋊C414yes—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8C22×C412yes—
222C4×Q8C22×C412yes—
223C4×Q8C2×Q812yes—
224C4×Q8C2×Q812yes—
225#117—24no—
226#117—24no—
227#117—81no—
228#117—81no—
229#117—82no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42C22×C412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×Q812yes—
264C2×C42#4111yescharacteristic
265C2×C42#4112yes—
266C2×C42C2411yescharacteristic
267#107—42no—
268#107—42no—
269#106—42no—
270#106—42no—
271C4⋊C8—24no—
272C4⋊C8—24no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—42no—
287C4⋊C8C22×C412yes—
288C4⋊C8C22×C412yes—
289#103—81no—
290#103—81no—
291#103—81no—
292#103—81no—
293#103—81no—
294#103—81no—
295#103—81no—
296#103—81no—
297C22⋊C8—42no—
298#97—24no—
299#97—24no—
300#97—42no—
301#97C22⋊C414yes—
302#97C22⋊C414yes—
303C4×C8—24no—
304C4×C8—24no—
305C4×C8C22×C412yes—
306C4×C8C22×C412yes—
307C4×C8C2×D412yes—
308C4×C8C2×D412yes—
309#95—41no—
310#95—41no—
311#95—41no—
312#95—41no—
313C24#12111yescharacteristic
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—24no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—81no—
326C2×Q8—81no—
327C2×Q8C2×C4⋊C412yes—
328C2×Q8C2×C4⋊C412yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C4⋊C412yes—
349C22×C4#11712yes—
350C22×C4#12112yes—
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359M16—28no—
360M16—44no—
361M16—84no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—48no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8C2×C22⋊C412yes—
388C2×C8C2×C22⋊C412yes—
389C2×C8C2×C22⋊C412yes—
390C2×C8C2×C22⋊C412yes—
391C2×C8#13712yes—
392C2×C8#13712yes—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4C2×C4⋊C412yes—
425C4⋊C4C2×C4⋊C412yes—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—82no—
435C42—82no—
436C42C2×C22⋊C412yes—
437C42C2×C4⋊C412yes—
438C42#11712yes—
439C42#12112yes—
440C42C22⋊Q812yes—
441C42C22⋊Q812yes—
442C42C23×C411yescharacteristic
443C42C22×D411yescharacteristic
444C42C22×Q811yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23#38012yes—
453C23C2×#12111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—84no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4#38012yes—
506C2×C4#38012yes—
507C2×C4#38012yes—
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C22×C4⋊C411yescharacteristic
510C2×C4C2×#11711yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×C22⋊Q811yescharacteristic
514C2×C4C2×#13711yescharacteristic
515C2×C4C2×#13711yescharacteristic
516C2×C4#57511yescharacteristic
517C2×C4#57511yescharacteristic
518C2×C4#57611yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57711yescharacteristic
521C2×C4#57811yescharacteristic
522C8—22no—
523C8—22no—
524C8—22no—
525C8—42no—
526C8—42no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8—81no—
531C8#41914yes—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22C2×#38011yescharacteristic
540C22#303711yescharacteristic
541C22#303711yescharacteristic
542C22#306211yescharacteristic
543C22#306311yescharacteristic
544C22#306811yescharacteristic
545C22#306911yescharacteristic
546C4—22no—
547C4—22no—
548C4—21no—
549C4—21no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—161no—
556C4—161no—
557C4—161no—
558C4—161no—
559C4C2×#41911yescharacteristic
560C4#289611yescharacteristic
561C4#291211yescharacteristic
562C4#291311yescharacteristic
563C4#306211yescharacteristic
564C4#306311yescharacteristic
565C4#307811yescharacteristic
566C4#307811yescharacteristic
567C2—42no—
568C2#1937211yescharacteristic
569C2#1937311yescharacteristic
570C2#1937411yescharacteristic
571C2#1937511yescharacteristic
572C2#2064711yescharacteristic
573C2#2066411yescharacteristic
574C2#2067911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ 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
6. #8657C2semidirect
7. #8655C2semidirect
8. #8653C2semidirect
9. #8651C2semidirect
10. #8649C2semidirect
11. #8647C2semidirect
12. #8645C2semidirect
13. #8643C2semidirect
18. #2962C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#218942
#218702
#208242
C2×#15692
#86572
#86552
#86532
#86512
#86492
#86472
#86452
#86432
#80452
#80442