Finite Group Databasebeta

← #30827700

Subgroups — #30827700

Order 17152.

588 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570809C211yesmaximal, characteristic
2#22569213C211yesmaximal, characteristic
3#22559109C211yesmaximal, characteristic, complemented
4#22559104C211yesmaximal, characteristic, complemented
5#22559087C211yesmaximal, characteristic, complemented
6#22559078C211yesmaximal, characteristic, complemented
7#22559059C211yesmaximal, characteristic, complemented
8#22559043C211yesmaximal, characteristic, complemented
9#22559026C211yesmaximal, characteristic, complemented
10C8×#11761615C211yesmaximal, characteristic, complemented
11#22558959C211yesmaximal, characteristic, complemented
12#22558630C211yesmaximal, characteristic, complemented
13#22558620C211yesmaximal, characteristic
14#22558512C211yesmaximal, characteristic, complemented
15C67×#2962C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863802C411yescharacteristic
18#16863788C411yescharacteristic
19#16863764C411yescharacteristic
20#16863758C411yescharacteristic
21#16863652C411yescharacteristic
22#16863652C411yescharacteristic
23#16863644C411yescharacteristic
24#16863644C411yescharacteristic
25#16863526C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C4×#11761618C2211yescharacteristic
28C4×#11761618C2211yescharacteristic
29C4×#11761615C2211yescharacteristic
30C4×#11761615C2211yescharacteristic
31#16863475C2211yescharacteristic
32#16863472—21no—
33#16863465—21no—
34C8×D268C2211yescharacteristic
35#16863458—21no—
36#16863449C2211yescharacteristic
37#16863353—21no—
38C4×#11761612C2211yescharacteristic
39C4×#11761612C2211yescharacteristic
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46Q8⋊(C67⋊C8)C2211yescharacteristic
47Q8⋊(C67⋊C8)C2211yescharacteristic
48#16863126C2211yescharacteristic
49#16863126C2211yescharacteristic
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863117C2211yescharacteristic
55C8×C67⋊C8C2211yescharacteristic
56#533×C67C2211yescharacteristic
57#446×C67—21no—
58#446×C67—21no—
59#446×C67C2211yescharacteristic
60#446×C67C2211yescharacteristic
61C8×Q8×C67C2211yescharacteristic
62C8×Q8×C67C2211yescharacteristic
63#424×C67—21no—
64C2×C4⋊C8×C67C2211yescharacteristic
65#405×C67—21no—
66C4×M16×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761612—21no—
72C2×#11761612C2×C411yescharacteristic
73C2×#11761612C2×C411yescharacteristic
74C2×#11761611—21no—
75C2×C8×D67—21no—
76#14524037—21no—
77#14524035C2×C411yescharacteristic
78#14524031C2×C411yescharacteristic
79#14524026—21no—
80#14524022C2×C411yescharacteristic
81#14524017C2×C411yescharacteristic
82C4×Dic134C2311yescharacteristic
83C4×D268C2311yescharacteristic
84C4×C22⋊D67—21no—
85#14524000—21no—
86#14523976C2×C411yescharacteristic
87#14523976C2×C411yescharacteristic
88#14523974C2×C411yescharacteristic
89#14523974C2×C411yescharacteristic
90C4⋊(C67⋊C8)C2311yescharacteristic
91C4⋊(C67⋊C8)C2311yescharacteristic
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94#14523967C2×C411yescharacteristic
95#14523967C2×C411yescharacteristic
96#14523967C2×C411yescharacteristic
97#14523967C2×C411yescharacteristic
98#14523966C2×C411yescharacteristic
99#14523966C2×C411yescharacteristic
100#14523963C2×C411yescharacteristic
101#14523963C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103(C2×D67)⋊C8—21no—
104(C2×D67)⋊C8—21no—
105#14523954—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×M16×C67D411yescharacteristic
116C2×M16×C67D411yescharacteristic
117C22×C8×C67—21no—
118C4⋊Q8×C67C2×C411yescharacteristic
119C4⋊Q8×C67C2×C411yescharacteristic
120#125×C67—21no—
121C22⋊Q8×C67—21no—
122C22⋊Q8×C67C2×C411yescharacteristic
123C22⋊Q8×C67C2×C411yescharacteristic
124C4×Q8×C67—21no—
125C4×Q8×C67C2311yescharacteristic
126C4×Q8×C67C2311yescharacteristic
127#117×C67—21no—
128C2×C42×C67C2311yescharacteristic
129C4⋊C8×C67—21no—
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67—21no—
133C4⋊C8×C67—21no—
134C4⋊C8×C67C2311yescharacteristic
135C4⋊C8×C67C2311yescharacteristic
136#97×C67—21no—
137#97×C67—21no—
138#97×C67D411yescharacteristic
139#97×C67D411yescharacteristic
