Finite Group Databasebeta

← #387450

Subgroups — #387450

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2569C211yesmaximal, characteristic
2#13883C212yesmaximal
3#11353C212yesmaximal
4#11353C212yesmaximal
5#11203C212yesmaximal
6#10912C212yesmaximal
7#10854C212yesmaximal
8#10801C211yesmaximal, characteristic
9#10801C211yesmaximal, characteristic
10#2569—22no—
11#2569—22no—
12#2569—22no—
13#2569—22no—
14C2×#400C412yescomplemented
15C2×#398C412yescomplemented
16C2×#387C412yescomplemented
17C2×#387C412yescomplemented
18C2×#385C2212yes—
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24C22×#95C2211yescharacteristic
25C2×#342C2212yes—
26C2×#342C2212yes—
27C2×#342C2212yes—
28C2×#342C2212yes—
29C2×#342C2211yescharacteristic
30C2×#342C2211yescharacteristic
31C2×#342C2211yescharacteristic
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2212yes—
35#1443—22no—
36#1442—22no—
37#1436—22no—
38#1434—22no—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C22⋊#95C2212yes—
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C22⋊C4C2311yescharacteristic
55C22×C22⋊C4C2312yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2312yes—
58C22×C42—22no—
59C22×C42C2×C412yes—
60#400—22no—
61#400—22no—
62#400—22no—
63#400—22no—
64#398—22no—
65#398—22no—
66#398—22no—
67#398—22no—
68#387—22no—
69#387—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#385—22no—
77#385—22no—
78#385—22no—
79#385—22no—
80#380—22no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—24no—
87#380—42no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—42no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C412yes—
112C2×#95D412yes—
113C2×#95D412yes—
114C2×#95D412yes—
115C2×#95D412yes—
116#342—24no—
117#342—24no—
118#342—24no—
119#342—24no—
120#342—41no—
121#342—42no—
122#342—41no—
123#342—41no—
124#342—42no—
125#342—41no—
126C25C2411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
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#4114yes—
210C2×C22⋊C4#4114yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—82no—
216C2×C42—82no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—82no—
237#95—82no—
238#95—82no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C411yescharacteristic
244#95C22⋊C412yes—
245#95C22⋊C411yescharacteristic
246#95C22⋊C411yescharacteristic
247#95C22⋊C412yes—
248#95C22⋊C411yescharacteristic
249C24—42no—
250C24—42no—
251C24C23×C411yescharacteristic
252C24C22×D412yes—
253C24C2×#4112yes—
254C24C2×#4112yes—
255C24H(5,𝔽2)11yescharacteristic
256C24H(5,𝔽2)11yescharacteristic
257C242−1+412yes—
258C242−1+411yescharacteristic
259C242−1+411yescharacteristic
260C242−1+411yescharacteristic
261C242−1+411yescharacteristic
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
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—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
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—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—88no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C42—82no—
398C42—82no—
399C42—88no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23C22×C22⋊C411yescharacteristic
421C23C4×#4111yescharacteristic
422C23C4×#4111yescharacteristic
423C23#51911yescharacteristic
424C23#51911yescharacteristic
425C23#52011yescharacteristic
426C23#52011yescharacteristic
427C23#53612yes—
428C23#53612yes—
429C23#53912yes—
430C23#53912yes—
431C23#54112yes—
432C23#54112yes—
433C23#54612yes—
434C23#54712yes—
435C23#54812yes—
436C23#55012yes—
437C23#55212yes—
438C23#55212yes—
439C23#56211yescharacteristic
440C23#56311yescharacteristic
441C23#56311yescharacteristic
442C23#56411yescharacteristic
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
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—22no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
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—82no—
490C2×C4—82no—
491C2×C4—82no—
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—82no—
503C2×C4—82no—
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×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#41014yes—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22#231011yescharacteristic
561C22#233611yescharacteristic
562C22#233611yescharacteristic
563C22#233911yescharacteristic
564C22#234111yescharacteristic
565C22#237111yescharacteristic
566C22#238211yescharacteristic
567C22#269012yes—
568C22#269412yes—
569C22#271912yes—
570C22#272112yes—
571C22C2×#41012yes—
572C22#288312yes—
573C4—24no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—84no—
581C4—84no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—162no—
587C4—162no—
588C2—22no—
589C2—22no—
590C2—22no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—22no—
596C2#1447411yescharacteristic
597C2#2182611yescharacteristic
598C2#2182711yescharacteristic
599C2#2247712yes—
600C2#2253512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C2×#400C4semidirect
15. C2×#398C4semidirect
16. C2×#387C4semidirect
17. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25692
#138832
#113532
#112032
#109122
#108542
#108012