Finite Group Databasebeta

← #30877855

Subgroups — #30877855

Order 17152.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863676C211yesmaximal, characteristic
2#22563298C211yesmaximal, characteristic, complemented
3#22561181C211yesmaximal, characteristic, complemented
4#22560975C211yesmaximal, characteristic, complemented
5#22560944C211yesmaximal, characteristic, complemented
6#22560939C211yesmaximal, characteristic, complemented
7#22560798C211yesmaximal, characteristic, complemented
8#22560777C211yesmaximal, characteristic, complemented
9#22560470C211yesmaximal, characteristic
10#22560456C211yesmaximal, characteristic
11#22560084C211yesmaximal, characteristic, complemented
12#22560034C211yesmaximal, characteristic
13#22560034C211yesmaximal, characteristic
14C4×#14523964C211yesmaximal, characteristic
15C67×#2349C211yesmaximal, characteristic
16C2×#14524043C2211yescharacteristic
17#16863676C418yescomplemented
18C2×#14523976C2211yescharacteristic
19C2×#14523976C2211yescharacteristic
20C2×#14523976C2211yescharacteristic
21C2×#14523974C2211yescharacteristic
22C2×C4⋊(C67⋊C8)C2211yescharacteristic
23C2×#14523964C2211yescharacteristic
24C2×#14523964C2211yescharacteristic
25C2×#14523964C2211yescharacteristic
26C2×#14523964C2211yescharacteristic
27C2×#14523954C2211yescharacteristic
28C2×#14523954C2211yescharacteristic
29C2×C4×C67⋊C8C2211yescharacteristic
30#16863300—21no—
31#16863300—21no—
32#16863294C2211yescharacteristic
33#16863294C2211yescharacteristic
34#16863292—21no—
35C4⋊C4×Dic67—21no—
36C4×C4⋊Dic67C2211yescharacteristic
37C4×C4⋊Dic67C2211yescharacteristic
38C4×C4⋊Dic67C2211yescharacteristic
39#16863158C2211yescharacteristic
40#16863158C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863151C2211yescharacteristic
48#16863151C2211yescharacteristic
49C2×C4×D4×C67C2211yescharacteristic
50C2×C4×D4×C67C2211yescharacteristic
51C2×C4×D4×C67C2211yescharacteristic
52#389×C67C2211yescharacteristic
53#389×C67C2211yescharacteristic
54#385×C67—21no—
55#385×C67—21no—
56#382×C67C2211yescharacteristic
57#380×C67—21no—
58C4×C4⋊C4×C67C2211yescharacteristic
59C2×#95×C67—21no—
60C22×Dic134—21no—
61C22×C67⋊C8C2311yescharacteristic
62C22×C67⋊C8C2311yescharacteristic
63C22×C67⋊C8C2311yescharacteristic
64C22×C67⋊C8C2311yescharacteristic
65#14524043—22no—
66#14524043C2×C414yescomplemented
67C2×C4⋊Dic67—21no—
68C2×C4⋊Dic67C2311yescharacteristic
69C2×C4⋊Dic67C2311yescharacteristic
70C2×C4⋊Dic67C2311yescharacteristic
71C2×C4⋊Dic67C2311yescharacteristic
72C2×Dic67⋊C4—21no—
73C2×Dic67⋊C4—21no—
74C2×C4×Dic67—21no—
75C2×C4×Dic67—21no—
76C2×C4×Dic67—21no—
77#14523976—41no—
78#14523976C2×C414yescomplemented
79#14523976C2×C414yescomplemented
80#14523974—41no—
81C4⋊(C67⋊C8)C2×C414yescomplemented
82#14523964—22no—
83#14523964C2×C414yes—
84#14523964C2×C414yes—
85#14523964Q814yescomplemented
86#14523954—22no—
87#14523954—41no—
88C4×C67⋊C8—41no—
89C67⋊#95—21no—
90C67⋊#95—21no—
91C67⋊#95—21no—
92C67⋊#95—21no—
93C67⋊#95—21no—
94C22×D4×C67C2311yescharacteristic
95C23×C4×C67—21no—
96C23×C4×C67—21no—
97C23×C4×C67—21no—
98C4×D4×C67—22no—
99C4×D4×C67—22no—
100C4×D4×C67—22no—
101C4×D4×C67—41no—
102C4×D4×C67—41no—
103C4×D4×C67C2×C414yes—
104C2×C4⋊C4×C67—21no—
105C2×C4⋊C4×C67—21no—
106C2×C4⋊C4×C67D411yescharacteristic
