Finite Group Databasebeta

← #387102

Subgroups — #387102

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2#13924C211yesmaximal, characteristic
3#13923C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11203C212yesmaximal
6#11203C212yesmaximal
7#10912C212yesmaximal
8#10912C212yesmaximal
9#10801C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12#2583—22no—
13#2583—22no—
14#2583—22no—
15#2583—22no—
16C2×#398C412yescomplemented
17C2×#395C412yescomplemented
18C2×#389C414yescomplemented
19C2×#385C2212yes—
20C2×#385C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×#342C2212yes—
27C2×#342C2212yes—
28C2×#342C2212yes—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2212yes—
34#1445—22no—
35#1441—22no—
36#1436—22no—
37C4⋊#95—22no—
38C22⋊#95C2212yes—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C23×Q8C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—41no—
49C2×C22⋊Q8—41no—
50C2×C22⋊Q8—41no—
51C2×C22⋊Q8—41no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D412yes—
68C22×C22⋊C4C2312yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2312yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2×C412yes—
77#398—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#395—22no—
82#395—22no—
83#395—22no—
84#395—22no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—24no—
89#385—22no—
90#385—24no—
91#385—22no—
92#385—22no—
93#385—24no—
94#385—22no—
95#380—41no—
96#380—41no—
97#380—41no—
98#380—41no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—41no—
102C4×C22⋊C4—41no—
103C4×C22⋊C4—41no—
104C4×C22⋊C4—41no—
105C2×#95—24no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—24no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95C2×C412yes—
114C2×#95C2×C414yes—
115C2×#95C2×C414yes—
116#342—24no—
117#342—24no—
118#342—24no—
119#342—42no—
120#342—42no—
121#342—42no—
122#342—42no—
123#342Q818yescomplemented
124C25C2411yescharacteristic
125C22×Q8—22no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—82no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—44no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C414yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C22⋊Q8—44no—
164C22⋊Q8—44no—
165C22⋊Q8—44no—
166C22⋊Q8—44no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C2×Q814yes—
236C2×C22⋊C4C2×Q814yes—
237C2×C22⋊C4C2×Q814yes—
238C2×C22⋊C4#4114yes—
239C2×C42—24no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42—82no—
244C2×C42—82no—
245#95—22no—
246#95—22no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—48no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—84no—
257#95—84no—
258#95—84no—
259C24—42no—
260C24—42no—
261C24C23×C411yescharacteristic
262C24C22×D411yescharacteristic
263C24C22×D411yescharacteristic
264C24C22×Q812yes—
265C24C2×#4112yes—
266C24H(5,𝔽2)11yescharacteristic
267C24H(5,𝔽2)11yescharacteristic
268C24H(5,𝔽2)11yescharacteristic
269C24H(5,𝔽2)11yescharacteristic
270C242−1+412yes—
271C242−1+412yes—
272C2×Q8—44no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C4×Q818yes—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—84no—
406C42—84no—
407C42—84no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23C22×C22⋊C411yescharacteristic
427C23C2×C4×Q812yes—
428C23#51811yescharacteristic
429C23#51811yescharacteristic
430C23#51912yes—
431C23#53411yescharacteristic
432C23#53411yescharacteristic
433C23#53411yescharacteristic
434C23#53411yescharacteristic
435C23#54212yes—
436C23#54212yes—
437C23#54412yes—
438C23#54412yes—
439C23#54812yes—
440C23#54812yes—
441C23D4×Q812yes—
442C23D4×Q812yes—
443C23#55412yes—
444C23#55511yescharacteristic
445C23Q8211yescharacteristic
446C23#56312yes—
447C23#56311yescharacteristic
448C23#56311yescharacteristic
449Q8—88no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—84no—
503C2×C4—84no—
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—84no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#41014yes—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22#230811yescharacteristic
550C22#233112yes—
551C22Q8×C22⋊C412yes—
552C22#236911yescharacteristic
553C22#238111yescharacteristic
554C22#272912yes—
555C22#272912yes—
556C22#273111yescharacteristic
557C22#273111yescharacteristic
558C22#273411yescharacteristic
559C22#273411yescharacteristic
560C22C2×#41012yes—
561C22#288312yes—
562C4—24no—
563C4—48no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—82no—
573C4—162no—
574C4—162no—
575C2—22no—
576C2—22no—
577C2—21no—
578C2—21no—
579C2—22no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2#1445011yescharacteristic
585C2#2174811yescharacteristic
586C2#2174911yescharacteristic
587C2#2251314yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#398C4semidirect
17. C2×#395C4semidirect
18. C2×#389C4semidirect
123. #342Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
#139242
#139232
#113812
#112032
#109122
#108012