Finite Group Databasebeta

← #30855354

Subgroups — #30855354

Order 17152.

1462 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×#16863876C211yesmaximal, characteristic
2C2×#16863423C211yesmaximal, characteristic, complemented
3C2×#16863395C211yesmaximal, characteristic
4C2×#16863390C211yesmaximal, characteristic
5#22560221C212yesmaximal, complemented
6#22560216C212yesmaximal, complemented
7#22560211C212yesmaximal, complemented
8#22560206C212yesmaximal, complemented
9#22559930C211yesmaximal, characteristic, complemented
10C2×#16863158C211yesmaximal, characteristic, complemented
11C67×#2317C211yesmaximal, characteristic, complemented
12C22×#11761628C2211yescharacteristic
13C22×#11761612C2211yescharacteristic
14C23×C67⋊C8C2211yescharacteristic
15#16863876—24no—
16#16863876C418yescomplemented
17C2×C4×Dic134C2212yes—
18C2×C4×D268C2212yes—
19C2×(C22⋊D67)⋊C4—22no—
20C2×#14524001—21no—
21C22×C4⋊Dic67C2211yescharacteristic
22C2×C4⋊C4×D67—21no—
23#16863423—24no—
24C2×C4⋊(C67⋊C8)C2212yes—
25C2×C4⋊(C67⋊C8)C2212yes—
26#16863395—24no—
27#16863390—24no—
28C2×#14523966C2212yes—
29C2×#14523966C2211yescharacteristic
30C2×#14523966C2211yescharacteristic
31C2×#14523963C2212yes—
32C2×#14523963C2211yescharacteristic
33C2×#14523963C2211yescharacteristic
34#16863158C2212yes—
35#16863158C2212yes—
36#16863158C2212yes—
37#16863158C2212yes—
38C2×#117×C67C2211yescharacteristic
39C2×#117×C67C2211yescharacteristic
40C22×C4⋊C4×C67C2211yescharacteristic
41#389×C67—21no—
42#389×C67—21no—
43#385×C67—22no—
44#384×C67C2212yes—
45#382×C67C2212yes—
46C2×#11761628—22no—
47C2×#11761628—22no—
48C2×#11761628—22no—
49C2×#11761628C2×C414yes—
50C2×#11761628D414yes—
51C2×#11761628Q814yes—
52C22×Dic134C2311yescharacteristic
53C22×D268C2311yescharacteristic
54C22×C22⋊D67—21no—
55C23×Dic67—21no—
56C22×C4×D67—21no—
57C22×C4×D67—22no—
58C2×#11761612—22no—
59C2×#11761612—22no—
60C2×#11761612—22no—
61C22×C67⋊C8—22no—
62C22×C67⋊C8—22no—
63C22×C67⋊C8—22no—
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C22×C67⋊C8C2312yes—
67C4×Dic134—24no—
68C4×Dic134C2×C418yescomplemented
69C4×D268—24no—
70C4×D268C2×C418yescomplemented
71(C22⋊D67)⋊C4—216no—
72#14524001—28no—
73C2×C4⋊Dic67—21no—
74C2×C4⋊Dic67—22no—
75C2×C4⋊Dic67C2×C414yes—
76C2×C4⋊Dic67C2312yes—
77C2×Dic67⋊C4—22no—
78C4⋊C4×D67—28no—
79C2×(C2×D67)⋊C4—22no—
80C2×C4×Dic67—22no—
81C4⋊(C67⋊C8)—24no—
82C4⋊(C67⋊C8)—24no—
83#14523966—22no—
84#14523966—22no—
85#14523966—24no—
86#14523963—22no—
87#14523963—22no—
88#14523963—24no—
89C23×C4×C67—21no—
90C23×C4×C67C2311yescharacteristic
91#117×C67—22no—
92#117×C67—22no—
93#117×C67C2×C414yes—
94#117×C67D414yes—
95C2×C4⋊C4×C67—22no—
96C2×C4⋊C4×C67—22no—
97C2×C4⋊C4×C67—21no—
98C2×C4⋊C4×C67—22no—
