Finite Group Databasebeta

← #30863595

Subgroups — #30863595

Order 17152.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863683C211yesmaximal, characteristic
2#22562571C211yesmaximal, characteristic
3#22561162C211yesmaximal, characteristic, complemented
4#22560975C211yesmaximal, characteristic, complemented
5#22560791C211yesmaximal, characteristic, complemented
6#22560784C211yesmaximal, characteristic, complemented
7#22560469C211yesmaximal, characteristic
8#22560455C211yesmaximal, characteristic
9#22560216C211yesmaximal, characteristic
10#22560107C211yesmaximal, characteristic, complemented
11C4×#14523974C211yesmaximal, characteristic, complemented
12#22560066C211yesmaximal, characteristic, complemented
13#22560037C211yesmaximal, characteristic
14C4×#14523963C211yesmaximal, characteristic
15C67×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D268C2211yescharacteristic
17C2×C4×D268C2211yescharacteristic
18C2×C4×D268C2211yescharacteristic
19#16863683C418yescomplemented
20C2×#14523976C2211yescharacteristic
21C2×#14523974C2211yescharacteristic
22C2×#14523974C2211yescharacteristic
23C2×#14523974C2211yescharacteristic
24C2×C4⋊(C67⋊C8)C2211yescharacteristic
25C2×C4⋊(C67⋊C8)C2211yescharacteristic
26C2×#14523963C2211yescharacteristic
27C2×#14523963C2211yescharacteristic
28C2×#14523963C2211yescharacteristic
29C2×#14523963C2211yescharacteristic
30C2×#14523954C2211yescharacteristic
31C2×C4×C67⋊C8C2211yescharacteristic
32C4×C4⋊Dic67C2211yescharacteristic
33C4×C4⋊Dic67C2211yescharacteristic
34C4×C4⋊Dic67C2211yescharacteristic
35#16863243—21no—
36#16863243—21no—
37#16863243—21no—
38#95×D67—21no—
39#16863158C2211yescharacteristic
40#16863158C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863151C2211yescharacteristic
48#16863151C2211yescharacteristic
49C2×C4⋊Q8×C67C2211yescharacteristic
50#391×C67—21no—
51#391×C67—21no—
52#389×C67C2211yescharacteristic
53#389×C67C2211yescharacteristic
54#387×C67—21no—
55#387×C67—21no—
56#382×C67C2211yescharacteristic
57C4×C4⋊C4×C67C2211yescharacteristic
58C4×C4⋊C4×C67C2211yescharacteristic
59#376×C67C2211yescharacteristic
60C22×D268C2311yescharacteristic
61C22×C4×D67—21no—
62C22×C4×D67—21no—
63C22×C4×D67—21no—
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C22×C67⋊C8C2311yescharacteristic
67C22×C67⋊C8C2311yescharacteristic
68C4×D268—22no—
69C4×D268—22no—
70C4×D268—22no—
71C4×D268—22no—
72C4×D268C2×C414yes—
73C4×D268Q814yescomplemented
74C2×C4⋊Dic67C2311yescharacteristic
75C2×C4⋊Dic67C2311yescharacteristic
76C2×C4⋊Dic67C2311yescharacteristic
77C2×(C2×D67)⋊C4—21no—
78C2×(C2×D67)⋊C4—21no—
79C2×(C2×D67)⋊C4—21no—
80C2×C4×Dic67—21no—
81C2×C4×Dic67—21no—
82C2×C4×Dic67—21no—
83#14523976—22no—
84#14523974—22no—
85#14523974C2×C414yescomplemented
86#14523974C2×C414yescomplemented
87C4⋊(C67⋊C8)—22no—
88C4⋊(C67⋊C8)C2×C414yescomplemented
89#14523963—22no—
90#14523963—22no—
91#14523963C2×C414yes—
92#14523963C2×C414yes—
93#14523954—22no—
94C4×C67⋊C8—22no—
95C67⋊#95—21no—
96C67⋊#95—21no—
97C67⋊#95—21no—
98C22×Q8×C67—21no—
99C4⋊Q8×C67—22no—
100C4⋊Q8×C67C2×C414yescomplemented
101C2×C4⋊C4×C67—21no—
102C2×C4⋊C4×C67—21no—
103C2×C4⋊C4×C67—21no—
104C2×C4⋊C4×C67C2311yescharacteristic
105C2×C4⋊C4×C67C2311yescharacteristic
106C2×C4⋊C4×C67C2311yescharacteristic
107C2×C4⋊C4×C67C2311yescharacteristic
108C2×C42×C67—21no—
109C2×C42×C67—21no—
