Finite Group Databasebeta

← #30826461

Subgroups — #30826461

Order 17152.

622 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
1#22570798C211yesmaximal, characteristic
2#22569360C211yesmaximal, characteristic, complemented
3#22559121C211yesmaximal, characteristic, complemented
4#22559111C211yesmaximal, characteristic, complemented
5#22559097C211yesmaximal, characteristic, complemented
6#22559092C211yesmaximal, characteristic, complemented
7#22559031C211yesmaximal, characteristic, complemented
8#22559021C211yesmaximal, characteristic, complemented
9C8×#11761616C211yesmaximal, characteristic, complemented
10C8×#11761613C211yesmaximal, characteristic, complemented
11#22558977C211yesmaximal, characteristic, complemented
12#22558667C211yesmaximal, characteristic, complemented
13#22558660C211yesmaximal, characteristic, complemented
14#22558490C211yesmaximal, characteristic, complemented
15C67×#2981C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863802C411yescharacteristic
18#16863783C411yescharacteristic
19#16863760C411yescharacteristic
20#16863758C411yescharacteristic
21#16863647C411yescharacteristic
22#16863647C411yescharacteristic
23#16863640C411yescharacteristic
24#16863640C411yescharacteristic
25#16863526C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C4×#11761616C2211yescharacteristic
28C4×#11761616C2211yescharacteristic
29C4×#11761613C2211yescharacteristic
30C4×#11761613C2211yescharacteristic
31#16863475C2211yescharacteristic
32#16863471—21no—
33#16863470—21no—
34C8×D268C2211yescharacteristic
35#16863456—21no—
36#16863421—21no—
37#16863420C2211yescharacteristic
38C4×#11761612C2211yescharacteristic
39C2×C4×C67⋊C8C2211yescharacteristic
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44#16863126C2211yescharacteristic
45#16863126C2211yescharacteristic
46D4⋊(C67⋊C8)C2211yescharacteristic
47D4⋊(C67⋊C8)C2211yescharacteristic
48D4⋊(C67⋊C8)C2211yescharacteristic
49D4⋊(C67⋊C8)C2211yescharacteristic
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54C8×C67⋊C8C2211yescharacteristic
55C8×C67⋊C8C2211yescharacteristic
56#532×C67C2211yescharacteristic
57#436×C67C2211yescharacteristic
58#436×C67C2211yescharacteristic
59#435×C67—21no—
60#435×C67—21no—
61C8×D4×C67C2211yescharacteristic
62C8×D4×C67C2211yescharacteristic
63#432×C67C2211yescharacteristic
64#431×C67C2211yescharacteristic
65#408×C67—21no—
66#408×C67—21no—
67C2×#11761616C2×C411yescharacteristic
68C2×#11761616C2×C411yescharacteristic
69C2×#11761613C2×C411yescharacteristic
70C2×#11761613C2×C411yescharacteristic
71C2×#11761612—21no—
72C2×#11761612C2×C411yescharacteristic
73C2×#11761612C2×C411yescharacteristic
74C2×#11761611—21no—
75C22×C67⋊C8—21no—
76C2×C8×D67—21no—
77#14524037—21no—
78#14524035C2×C411yescharacteristic
79#14524031C2×C411yescharacteristic
80#14524026—21no—
81#14524022C2×C411yescharacteristic
82#14524017C2×C411yescharacteristic
83C4×Dic134C2311yescharacteristic
84C4×D268C2311yescharacteristic
85C4×C22⋊D67—21no—
86#14524000—21no—
87#14523976C2×C411yescharacteristic
88#14523976C2×C411yescharacteristic
89#14523974C2×C411yescharacteristic
90#14523974C2×C411yescharacteristic
91C4⋊(C67⋊C8)—21no—
92C4⋊(C67⋊C8)C2311yescharacteristic
93C4⋊(C67⋊C8)C2311yescharacteristic
94Dic67⋊C8—21no—
95Dic67⋊C8—21no—
96#14523966C2×C411yescharacteristic
97#14523966C2×C411yescharacteristic
98#14523964C2×C411yescharacteristic
99#14523964C2×C411yescharacteristic
100#14523964C2×C411yescharacteristic
101#14523964C2×C411yescharacteristic
102#14523963C2×C411yescharacteristic
