Finite Group Databasebeta

← #30827163

Subgroups — #30827163

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570218C211yesmaximal, characteristic
2#22569310C211yesmaximal, characteristic, complemented
3#22569246C211yesmaximal, characteristic
4#22559110C211yesmaximal, characteristic, complemented
5#22559098C211yesmaximal, characteristic, complemented
6#22559086C211yesmaximal, characteristic, complemented
7#22559065C211yesmaximal, characteristic, complemented
8#22559051C211yesmaximal, characteristic, complemented
9#22559044C211yesmaximal, characteristic, complemented
10#22559020C211yesmaximal, characteristic, complemented
11#22559008C211yesmaximal, characteristic, complemented
12#22558681C211yesmaximal, characteristic, complemented
13#22558599C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic, complemented
15C67×#1572C211yesmaximal, characteristic, complemented
16#16863953C2211yescharacteristic
17C2×C4×D268C2211yescharacteristic
18#16863727C411yescharacteristic
19#16863727C411yescharacteristic
20#16863687C411yescharacteristic
21#16863687C411yescharacteristic
22#16863602C411yescharacteristic
23#16863602C411yescharacteristic
24#16863549C411yescharacteristic
25#16863549C411yescharacteristic
26#16863526—21no—
27#16863523C2211yescharacteristic, complemented
28#16863523C2211yescharacteristic, complemented
29C8×Dic134—21no—
30#16863520C2211yescharacteristic, complemented
31#16863520C2211yescharacteristic, complemented
32C4×#11761617C2211yescharacteristic
33C4×#11761617C2211yescharacteristic
34C4×#11761617C2211yescharacteristic
35C4×#11761617C2211yescharacteristic
36#16863473C2211yescharacteristic
37#16863473C2211yescharacteristic
38#16863469—21no—
39#16863469—21no—
40#16863461C2211yescharacteristic
41#16863461C2211yescharacteristic
42#16863454—21no—
43#16863454—21no—
44#16863449C2211yescharacteristic
45#16863449C2211yescharacteristic
46#16863358—21no—
47C22⋊C8×D67—21no—
48#16863126C2211yescharacteristic, complemented
49#16863126C2211yescharacteristic, complemented
50#16863126C2211yescharacteristic, complemented
51#16863126C2211yescharacteristic, complemented
52#16863123C2211yescharacteristic
53#16863123C2211yescharacteristic
54#16863123C2211yescharacteristic
55#16863123C2211yescharacteristic
56#16863118C2211yescharacteristic, complemented
57#16863118C2211yescharacteristic, complemented
58#16863117C2211yescharacteristic, complemented
59C8×C67⋊C8C2211yescharacteristic, complemented
60#432×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63#335×C67C2211yescharacteristic, complemented
64#335×C67C2211yescharacteristic, complemented
65#334×C67C2211yescharacteristic, complemented
66#334×C67C2211yescharacteristic, complemented
67C22×D268C2×C411yescharacteristic
68C22×C4×D67—21no—
69C2×#11761617C2×C411yescharacteristic
70C2×#11761617C2×C411yescharacteristic
71C2×#11761617C2×C411yescharacteristic
72C2×#11761617C2×C411yescharacteristic
73C2×#11761611—21no—
74C2×#11761611—21no—
75C2×C8×D67—21no—
76C2×C8×D67—21no—