99C2×C4⋊C4×C67—22no—
100C2×C4⋊C4×C67D411yescharacteristic
101C2×C4⋊C4×C67D411yescharacteristic
102C2×C4⋊C4×C67C2312yes—
103C2×C4⋊C4×C67C2311yescharacteristic
104C2×C4⋊C4×C67C2311yescharacteristic
105C2×C22⋊C4×C67—21no—
106C2×C22⋊C4×C67D412yes—
107C2×C42×C67—21no—
108C2×C42×C67C2312yes—
109#95×C67—22no—
110#95×C67—22no—
111C23×D67—21no—
112#11761628—28no—
113#11761628—28no—
114#11761628—28no—
115#11761628C4⋊C4116yescomplemented
116C2×Dic134—21no—
117C2×Dic134—21no—
118C2×Dic134—21no—
119C2×Dic134C22×C412yes—
120C2×Dic134C2×D412yes—
121C2×Dic134C2×Q812yes—
122C2×D268—21no—
123C2×D268—21no—
124C2×D268—21no—
125C2×D268C22×C412yes—
126C2×D268C2×D412yes—
127C2×D268C2×Q812yes—
128C2×C22⋊D67—24no—
129C2×C22⋊D67—24no—
130C2×C22⋊D67—24no—
131C22×Dic67—21no—
132C22×Dic67—22no—
133C22×Dic67—24no—
134C22×Dic67—81no—
135C22×Dic67—81no—
136C2×C4×D67—24no—
137C2×C4×D67—24no—
138C2×C4×D67—24no—
139C2×C4×D67—28no—
140C2×C4×D67—82no—
141C2×C4×D67—82no—
142#11761612—44no—
143C2×C67⋊C8—21no—
144C2×C67⋊C8—21no—
145C2×C67⋊C8—22no—
146C2×C67⋊C8—21no—
147C2×C67⋊C8—21no—
148C2×C67⋊C8—22no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—22no—
152C2×C67⋊C8—44no—
153C4⋊Dic67—24no—
154C4⋊Dic67—22no—
155C4⋊Dic67C22×C414yes—
156Dic67⋊C4—28no—
157(C2×D67)⋊C4—28no—
158C4×Dic67—28no—
159C24×C67C2×D411yescharacteristic
160C22×C4×C67—21no—
161C22×C4×C67—21no—
162C22×C4×C67—22no—
163C22×C4×C67—42no—
164C22×C4×C67—81no—
165C22×C4×C67—81no—
166C22×C4×C67C22×C412yes—
167C22×C4×C67C2×D412yes—
168C22×C4×C67C2×D412yes—
169C22×C4×C67C2×D412yes—
170C22×C4×C67C2×D411yescharacteristic
171C22×C4×C67C2×D412yes—
172C22×C4×C67C2×Q812yes—
173C22×C4×C67#4112yes—
174C22×C4×C67#4112yes—
175C22×C4×C67C2411yescharacteristic
176C4⋊C4×C67—22no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—21no—
179C4⋊C4×C67—22no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—42no—
183C4⋊C4×C67—82no—
184C4⋊C4×C67—82no—
185C4⋊C4×C67C22⋊C414yes—
186C4⋊C4×C67C22⋊C414yes—
187C22⋊C4×C67—42no—
188C22⋊C4×C67C22⋊C418yes—
189C42×C67—42no—
190C42×C67C22×C414yes—
191C42×C67C2×D414yes—
192C23×C67—21no—
193C23×C67—21no—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67C2×C22⋊C412yes—
197C23×C67#12112yes—
198C23×C67C22⋊Q812yes—
199C23×C67C22×D411yescharacteristic
200C22×D67—22no—
201C22×D67—22no—
202C22×D67—22no—
203C22×D67—41no—
204C22×D67—41no—
205C22×D67—41no—
206C22×D67—41no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—22no—
210C2×C4×C67—22no—
211C2×C4×C67—41no—
212C2×C4×C67—41no—
213C2×C4×C67—42no—
214C2×C4×C67—41no—
215C2×C4×C67—42no—
216C2×C4×C67—41no—
217C2×C4×C67—42no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—84no—
221C2×C4×C67C2×C22⋊C414yes—
222C2×C4×C67C2×C22⋊C414yes—
223C2×C4×C67C2×C22⋊C412yes—