140C4×C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134C22×C411yescharacteristic
145C2×D268C22×C411yescharacteristic
146C2×C22⋊D67—21no—
147C2×C4×D67—21no—
148#11761618—22no—
149#11761618—41no—
150#11761615—22no—
151#11761615—41no—
152#11761612—22no—
153#11761612—41no—
154#11761612—42no—
155#11761611—42no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C8×D67—42no—
165C4⋊Dic67C22×C411yescharacteristic
166C4⋊Dic67C22×C411yescharacteristic
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171C22⋊Dic67—21no—
172(C2×D67)⋊C4—21no—
173(C2×D67)⋊C4—21no—
174(C2×D67)⋊C4—21no—
175(C2×D67)⋊C4—21no—
176C4×Dic67—21no—
177C2×Q8×C67—21no—
178C2×Q8×C67C22×C411yescharacteristic
179C2×Q8×C67C22×C411yescharacteristic
180C22×C4×C67C22×C411yescharacteristic
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67#4111yescharacteristic
183M16×C67—22no—
184M16×C67—41no—
185C2×C8×C67—21no—
186C2×C8×C67—21no—
187C2×C8×C67—42no—
188C2×C8×C67C2×D411yescharacteristic
189C2×C8×C67C2×D411yescharacteristic
190C2×C8×C67C2×D411yescharacteristic
191C2×C8×C67C2×D411yescharacteristic
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67—21no—
199C4⋊C4×C67C22×C411yescharacteristic
200C4⋊C4×C67C22×C411yescharacteristic
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67C4×D411yescharacteristic
210C22×D67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—81no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
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×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C23×C411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C22⋊D67—81no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235D268—21no—
236D268—21no—
237C4×D67—41no—
238C4×D67—41no—
239Dic134—21no—
240Dic134—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—41no—
245C8×C67—41no—
246C8×C67#13712yes—
247C67⋊C8—22no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8#13112yes—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67C2×C4×D411yescharacteristic
259C2×D67—41no—
260C2×D67—41no—
261C2×D67—41no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67C2×#13111yescharacteristic
271C4×C67#56811yescharacteristic
272C4×C67C2×#13711yescharacteristic
273C4×C67#57511yescharacteristic
274Dic67—41no—
275Dic67—41no—
276Dic67—41no—
277#18077—671nomaximal, Sylow
278C2×C67—41no—
279C2×C67#293511yescharacteristic
280C2×C67C4×#13711yescharacteristic
281C2×C67#295611yescharacteristic
282D67—81no—
283#3172—671no—
284#2969—671no—
285#2962D6711yescharacteristic, complemented
286#1592—671no—
287#1591—671no—
288#1588—671no—
289#1587—671no—
290#1584—671no—
291#1582—671no—
292#1579—671no—
293C8×SD16—671no—
294#1565—671no—
295#1485—671no—
296#1483—671no—
297#1455—671no—
298C67#1807711yescharacteristic, complemented, Sylow
299#533C2×D6711yescharacteristic
300#529—671no—
301#501—671no—
302#497—671no—
303#491—671no—
304#489—671no—
305#470—671no—
306#470—671no—
307#468—671no—
308#468—671no—
309#446—21no—
310#446—21no—
311#446—671no—
312#446C2×D6711yescharacteristic
313#446C2×D6711yescharacteristic
314C8×Q8—671no—
315C8×Q8C2×D6711yescharacteristic
316C8×Q8C2×D6711yescharacteristic
317C4×Q16—671no—
318C4×Q16—671no—
319C4×SD16—671no—
320C4×SD16—671no—
321#436—671no—
322#436—1341no—
323#435—1341no—
324C8×D4—671no—
325#433—1341no—
326#431—671no—
327#424—21no—
328C2×C4⋊C8C2×D6711yescharacteristic
329#405—21no—
330#405—1341no—
331C4×M16—671no—
332C4×M16—671no—
333C4×M16C2×D6711yescharacteristic
334#335—671no—
335#335—671no—
336#334—671no—
337#334—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344D4⋊C8—671no—
345D4⋊C8—671no—
346#322—671no—
347#322—671no—
348#322—671no—
349C82—671no—
350C2×Q16—671no—
351C2×Q16—671no—
352C2×SD16—671no—
353C2×SD16—671no—
354C2×M16—671no—
355C2×M16—671no—
356C2×M16—1341no—
357C2×M16—1341no—
358C2×M16C22⋊D6711yescharacteristic
359C2×M16C22⋊D6711yescharacteristic
360C22×C8—21no—
361C22×C8—1341no—
362C4⋊Q8C4×D6711yescharacteristic
363C4⋊Q8C4×D6711yescharacteristic
364#126—1341no—
365#125—21no—
366#125—671no—
367#124—671no—
368#123—1341no—
369C22⋊Q8—21no—
370C22⋊Q8—671no—
371C22⋊Q8C4×D6711yescharacteristic
372C22⋊Q8C4×D6711yescharacteristic
