Finite Group Databasebeta

← #30856835

Subgroups — #30856835

Order 17152.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863436C211yesmaximal, characteristic, complemented
2C2×#16863397C211yesmaximal, characteristic
3C2×#16863392C211yesmaximal, characteristic
4#22562825C211yesmaximal, characteristic, complemented
5#22560477C212yesmaximal, complemented
6#22560470C212yesmaximal, complemented
7#22560463C212yesmaximal, complemented
8#22560456C212yesmaximal, complemented
9C2×#16863158C211yesmaximal, characteristic, complemented
10#22559906C211yesmaximal, characteristic, complemented
11C2×#520×C67C211yesmaximal, characteristic
12C22×#11761612C2211yescharacteristic
13C23×C67⋊C8C2211yescharacteristic
14C2×#14524002C2211yescharacteristic
15C22×C4⋊Dic67C2211yescharacteristic
16#16863436—24no—
17C2×#14523976C2212yes—
18C2×#14523974C2212yes—
19#16863397C418yescomplemented
20#16863392—24no—
21C2×#14523967C2211yescharacteristic
22C2×#14523967C2211yescharacteristic
23C2×#14523967C2212yes—
24C2×#14523964C2211yescharacteristic
25C2×#14523964C2211yescharacteristic
26C2×#14523964C2212yes—
27#16863294C2212yes—
28#16863294C2212yes—
29#16863292—21no—
30#16863292—21no—
31#16863254—22no—
32#16863158C2212yes—
33#16863158C2212yes—
34#16863151C2212yes—
35#16863151C2212yes—
36C22×#41×C67C2211yescharacteristic
37#520×C67—24no—
38#520×C67—24no—
39C2×C4×Q8×C67C2212yes—
40C2×C4×D4×C67—22no—
41C2×C4×D4×C67C2212yes—
42C2×#117×C67—21no—
43C2×#117×C67C2211yescharacteristic
44C22×C4⋊C4×C67—21no—
45C22×C4⋊C4×C67C2211yescharacteristic
46C23×Dic67—21no—
47C2×#11761612—22no—
48C2×#11761612—22no—
49C2×#11761612C2×C414yescomplemented
50C22×C67⋊C8—22no—
51C22×C67⋊C8—22no—
52C22×C67⋊C8—22no—
53C22×C67⋊C8C2311yescharacteristic
54C22×C67⋊C8C2311yescharacteristic
55C22×C67⋊C8C2312yes—
56#14524002—22no—
57#14524002—22no—
58C2×C4⋊Dic67—21no—
59C2×C4⋊Dic67—22no—
60C2×C4⋊Dic67C2×C414yescomplemented
61C2×C4⋊Dic67C2311yescharacteristic
62C2×C4⋊Dic67C2311yescharacteristic
63C2×C4⋊Dic67C2312yes—
64C2×Dic67⋊C4—22no—
65C2×Dic67⋊C4—22no—
66C2×C22⋊Dic67—21no—
67C2×C4×Dic67—21no—
68#14523976—24no—
69#14523974—24no—
70#14523967—22no—
71#14523967—22no—
72#14523967C2×C418yescomplemented
73#14523964—22no—
74#14523964—22no—
75#14523964C2×C418yescomplemented
76C67⋊#95—22no—
77C67⋊#95—22no—
78C2×#41×C67—22no—
79C2×#41×C67—22no—
80C2×#41×C67—22no—
81C2×#41×C67C2×C414yes—
82C2×#41×C67D414yes—
83C2×#41×C67Q814yes—
84C22×Q8×C67C2311yescharacteristic
85C22×D4×C67—21no—
86C22×D4×C67C2311yescharacteristic
87C23×C4×C67—21no—
88C23×C4×C67—21no—
89C23×C4×C67—22no—
90C23×C4×C67C2311yescharacteristic
91C4×Q8×C67—24no—
92C4×Q8×C67—24no—
93C4×D4×C67—216no—
94C4×D4×C67—24no—
95C4×D4×C67—24no—
96#117×C67—28no—
97#117×C67—22no—
98#117×C67—22no—
99C2×C4⋊C4×C67—28no—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67—22no—
102C2×C4⋊C4×C67—22no—
103C2×C4⋊C4×C67D411yescharacteristic
104C2×C4⋊C4×C67D411yescharacteristic
105C2×C4⋊C4×C67D414yes—
106C2×C4⋊C4×C67C2312yes—
107C2×C22⋊C4×C67—22no—
108C2×C22⋊C4×C67D412yes—
109C2×C42×C67—22no—
110C2×C42×C67C2312yes—
111C22×Dic67—22no—
112C22×Dic67—24no—
113C22×Dic67—21no—
114C22×Dic67—21no—
115C22×Dic67—81no—
116C22×Dic67—81no—
117#11761612—28no—
118C2×C67⋊C8—22no—
119C2×C67⋊C8—22no—
120C2×C67⋊C8—21no—
121C2×C67⋊C8—21no—
122C2×C67⋊C8—21no—
123C2×C67⋊C8—21no—