224C2×C4×C67C2×C4⋊C414yes—
225C2×C4×C67#11714yes—
226C2×C4×C67#12112yes—
227C2×C4×C67#12114yes—
228C2×C4×C67C22⋊Q812yes—
229C2×C4×C67C22⋊Q814yes—
230C2×C4×C67C23×C411yescharacteristic
231C2×C4×C67C22×D411yescharacteristic
232C2×C4×C67C22×D411yescharacteristic
233C2×C4×C67C22×Q811yescharacteristic
234C2×C4×C67C2×#4111yescharacteristic
235C2×C4×C67C2×#4111yescharacteristic
236C22⋊D67—216no—
237C2×Dic67—22no—
238C2×Dic67—22no—
239C2×Dic67—22no—
240C2×Dic67—24no—
241C2×Dic67—42no—
242C2×Dic67—42no—
243C2×Dic67—84no—
244D268—22no—
245D268—22no—
246D268—22no—
247D268C2×C4⋊C414yes—
248C4×D67—216no—
249C4×D67—88no—
250Dic134—22no—
251Dic134—22no—
252Dic134—22no—
253Dic134C2×C4⋊C414yes—
254C67⋊C8—41no—
255C67⋊C8—41no—
256C67⋊C8—42no—
257C22×C67—22no—
258C22×C67—22no—
259C22×C67—22no—
260C22×C67—22no—
261C22×C67—22no—
262C22×C67—22no—
263C22×C67#38014yes—
264C22×C67C22×C22⋊C411yescharacteristic
265C22×C67C2×#12111yescharacteristic
266C22×C67C2×C22⋊Q811yescharacteristic
267C22×C67#57611yescharacteristic
268C22×C67#57711yescharacteristic
269C22×C67#57711yescharacteristic
270C22×C67#57811yescharacteristic
271C2×D67—24no—
272C2×D67—42no—
273C2×D67—42no—
274C2×D67—42no—
275C2×D67—42no—
276C2×D67—42no—
277C2×D67—42no—
278C4×C67—24no—
279C4×C67—24no—
280C4×C67—81no—
281C4×C67—82no—
282C4×C67—81no—
283C4×C67#38014yes—
284C4×C67C22×C4⋊C411yescharacteristic
285C4×C67C2×#11711yescharacteristic
286C4×C67C2×#12111yescharacteristic
287C4×C67C2×C22⋊Q811yescharacteristic
288Dic67—24no—
289Dic67—44no—
290#20652—671nomaximal, Sylow
291C2×C67—24no—
292C2×C67C2×#38011yescharacteristic
293C2×C67#289711yescharacteristic
294C2×C67#289811yescharacteristic
295C2×C67#304511yescharacteristic
296C2×C67#304611yescharacteristic
297C2×C67#308311yescharacteristic
298C2×C67#308411yescharacteristic
299D67—44no—
300C2×#520—671no—
301C2×#424—671no—
302C2×#417—671no—
303C2×#416—671no—
304#2317D6711yescharacteristic, complemented
305#1805—672no—
306#1804—672no—
307#1803—672no—
308#1802—672no—
309#1736—671no—
310C2×#340—671no—
311C67#2065211yescharacteristic, complemented, Sylow
312C22×#41—671no—
313C22×M16—671no—
314C23×C8—671no—
315#520—678no—
316#520—1344no—
317C2×C4×Q8—672no—
318C2×C4×D4—672no—
319C2×C4×D4—1342no—
320C2×#117—1341no—
321C2×#117C2×D6711yescharacteristic
322C2×#117C2×D6711yescharacteristic
323C22×C4⋊C4—671no—
324C22×C4⋊C4—1341no—
325C22×C4⋊C4C2×D6711yescharacteristic
326#424—1344no—
327C2×C4⋊C8—672no—
328C2×C4⋊C8—672no—
329#417—1344no—
330#416—1344no—
331C2×#103—671no—
332C2×#103—672no—
333C2×#103—671no—
334C2×#102—671no—
335C2×#102—672no—
336C2×#102—671no—
337#389—21no—
338#389—21no—
339#385—22no—
340#384C2×D6712yes—
341#382C2×D6712yes—
342#340—672no—