77#14524043C2×C411yescharacteristic
78#14524043C2×C411yescharacteristic
79#14524035—21no—
80#14524022—21no—
81#14524022C2×C411yescharacteristic
82#14524022C2×C411yescharacteristic
83C4×Dic134—21no—
84C4×Dic134C2311yescharacteristic
85C4×Dic134C2311yescharacteristic
86C4×D268—21no—
87C4×D268—21no—
88C4×D268C2×C411yescharacteristic
89C4×D268C2×C411yescharacteristic
90C4×D268C2311yescharacteristic
91C4×D268C2311yescharacteristic
92#14524002—21no—
93C2×C4⋊Dic67C2×C411yescharacteristic
94C2×(C2×D67)⋊C4—21no—
95#14523975C2×C411yescharacteristic
96#14523975C2×C411yescharacteristic
97#14523975C2×C411yescharacteristic
98#14523975C2×C411yescharacteristic
99C4⋊(C67⋊C8)—21no—
100C4⋊(C67⋊C8)D411yescharacteristic
101C4⋊(C67⋊C8)D411yescharacteristic
102Dic67⋊C8—21no—
103Dic67⋊C8—21no—
104Dic67⋊C8—21no—
105Dic67⋊C8—21no—
106#14523968C2×C411yescharacteristic
107#14523968C2×C411yescharacteristic
108#14523968C2×C411yescharacteristic
109#14523968C2×C411yescharacteristic
110#14523965C2×C411yescharacteristic
111#14523965C2×C411yescharacteristic
112#14523965C2×C411yescharacteristic
113#14523965C2×C411yescharacteristic
114C22⋊(C67⋊C8)—21no—
115C22⋊(C67⋊C8)D411yescharacteristic
116C22⋊(C67⋊C8)D411yescharacteristic
117(C2×D67)⋊C8—21no—
118(C2×D67)⋊C8—21no—
119(C2×D67)⋊C8—21no—
120(C2×D67)⋊C8—21no—
121#14523953—21no—
122#14523953—21no—
123C4×C67⋊C8C2311yescharacteristic
124C4×C67⋊C8C2311yescharacteristic
125C4×C67⋊C8C2311yescharacteristic
126C4×C67⋊C8C2311yescharacteristic
127C8×Dic67—21no—
128C8×Dic67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67C2×C411yescharacteristic
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C4⋊C8×C67C2311yescharacteristic
135C4⋊C8×C67C2311yescharacteristic
136C22⋊C8×C67—21no—
137C22⋊C8×C67—21no—
138C22⋊C8×C67C2×C411yescharacteristic
139C22⋊C8×C67C2×C411yescharacteristic
140#97×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C23×D67—21no—
145C2×Dic134—21no—
146C2×Dic134C22×C411yescharacteristic
147C2×Dic134C22×C411yescharacteristic
148C2×D268—22no—
149C2×D268—21no—
150C2×D268—41no—
151C2×D268C22×C411yescharacteristic
152C2×D268C22×C411yescharacteristic
153C22×Dic67—21no—
154C2×C4×D67—21no—
155C2×C4×D67—21no—
156C2×C4×D67—21no—
157C2×C4×D67—21no—
158C2×C4×D67—81no—
159C2×C4×D67—81no—
160#11761617—22no—
161#11761617—22no—
162#11761617—41no—
163#11761617—41no—
164#11761611—81no—
165#11761611—81no—
166C2×C67⋊C8—21no—
167C2×C67⋊C8—21no—
168C2×C67⋊C8C2×D411yescharacteristic
169C2×C67⋊C8C2×D411yescharacteristic
170C2×C67⋊C8C2×D411yescharacteristic
171C2×C67⋊C8C2×D411yescharacteristic
172C8×D67—81no—
173C8×D67—81no—
174C4⋊Dic67—21no—
175C4⋊Dic67—21no—
176C4⋊Dic67—21no—
177C4⋊Dic67—21no—
178C4⋊Dic67C22×C411yescharacteristic
179C4⋊Dic67C22×C411yescharacteristic
180C4⋊Dic67C22×C411yescharacteristic
181C4⋊Dic67C22×C411yescharacteristic
182Dic67⋊C4—21no—
183Dic67⋊C4—21no—
184Dic67⋊C4—21no—
185Dic67⋊C4—21no—
186C22⋊Dic67—21no—
187C22⋊Dic67—21no—
188(C2×D67)⋊C4—21no—
189(C2×D67)⋊C4—21no—
190(C2×D67)⋊C4—21no—