110C2×C42×C67C2311yescharacteristic
111C2×C42×C67C2311yescharacteristic
112C2×C42×C67C2311yescharacteristic
113#95×C67—21no—
114#95×C67—21no—
115#95×C67—21no—
116#95×C67—21no—
117#95×C67—21no—
118#95×C67—21no—
119#95×C67D411yescharacteristic
120#95×C67D411yescharacteristic
121#95×C67D411yescharacteristic
122#95×C67D411yescharacteristic
123C23×D67—21no—
124C2×D268—21no—
125C2×D268—21no—
126C2×D268—21no—
127C2×D268C22×C412yes—
128C2×D268C2×Q812yes—
129C2×D268#4112yes—
130C22×Dic67—21no—
131C22×Dic67—21no—
132C22×Dic67—21no—
133C2×C4×D67—24no—
134C2×C4×D67—24no—
135C2×C4×D67—24no—
136C2×C4×D67—81no—
137C2×C4×D67—81no—
138C2×C4×D67—81no—
139C2×C4×D67—81no—
140C2×C4×D67—81no—
141C2×C4×D67—81no—
142C2×C67⋊C8—21no—
143C2×C67⋊C8—21no—
144C2×C67⋊C8—21no—
145C2×C67⋊C8—21no—
146C2×C67⋊C8—21no—
147C2×C67⋊C8—21no—
148C2×C67⋊C8—21no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8C22×C412yes—
153C2×C67⋊C8C22×C412yes—
154C4⋊Dic67—21no—
155C4⋊Dic67—21no—
156C4⋊Dic67—21no—
157C4⋊Dic67C22×C412yes—
158C4⋊Dic67C2×Q812yes—
159C4⋊Dic67#4112yes—
160(C2×D67)⋊C4—24no—
161(C2×D67)⋊C4—24no—
162(C2×D67)⋊C4—24no—
163C4×Dic67—81no—
164C4×Dic67—81no—
165C4×Dic67—81no—
166C2×Q8×C67—24no—
167C2×Q8×C67—81no—
168C2×Q8×C67—81no—
169C22×C4×C67—21no—
170C22×C4×C67—21no—
171C22×C4×C67—21no—
172C22×C4×C67—21no—
173C22×C4×C67C2×D411yescharacteristic
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2411yescharacteristic
180C4⋊C4×C67—24no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—81no—
188C4⋊C4×C67—81no—
189C4⋊C4×C67C22×C412yes—
190C4⋊C4×C67C22×C412yes—
191C42×C67—21no—
192C42×C67—21no—
193C42×C67—81no—
194C42×C67—81no—
195C42×C67C22×C412yes—
196C42×C67C2×Q812yes—
197C42×C67#4112yes—
198C42×C67#4112yes—
199C23×C67C22×D411yescharacteristic
200C22×D67—22no—
201C22×D67—22no—
202C22×D67—22no—
203C22×D67—41no—
204C22×D67—41no—
205C22×D67—41no—
206C22×D67—41no—
207Q8×C67—84no—
208C2×C4×C67—21no—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—22no—
217C2×C4×C67—22no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67C4×D412yes—
235C2×C4×C67C4×D412yes—
236C2×C4×C67C22⋊Q812yes—
237C2×C4×C67C22⋊Q812yes—
238C2×C4×C67C23×C411yescharacteristic
239C2×C4×C67C22×Q811yescharacteristic
240C2×C4×C67C2×#4111yescharacteristic
241C2×C4×C67C2×#4111yescharacteristic
242C2×C4×C67C2×#4111yescharacteristic
243C2×C4×C67C2×#4111yescharacteristic
244C2×Dic67—22no—
245C2×Dic67—22no—
246C2×Dic67—22no—
247C2×Dic67—41no—
248C2×Dic67—41no—
249C2×Dic67—41no—
250C2×Dic67—41no—
251C2×Dic67—41no—
252C2×Dic67—41no—
253D268—22no—
254D268—22no—
255D268—22no—
256D268C4×Q814yes—
257C4×D67—84no—
258C4×D67—84no—
259C4×D67—84no—
260C67⋊C8—22no—
261C67⋊C8—22no—
262C67⋊C8—41no—
263C67⋊C8—41no—
264C22×C67C2×C4×D411yescharacteristic
265C22×C67C2×C22⋊Q811yescharacteristic
266C22×C67#52511yescharacteristic
267C22×C67#57611yescharacteristic
268C22×C67#57711yescharacteristic
269C22×C67#57711yescharacteristic
270C22×C67#57711yescharacteristic
271C2×D67—24no—
272C2×D67—42no—
273C2×D67—42no—
274C2×D67—42no—
275C2×D67—42no—
276C2×D67—42no—
277C2×D67—42no—
278C4×C67—22no—
279C4×C67—22no—
280C4×C67—22no—
281C4×C67—22no—
282C4×C67—41no—
283C4×C67—41no—
284C4×C67—42no—
285C4×C67—42no—
286C4×C67—81no—
287C4×C67—81no—
288C4×C67C2×C4×Q811yescharacteristic
289C4×C67C4×#4111yescharacteristic
290C4×C67#52911yescharacteristic