343#340—672no—
344#340—672no—
345#340—672no—
346C2×#41—674no—
347C2×#41—674no—
348C2×#41—674no—
349C2×#41—1342no—
350C2×#41—1342no—
351C2×#41—1342no—
352C22×Q8—671no—
353C22×D4—671no—
354C22×D4—1341no—
355C23×C4—21no—
356C23×C4—1341no—
357C23×C4—1342no—
358C23×C4—1341no—
359C23×C4C22×D6711yescharacteristic
360C2×M16—1342no—
361C2×M16—1342no—
362C2×M16—1342no—
363C22×C8—671no—
364C22×C8—671no—
365C22×C8—672no—
366C22×C8—1342no—
367C22×C8—1342no—
368C22×C8—1342no—
369C4×Q8—678no—
370C4×Q8—1344no—
371C4×D4—678no—
372C4×D4—13416no—
373C4×D4—1344no—
374#117—22no—
375#117—22no—
376#117—1348no—
377#117C4×D6714yes—
378#117D26814yes—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—674no—
385C2×C4⋊C4—672no—
386C2×C4⋊C4—1341no—
387C2×C4⋊C4—1342no—
388C2×C4⋊C4—1342no—
389C2×C4⋊C4—1348no—
390C2×C4⋊C4C22⋊D6711yescharacteristic
391C2×C4⋊C4C22⋊D6711yescharacteristic
392C2×C4⋊C4C22×D6712yes—
393C2×C4⋊C4C22×D6711yescharacteristic
394C2×C4⋊C4C22×D6711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1342no—
397C2×C22⋊C4C22⋊D6712yes—
398C2×C42—21no—
399C2×C42—1342no—
400C2×C42C22×D6712yes—
401C4⋊C8—1344no—
402C4⋊C8—1344no—
403#103—1342no—
404#103—1342no—
405#103—1344no—
406#102—1342no—
407#102—1342no—
408#102—1344no—
409#95—22no—
410#95—22no—
411C24—1341no—
412C24C2×C22⋊D6711yescharacteristic
413#41—6716no—
414#41—1348no—
415#41—1348no—
416#41—1348no—
417C2×Q8—672no—
418C2×Q8—672no—
419C2×Q8—672no—
420C2×Q8—1341no—
421C2×Q8—1341no—
422C2×Q8—1341no—
423C2×D4—672no—
424C2×D4—672no—
425C2×D4—672no—
426C2×D4—1344no—
427C2×D4—1344no—
428C2×D4—1344no—
429C2×D4—1341no—
430C2×D4—1341no—
431C2×D4—1341no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—5362no—
439C22×C4—5362no—
440C22×C4—5361no—
441C22×C4—5361no—
442C22×C4—1342no—
443C22×C4—1344no—
444C22×C4—1348no—
445C22×C4—1341no—
446C22×C4—1344no—
447C22×C4—1344no—
448C22×C4—1344no—
449C22×C4C2×C4×D6712yes—
450C22×C4D4×D6712yes—
451C22×C4C2×C22⋊D6712yes—
452C22×C4C2×C22⋊D6712yes—
453C22×C4C2×C22⋊D6711yescharacteristic
454C22×C4C2×D26812yes—
455C22×C4Q8×D6712yes—
456C22×C4#1176162712yes—
457C22×C4#1176162912yes—
458C22×C4C23×D6711yescharacteristic
459M16—2684no—
460C2×C8—2684no—
461C2×C8—1341no—
462C2×C8—1341no—
463C2×C8—1342no—
464C2×C8—1341no—
465C2×C8—1341no—
466C2×C8—1342no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8—1342no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—674no—
480C4⋊C4—1344no—
481C4⋊C4—1348no—
482C4⋊C4—1342no—
483C4⋊C4(C2×D67)⋊C414yes—
484C4⋊C4(C2×D67)⋊C414yes—
485C22⋊C4—42no—
486C22⋊C4—1348no—
487C22⋊C4(C2×D67)⋊C418yes—
488C42—42no—
489C42—1348no—
490C42C2×C4×D6714yes—
491C42C2×D26814yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—2681no—
497C23—2681no—
498C23—2681no—