124C2×C67⋊C8—21no—
125C2×C67⋊C8—21no—
126C2×C67⋊C8—44no—
127C2×C67⋊C8C22×C414yes—
128C4⋊Dic67—24no—
129C4⋊Dic67—22no—
130C4⋊Dic67—21no—
131C4⋊Dic67—21no—
132C4⋊Dic67—21no—
133C4⋊Dic67—21no—
134C4⋊Dic67C22×C414yes—
135Dic67⋊C4—82no—
136Dic67⋊C4—82no—
137C22⋊Dic67—42no—
138C4×Dic67—42no—
139C24×C67—21no—
140C24×C67C2×D411yescharacteristic
141#41×C67—28no—
142#41×C67—28no—
143#41×C67—28no—
144#41×C67C4⋊C4116yescomplemented
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67C22×C412yes—
149C2×Q8×C67C2×D412yes—
150C2×Q8×C67C2×Q812yes—
151C2×D4×C67—24no—
152C2×D4×C67—24no—
153C2×D4×C67—24no—
154C2×D4×C67—21no—
155C2×D4×C67—21no—
156C2×D4×C67—21no—
157C2×D4×C67C22×C412yes—
158C2×D4×C67C2×D412yes—
159C2×D4×C67C2×Q812yes—
160C22×C4×C67—21no—
161C22×C4×C67—24no—
162C22×C4×C67—24no—
163C22×C4×C67—24no—
164C22×C4×C67—22no—
165C22×C4×C67—28no—
166C22×C4×C67—24no—
167C22×C4×C67—82no—
168C22×C4×C67—82no—
169C22×C4×C67—81no—
170C22×C4×C67—81no—
171C22×C4×C67C22×C412yes—
172C22×C4×C67C2×D412yes—
173C22×C4×C67C2×D412yes—
174C22×C4×C67C2×D412yes—
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D412yes—
177C22×C4×C67C2×Q812yes—
178C22×C4×C67#4112yes—
179C22×C4×C67#4112yes—
180C22×C4×C67C2411yescharacteristic
181C4⋊C4×C67—28no—
182C4⋊C4×C67—22no—
183C4⋊C4×C67—24no—
184C4⋊C4×C67—22no—
185C4⋊C4×C67—22no—
186C4⋊C4×C67C2×D414yes—
187C22⋊C4×C67—28no—
188C22⋊C4×C67—24no—
189C42×C67—28no—
190C42×C67—22no—
191C42×C67—22no—
192C23×C67—21no—
193C23×C67—21no—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67—22no—
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—41no—
200C23×C67—41no—
201C23×C67—41no—
202C23×C67—41no—
203C23×C67C4×D412yes—
204C23×C67#12012yes—
205C23×C67C22⋊Q812yes—
206C23×C67C22×D411yescharacteristic
207Q8×C67—22no—
208Q8×C67—22no—
209Q8×C67—22no—
210Q8×C67C2×C4⋊C414yes—
211D4×C67—216no—
212D4×C67—22no—
213D4×C67—22no—
214D4×C67—22no—
215D4×C67C2×C4⋊C414yes—
216C2×C4×C67—22no—
217C2×C4×C67—22no—
218C2×C4×C67—22no—
219C2×C4×C67—216no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—24no—
223C2×C4×C67—22no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—22no—
227C2×C4×C67—42no—
228C2×C4×C67—42no—
229C2×C4×C67—84no—
230C2×C4×C67—88no—
231C2×C4×C67C2×C4⋊C414yes—
232C2×C4×C67C4×D414yes—
233C2×C4×C67C4×D412yes—
234C2×C4×C67#12012yes—
235C2×C4×C67C22⋊Q814yes—
236C2×C4×C67C22⋊Q812yes—
237C2×C4×C67#12314yes—
238C2×C4×C67C23×C411yescharacteristic
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C22×D411yescharacteristic
241C2×C4×C67C22×Q811yescharacteristic
242C2×C4×C67C2×#4111yescharacteristic
243C2×C4×C67C2×#4111yescharacteristic
244C2×Dic67—24no—
245C2×Dic67—42no—
246C2×Dic67—42no—
247C2×Dic67—41no—
248C2×Dic67—41no—
249C2×Dic67—41no—
250C2×Dic67—41no—
251C2×Dic67—41no—
252C2×Dic67—41no—
253C2×Dic67—84no—
254C67⋊C8—24no—
255C67⋊C8—41no—
256C67⋊C8—41no—
257C22×C67—22no—
258C22×C67—22no—
259C22×C67—22no—
260C22×C67—22no—
261C22×C67—22no—
262C22×C67—22no—
263C22×C67—24no—
264C22×C67—42no—
265C22×C67—42no—
266C22×C67—42no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—42no—
270C22×C67#38514yes—
271C22×C67C2×C4×D411yescharacteristic
272C22×C67C2×#12011yescharacteristic
273C22×C67C2×C22⋊Q811yescharacteristic
274C22×C67#57611yescharacteristic
275C22×C67#57711yescharacteristic
276C22×C67#57711yescharacteristic
