Finite Group Databasebeta

← #30827189

Subgroups — #30827189

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570240C211yesmaximal, characteristic
2#22569298C211yesmaximal, characteristic, complemented
3#22569275C211yesmaximal, characteristic
4#22559114C211yesmaximal, characteristic, complemented
5#22559096C211yesmaximal, characteristic, complemented
6#22559084C211yesmaximal, characteristic, complemented
7#22559069C211yesmaximal, characteristic, complemented
8#22559049C211yesmaximal, characteristic, complemented
9#22559048C211yesmaximal, characteristic, complemented
10#22559024C211yesmaximal, characteristic, complemented
11#22559006C211yesmaximal, characteristic, complemented
12#22558987C211yesmaximal, characteristic, complemented
13#22558679C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic, complemented
15C67×#1479C211yesmaximal, characteristic, complemented
16#16863946C2211yescharacteristic
17#16863714C411yescharacteristic
18#16863714C411yescharacteristic
19#16863681C411yescharacteristic
20#16863681C411yescharacteristic
21#16863596C411yescharacteristic
22#16863596C411yescharacteristic
23#16863535C411yescharacteristic
24#16863535C411yescharacteristic
25#16863527—21no—
26#16863527—21no—
27#16863527C2211yescharacteristic
28#16863527C2211yescharacteristic
29Q8×C67⋊C8C2211yescharacteristic
30Q8×C67⋊C8C2211yescharacteristic
31C4×#11761615C2211yescharacteristic
32C4×#11761615C2211yescharacteristic
33C4×#11761615C2211yescharacteristic
34C4×#11761615C2211yescharacteristic
35#16863475—21no—
36#16863473C2211yescharacteristic
37#16863473C2211yescharacteristic
38C8×D268—21no—
39#16863461C2211yescharacteristic
40#16863461C2211yescharacteristic
41#16863453—21no—
42#16863453C2211yescharacteristic
43#16863449—21no—
44#16863449C2211yescharacteristic
45#16863449C2211yescharacteristic
46#16863361—21no—
47#16863361—21no—
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863125C2211yescharacteristic
53#16863125C2211yescharacteristic
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863118C2211yescharacteristic
57#16863118C2211yescharacteristic
58#16863117C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60C2×C4×Q8×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65Q8⋊C8×C67C2211yescharacteristic
66Q8⋊C8×C67C2211yescharacteristic
67C2×#11761628—21no—
68C2×#11761615C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761615C2×C411yescharacteristic
72C2×#11761611—21no—
73C2×#11761611—21no—
74C2×C8×D67—21no—
75C2×C8×D67—21no—
76#14524043C2×C411yescharacteristic
77#14524040C2×C411yescharacteristic
78#14524031—21no—
79#14524017—21no—
80#14524017C2×C411yescharacteristic
81#14524017C2×C411yescharacteristic
82C4×D268—21no—
83C4×D268C2311yescharacteristic
84C4×D268C2311yescharacteristic
85#14523974C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87#14523974C2×C411yescharacteristic
88#14523974C2×C411yescharacteristic
89C4⋊(C67⋊C8)—21no—
90C4⋊(C67⋊C8)—21no—
91C4⋊(C67⋊C8)—21no—
92C4⋊(C67⋊C8)—21no—
93C4⋊(C67⋊C8)C2×C411yescharacteristic
94C4⋊(C67⋊C8)C2×C411yescharacteristic
95C4⋊(C67⋊C8)D411yescharacteristic
96C4⋊(C67⋊C8)D411yescharacteristic
97C4⋊(C67⋊C8)C2311yescharacteristic
98C4⋊(C67⋊C8)C2311yescharacteristic
99#14523967C2×C411yescharacteristic
100#14523967C2×C411yescharacteristic
101#14523967C2×C411yescharacteristic
102#14523967C2×C411yescharacteristic
103#14523963C2×C411yescharacteristic
104#14523963C2×C411yescharacteristic
105#14523963C2×C411yescharacteristic
106#14523963C2×C411yescharacteristic
107C22⋊(C67⋊C8)—21no—
108C22⋊(C67⋊C8)C2×C411yescharacteristic
109C22⋊(C67⋊C8)C2×C411yescharacteristic
110C22⋊(C67⋊C8)D411yescharacteristic
111C22⋊(C67⋊C8)D411yescharacteristic
112(C2×D67)⋊C8—21no—
113(C2×D67)⋊C8—21no—
114(C2×D67)⋊C8—21no—
