Finite Group Databasebeta

← #30826480

Subgroups — #30826480

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570815C211yesmaximal, characteristic
2#22569360C211yesmaximal, characteristic
3#22569303C211yesmaximal, characteristic, complemented
4#22559119C211yesmaximal, characteristic, complemented
5#22559113C211yesmaximal, characteristic, complemented
6#22559107C211yesmaximal, characteristic, complemented
7#22559101C211yesmaximal, characteristic, complemented
8#22559029C211yesmaximal, characteristic, complemented
9#22559023C211yesmaximal, characteristic, complemented
10C8×Dic268C211yesmaximal, characteristic, complemented
11C8×#11761617C211yesmaximal, characteristic, complemented
12#22558976C211yesmaximal, characteristic, complemented
13#22558670C211yesmaximal, characteristic
14#22558665C211yesmaximal, characteristic, complemented
15C8×M16×C67C211yesmaximal, characteristic, complemented
16#16863953C2211yescharacteristic
17#16863930C2211yescharacteristic
18#16863805C411yescharacteristic
19#16863790C411yescharacteristic
20#16863765C411yescharacteristic
21#16863756C411yescharacteristic
22#16863654C411yescharacteristic
23#16863654C411yescharacteristic
24#16863642C411yescharacteristic
25#16863642C411yescharacteristic
26#16863526C2211yescharacteristic
27#16863526C2211yescharacteristic
28#16863526C2211yescharacteristic
29#16863525—21no—
30#16863525—21no—
31C8×Dic134C2211yescharacteristic
32C8×Dic134C2211yescharacteristic
33C8×Dic134C2211yescharacteristic
34C4×Dic268C2211yescharacteristic
35C4×Dic268C2211yescharacteristic
36C4×#11761617C2211yescharacteristic
37C4×#11761617C2211yescharacteristic
38#16863475C2211yescharacteristic
39#16863471—21no—
40#16863470—21no—
41C8×D268C2211yescharacteristic
42#16863456—21no—
43#16863454—21no—
44#16863454—21no—
45#16863453C2211yescharacteristic
46#16863421—21no—
47#16863420C2211yescharacteristic
48#16863149C2211yescharacteristic
49#16863149C2211yescharacteristic
50#16863147C2211yescharacteristic
51#16863147C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863128C2211yescharacteristic
55#16863128C2211yescharacteristic
56#16863128C2211yescharacteristic
57#16863128C2211yescharacteristic
58#16863125C2211yescharacteristic
59#16863125C2211yescharacteristic
60#431×C67C2211yescharacteristic
61C4×M16×C67C2211yescharacteristic
62C2×C4×C8×C67C2211yescharacteristic
63#322×C67C2211yescharacteristic
64#322×C67C2211yescharacteristic
65C82×C67C2211yescharacteristic
66C82×C67C2211yescharacteristic
67C2×Dic268C2×C411yescharacteristic
68C2×Dic268C2×C411yescharacteristic
69C2×#11761617C2×C411yescharacteristic
70C2×#11761617C2×C411yescharacteristic
71C2×#11761612—21no—
72C2×#11761611—21no—
73C2×C8×D67—21no—
74#14524043C2×C411yescharacteristic
75#14524043C2×C411yescharacteristic
76#14524037—21no—
77#14524035—21no—
78#14524035C2×C411yescharacteristic
79#14524031C2×C411yescharacteristic
80#14524026—21no—
81#14524022—21no—
82#14524022C2×C411yescharacteristic
83#14524022C2×C411yescharacteristic
84#14524022C2×C411yescharacteristic
85#14524017C2×C411yescharacteristic
86C4×Dic134—21no—
87C4×Dic134C2311yescharacteristic
88C4×Dic134C2311yescharacteristic
89C4×Dic134C2311yescharacteristic
90C4×D268C2311yescharacteristic
91C4×C22⋊D67—21no—
92#14524002—21no—
93#14524000—21no—
94#14523977C2×C411yescharacteristic
95#14523977C2×C411yescharacteristic
96#14523975C2×C411yescharacteristic
97#14523975C2×C411yescharacteristic
98C4⋊(C67⋊C8)—21no—