103#14523963C2×C411yescharacteristic
104(C2×D67)⋊C8—21no—
105(C2×D67)⋊C8—21no—
106#14523954C2×C411yescharacteristic
107#14523954C2×C411yescharacteristic
108#14523953—21no—
109C4×C67⋊C8—21no—
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C4×C67⋊C8C2311yescharacteristic
114C8×Dic67—21no—
115C2×#41×C67—21no—
116C2×M16×C67—21no—
117C2×M16×C67—21no—
118C22×C8×C67—21no—
119C22×C8×C67—21no—
120C4⋊Q8×C67C2×C411yescharacteristic
121#127×C67C2×C411yescharacteristic
122#124×C67—21no—
123#121×C67—21no—
124#121×C67C2×C411yescharacteristic
125#121×C67C2×C411yescharacteristic
126C4×D4×C67—21no—
127C4×D4×C67C2311yescharacteristic
128C4×D4×C67C2311yescharacteristic
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67D411yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67—21no—
137C22⋊C8×C67—21no—
138C22⋊C8×C67—21no—
139C22⋊C8×C67D411yescharacteristic
140C22⋊C8×C67D411yescharacteristic
141#97×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134C22×C411yescharacteristic
145C2×D268C22×C411yescharacteristic
146C2×C22⋊D67—21no—
147C2×C4×D67—21no—
148#11761616—22no—
149#11761616—22no—
150#11761613—22no—
151#11761613—22no—
152#11761612—22no—
153#11761612—22no—
154#11761612—81no—
155#11761611—81no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8—42no—
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C2×C67⋊C8C22×C411yescharacteristic
165C8×D67—81no—
166C4⋊Dic67C22×C411yescharacteristic
167C4⋊Dic67C22×C411yescharacteristic
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171Dic67⋊C4—21no—
172C22⋊Dic67—21no—
173(C2×D67)⋊C4—21no—
174(C2×D67)⋊C4—21no—
175(C2×D67)⋊C4—21no—
176(C2×D67)⋊C4—21no—
177C4×Dic67—21no—
178#41×C67—81no—
179#41×C67—81no—
180C2×Q8×C67—21no—
181C2×D4×C67—21no—
182C2×D4×C67—21no—
183C2×D4×C67—21no—
184C2×D4×C67—21no—
185C2×D4×C67C22×C411yescharacteristic
186C2×D4×C67C22×C411yescharacteristic
187C22×C4×C67—21no—
188C22×C4×C67—21no—
189C22×C4×C67C22×C411yescharacteristic
190C22×C4×C67C2×D411yescharacteristic
191C22×C4×C67#4111yescharacteristic
192M16×C67—81no—
193M16×C67—81no—
194C2×C8×C67—21no—
195C2×C8×C67—21no—
196C2×C8×C67—81no—
197C2×C8×C67—81no—
198C2×C8×C67C2×D411yescharacteristic
199C2×C8×C67C2×D411yescharacteristic
200C2×C8×C67C2×D411yescharacteristic
201C2×C8×C67C2×D411yescharacteristic
202C4⋊C4×C67—21no—
203C4⋊C4×C67C22×C411yescharacteristic
204C4⋊C4×C67C22×C411yescharacteristic
205C22⋊C4×C67—21no—
206C22⋊C4×C67—21no—
207C22⋊C4×C67—21no—
208C22⋊C4×C67—21no—
209C42×C67C22×C411yescharacteristic
210C42×C67C2×D411yescharacteristic
211C42×C67#4111yescharacteristic
212C42×C67C2411yescharacteristic
213C23×C67—21no—
214C23×C67—21no—
215C23×C67C4×D411yescharacteristic
216C22×D67—21no—
217Q8×C67—81no—
218D4×C67—21no—
219D4×C67—21no—
220D4×C67—21no—
221D4×C67—21no—
222D4×C67—81no—
223D4×C67—81no—
224D4×C67—81no—
225D4×C67—81no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67C4×D411yescharacteristic
238C2×C4×C67C4×D411yescharacteristic
239C2×C4×C67C4×D411yescharacteristic
240C2×C4×C67C23×C411yescharacteristic
241C2×C4×C67C22×D411yescharacteristic
242C2×C4×C67C2×#4111yescharacteristic
243C22⋊D67—81no—
244C2×Dic67—21no—
245C2×Dic67—21no—
246C2×Dic67—21no—
247D268—21no—
248D268—21no—
249C4×D67—41no—
250C4×D67—41no—
251Dic134—21no—
252Dic134—21no—
253C8×C67—21no—
254C8×C67—21no—
255C8×C67—21no—
256C8×C67—21no—
257C8×C67—41no—
258C8×C67—41no—
259C67⋊C8—22no—
260C67⋊C8—21no—
261C67⋊C8—21no—
262C67⋊C8—41no—
263C67⋊C8—41no—
264C67⋊C8—41no—
265C67⋊C8#13112yes—
266C67⋊C8#13112yes—
267C22×C67—21no—
268C22×C67—21no—