115(C2×D67)⋊C8—21no—
116#14523954—21no—
117#14523954—21no—
118#14523954—21no—
119C4×C67⋊C8—21no—
120C4×C67⋊C8—21no—
121C4×C67⋊C8C2311yescharacteristic
122C4×C67⋊C8C2311yescharacteristic
123C4×C67⋊C8C2311yescharacteristic
124C4×C67⋊C8C2311yescharacteristic
125C22×Q8×C67C2×C411yescharacteristic
126C4×Q8×C67—21no—
127C4×Q8×C67—21no—
128C4×Q8×C67C2×C411yescharacteristic
129C4×Q8×C67C2×C411yescharacteristic
130C4×Q8×C67C2311yescharacteristic
131C4×Q8×C67C2311yescharacteristic
132C2×C4⋊C4×C67—21no—
133C2×C4⋊C4×C67C2×C411yescharacteristic
134C2×C42×C67—21no—
135C2×C42×C67C2311yescharacteristic
136C4⋊C8×C67—21no—
137C4⋊C8×C67C2311yescharacteristic
138C4⋊C8×C67C2311yescharacteristic
139C22⋊C8×C67—21no—
140C22⋊C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#11761628—81no—
145#11761628—81no—
146C2×Dic134—21no—
147C2×D268—21no—
148C2×D268—21no—
149C2×D268C22×C411yescharacteristic
150C2×D268C22×C411yescharacteristic
151C2×C22⋊D67—21no—
152C2×C22⋊D67—21no—
153C2×C4×D67—21no—
154C2×C4×D67—21no—
155#11761615—22no—
156#11761615—22no—
157#11761615—41no—
158#11761615—41no—
159#11761611—81no—
160#11761611—81no—
161C2×C67⋊C8—21no—
162C2×C67⋊C8—21no—
163C2×C67⋊C8C22×C411yescharacteristic
164C2×C67⋊C8C22×C411yescharacteristic
165C2×C67⋊C8C22×C411yescharacteristic
166C2×C67⋊C8C22×C411yescharacteristic
167C2×C67⋊C8C2×D411yescharacteristic
168C2×C67⋊C8C2×D411yescharacteristic
169C2×C67⋊C8C2×D411yescharacteristic
170C2×C67⋊C8C2×D411yescharacteristic
171C8×D67—81no—
172C8×D67—81no—
173C4⋊Dic67—21no—
174C4⋊Dic67C22×C411yescharacteristic
175C4⋊Dic67C22×C411yescharacteristic
176(C2×D67)⋊C4—21no—
177(C2×D67)⋊C4—21no—
178(C2×D67)⋊C4—21no—
179(C2×D67)⋊C4—21no—
180C2×Q8×C67—21no—
181C2×Q8×C67—22no—
182C2×Q8×C67—41no—
183C2×Q8×C67C22×C411yescharacteristic
184C2×Q8×C67C22×C411yescharacteristic
185C22×C4×C67—21no—
186C22×C4×C67—21no—
187C22×C4×C67C22×C411yescharacteristic
188C22×C4×C67C2×D411yescharacteristic
189C22×C4×C67#4111yescharacteristic
190C2×C8×C67—21no—
191C2×C8×C67—21no—
192C2×C8×C67—21no—
193C2×C8×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
201C42×C67—21no—
202C42×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—
211C22×D67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—42no—
216Q8×C67#13112yes—
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—21no—
225C2×C4×C67—41no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67C4×D411yescharacteristic
229C2×C4×C67C4×D411yescharacteristic
230C2×C4×C67C4×D411yescharacteristic
231C2×C4×C67C23×C411yescharacteristic
232C2×C4×C67C22×D411yescharacteristic
233C2×C4×C67C2×#4111yescharacteristic
234C22⋊D67—81no—
235C22⋊D67—81no—
236C2×Dic67—21no—
237C2×Dic67—21no—
238D268—21no—
239D268—21no—
240D268—21no—
241D268—21no—
242D268—42no—
243D268—81no—
244C4×D67—41no—
245C4×D67—41no—
246C4×D67—41no—
247C4×D67—41no—
248Dic134—81no—
249C8×C67—41no—
250C8×C67—41no—
251C8×C67—41no—
252C8×C67—41no—
253C67⋊C8—21no—
254C67⋊C8—21no—
255C67⋊C8—21no—
256C67⋊C8—21no—
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—41no—
268C2×D67—41no—
269C2×D67—41no—
270C2×D67—41no—
271C2×D67—41no—
272C2×D67—41no—
273C4×C67—21no—
274C4×C67—22no—
275C4×C67—21no—
276C4×C67—21no—
277C4×C67—21no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67—41no—
281C4×C67C2×#13111yescharacteristic
282C4×C67#56811yescharacteristic
283C4×C67#57711yescharacteristic
284C4×C67#57711yescharacteristic
285Dic67—41no—
286Dic67—41no—
287#18030—671nomaximal, Sylow
288C2×C67—41no—
289C2×C67#293311yescharacteristic
290C2×C67#294911yescharacteristic
291C2×C67#295511yescharacteristic
292D67—81no—
293D67—81no—
294#3103—671no—
295#2984—671no—
296#2979—671no—
297#1592—671no—
298#1590—671no—
299#1588—671no—
300#1585—671no—
301#1583—671no—
302#1582—671no—