191(C2×D67)⋊C4—21no—
192C4×Dic67—21no—
193C4×Dic67—21no—
194C22×C4×C67C22×C411yescharacteristic
195C22×C4×C67C2×D411yescharacteristic
196C22×C4×C67#4111yescharacteristic
197C2×C8×C67—21no—
198C2×C8×C67—21no—
199C2×C8×C67—21no—
200C2×C8×C67—21no—
201C2×C8×C67C22×C411yescharacteristic
202C2×C8×C67C22×C411yescharacteristic
203C2×C8×C67C22×C411yescharacteristic
204C2×C8×C67C22×C411yescharacteristic
205C42×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67C4×D411yescharacteristic
210C22×D67—21no—
211C22×D67—21no—
212C22×D67—42no—
213C22×D67—42no—
214C22×D67—81no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—41no—
220C2×C4×C67C4×D411yescharacteristic
221C2×C4×C67C4×D411yescharacteristic
222C2×C4×C67C4×D411yescharacteristic
223C2×C4×C67C23×C411yescharacteristic
224C2×C4×C67C22×D411yescharacteristic
225C2×C4×C67C2×#4111yescharacteristic
226C2×Dic67—21no—
227C2×Dic67—21no—
228C2×Dic67—21no—
229C2×Dic67—21no—
230C2×Dic67—21no—
231C2×Dic67—21no—
232C2×Dic67—81no—
233D268—21no—
234D268—21no—
235D268—21no—
236D268—42no—
237D268#13112yes—
238C4×D67—41no—
239C4×D67—41no—
240C4×D67—41no—
241C4×D67—41no—
242C4×D67—81no—
243C4×D67—81no—
244Dic134—21no—
245Dic134—21no—
246Dic134—21no—
247Dic134—21no—
248Dic134—42no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—41no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C67⋊C8—21no—
258C67⋊C8—21no—
259C67⋊C8—21no—
260C67⋊C8—21no—
261C67⋊C8—41no—
262C67⋊C8—41no—
263C22×C67—21no—
264C22×C67—21no—
265C22×C67—21no—
266C22×C67C2×C4×D411yescharacteristic
267C2×D67—22no—
268C2×D67—41no—
269C2×D67—41no—
270C2×D67—41no—
271C2×D67—41no—
272C2×D67—41no—
273C2×D67—84no—
274C4×C67—21no—
275C4×C67—21no—
276C4×C67—21no—
277C4×C67—21no—
278C4×C67C2×#13111yescharacteristic
279C4×C67#56811yescharacteristic
280C4×C67#57711yescharacteristic
281C4×C67#57711yescharacteristic
282Dic67—41no—
283Dic67—41no—
284Dic67—41no—
285Dic67—41no—
286Dic67—41no—
287Dic67—41no—
288#18028—671nomaximal, Sylow
289C2×C67—41no—
290C2×C67#293311yescharacteristic
291C2×C67#294911yescharacteristic
292C2×C67#295511yescharacteristic
293D67—42no—
294D67—81no—
295#3101—671no—
296#2986—671no—
297#2974—671no—
298#1592—671no—
299#1590—671no—
300#1588—671no—
301#1585—671no—
302#1583—671no—
303#1582—671no—
304SD16⋊C8—671no—
305C8×SD16—671no—
306#1572D6711yescharacteristic, complemented
307#1497—671no—
308#1478—671no—
309#1454—671no—
310C67#1802811yescharacteristic, complemented, Sylow
311#533—671no—
312C2×C4×D4—671no—
313#483—671no—
314#483—671no—
315#476—671no—
316#476—671no—
317#461—671no—
318#461—671no—
319#450—671no—
320#450—671no—
321#446—671no—
322#446—671no—
323#446—1341no—
324C8×Q8—671no—
325C8×Q8—671no—
326C8×Q8—1341no—
327C4×SD16—671no—
328C4×SD16—671no—
329C4×SD16—671no—
330C4×SD16—671no—
331#436—671no—
332#436—671no—
333#435—1341no—
334#435—1341no—
335C8×D4—671no—
336C8×D4—671no—
337#433—1341no—
338#433—1341no—
339#432C2×D6711yescharacteristic
340#431—671no—
341#431—671no—
342#431C2×D6711yescharacteristic