107C2×C4⋊C4×C67D411yescharacteristic
108C2×C4⋊C4×C67C2311yescharacteristic
109C2×C4⋊C4×C67C2311yescharacteristic
110C2×C4⋊C4×C67C2311yescharacteristic
111C2×C22⋊C4×C67—21no—
112C2×C22⋊C4×C67—21no—
113C2×C22⋊C4×C67—21no—
114C2×C42×C67C2311yescharacteristic
115C2×C42×C67C2311yescharacteristic
116C2×C42×C67C2311yescharacteristic
117#95×C67—21no—
118#95×C67—21no—
119#95×C67—21no—
120#95×C67—21no—
121#95×C67D411yescharacteristic
122#95×C67D411yescharacteristic
123C2×Dic134—24no—
124C2×Dic134—81no—
125C2×Dic134—81no—
126C22×Dic67—21no—
127C22×Dic67—21no—
128C22×Dic67—21no—
129C22×Dic67—21no—
130C2×C67⋊C8—21no—
131C2×C67⋊C8—21no—
132C2×C67⋊C8—21no—
133C2×C67⋊C8—21no—
134C2×C67⋊C8—21no—
135C2×C67⋊C8—21no—
136C2×C67⋊C8—21no—
137C2×C67⋊C8—21no—
138C2×C67⋊C8—21no—
139C2×C67⋊C8C22×C412yes—
140C2×C67⋊C8C22×C412yes—
141C2×C67⋊C8C2×Q812yes—
142C4⋊Dic67—24no—
143C4⋊Dic67—21no—
144C4⋊Dic67—21no—
145C4⋊Dic67—21no—
146C4⋊Dic67—21no—
147C4⋊Dic67—21no—
148C4⋊Dic67C22×C412yes—
149C4⋊Dic67C22×C412yes—
150C4⋊Dic67C2×Q812yes—
151Dic67⋊C4—81no—
152Dic67⋊C4—81no—
153C4×Dic67—42no—
154C4×Dic67—81no—
155C4×Dic67—81no—
156C24×C67—21no—
157C2×D4×C67—21no—
158C2×D4×C67—21no—
159C2×D4×C67—21no—
160C2×D4×C67C22×C412yes—
161C2×D4×C67C2×Q812yes—
162C2×D4×C67#4112yes—
163C22×C4×C67—21no—
164C22×C4×C67—24no—
165C22×C4×C67—21no—
166C22×C4×C67—21no—
167C22×C4×C67—42no—
168C22×C4×C67—42no—
169C22×C4×C67—42no—
170C22×C4×C67—42no—
171C22×C4×C67—81no—
172C22×C4×C67—81no—
173C22×C4×C67—81no—
174C22×C4×C67—81no—
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67C2411yescharacteristic
182C4⋊C4×C67—22no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—41no—
189C4⋊C4×C67—81no—
190C4⋊C4×C67—81no—
191C4⋊C4×C67C22×C412yes—
192C22⋊C4×C67—24no—
193C22⋊C4×C67—42no—
194C22⋊C4×C67—42no—
195C42×C67—21no—
196C42×C67—21no—
197C42×C67—21no—
198C42×C67—21no—
199C42×C67C22×C412yes—
200C42×C67#4112yes—
201C23×C67—22no—
202C23×C67—22no—
203C23×C67—22no—
204C23×C67—41no—
205C23×C67—41no—
206C23×C67—41no—
207C23×C67—41no—
208C23×C67C22×D411yescharacteristic
209D4×C67—22no—
210D4×C67—22no—
211D4×C67—22no—
212D4×C67C4×Q814yes—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—22no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—84no—
238C2×C4×C67—84no—
239C2×C4×C67—84no—
240C2×C4×C67C4×D412yes—
241C2×C4×C67C4×D412yes—
242C2×C4×C67#13712yes—
243C2×C4×C67C23×C411yescharacteristic
244C2×C4×C67C22×Q811yescharacteristic
245C2×C4×C67C2×#4111yescharacteristic
246C2×C4×C67C2×#4111yescharacteristic
247C2×C4×C67H(5,𝔽2)11yescharacteristic
248C2×C4×C672−1+411yescharacteristic
249C2×Dic67—22no—
250C2×Dic67—22no—
251C2×Dic67—22no—
252C2×Dic67—41no—
253C2×Dic67—41no—
254C2×Dic67—41no—
255C2×Dic67—41no—
256C2×Dic67—41no—
257C2×Dic67—41no—
258C2×Dic67—41no—
259C2×Dic67—41no—
260C2×Dic67—41no—
261Dic134—84no—
262C67⋊C8—22no—
263C67⋊C8—22no—
264C67⋊C8—22no—
265C67⋊C8—41no—
266C22×C67—24no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—42no—
270C22×C67—42no—
271C22×C67—42no—
272C22×C67—42no—
273C22×C67C2×C4×D411yescharacteristic
274C22×C67#54611yescharacteristic
275C22×C67D4×Q811yescharacteristic
276C22×C67C2×#13711yescharacteristic
277C22×C67#57511yescharacteristic
278C22×C67#57611yescharacteristic
279C22×C67#57611yescharacteristic
280C4×C67—22no—
281C4×C67—22no—
