Finite Group Databasebeta

← #461641

Subgroups — #461641

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2093C211yesmaximal, characteristic, complemented
2C2×#1798C211yesmaximal, characteristic, complemented
3#14340C211yesmaximal, characteristic, complemented
4#14257C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11232C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#11051C211yesmaximal, characteristic, complemented
11#10969C212yesmaximal
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25#2093—24no—
26#1798—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1441—22no—
35#1439—22no—
36#1439—22no—
37#1438—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#121—22no—
47C2×#121—42no—
48C2×#121—42no—
49C2×#121Q814yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—42no—
71C2×#107—42no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—42no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84#390—22no—
85#390—22no—
86#390—24no—
87#384—22no—
88#384—22no—
89#382—42no—
90#380—42no—
91#380—42no—
92#380—42no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336D414yescomplemented
107C25—21no—
108C25C2×D411yescharacteristic
109C22×D4—24no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—42no—
120C22×D4—42no—
121C22×D4—82no—
122C22×D4—82no—
123C22×D4C2×Q812yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C2×D414yes—
157#121—48no—
158#121—48no—
159#117—48no—
160#117—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
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—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×Q812yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215C2×C42#4114yes—
216#107—48no—
217C4⋊C8—48no—
218#102—48no—
219#102—48no—
220C22⋊C8—48no—
221#95—44no—
222#95—42no—
223#95—44no—
224#95—82no—
225#95—82no—
226#95—82no—
227#95—82no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C22⋊Q812yes—
250C24C22×D411yescharacteristic
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—48no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—88no—
269C2×D4—88no—
270C2×D4—84no—
271C2×D4—84no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
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—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
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#12114yes—
336C22×C4C22⋊Q812yes—
337C22×C4#12314yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C42−1+411yescharacteristic
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—48no—
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—42no—
376C4⋊C4—42no—
377C4⋊C4—88no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C22⋊C4—84no—
383C22⋊C4—88no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C23—24no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—82no—
417C23—82no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—82no—
422C23—82no—
423C23—81no—
424C23—81no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—81no—
434C23C2×#12111yescharacteristic
435C23C2×C22⋊Q811yescharacteristic
436C23C2×#12311yescharacteristic
437C23#53411yescharacteristic
438C23#53511yescharacteristic
439C23#53511yescharacteristic
440C23#53711yescharacteristic
441C23#57612yes—
442C23#57612yes—
443C23#57712yes—
444C23#57812yes—
445D4—44no—
446D4—44no—
447D4—816no—
448D4—82no—
449D4—82no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
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#53512yes—
505C2×C4#53812yes—
506C2×C4#54012yes—
507C2×C4#54012yes—
508C2×C4#54212yes—
509C2×C4#54312yes—
510C2×C4#54612yes—
511C2×C4#54611yescharacteristic
512C2×C4D4×Q811yescharacteristic
513C2×C4#55112yes—
514C2×C4#55211yescharacteristic
515C2×C4#55312yes—
516C2×C4#55412yes—
517C2×C4#55411yescharacteristic
518C2×C4#56012yes—
519C2×C4#56212yes—
520C8—44no—
521C8—82no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—81no—
528C22—81no—
529C22—82no—
530C22—81no—
531C22—81no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22—161no—
539C22—161no—
540C22—162no—
541C22—161no—
542C22—161no—
543C22—162no—
544C22#241211yescharacteristic
545C22#241711yescharacteristic
546C22#248711yescharacteristic
547C22#261911yescharacteristic
548C22#262411yescharacteristic
549C22#262611yescharacteristic
550C22#281311yescharacteristic
551C22#305812yes—
552C22#306312yes—
553C22#308112yes—
554C22#308312yes—
555C22#309612yes—
556C22#309812yes—
557C22#319812yes—
558C22#320312yes—
559C22#322011yescharacteristic
560C22#322011yescharacteristic
561C22#322012yes—
562C22#322211yescharacteristic
563C22#322311yescharacteristic
564C22#322512yes—
565C22#325712yes—
566C22#326212yes—
567C4—42no—
568C4—44no—
569C4—44no—
570C4—82no—
571C4—161no—
572C4—162no—
573C4—162no—
574C4—161no—
575C4#269212yes—
576C4#270312yes—
577C4#274112yes—
578C4#274812yes—
579C2—82no—
580C2—162no—
581C2#1639811yescharacteristic
582C2#2456212yes—
583C2#2484712yes—
584C2#2768712yes—
585C2#2981011yescharacteristic
586C2#2981711yescharacteristic
587C2#2993111yescharacteristic
588C2#2993311yescharacteristic
589C2#2996211yescharacteristic
590C2#2996411yescharacteristic
591C2#3273512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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×#2093C2semidirect
2. C2×#1798C2semidirect
3. #14340C2semidirect
5. #11371C2semidirect
6. #11268C2semidirect
7. #11232C2semidirect
8. #11096C2semidirect
9. #11076C2semidirect
10. #11051C2semidirect
106. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20932
C2×#17982
#143402
#142572
#113712
#112682
#112322
#110962
#110762
#110512
#109692