291C4×C67#53311yescharacteristic
292Dic67—42no—
293Dic67—42no—
294Dic67—42no—
295#21623—671nomaximal, Sylow
296C2×C67C4×C22⋊Q811yescharacteristic
297C2×C67#294611yescharacteristic
298C2×C67#294911yescharacteristic
299C2×C67#308111yescharacteristic
300C2×C67#308411yescharacteristic
301C2×C67#311811yescharacteristic
302C2×C67#312511yescharacteristic
303D67—44no—
304C2×#476—671no—
305#2331D6711yescharacteristic, complemented
306#2275—671no—
307#1988—671no—
308#1949—671no—
309#1920—671no—
310#1919—671no—
311#1864—671no—
312#1862—671no—
313#1804—671no—
314#1776—671no—
315C4×#106—671no—
316#1768—671no—
317#1762—671no—
318C4×#102—671no—
319C67#2162311yescharacteristic, complemented, Sylow
320C2×C4⋊Q8C2×D6711yescharacteristic
321C2×C4×D4—671no—
322C2×C4×D4—671no—
323C2×C4×D4—671no—
324#476—678no—
325C2×#107—671no—
326C2×#106—671no—
327C2×#106—671no—
328C2×#106—671no—
329C2×C4⋊C8—671no—
330C2×C4⋊C8—671no—
331C2×#102—671no—
332C2×#102—671no—
333C2×#102—671no—
334C2×#102—671no—
335C2×#97—671no—
336C2×C4×C8—671no—
337#391—21no—
338#391—21no—
339#389C2×D6711yescharacteristic
340#389C2×D6711yescharacteristic
341#387—21no—
342#387—21no—
343#382C2×D6711yescharacteristic
344C4×C4⋊C4—671no—
345C4×C4⋊C4—671no—
346C4×C4⋊C4—671no—
347C4×C4⋊C4C2×D6711yescharacteristic
348C4×C4⋊C4C2×D6711yescharacteristic
349C4×C22⋊C4—1341no—
350C4×C22⋊C4—1341no—
351C4×C22⋊C4—1341no—
352#376C2×D6711yescharacteristic
353C2×#95—1341no—
354#340—671no—
355#340—671no—
356#340—671no—
357#340—671no—
358#340—671no—
359#340—671no—
360#340—671no—
361#340—671no—
362#336—671no—
363#336—671no—
364C22×Q8—21no—
365C22×D4—671no—
366C23×C4—1341no—
367C23×C4—1341no—
368C23×C4—1341no—
369C22×C8—671no—
370C22×C8—671no—
371C22×C8—671no—
372C22×C8—671no—
373C4⋊Q8—22no—
374C4⋊Q8C4×D6714yescomplemented
375C4×D4—674no—
376C4×D4—674no—
377C4×D4—1342no—
378C4×D4—1342no—
379C4×D4—1342no—
380C4×D4—1342no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—671no—
385C2×C4⋊C4—671no—
386C2×C4⋊C4—671no—
387C2×C4⋊C4C22×D6711yescharacteristic
388C2×C4⋊C4C22×D6711yescharacteristic
389C2×C4⋊C4C22×D6711yescharacteristic
390C2×C4⋊C4C22×D6711yescharacteristic
391C2×C22⋊C4—1341no—
392C2×C22⋊C4—1341no—
393C2×C22⋊C4—1341no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—1341no—
397C2×C42—1341no—
398C2×C42—1341no—
399C2×C42C22×D6711yescharacteristic
400C2×C42C22×D6711yescharacteristic
401C2×C42C22×D6711yescharacteristic
402#107—1342no—
403#106—674no—
404#106—674no—
405#106—1342no—
406C4⋊C8—674no—
407C4⋊C8—1342no—
408#102—674no—
409#102—674no—
410#102—1342no—
411#102—1342no—
412#97—1342no—
413C4×C8—1342no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—1341no—
421#95—1341no—
422#95—1341no—
423#95C22⋊D6711yescharacteristic
424#95C22⋊D6711yescharacteristic
425#95C22⋊D6711yescharacteristic
426#95C22⋊D6711yescharacteristic
427C24—1341no—
428C2×Q8—24no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×D4—672no—
432C2×D4—672no—
433C2×D4—672no—
434C2×D4—1341no—
435C2×D4—1341no—
436C2×D4—1341no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—5361no—
442C22×C4—5361no—
443C22×C4—5361no—
444C22×C4—5361no—
445C22×C4—5361no—
446C22×C4—5361no—
447C22×C4—1344no—
448C22×C4—1341no—
449C22×C4—1341no—
450C22×C4—1341no—
451C22×C4—1344no—
452C22×C4—1344no—
453C22×C4C2×C22⋊D6711yescharacteristic
454C22×C4C2×C22⋊D6711yescharacteristic