277C22×C67#57811yescharacteristic
278C4×C67—24no—
279C4×C67—42no—
280C4×C67—44no—
281C4×C67—42no—
282C4×C67#38514yes—
283C4×C67C22×C4⋊C411yescharacteristic
284C4×C67C2×C4×D411yescharacteristic
285C4×C67C2×C22⋊Q811yescharacteristic
286C4×C67C2×#12311yescharacteristic
287Dic67—44no—
288Dic67—81no—
289Dic67—81no—
290#20878—671nomaximal, Sylow
291C2×C67—24no—
292C2×C67—44no—
293C2×C67C2×#38511yescharacteristic
294C2×C67#294611yescharacteristic
295C2×C67#294711yescharacteristic
296C2×C67#301211yescharacteristic
297C2×C67#301311yescharacteristic
298C2×C67#308311yescharacteristic
299C2×C67#308411yescharacteristic
300C2×#520D6711yescharacteristic
301C2×#428—671no—
302C2×#417—671no—
303C2×#416—671no—
304#2316—671no—
305#1865—672no—
306#1864—672no—
307#1863—672no—
308#1862—672no—
309C2×#340—671no—
310#1729—671no—
311C67#2087811yescharacteristic, complemented, Sylow
312C22×#41C2×D6711yescharacteristic
313C22×M16—671no—
314C23×C8—671no—
315#520—24no—
316#520—24no—
317C2×C4×Q8C2×D6712yes—
318C2×C4×D4—22no—
319C2×C4×D4C2×D6712yes—
320C2×#117—21no—
321C2×#117—671no—
322C2×#117C2×D6711yescharacteristic
323C22×C4⋊C4—21no—
324C22×C4⋊C4—671no—
325C22×C4⋊C4C2×D6711yescharacteristic
326#428—1344no—
327C2×#107—672no—
328C2×#106—672no—
329#417—678no—
330#416—1344no—
331C2×#103—672no—
332C2×#103—671no—
333C2×#103—671no—
334C2×#102—672no—
335C2×#102—671no—
336C2×#102—671no—
337#389—672no—
338#389—672no—
339#389—1341no—
340#389—1341no—
341#380—1342no—
342#340—672no—
343#340—672no—
344#336—672no—
345#336—672no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41Dic13414yes—
350C2×#41C4×D6714yes—
351C2×#41C22⋊D6714yes—
352C22×Q8C22×D6711yescharacteristic
353C22×D4—21no—
354C22×D4C22×D6711yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—22no—
358C23×C4—1341no—
359C23×C4C22×D6711yescharacteristic
360C2×M16—674no—
361C2×M16—1342no—
362C2×M16—1342no—
363C22×C8—672no—
364C22×C8—671no—
365C22×C8—671no—
366C22×C8—1342no—
367C22×C8—1342no—
368C22×C8—1342no—
369C4×Q8—24no—
370C4×Q8—24no—
371C4×D4—216no—
372C4×D4—24no—
373C4×D4—24no—
374#117—28no—
375#117—22no—
376#117—22no—
377#117—1342no—
378#117—1342no—
379C2×C4⋊C4—28no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—674no—
384C2×C4⋊C4—672no—
385C2×C4⋊C4—671no—
386C2×C4⋊C4—671no—
387C2×C4⋊C4—1342no—
388C2×C4⋊C4—1342no—
389C2×C4⋊C4—1341no—
390C2×C4⋊C4—1342no—
391C2×C4⋊C4C22⋊D6711yescharacteristic
392C2×C4⋊C4C22⋊D6711yescharacteristic
393C2×C4⋊C4C22⋊D6714yes—
394C2×C4⋊C4C22×D6712yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1341no—
397C2×C22⋊C4C22⋊D6712yes—
398C2×C42—22no—
399C2×C42—1341no—
400C2×C42C22×D6712yes—
401#107—1344no—
402#106—1344no—
403#103—678no—
404#103—1342no—
405#103—1342no—
406#102—678no—
407#102—1342no—
408#102—1342no—
409#95—1342no—
410#95—1342no—
411C24—21no—
412C24C2×C22⋊D6711yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41Dic67⋊C4116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×C4×D6712yes—
421C2×Q8C2×C22⋊D6712yes—
422C2×Q8C2×Dic13412yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×C4×D6712yes—
430C2×D4C2×C22⋊D6712yes—
431C2×D4C2×Dic13412yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—5361no—
444C22×C4—5361no—
445C22×C4—1341no—
446C22×C4—1341no—
447C22×C4—1342no—
448C22×C4—1344no—
449C22×C4C2×C4×D6712yes—
450C22×C4C2×C22⋊D6712yes—
451C22×C4C2×C22⋊D6712yes—