99C4⋊(C67⋊C8)C2311yescharacteristic
100C4⋊(C67⋊C8)C2311yescharacteristic
101C4⋊(C67⋊C8)C2311yescharacteristic
102C4⋊(C67⋊C8)C2311yescharacteristic
103Dic67⋊C8—21no—
104Dic67⋊C8—21no—
105Dic67⋊C8—21no—
106Dic67⋊C8—21no—
107Dic67⋊C8—21no—
108Dic67⋊C8—21no—
109#14523968C2×C411yescharacteristic
110#14523968C2×C411yescharacteristic
111#14523968C2×C411yescharacteristic
112#14523968C2×C411yescharacteristic
113#14523968C2×C411yescharacteristic
114#14523968C2×C411yescharacteristic
115#14523965C2×C411yescharacteristic
116#14523965C2×C411yescharacteristic
117C22⋊(C67⋊C8)—21no—
118(C2×D67)⋊C8—21no—
119(C2×D67)⋊C8—21no—
120#14523954—21no—
121#14523953—21no—
122#14523953—21no—
123#14523953—21no—
124C8×Dic67—21no—
125C8×Dic67—21no—
126C8×Dic67—21no—
127C2×M16×C67C2×C411yescharacteristic
128C2×M16×C67C2×C411yescharacteristic
129C22×C8×C67—21no—
130C2×C42×C67C2311yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67D411yescharacteristic
133C22⋊C8×C67D411yescharacteristic
134C22⋊C8×C67D411yescharacteristic
135#97×C67C2×C411yescharacteristic
136#97×C67C2×C411yescharacteristic
137C4×C8×C67—21no—
138C4×C8×C67C2311yescharacteristic
139C4×C8×C67C2311yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×Dic134C22×C411yescharacteristic
147C2×Dic134C22×C411yescharacteristic
148C2×D268C22×C411yescharacteristic
149C2×C22⋊D67—21no—
150C2×C4×D67—21no—
151Dic268—22no—
152Dic268—22no—
153#11761617—22no—
154#11761617—22no—
155#11761612—81no—
156#11761611—81no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8—21no—
161C8×D67—81no—
162C4⋊Dic67—21no—
163C4⋊Dic67—21no—
164C4⋊Dic67—21no—
165C4⋊Dic67C22×C411yescharacteristic
166C4⋊Dic67C22×C411yescharacteristic
167C4⋊Dic67C22×C411yescharacteristic
168C4⋊Dic67C22×C411yescharacteristic
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171Dic67⋊C4—21no—
172Dic67⋊C4—21no—
173Dic67⋊C4—21no—
174Dic67⋊C4—21no—
175Dic67⋊C4—21no—
176Dic67⋊C4—21no—
177C22⋊Dic67—21no—
178C22⋊Dic67—21no—
179C22⋊Dic67—21no—
180(C2×D67)⋊C4—21no—
181(C2×D67)⋊C4—21no—
182(C2×D67)⋊C4—21no—
183(C2×D67)⋊C4—21no—
184C4×Dic67—21no—
185C4×Dic67—21no—
186C4×Dic67—21no—
187C22×C4×C67C22×C411yescharacteristic
188C22×C4×C67C2×D411yescharacteristic
189C22×C4×C67#4111yescharacteristic
190M16×C67—22no—
191M16×C67—22no—
192C2×C8×C67—21no—
193C2×C8×C67—21no—
194C2×C8×C67—42no—
195C2×C8×C67C22×C411yescharacteristic
196C2×C8×C67C22×C411yescharacteristic
197C2×C8×C67C22×C411yescharacteristic
198C2×C8×C67C22×C411yescharacteristic
199C2×C8×C67C2×D411yescharacteristic
200C2×C8×C67C2×D411yescharacteristic
201C2×C8×C67C2×D411yescharacteristic
202C2×C8×C67C2×D411yescharacteristic
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67C4×D411yescharacteristic
208C22×D67—21no—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—41no—
214C2×C4×C67C4×D411yescharacteristic
215C2×C4×C67C4×D411yescharacteristic
216C2×C4×C67C4×D411yescharacteristic
217C2×C4×C67C23×C411yescharacteristic
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C2×#4111yescharacteristic
220C22⋊D67—81no—
221C2×Dic67—21no—
222C2×Dic67—21no—
223C2×Dic67—21no—
224C2×Dic67—21no—
225C2×Dic67—21no—
226C2×Dic67—21no—
227C2×Dic67—21no—
228D268—21no—
229D268—21no—
230C4×D67—41no—
231C4×D67—41no—
232Dic134—21no—
233Dic134—21no—
234Dic134—21no—
235Dic134—21no—
236Dic134—21no—
237Dic134—21no—
238Dic134—81no—
239C8×C67—22no—
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—41no—
247C8×C67#13112yes—
248C8×C67#13112yes—
