Finite Group Databasebeta

← #381866

Subgroups — #381866

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1969C211yesmaximal, characteristic, complemented
3C2×#1750C211yesmaximal, characteristic
4#14258C212yesmaximal
5#11256C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11024C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10928C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#380C2212yes—
26#2001C418yescomplemented
27#1969—24no—
28#1750—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1441—22no—
37#1441—22no—
38#1440—22no—
39#1440—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8C2312yes—
43C24×C4C2311yescharacteristic
44C22×Q16—22no—
45C22×Q16—22no—
46C22×Q16—42no—
47C22×Q16—42no—
48C22×Q16C2×C414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8C2×C418yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—42no—
62C2×#117—42no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—22no—
76C2×#107—42no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103C2×C418yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2×C414yes—
92C2×#97—22no—
93C2×#97—42no—
94C2×C4×C8—42no—
95C2×C4×C8D414yes—
96#391—24no—
97#391—24no—
98#391—42no—
99#391—42no—
100#380—42no—
101#380—42no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340C2×C414yes—
116#336—22no—
117#336—22no—
118C25C2×D411yescharacteristic
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8C22×C414yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C22×C412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×Q16—28no—
154C2×Q16—48no—
155C2×Q16—44no—
156C2×Q16—44no—
157C2×Q16—48no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C22×C412yes—
173C22×C8C22×C412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C22×C8#4112yes—
177C22×C8#4112yes—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182#117—44no—
183#117—44no—
184#117—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C22×C414yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42C2×D412yes—
227C2×C42#4112yes—
228#107—44no—
229#103—48no—
230#103—48no—
231#103—48no—
232C22⋊C8—28no—
233C22⋊C8—28no—
234#97—44no—
235C4×C8—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—82no—
242#95—82no—
243#95—82no—
244#95—82no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24C4×D412yes—
256C24C22×D411yescharacteristic
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
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—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4C4×D412yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366Q16—416no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8C4×D414yes—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—416no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C42—22no—
407C42—22no—
408C42—24no—
409C42—41no—
410C42—41no—
411C42—42no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23C2×C4×D411yescharacteristic
436C23C2×#12111yescharacteristic
437C23#53211yescharacteristic
438C23#53412yes—
439C23#53512yes—
440C23#57611yescharacteristic
441C23#57712yes—
442C23#57811yescharacteristic
443C23#57811yescharacteristic
444C23#57811yescharacteristic
445C23#57812yes—
446Q8—44no—
447Q8—44no—
448Q8—44no—
449Q8—44no—
450Q8—82no—
451Q8—82no—
452Q8—82no—
453Q8—82no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
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—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
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×C4C2×C4×D411yescharacteristic
514C2×C4C4×#4111yescharacteristic
515C2×C4#51812yes—
516C2×C4#51812yes—
517C2×C4#52511yescharacteristic
518C2×C4#52911yescharacteristic
519C2×C4#53411yescharacteristic
520C2×C4#53511yescharacteristic
521C2×C4#53511yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53812yes—
524C2×C4#53812yes—
525C2×C4#53812yes—
526C2×C4#53812yes—
527C2×C4#53812yes—
528C2×C4#54012yes—
529C2×C4#54012yes—
530C2×C4#54012yes—
531C8—24no—
532C8—42no—
533C8—42no—
534C8—42no—
535C22—42no—
536C22—42no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—82no—
545C22—82no—
546C22C4×#12111yescharacteristic
547C22#232112yes—
548C22#261711yescharacteristic
549C22#261911yescharacteristic
550C22#266112yes—
551C22#294712yes—
552C22#295011yescharacteristic
553C22#295111yescharacteristic
554C22#305911yescharacteristic
555C22#305911yescharacteristic
556C22#306011yescharacteristic
557C22#306111yescharacteristic
558C22#317511yescharacteristic
559C22#317711yescharacteristic
560C22#317812yes—
561C22#320112yes—
562C22#320112yes—
563C22#320212yes—
564C22#320312yes—
565C22#322312yes—
566C22#322312yes—
567C22#322412yes—
568C22#322512yes—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—84no—
573C4—81no—
574C4—81no—
575C4—84no—
576C4—162no—
577C4—162no—
578C4#232111yescharacteristic
579C4#232811yescharacteristic
580C4#233312yes—
581C4#263111yescharacteristic
582C4#263511yescharacteristic
583C4#264812yes—
584C2—82no—
585C2#1431111yescharacteristic
586C2#2152511yescharacteristic
587C2#2153711yescharacteristic
588C2#2205412yes—
589C2#2205712yes—
590C2#2973711yescharacteristic
591C2#2973911yescharacteristic
592C2#2981211yescharacteristic
593C2#2981311yescharacteristic
594C2#3033012yes—
595C2#3033112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
2. C2×#1969C2semidirect
5. #11256C2semidirect
6. #11190C2semidirect
8. #11024C2semidirect
10. #10928C2semidirect
11. #10820C2semidirect
26. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#19692
C2×#17502
#142582
#112562
#111902
#110962
#110242
#109702
#109282
#108202