Finite Group Databasebeta

← #416809

Subgroups — #416809

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1836C211yesmaximal, characteristic, complemented
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10843C211yesmaximal, characteristic, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23#1836—24no—
24#1750—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1441—22no—
36#1441—22no—
37#1439—22no—
38#1439—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97D414yes—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76#384—22no—
77#384—22no—
78#384—22no—
79#384—22no—
80#382—22no—
81#382—22no—
82#382—22no—
83#382—22no—
84#380—24no—
85#380—24no—
86#380—42no—
87#380—42no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—22no—
105#336—22no—
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157#117—44no—
158#117—44no—
159#117—44no—
160#117—44no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C2×D412yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C22⋊C8—28no—
216C22⋊C8—28no—
217#97—28no—
218C4×C8—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95—82no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24#12412yes—
238C24C22×D411yescharacteristic
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
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#12114yes—
313C22×C4#12314yes—
314C22×C4#12314yes—
315C22×C4#12412yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C42−1+411yescharacteristic
329C22×C42−1+411yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
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—42no—
374C4⋊C4—42no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C22⋊C4—48no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42#12314yes—
385C42#12414yes—
386C42#12614yes—
387C42#12614yes—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23C2×#12111yescharacteristic
412C23C2×#12311yescharacteristic
413C23C2×#12311yescharacteristic
414C23C2×#12411yescharacteristic
415C23#53211yescharacteristic
416C23#53511yescharacteristic
417C23#53611yescharacteristic
418C23#57612yes—
419C23#57712yes—
420C23#57712yes—
421C23#57812yes—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#39718yes—
479C2×C4C2×#12311yescharacteristic
480C2×C4#52511yescharacteristic
481C2×C4#52511yescharacteristic
482C2×C4C2×#12411yescharacteristic
483C2×C4C2×#12611yescharacteristic
484C2×C4C2×#12611yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#52911yescharacteristic
487C2×C4#53711yescharacteristic
488C2×C4#53711yescharacteristic
489C2×C4#54211yescharacteristic
490C2×C4#54211yescharacteristic
491C2×C4#55211yescharacteristic
492C2×C4#55211yescharacteristic
493C2×C4#55211yescharacteristic
494C2×C4#55211yescharacteristic
495C2×C4#55211yescharacteristic
496C2×C4#55211yescharacteristic
497C2×C4#55311yescharacteristic
498C2×C4#55311yescharacteristic
499C2×C4#55311yescharacteristic
500C2×C4#55311yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4#55511yescharacteristic
504C2×C4#55511yescharacteristic
505C2×C4#55611yescharacteristic
506C2×C4#55611yescharacteristic
507C8—42no—
508C8—42no—
509C8—42no—
510C8—42no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C2×#39711yescharacteristic
523C22#249911yescharacteristic
524C22#250011yescharacteristic
525C22#250311yescharacteristic
526C22#250411yescharacteristic
527C22#262011yescharacteristic
528C22#285011yescharacteristic
529C22#306212yes—
530C22#306312yes—
531C22#309612yes—
532C22#309712yes—
533C22#309812yes—
534C22#309912yes—
535C22#313612yes—
536C22#313712yes—
537C22#317411yescharacteristic
538C22#317411yescharacteristic
539C22#317611yescharacteristic
540C22#317711yescharacteristic
541C22#322412yes—
542C22#322512yes—
543C22#324312yes—
544C22#324412yes—
545C4—28no—
546C4—42no—
547C4—82no—
548C4—162no—
549C4—162no—
550C4—162no—
551C4—162no—
552C4#248511yescharacteristic
553C4#248511yescharacteristic
554C4#249411yescharacteristic
555C4#249411yescharacteristic
556C4#249511yescharacteristic
557C4#249511yescharacteristic
558C4#256211yescharacteristic
559C4#256211yescharacteristic
560C2—82no—
561C2#1523311yescharacteristic
562C2#2364312yes—
563C2#2788411yescharacteristic
564C2#2788511yescharacteristic
565C2#2790011yescharacteristic
566C2#2790111yescharacteristic
567C2#2797112yes—
568C2#2798412yes—
569C2#2985712yes—
570C2#3305511yescharacteristic
571C2#3305611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1836C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
9. #10843C2semidirect
10. #10842C2semidirect
11. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18362
C2×#17502
#143402
#113162
#113152
#112002
#111992
#108432
#108422
#108412