249C67⋊C8—41no—
250C67⋊C8—41no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C22×C67—21no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67C2×C4×D411yescharacteristic
257C2×D67—41no—
258C2×D67—41no—
259C2×D67—41no—
260C4×C67—21no—
261C4×C67—21no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67C2×#13111yescharacteristic
265C4×C67C2×#13111yescharacteristic
266C4×C67#57711yescharacteristic
267C4×C67#57811yescharacteristic
268Dic67—41no—
269Dic67—41no—
270Dic67—41no—
271Dic67—41no—
272Dic67—41no—
273Dic67—41no—
274Dic67—41no—
275#17969—671nomaximal, Sylow
276C2×C67—41no—
277C2×C67#293111yescharacteristic
278C2×C67#294711yescharacteristic
279C2×C67#295611yescharacteristic
280D67—81no—
281#3172—671no—
282#2993—671no—
283#2985—671no—
284#1593—671no—
285#1592—671no—
286#1591—671no—
287#1590—671no—
288#1579—671no—
289SD16⋊C8—671no—
290C8×Q16—671no—
291C8×SD16—671no—
292#1569—671no—
293#1494—671no—
294#1493—671no—
295C8×M16D6711yescharacteristic, complemented
296C67#1796911yescharacteristic, complemented, Sylow
297#533—671no—
298#529—671no—
299#501—671no—
300#497—671no—
301#491—671no—
302#489—671no—
303#470—671no—
304#470—671no—
305#468—671no—
306#468—671no—
307#446—671no—
308#446—671no—
309#446—671no—
310#446—1341no—
311#446—1341no—
312C8×Q8—671no—
313C8×Q8—671no—
314C8×Q8—671no—
315C4×Q16—671no—
316C4×Q16—671no—
317C4×SD16—671no—
318C4×SD16—671no—
319#436—671no—
320#436—1341no—
321#435—1341no—
322C8×D4—671no—
323#433—1341no—
324#433—1341no—
325#433—1341no—
326#432—671no—
327#431C2×D6711yescharacteristic
328#424—1341no—
329#423—671no—
330C4×M16C2×D6711yescharacteristic
331C2×C4×C8C2×D6711yescharacteristic
332#335—671no—
333#335—671no—
334#334—671no—
335#334—671no—
336Q8⋊C8—671no—
337Q8⋊C8—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342D4⋊C8—671no—
343D4⋊C8—671no—
344#322C2×D6711yescharacteristic
345#322C2×D6711yescharacteristic
346C82C2×D6711yescharacteristic
347C82C2×D6711yescharacteristic
348C2×Q16—671no—
349C2×Q16—671no—
350C2×SD16—671no—
351C2×SD16—671no—
352C2×M16—1341no—
353C2×M16—1341no—
354C2×M16C4×D6711yescharacteristic
355C2×M16C4×D6711yescharacteristic
356C22×C8—21no—
357C22×C8—1341no—
358C4⋊Q8—671no—
359C4⋊Q8—671no—
360#126—1341no—
361#125—671no—
362#125—1341no—
363#124—671no—
364#123—1341no—
365C22⋊Q8—671no—
366C22⋊Q8—671no—
367C22⋊Q8—671no—
368C22⋊Q8—1341no—
369#121—671no—
370C4×Q8—671no—
371C4×Q8—671no—
372C4×Q8—671no—
373C4×Q8—1341no—
374C4×D4—671no—
375C4×D4—1341no—
376#117—1341no—
377#117—1341no—
378C2×C42C22×D6711yescharacteristic
379#107—671no—
380#107—671no—
381#106—671no—
382#106—671no—
383C4⋊C8—671no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—1341no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8D26811yescharacteristic
395C4⋊C8D26811yescharacteristic
396#103—671no—
397#103—671no—
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#102—671no—
403#102—671no—
404C22⋊C8—1341no—
405C22⋊C8—1341no—
406C22⋊C8—1341no—
407C22⋊C8D26811yescharacteristic
408C22⋊C8D26811yescharacteristic
409#97—1341no—
410#97—1341no—
411#97—1341no—
412#97—1341no—
413#97C4×D6711yescharacteristic
414#97C4×D6711yescharacteristic
415C4×C8—21no—
416C4×C8—1341no—
417C4×C8—1341no—
418C4×C8—1341no—
419C4×C8C22×D6711yescharacteristic
420C4×C8C22×D6711yescharacteristic
421C4×C8C22×D6711yescharacteristic
422C4×C8C22×D6711yescharacteristic
423C4×C8C22×D6711yescharacteristic
424C4×C8C22×D6711yescharacteristic
425C2×Q8—671no—
426C2×Q8—671no—