282C4×C67—22no—
283C4×C67—42no—
284C4×C67—41no—
285C4×C67—41no—
286C4×C67—41no—
287C4×C67—81no—
288C4×C67—81no—
289C4×C67C2×C4×Q811yescharacteristic
290C4×C67#52011yescharacteristic
291C4×C67#55211yescharacteristic
292C4×C67#55411yescharacteristic
293Dic67—42no—
294Dic67—42no—
295Dic67—42no—
296Dic67—81no—
297#22979—671nomaximal, Sylow
298C2×C67—44no—
299C2×C67#236011yescharacteristic
300C2×C67C4×#13711yescharacteristic
301C2×C67#294811yescharacteristic
302C2×C67#331411yescharacteristic
303C2×C67#331911yescharacteristic
304C2×C67#338511yescharacteristic
305C2×C67#338911yescharacteristic
306C2×#474—671no—
307#2369—671no—
308#2349D6711yescharacteristic
309#1993—671no—
310#1949—671no—
311#1943—671no—
312#1942—671no—
313#1921—671no—
314#1918—671no—
315#1864—671no—
316#1862—671no—
317#1771—671no—
318#1762—671no—
319#1762—671no—
320C4×#102—671no—
321C67#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8—671no—
323C2×C4×D4C2×D6711yescharacteristic
324C2×C4×D4C2×D6711yescharacteristic
325C2×C4×D4C2×D6711yescharacteristic
326#474—678no—
327C2×#107—671no—
328C2×#107—671no—
329C2×#107—671no—
330C2×#106—671no—
331C2×C4⋊C8—671no—
332C2×#102—671no—
333C2×#102—671no—
334C2×#102—671no—
335C2×#102—671no—
336C2×#97—671no—
337C2×#97—671no—
338C2×C4×C8—671no—
339#391—1341no—
340#391—1341no—
341#389—671no—
342#389—671no—
343#389—1341no—
344#389C2×D6711yescharacteristic
345#389C2×D6711yescharacteristic
346#385—21no—
347#385—21no—
348#382C2×D6711yescharacteristic
349#380—21no—
350C4×C4⋊C4—671no—
351C4×C4⋊C4—671no—
352C4×C4⋊C4—671no—
353C4×C4⋊C4—1341no—
354C4×C4⋊C4C2×D6711yescharacteristic
355C2×#95—21no—
356#340—671no—
357#340—671no—
358#340—671no—
359#340—671no—
360#340—671no—
361#340—671no—
362#340—671no—
363#340—671no—
364#336—671no—
365#336—671no—
366C22×Q8—1341no—
367C22×D4C22×D6711yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—21no—
371C22×C8—671no—
372C22×C8—671no—
373C22×C8—671no—
374C22×C8—671no—
375C4⋊Q8—674no—
376C4⋊Q8—1342no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—41no—
381C4×D4—41no—
382C4×D4C4×D6714yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—671no—
386C2×C4⋊C4—671no—
387C2×C4⋊C4—671no—
388C2×C4⋊C4—671no—
389C2×C4⋊C4—1341no—
390C2×C4⋊C4—1341no—
391C2×C4⋊C4—1341no—
392C2×C4⋊C4C22⋊D6711yescharacteristic
393C2×C4⋊C4C22⋊D6711yescharacteristic
394C2×C4⋊C4C22×D6711yescharacteristic
395C2×C4⋊C4C22×D6711yescharacteristic
396C2×C4⋊C4C22×D6711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1341no—
401C2×C42—1341no—
402C2×C42—1341no—
403C2×C42C22×D6711yescharacteristic
404C2×C42C22×D6711yescharacteristic
405C2×C42C22×D6711yescharacteristic
406#107—2681no—
407#107—674no—
408#107—674no—
409#106—2681no—
410C4⋊C8—674no—
411#102—674no—
412#102—674no—
413#102—674no—
414#102—1342no—
415#97—2681no—
416#97—1342no—
417C4×C8—2681no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—1341no—
423#95—1341no—
424#95—1341no—
425#95—1341no—
426#95—1341no—
427#95C22⋊D6711yescharacteristic
428#95C22⋊D6711yescharacteristic
429C24—21no—
430C2×Q8—5361no—
431C2×Q8—5361no—
432C2×Q8—1344no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×C4×D6712yes—
437C2×D4Q8×D6712yes—
438C2×D4#1176162712yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—1341no—
452C22×C4—1341no—
453C22×C4—1341no—