343#431C2×D6711yescharacteristic
344#407—1341no—
345C2×C22⋊C8—1341no—
346#335C2×D6711yescharacteristic, complemented
347#335C2×D6711yescharacteristic, complemented
348#334C2×D6711yescharacteristic, complemented
349#334C2×D6711yescharacteristic, complemented
350Q8⋊C8—671no—
351Q8⋊C8—671no—
352Q8⋊C8—671no—
353Q8⋊C8—671no—
354D4⋊C8—671no—
355D4⋊C8—671no—
356D4⋊C8—671no—
357D4⋊C8—671no—
358#322—671no—
359#322—671no—
360#322—671no—
361C82—671no—
362C22×D4—671no—
363C23×C4—1341no—
364C2×SD16—671no—
365C2×SD16—671no—
366C2×SD16—671no—
367C2×SD16—671no—
368C2×M16—1341no—
369C2×M16—1341no—
370C22×C8—1341no—
371C22×C8—1341no—
372C4⋊Q8—671no—
373C4⋊Q8—671no—
374#125—1341no—
375C22⋊Q8—671no—
376C22⋊Q8—671no—
377C22⋊Q8—1341no—
378C4×Q8—671no—
379C4×Q8—671no—
380C4×Q8—1341no—
381C4×D4—671no—
382C4×D4—671no—
383C4×D4—671no—
384C4×D4—671no—
385C4×D4—1341no—
386C4×D4—1341no—
387#117—1341no—
388C2×C4⋊C4—671no—
389C2×C22⋊C4—1341no—
390C2×C42C22×D6711yescharacteristic
391#106—671no—
392#106—671no—
393#106—671no—
394#106—671no—
395C4⋊C8—671no—
396C4⋊C8—671no—
397C4⋊C8—1341no—
398C4⋊C8—1341no—
399C4⋊C8—1341no—
400C4⋊C8—1341no—
401C4⋊C8—1341no—
402C4⋊C8C4×D6711yescharacteristic
403C4⋊C8C4×D6711yescharacteristic
404C4⋊C8C22×D6711yescharacteristic
405C4⋊C8C22×D6711yescharacteristic
406C4⋊C8C22×D6711yescharacteristic
407C4⋊C8C22×D6711yescharacteristic
408#103—671no—
409#103—671no—
410#103—671no—
411#103—671no—
412#102—671no—
413#102—671no—
414#102—671no—
415#102—671no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—671no—
419C22⋊C8—671no—
420C22⋊C8—1341no—
421C22⋊C8—1341no—
422C22⋊C8—1341no—
423C22⋊C8—1341no—
424C22⋊C8—1341no—
425C22⋊C8C4×D6711yescharacteristic
426C22⋊C8C4×D6711yescharacteristic
427#97—21no—
428#97—1341no—
429#97—1341no—
430C4×C8—21no—
431C4×C8—671no—
432C4×C8—671no—
433C4×C8—671no—
434C4×C8—671no—
435C4×C8—1341no—
436C4×C8—1341no—
437C4×C8C22×D6711yescharacteristic
438C4×C8C22×D6711yescharacteristic
439C24—1341no—
440C2×Q8—671no—
441C2×Q8—671no—
442C2×Q8—1341no—
443C2×D4—2681no—
444C2×D4—671no—
445C2×D4—671no—
446C2×D4—1342no—
447C2×D4—1341no—
448C22×C4—5361no—
449C22×C4—5361no—
450C22×C4—1341no—
451C22×C4—1341no—
452C22×C4—1341no—
453C22×C4—1341no—
454C22×C4—1341no—
455C22×C4C2×C4×D6711yescharacteristic
456C22×C4D4×D6711yescharacteristic
457C22×C4#1176162911yescharacteristic
458SD16—2681no—
459SD16—2681no—
460SD16—1342no—
461SD16—1342no—
462M16—5361no—
463M16—5361no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—5361no—
469C2×C8—5361no—
470C2×C8—671no—
471C2×C8—671no—
472C2×C8—671no—
473C2×C8—671no—
474C2×C8—1341no—
475C2×C8—1341no—
476C2×C8C2×C4×D6711yescharacteristic
477C2×C8C2×C4×D6711yescharacteristic
478C2×C8C2×C4×D6711yescharacteristic
479C2×C8C2×C4×D6711yescharacteristic
480C4⋊C4—671no—
481C4⋊C4—671no—
482C4⋊C4—671no—
483C4⋊C4—671no—
484C4⋊C4—1341no—
485C4⋊C4—1341no—
486C4⋊C4—1341no—
487C4⋊C4—1341no—
488C4⋊C4—1341no—
489C4⋊C4—1341no—
490C4⋊C4—1341no—
491C4⋊C4—1341no—
492C22⋊C4—1341no—
493C22⋊C4—1341no—