427C2×Q8—671no—
428C2×Q8—1341no—
429C2×D4—671no—
430C2×D4—1341no—
431C22×C4—1341no—
432C22×C4C2×C4×D6711yescharacteristic
433C22×C4C2×D26811yescharacteristic
434C22×C4#1176162811yescharacteristic
435Q16—1342no—
436Q16—1342no—
437SD16—1342no—
438SD16—1342no—
439M16—22no—
440M16—22no—
441M16—5361no—
442M16—5361no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—42no—
446C2×C8—5361no—
447C2×C8—1341no—
448C2×C8—1341no—
449C2×C8—1341no—
450C2×C8—1341no—
451C2×C8C2×C4×D6711yescharacteristic
452C2×C8C2×C4×D6711yescharacteristic
453C2×C8C2×C4×D6711yescharacteristic
454C2×C8C2×C4×D6711yescharacteristic
455C2×C8C2×D26811yescharacteristic
456C2×C8C2×D26811yescharacteristic
457C2×C8C2×D26811yescharacteristic
458C2×C8C2×D26811yescharacteristic
459C4⋊C4—671no—
460C4⋊C4—671no—
461C4⋊C4—671no—
462C4⋊C4—671no—
463C4⋊C4—1341no—
464C4⋊C4—1341no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1341no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C22⋊C4—1341no—
475C22⋊C4—1341no—
476C22⋊C4—1341no—
477C22⋊C4—1341no—
478C22⋊C4—1341no—
479C22⋊C4—1341no—
480C22⋊C4—1341no—
481C42—1341no—
482C42—1341no—
483C42—1341no—
484C42C2×C4×D6711yescharacteristic
485C42C2×D26811yescharacteristic
486C42#1176162811yescharacteristic
487C42C23×D6711yescharacteristic
488C23—1341no—
489C23C4×D26811yescharacteristic
490Q8—5361no—
491Q8—1341no—
492Q8—1341no—
493Q8—1341no—
494Q8—1341no—
495Q8—1341no—
496Q8—1341no—
497D4—5361no—
498D4—1341no—
499D4—1341no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—2681no—
506C2×C4—2681no—
507C2×C4—1341no—
508C2×C4—1341no—
509C2×C4—1341no—
510C2×C4—1341no—
511C2×C4—1341no—
512C2×C4—1341no—
513C2×C4—1341no—
514C2×C4C4×D26811yescharacteristic
515C2×C4C4×D26811yescharacteristic
516C2×C4C4×D26811yescharacteristic
517C2×C4C22×C4×D6711yescharacteristic
518C2×C4C22×D26811yescharacteristic
519C2×C4C2×#1176162811yescharacteristic
520C8—22no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—2681no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8#1452404912yes—
533C8#1452404912yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2681no—
538C22—2681no—
539C22—2681no—
540C22C2×C4×D26811yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—2681no—
546C4—2681no—
547C4—2681no—
548C4—2681no—
549C4—2681no—
550C4—2681no—
551C4—2681no—
552C4C2×#1452404911yescharacteristic
553C4C2×#1452404911yescharacteristic
554C4#1686420811yescharacteristic
555C4#1686421311yescharacteristic
556C2—41no—
557C2—5361no—
558C2#2256892211yescharacteristic
559C2#2256905411yescharacteristic
560C2#2256912011yescharacteristic
subgroupG/H
centerC2×C4C22×D268
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC8×M16×C67C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenterC8×M16D67
hypocenterC67#17969
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
3. #22569303C2semidirect
4. #22559119C2semidirect
5. #22559113C2semidirect
6. #22559107C2semidirect
7. #22559101C2semidirect
8. #22559029C2semidirect
9. #22559023C2semidirect
10. C8×Dic268C2semidirect
11. C8×#11761617C2semidirect
12. #22558976C2semidirect
14. #22558665C2semidirect
15. C8×M16×C67C2semidirect
295. C8×M16D67semidirect
296. C67#17969semidirect

Maximal subgroups

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

subgroupindex
#225708152
#225693602
#225693032
#225591192
#225591132
#225591072
#225591012
#225590292
#225590232
C8×Dic2682
C8×#117616172
#225589762
#225586702
#225586652
C8×M16×C672
#1796967