373#121—671no—
374C4×Q8—21no—
375C4×Q8—671no—
376C4×Q8C22×D6711yescharacteristic
377C4×Q8C22×D6711yescharacteristic
378C4×D4—671no—
379C4×D4—1341no—
380#117—21no—
381#117—1341no—
382C2×C42C22×D6711yescharacteristic
383#107—671no—
384#107—671no—
385#106—671no—
386#106—671no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8—671no—
393C4⋊C8—671no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8C22×D6711yescharacteristic
397C4⋊C8C22×D6711yescharacteristic
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#103—671no—
404#102—671no—
405#102—671no—
406C22⋊C8—1341no—
407C22⋊C8—1341no—
408C22⋊C8—1341no—
409#97—21no—
410#97—21no—
411#97—671no—
412#97—671no—
413#97—1341no—
414#97—1341no—
415#97C22⋊D6711yescharacteristic
416#97C22⋊D6711yescharacteristic
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8C22×D6711yescharacteristic
426C4×C8C22×D6711yescharacteristic
427C2×Q8—21no—
428C2×Q8—671no—
429C2×Q8C2×C4×D6711yescharacteristic
430C2×Q8C2×C4×D6711yescharacteristic
431C2×D4—671no—
432C2×D4—1341no—
433C22×C4—1341no—
434C22×C4C2×C4×D6711yescharacteristic
435C22×C4C2×C22⋊D6711yescharacteristic
436C22×C4#1176162811yescharacteristic
437Q16—2681no—
438Q16—1342no—
439SD16—2681no—
440SD16—1342no—
441M16—22no—
442M16—41no—
443M16—2682no—
444M16—2682no—
445M16—2681no—
446M16—1342no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—42no—
450C2×C8—2682no—
451C2×C8—671no—
452C2×C8—671no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—1341no—
456C2×C8—1341no—
457C2×C8—1341no—
458C2×C8—1341no—
459C2×C8C2×C22⋊D6711yescharacteristic
460C2×C8C2×C22⋊D6711yescharacteristic
461C2×C8C2×C22⋊D6711yescharacteristic
462C2×C8C2×C22⋊D6711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—671no—
471C4⋊C4—671no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C4⋊C4—1341no—
475C4⋊C4—1341no—
476C4⋊C4C2×C4×D6711yescharacteristic
477C4⋊C4C2×C4×D6711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1341no—
481C22⋊C4—1341no—
482C22⋊C4—1341no—
483C22⋊C4—1341no—
484C22⋊C4—1341no—
485C42—21no—
486C42—21no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42C2×C22⋊D6711yescharacteristic
490C42#1176162811yescharacteristic
491C42C23×D6711yescharacteristic
492C23—1341no—
493C23C4×C22⋊D6711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—1341no—
500Q8—1341no—
501D4—5361no—
502D4—1341no—
503D4—1341no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—2681no—
514C2×C4—2681no—
515C2×C4—1341no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4C4×C22⋊D6711yescharacteristic
519C2×C4C4×C22⋊D6711yescharacteristic
520C2×C4C4×C22⋊D6711yescharacteristic
521C2×C4C22×C4×D6711yescharacteristic
522C2×C4C22×C22⋊D6711yescharacteristic
523C2×C4C2×#1176162811yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8—672no—
533C8—1342no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8#1452407412yes—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—2681no—
542C22—2681no—
543C22—2681no—
544C22C2×C4×C22⋊D6711yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—2681no—
554C4—2681no—
555C4—2681no—
556C4#131×D6711yescharacteristic
557C4#1686416111yescharacteristic
558C4C2×#1452407411yescharacteristic
559C4#1686419411yescharacteristic
560C2—41no—
561C2—5361no—
562C2#2256895111yescharacteristic
563C2C4×#1452407411yescharacteristic
564C2#2256911611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2962C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenter#2962D67
hypocenterC67#18077
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #22559109C2semidirect
4. #22559104C2semidirect
5. #22559087C2semidirect
6. #22559078C2semidirect
7. #22559059C2semidirect
8. #22559043C2semidirect
9. #22559026C2semidirect
10. C8×#11761615C2semidirect
11. #22558959C2semidirect
12. #22558630C2semidirect
14. #22558512C2semidirect
15. C67×#2962C2semidirect
285. #2962D67semidirect
298. C67#18077semidirect

Maximal subgroups

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

subgroupindex
#225708092
#225692132
#225591092
#225591042
#225590872
#225590782
#225590592
#225590432
#225590262
C8×#117616152
#225589592
#225586302
#225586202
#225585122
C67×#29622
#1807767