452C22×C4C2×C22⋊D6712yes—
453C22×C4C2×C22⋊D6711yescharacteristic
454C22×C4C2×C22⋊D6712yes—
455C22×C4C2×Dic13412yes—
456C22×C4#1176162812yes—
457C22×C4#1176162812yes—
458C22×C4C23×D6711yescharacteristic
459M16—1348no—
460C2×C8—2684no—
461C2×C8—674no—
462C2×C8—1341no—
463C2×C8—1341no—
464C2×C8—1341no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1342no—
469C2×C8—1342no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—5362no—
476C4⋊C4—5362no—
477C4⋊C4—674no—
478C4⋊C4—1341no—
479C4⋊C4—1341no—
480C4⋊C4—1341no—
481C4⋊C4—1341no—
482C4⋊C4—1344no—
483C4⋊C4—1342no—
484C4⋊C4C2×C22⋊D6714yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—2682no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—2682no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C4×C22⋊D6712yes—
504C23#1452401212yes—
505C23#1452402212yes—
506C23C22×C22⋊D6711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×Dic67⋊C414yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×Dic67⋊C414yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—2682no—
532C2×C4—2682no—
533C2×C4—2681no—
534C2×C4—2681no—
535C2×C4—2681no—
536C2×C4—2681no—
537C2×C4—2681no—
538C2×C4—2681no—
539C2×C4—5364no—
540C2×C4—1344no—
541C2×C4C2×Dic67⋊C414yes—
542C2×C4C4×C22⋊D6712yes—
543C2×C4C4×C22⋊D6714yes—
544C2×C4#1452401212yes—
545C2×C4#1452402212yes—
546C2×C4#1452402214yes—
547C2×C4#1452402614yes—
548C2×C4C22×C4×D6711yescharacteristic
549C2×C4C22×C22⋊D6711yescharacteristic
550C2×C4C22×C22⋊D6711yescharacteristic
551C2×C4C22×Dic13411yescharacteristic
552C2×C4C2×#1176162811yescharacteristic
553C2×C4C2×#1176162811yescharacteristic
554C8—2681no—
555C8—2681no—
556C8—1344no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1686326914yes—
571C22C2×C4×C22⋊D6711yescharacteristic
572C22C2×#1452401211yescharacteristic
573C22C2×#1452402211yescharacteristic
574C22#1686419911yescharacteristic
575C22#1686420411yescharacteristic
576C22#1686420711yescharacteristic
577C22#1686421111yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—2684no—
583C4—5361no—
584C4—5361no—
585C4#1686326914yes—
586C4C22×Dic67⋊C411yescharacteristic
587C4C2×C4×C22⋊D6711yescharacteristic
588C4C2×#1452402211yescharacteristic
589C4C2×#1452402611yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1686326911yescharacteristic
593C2#2256903711yescharacteristic
594C2#2256904811yescharacteristic
595C2#2256951311yescharacteristic
596C2#2256952411yescharacteristic
597C2#2257006011yescharacteristic
598C2#2257007111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#520×C67C2
socleC23×C67C22×D4
prime radicalC22×#41×C67C22
cosocleC22×C4×C67C24
hypercenterC2×#520D67
hypocenterC67#20878
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16863436C2semidirect
4. #22562825C2semidirect
5. #22560477C2semidirect
6. #22560470C2semidirect
7. #22560463C2semidirect
8. #22560456C2semidirect
9. C2×#16863158C2semidirect
10. #22559906C2semidirect
19. #16863397C4semidirect
49. C2×#11761612C2×C4semidirect
60. C2×C4⋊Dic67C2×C4semidirect
72. #14523967C2×C4semidirect
75. #14523964C2×C4semidirect
144. #41×C67C4⋊C4semidirect
311. C67#20878semidirect
416. #41Dic67⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168634362
C2×#168633972
C2×#168633922
#225628252
#225604772
#225604702
#225604632
#225604562
C2×#168631582
#225599062
C2×#520×C672
#2087867