455C22×C4C2×C22⋊D6711yescharacteristic
456C22×C4C2×C22⋊D6711yescharacteristic
457C22×C4C2×C22⋊D6711yescharacteristic
458C22×C4C2×C22⋊D6711yescharacteristic
459C22×C4C23×D6711yescharacteristic
460C2×C8—672no—
461C2×C8—672no—
462C2×C8—1341no—
463C2×C8—1341no—
464C2×C8—1341no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8—1341no—
470C2×C8—1341no—
471C2×C8—1341no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—672no—
482C4⋊C4—672no—
483C4⋊C4—672no—
484C4⋊C4—1341no—
485C4⋊C4—1341no—
486C4⋊C4—1341no—
487C4⋊C4C2×C4×D6712yes—
488C4⋊C4C2×C4×D6712yes—
489C22⋊C4—1344no—
490C22⋊C4—1344no—
491C22⋊C4—1344no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—5361no—
497C42—5361no—
498C42—5361no—
499C42C2×C4×D6712yes—
500C42Q8×D6712yes—
501C42#1176162812yes—
502C42#1176162912yes—
503C23—2681no—
504C23—2681no—
505C23—2681no—
506C23—2681no—
507C23—1342no—
508C23—1342no—
509C23—1342no—
510C23C22×C22⋊D6711yescharacteristic
511Q8—84no—
512D4—674no—
513D4—1342no—
514D4—1342no—
515D4—1342no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—2681no—
543C2×C4—2681no—
544C2×C4—2681no—
545C2×C4—2681no—
546C2×C4—2681no—
547C2×C4—2681no—
548C2×C4—5364no—
549C2×C4—5364no—
550C2×C4—5364no—
551C2×C4—1342no—
552C2×C4—1342no—
553C2×C4—1342no—
554C2×C4C4×C22⋊D6712yes—
555C2×C4C4×C22⋊D6712yes—
556C2×C4#1452401912yes—
557C2×C4#1452401912yes—
558C2×C4C22×C4×D6711yescharacteristic
559C2×C4C2×Q8×D6711yescharacteristic
560C2×C4#41×D6711yescharacteristic
561C2×C4#41×D6711yescharacteristic
562C2×C4C2×#1176162811yescharacteristic
563C2×C4C2×#1176162911yescharacteristic
564C8—2681no—
565C8—2681no—
566C8—1342no—
567C8—1342no—
568C22—2682no—
569C22—2682no—
570C22—2682no—
571C22—2682no—
572C22—2682no—
573C22—2682no—
574C22—1344no—
575C22C2×C4×C22⋊D6711yescharacteristic
576C22C2×#1452401911yescharacteristic
577C22#1686390311yescharacteristic
578C22#1686419911yescharacteristic
579C22#1686420411yescharacteristic
580C22#1686420411yescharacteristic
581C22#1686420711yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—2682no—
593C4—2682no—
594C4—2682no—
595C4C4×Q8×D6711yescharacteristic
596C4C4×#1176162911yescharacteristic
597C4#1686393311yescharacteristic
598C4#1686395011yescharacteristic
599C2—2684no—
600C2C4×#1452401911yescharacteristic
601C2#2256904111yescharacteristic
602C2#2256906811yescharacteristic
603C2#2257003411yescharacteristic
604C2#2257006711yescharacteristic
605C2#2257038711yescharacteristic
606C2#2257048011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2331C2
socleC23×C67C22×D4
prime radicalC22×D268C23
cosocleC22×C4×C67C24
hypercenter#2331D67
hypocenterC67#21623
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #22561162C2semidirect
4. #22560975C2semidirect
5. #22560791C2semidirect
6. #22560784C2semidirect
10. #22560107C2semidirect
11. C4×#14523974C2semidirect
12. #22560066C2semidirect
15. C67×#2331C2semidirect
19. #16863683C4semidirect
73. C4×D268Q8semidirect
85. #14523974C2×C4semidirect
86. #14523974C2×C4semidirect
88. C4⋊(C67⋊C8)C2×C4semidirect
100. C4⋊Q8×C67C2×C4semidirect
305. #2331D67semidirect
319. C67#21623semidirect
374. C4⋊Q8C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168636832
#225625712
#225611622
#225609752
#225607912
#225607842
#225604692
#225604552
#225602162
#225601072
C4×#145239742
#225600662
#225600372
C4×#145239632
C67×#23312
#2162367