269C22×C67—21no—
270C22×C67—41no—
271C22×C67—41no—
272C22×C67—41no—
273C22×C67—41no—
274C22×C67—41no—
275C22×C67—41no—
276C22×C67C2×C4×D411yescharacteristic
277C2×D67—41no—
278C2×D67—41no—
279C2×D67—41no—
280C4×C67—21no—
281C4×C67—21no—
282C4×C67—21no—
283C4×C67—21no—
284C4×C67—41no—
285C4×C67—41no—
286C4×C67C2×#13111yescharacteristic
287C4×C67C2×#13111yescharacteristic
288C4×C67#57611yescharacteristic
289C4×C67#57711yescharacteristic
290Dic67—41no—
291Dic67—41no—
292Dic67—41no—
293#17968—671nomaximal, Sylow
294C2×C67—41no—
295C2×C67—81no—
296C2×C67—81no—
297C2×C67#293111yescharacteristic
298C2×C67#294611yescharacteristic
299C2×C67#295611yescharacteristic
300D67—81no—
301#3171—671no—
302#2993—671no—
303#2981D6711yescharacteristic, complemented
304#1594—671no—
305#1592—671no—
306#1590—671no—
307#1589—671no—
308#1580—671no—
309SD16⋊C8—671no—
310C8×SD16—671no—
311C8×D8—671no—
312#1569—671no—
313#1494—671no—
314#1492—671no—
315C8×M16—671no—
316C67#1796811yescharacteristic, complemented, Sylow
317#532C2×D6711yescharacteristic
318#529—671no—
319#501—671no—
320#496—671no—
321#490—671no—
322#489—671no—
323#469—671no—
324#469—671no—
325#468—671no—
326#468—671no—
327#446—671no—
328C8×Q8—671no—
329C4×SD16—671no—
330C4×SD16—671no—
331C4×D8—671no—
332C4×D8—671no—
333#436—671no—
334#436—1341no—
335#436C2×D6711yescharacteristic
336#436C2×D6711yescharacteristic
337#435—21no—
338#435—21no—
339#435—1341no—
340C8×D4—671no—
341C8×D4C2×D6711yescharacteristic
342C8×D4C2×D6711yescharacteristic
343#433—1341no—
344#432C2×D6711yescharacteristic
345#431C2×D6711yescharacteristic
346#424—1341no—
347#423—671no—
348#408—21no—
349#408—21no—
350C4×M16—671no—
351C2×C4×C8—671no—
352#335—671no—
353#335—671no—
354#334—671no—
355#334—671no—
356Q8⋊C8—671no—
357Q8⋊C8—671no—
358D4⋊C8—671no—
359D4⋊C8—671no—
360D4⋊C8—671no—
361D4⋊C8—671no—
362D4⋊C8—671no—
363D4⋊C8—671no—
364#322—671no—
365#322—671no—
366C82—671no—
367C82—671no—
368C2×#41—21no—
369C2×SD16—671no—
370C2×SD16—671no—
371C2×D8—671no—
372C2×D8—671no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—671no—
376C2×M16—671no—
377C2×M16—1341no—
378C2×M16—1341no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—1341no—
382C22×C8—1341no—
383C4⋊Q8C4×D6711yescharacteristic
384#127C4×D6711yescharacteristic
385#126—1341no—
386#125—671no—
387#124—21no—
388#124—671no—
389#123—1341no—
390C22⋊Q8—671no—
391#121—21no—
392#121—671no—
393#121C4×D6711yescharacteristic
394#121C4×D6711yescharacteristic
395C4×Q8—671no—
396C4×D4—21no—
397C4×D4—671no—
398C4×D4—1341no—
399C4×D4C22×D6711yescharacteristic
400C4×D4C22×D6711yescharacteristic
401#117—1341no—
402C2×C42C22×D6711yescharacteristic
403#107—671no—
404#107—671no—
405#106—671no—
406#106—671no—
407C4⋊C8—671no—
408C4⋊C8—671no—
409C4⋊C8—1341no—
410C4⋊C8—1341no—
411C4⋊C8—1341no—
412C4⋊C8C22⋊D6711yescharacteristic
413C4⋊C8C22⋊D6711yescharacteristic
414C4⋊C8C22×D6711yescharacteristic
415C4⋊C8C22×D6711yescharacteristic
416#103—671no—
417#103—671no—
418#102—671no—
419#102—671no—
420#102—671no—
421#102—671no—
422#102—671no—
423#102—671no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—1341no—
430C22⋊C8—1341no—
431C22⋊C8C22⋊D6711yescharacteristic
432C22⋊C8C22⋊D6711yescharacteristic
433#97—21no—
434#97—671no—
435#97—671no—
436#97—1341no—
437C4×C8—671no—
438C4×C8—671no—
439C4×C8—671no—
440C4×C8—671no—
441C4×C8—1341no—
442C4×C8—1341no—
443C4×C8C22×D6711yescharacteristic
444C4×C8C22×D6711yescharacteristic
445#41—81no—
446#41—81no—
447C2×Q8—21no—