494C22⋊C4—1341no—
495C22⋊C4—1341no—
496C22⋊C4—1341no—
497C22⋊C4—1341no—
498C42—1341no—
499C42—1341no—
500C42C2×C4×D6711yescharacteristic
501C42D4×D6711yescharacteristic
502C42#1176162911yescharacteristic
503C42C23×D6711yescharacteristic
504C23—2682no—
505C23—2682no—
506C23—5361no—
507C23—1341no—
508C23—1341no—
509C23#1452400511yescharacteristic
510Q8—2682no—
511Q8—1341no—
512Q8—1341no—
513Q8—1341no—
514Q8—1341no—
515D4—2682no—
516D4—672no—
517D4—1341no—
518D4—1341no—
519D4—1341no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—2681no—
526C2×C4—2681no—
527C2×C4—2681no—
528C2×C4—2681no—
529C2×C4—5361no—
530C2×C4—5361no—
531C2×C4—5361no—
532C2×C4—1341no—
533C2×C4—1341no—
534C2×C4—1341no—
535C2×C4—1341no—
536C2×C4—1341no—
537C2×C4—1341no—
538C2×C4#1452400511yescharacteristic
539C2×C4#1452400511yescharacteristic
540C2×C4#1452400511yescharacteristic
541C2×C4C22×C4×D6711yescharacteristic
542C2×C4C2×D4×D6711yescharacteristic
543C2×C4C2×#1176162911yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—2681no—
553C8—2681no—
554C8—1341no—
555C8—1341no—
556C8—1341no—
557C8—1341no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—2681no—
562C22—2681no—
563C22—2681no—
564C22—2681no—
565C22—2681no—
566C22—5364no—
567C22—1342no—
568C22C2×#1452400511yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—2681no—
574C4—2681no—
575C4—2681no—
576C4—2681no—
577C4—2681no—
578C4—2681no—
579C4#131×D6711yescharacteristic
580C4#1686416111yescharacteristic
581C4#1686420311yescharacteristic
582C4#1686420311yescharacteristic
583C2—41no—
584C2—2682no—
585C2—5361no—
586C2#2256893311yescharacteristic
587C2#2256906711yescharacteristic
588C2#2256911111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1572C2
socleC22×C67C2×C4×D4
prime radicalC22×D268C2×C4
cosocleC42×C67C24
hypercenter#1572D67
hypocenterC67#18028
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #22569310C2semidirect
4. #22559110C2semidirect
5. #22559098C2semidirect
6. #22559086C2semidirect
7. #22559065C2semidirect
8. #22559051C2semidirect
9. #22559044C2semidirect
10. #22559020C2semidirect
11. #22559008C2semidirect
12. #22558681C2semidirect
14. #22558509C2semidirect
15. C67×#1572C2semidirect
27. #16863523C22semidirect
28. #16863523C22semidirect
30. #16863520C22semidirect
31. #16863520C22semidirect
48. #16863126C22semidirect
49. #16863126C22semidirect
50. #16863126C22semidirect
51. #16863126C22semidirect
56. #16863118C22semidirect
57. #16863118C22semidirect
58. #16863117C22semidirect
59. C8×C67⋊C8C22semidirect
63. #335×C67C22semidirect
64. #335×C67C22semidirect
65. #334×C67C22semidirect
66. #334×C67C22semidirect
306. #1572D67semidirect
310. C67#18028semidirect
346. #335C2×D67semidirect
347. #335C2×D67semidirect
348. #334C2×D67semidirect
349. #334C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225702182
#225693102
#225692462
#225591102
#225590982
#225590862
#225590652
#225590512
#225590442
#225590202
#225590082
#225586812
#225585992
#225585092
C67×#15722
#1802867