499C23—2681no—
500C23—1342no—
501C23—1342no—
502C23—1342no—
503C23C2×(C2×D67)⋊C412yes—
504C23#1452401412yes—
505C23#1452401912yes—
506C23C22×C22⋊D6711yescharacteristic
507Q8—674no—
508Q8—1342no—
509Q8—1342no—
510Q8—1342no—
511D4—674no—
512D4—13416no—
513D4—1342no—
514D4—1342no—
515D4—1342no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—2682no—
531C2×C4—2682no—
532C2×C4—5368no—
533C2×C4—5364no—
534C2×C4—1344no—
535C2×C4—1342no—
536C2×C4—1342no—
537C2×C4—1342no—
538C2×C4—13416no—
539C2×C4C2×(C2×D67)⋊C414yes—
540C2×C4C2×(C2×D67)⋊C414yes—
541C2×C4C2×(C2×D67)⋊C412yes—
542C2×C4C4⋊C4×D6714yes—
543C2×C4#1452400114yes—
544C2×C4#1452401412yes—
545C2×C4#1452401814yes—
546C2×C4#1452401912yes—
547C2×C4#1452402314yes—
548C2×C4C22×C4×D6711yescharacteristic
549C2×C4C2×D4×D6711yescharacteristic
550C2×C4C22×D26811yescharacteristic
551C2×C4C2×Q8×D6711yescharacteristic
552C2×C4C2×#1176162711yescharacteristic
553C2×C4C2×#1176162911yescharacteristic
554C8—2681no—
555C8—2681no—
556C8—2682no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—2682no—
564C22—2682no—
565C22—2682no—
566C22—2682no—
567C22—2682no—
568C22—2682no—
569C22—1344no—
570C22#1686325314yes—
571C22C22×(C2×D67)⋊C411yescharacteristic
572C22C2×#1452401411yescharacteristic
573C22C2×#1452401911yescharacteristic
574C22#1686419911yescharacteristic
575C22#1686420411yescharacteristic
576C22#1686420711yescharacteristic
577C22#1686421111yescharacteristic
578C4—24no—
579C4—24no—
580C4—82no—
581C4—81no—
582C4—81no—
583C4—2684no—
584C4—1344no—
585C4#1686325314yes—
586C4C2×C4⋊C4×D6711yescharacteristic
587C4C2×#1452400111yescharacteristic
588C4C2×#1452401811yescharacteristic
589C4C2×#1452402311yescharacteristic
590C2—24no—
591C2—2684no—
592C2C2×#1686325311yescharacteristic
593C2#2256873211yescharacteristic
594C2#2256873911yescharacteristic
595C2#2256974411yescharacteristic
596C2#2256975511yescharacteristic
597C2#2257005611yescharacteristic
598C2#2257006711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2317C2
socleC23×C67C22×D4
prime radicalC22×#11761628C22
cosocleC22×C4×C67C24
hypercenter#2317D67
hypocenterC67#20652
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#16863423C2semidirect
5. #22560221C2semidirect
6. #22560216C2semidirect
7. #22560211C2semidirect
8. #22560206C2semidirect
9. #22559930C2semidirect
10. C2×#16863158C2semidirect
11. C67×#2317C2semidirect
16. #16863876C4semidirect
68. C4×Dic134C2×C4semidirect
70. C4×D268C2×C4semidirect
115. #11761628C4⋊C4semidirect
304. #2317D67semidirect
311. C67#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#168638762
C2×#168634232
C2×#168633952
C2×#168633902
#225602212
#225602162
#225602112
#225602062
#225599302
C2×#168631582
C67×#23172
#2065267