448C2×Q8—671no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—671no—
454C2×D4—1341no—
455C2×D4C2×C4×D6711yescharacteristic
456C2×D4C2×C4×D6711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—1341no—
460C22×C4C2×C4×D6711yescharacteristic
461C22×C4C2×C22⋊D6711yescharacteristic
462C22×C4#1176162811yescharacteristic
463SD16—1342no—
464SD16—1342no—
465D8—1342no—
466D8—1342no—
467M16—81no—
468M16—81no—
469M16—5361no—
470M16—5361no—
471M16—1342no—
472M16—1342no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—81no—
477C2×C8—2682no—
478C2×C8—5361no—
479C2×C8—671no—
480C2×C8—671no—
481C2×C8—671no—
482C2×C8—671no—
483C2×C8—1341no—
484C2×C8—1341no—
485C2×C8—1341no—
486C2×C8—1341no—
487C2×C8C2×C22⋊D6711yescharacteristic
488C2×C8C2×C22⋊D6711yescharacteristic
489C2×C8C2×C22⋊D6711yescharacteristic
490C2×C8C2×C22⋊D6711yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—671no—
493C4⋊C4—671no—
494C4⋊C4—1341no—
495C4⋊C4—1341no—
496C4⋊C4—1341no—
497C4⋊C4—1341no—
498C4⋊C4C2×C4×D6711yescharacteristic
499C4⋊C4C2×C4×D6711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—1341no—
505C22⋊C4—1341no—
506C22⋊C4—1341no—
507C22⋊C4—1341no—
508C22⋊C4—1341no—
509C42—1341no—
510C42C2×C4×D6711yescharacteristic
511C42C2×C22⋊D6711yescharacteristic
512C42#1176162811yescharacteristic
513C42C23×D6711yescharacteristic
514C23—21no—
515C23—21no—
516C23—1341no—
517C23C4×C22⋊D6711yescharacteristic
518Q8—81no—
519Q8—1341no—
520Q8—1341no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—81no—
526D4—81no—
527D4—81no—
528D4—81no—
529D4—5361no—
530D4—1341no—
531D4—1341no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—2681no—
544C2×C4—2681no—
545C2×C4—1341no—
546C2×C4—1341no—
547C2×C4—1341no—
548C2×C4C4×C22⋊D6711yescharacteristic
549C2×C4C4×C22⋊D6711yescharacteristic
550C2×C4C4×C22⋊D6711yescharacteristic
551C2×C4C22×C4×D6711yescharacteristic
552C2×C4C22×C22⋊D6711yescharacteristic
553C2×C4C2×#1176162811yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—2681no—
561C8—2681no—
562C8—2681no—
563C8—672no—
564C8—672no—
565C8—1342no—
566C8—1341no—
567C8—1341no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—2681no—
578C22—2681no—
579C22—2681no—
580C22C2×C4×C22⋊D6711yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—2681no—
588C4—2681no—
589C4—2681no—
590C4#131×D6711yescharacteristic
591C4#131×D6711yescharacteristic
592C4#1686419911yescharacteristic
593C4#1686420711yescharacteristic
594C2—41no—
595C2—81no—
596C2—81no—
597C2—5361no—
598C2#2256891911yescharacteristic
599C2#2256904111yescharacteristic
600C2#2256911611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2981C2
socleC22×C67C2×C4×D4
prime radical#22570798C2
cosocleC42×C67C24
hypercenter#2981D67
hypocenterC67#17968
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2981

Complements

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

NH ≅ G/Ntype
2. #22569360C2semidirect
3. #22559121C2semidirect
4. #22559111C2semidirect
5. #22559097C2semidirect
6. #22559092C2semidirect
7. #22559031C2semidirect
8. #22559021C2semidirect
9. C8×#11761616C2semidirect
10. C8×#11761613C2semidirect
11. #22558977C2semidirect
12. #22558667C2semidirect
13. #22558660C2semidirect
14. #22558490C2semidirect
15. C67×#2981C2semidirect
303. #2981D67semidirect
316. C67#17968semidirect

Maximal subgroups

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

subgroupindex
#225707982
#225693602
#225591212
#225591112
#225590972
#225590922
#225590312
#225590212
C8×#117616162
C8×#117616132
#225589772
#225586672
#225586602
#225584902
C67×#29812
#1796867