454C22×C4—1341no—
455C22×C4C2×C22⋊D6711yescharacteristic
456C22×C4C2×C22⋊D6711yescharacteristic
457C22×C4C2×C22⋊D6711yescharacteristic
458C22×C4C2×C22⋊D6711yescharacteristic
459C22×C4C2×C22⋊D6711yescharacteristic
460C22×C4C2×C22⋊D6711yescharacteristic
461C22×C4C23×D6711yescharacteristic
462C2×C8—672no—
463C2×C8—672no—
464C2×C8—672no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8—1341no—
470C2×C8—1341no—
471C2×C8—1341no—
472C2×C8—1341no—
473C2×C8—1341no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—5361no—
484C4⋊C4—5361no—
485C4⋊C4—672no—
486C4⋊C4—672no—
487C4⋊C4—672no—
488C4⋊C4—1341no—
489C4⋊C4—1341no—
490C4⋊C4—1344no—
491C4⋊C4—1341no—
492C4⋊C4—1341no—
493C4⋊C4—1341no—
494C4⋊C4C2×C4×D6712yes—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—2682no—
503C42—5361no—
504C42—5361no—
505C42C2×C4×D6712yes—
506C42#1176162812yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C22×C22⋊D6711yescharacteristic
515Q8—5364no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4#1452400814yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—2681no—
548C2×C4—2681no—
549C2×C4—2681no—
550C2×C4—2681no—
551C2×C4—2681no—
552C2×C4—2681no—
553C2×C4—2681no—
554C2×C4—2681no—
555C2×C4—2681no—
556C2×C4—1342no—
557C2×C4—1342no—
558C2×C4—1342no—
559C2×C4C4×C22⋊D6712yes—
560C2×C4C4×C22⋊D6712yes—
561C2×C4#1452407312yes—
562C2×C4C22×C4×D6711yescharacteristic
563C2×C4C2×Q8×D6711yescharacteristic
564C2×C4C2×#1176162711yescharacteristic
565C2×C4C2×#1176162811yescharacteristic
566C2×C4#1452408811yescharacteristic
567C2×C4#1452409111yescharacteristic
568C8—2681no—
569C8—1342no—
570C8—1342no—
571C8—1342no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D6711yescharacteristic
580C22#1686404011yescharacteristic
581C22Q8×C22⋊D6711yescharacteristic
582C22C2×#1452407311yescharacteristic
583C22#1686419411yescharacteristic
584C22#1686419911yescharacteristic
585C22#1686419911yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—2682no—
596C4—2682no—
597C4—2682no—
598C4—5361no—
599C4C2×#1452400811yescharacteristic
600C4#1686387711yescharacteristic
601C4#1686408111yescharacteristic
602C4#1686410111yescharacteristic
603C2—44no—
604C2#2256321411yescharacteristic
605C2C4×#1452407311yescharacteristic
606C2#2256905911yescharacteristic
607C2#2257229011yescharacteristic
608C2#2257236511yescharacteristic
609C2#2257311911yescharacteristic
610C2#2257316311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2349C2
socleC23×C67C22×D4
prime radicalC22×D4×C67C23
cosocleC22×C4×C67C24
hypercenter#2349D67
hypocenterC67#22979
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #22563298C2semidirect
3. #22561181C2semidirect
4. #22560975C2semidirect
5. #22560944C2semidirect
6. #22560939C2semidirect
7. #22560798C2semidirect
8. #22560777C2semidirect
11. #22560084C2semidirect
17. #16863676C4semidirect
66. #14524043C2×C4semidirect
78. #14523976C2×C4semidirect
79. #14523976C2×C4semidirect
81. C4⋊(C67⋊C8)C2×C4semidirect
85. #14523964Q8semidirect
321. C67#22979semidirect

Maximal subgroups

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

subgroupindex
C2×#168636762
#225632982
#225611812
#225609752
#225609442
#225609392
#225607982
#225607772
#225604702
#225604562
#225600842
#225600342
C4×#145239642
C67×#23492
#2297967