303SD16⋊C8—671no—
304C8×SD16—671no—
305#1572—671no—
306#1496—671no—
307#1479D6711yescharacteristic, complemented
308#1454—671no—
309C67#1803011yescharacteristic, complemented, Sylow
310#532—671no—
311C2×C4×Q8C2×D6711yescharacteristic
312#481—671no—
313#481—671no—
314#475—671no—
315#475—671no—
316#460—671no—
317#460—671no—
318#448—671no—
319#448—671no—
320#446—671no—
321#446—671no—
322#446—1341no—
323#446—1341no—
324C8×Q8—671no—
325C8×Q8—671no—
326C4×SD16—671no—
327C4×SD16—671no—
328C4×SD16—671no—
329C4×SD16—671no—
330#436—671no—
331#436—671no—
332#436—1341no—
333C8×D4—671no—
334C8×D4—671no—
335C8×D4—1341no—
336#432—671no—
337#432—1341no—
338#431—671no—
339#431—671no—
340#431—1341no—
341#431C2×D6711yescharacteristic
342#431C2×D6711yescharacteristic
343#408—1341no—
344#408—1341no—
345#335—671no—
346#335—671no—
347#334—671no—
348#334—671no—
349Q8⋊C8C2×D6711yescharacteristic
350Q8⋊C8C2×D6711yescharacteristic
351Q8⋊C8C2×D6711yescharacteristic
352Q8⋊C8C2×D6711yescharacteristic
353D4⋊C8—671no—
354D4⋊C8—671no—
355D4⋊C8—671no—
356D4⋊C8—671no—
357#322—671no—
358#322—671no—
359#322—671no—
360C82—671no—
361C2×#41—1341no—
362C22×Q8C4×D6711yescharacteristic
363C2×SD16—671no—
364C2×SD16—671no—
365C2×SD16—671no—
366C2×SD16—671no—
367C2×M16—1341no—
368C2×M16—1341no—
369C22×C8—1341no—
370C22×C8—1341no—
371C4⋊Q8—671no—
372#127—671no—
373#124—1341no—
374#121—671no—
375#121—671no—
376#121—1341no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C4×D6711yescharacteristic
380C4×Q8C4×D6711yescharacteristic
381C4×Q8C22×D6711yescharacteristic
382C4×Q8C22×D6711yescharacteristic
383C4×D4—671no—
384C4×D4—671no—
385C4×D4—1341no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4C4×D6711yescharacteristic
388C2×C42—21no—
389C2×C42C22×D6711yescharacteristic
390#106—671no—
391#106—671no—
392#106—671no—
393#106—671no—
394C4⋊C8—21no—
395C4⋊C8—671no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—671no—
399C4⋊C8—671no—
400C4⋊C8—671no—
401C4⋊C8—1341no—
402C4⋊C8—1341no—
403C4⋊C8—1341no—
404C4⋊C8—1341no—
405C4⋊C8C22×D6711yescharacteristic
406C4⋊C8C22×D6711yescharacteristic
407#103—671no—
408#103—671no—
409#103—671no—
410#103—671no—
411#102—671no—
412#102—671no—
413#102—671no—
414#102—671no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—671no—
418C22⋊C8—671no—
419C22⋊C8—671no—
420C22⋊C8—671no—
421C22⋊C8—1341no—
422C22⋊C8—1341no—
423C22⋊C8—1341no—
424C22⋊C8—1341no—
425C22⋊C8—1341no—
426#97—1341no—
427#97—1341no—
428#97—1341no—
429C4×C8—21no—
430C4×C8—671no—
431C4×C8—671no—
432C4×C8—671no—
433C4×C8—671no—
434C4×C8—1341no—
435C4×C8—1341no—
436C4×C8C22×D6711yescharacteristic
437C4×C8C22×D6711yescharacteristic
438#41—5361no—
439#41—5361no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—1341no—
444C2×Q8C2×C4×D6711yescharacteristic
445C2×Q8C2×C4×D6711yescharacteristic
446C2×D4—671no—
447C2×D4—671no—
448C2×D4—1341no—
449C2×D4—1341no—
450C2×D4—1341no—
451C2×D4—1341no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1341no—
455C22×C4—1341no—
456C22×C4C2×C4×D6711yescharacteristic
457C22×C4D4×D6711yescharacteristic
458C22×C4#1176162711yescharacteristic
459SD16—2681no—
460SD16—2681no—
461SD16—1342no—
462SD16—1342no—
463M16—5361no—
464M16—5361no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—5361no—
470C2×C8—5361no—
471C2×C8—671no—
472C2×C8—671no—
473C2×C8—671no—
474C2×C8—671no—
475C2×C8—671no—
476C2×C8—671no—
477C2×C8—671no—
478C2×C8—671no—
479C2×C8—1341no—
480C2×C8—1341no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—671no—
487C4⋊C4—671no—
488C4⋊C4—1341no—
489C4⋊C4C2×C4×D6711yescharacteristic
490C4⋊C4C2×C4×D6711yescharacteristic
491C22⋊C4—1341no—
492C22⋊C4—1341no—
493C22⋊C4—1341no—
494C22⋊C4—1341no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42C2×C4×D6711yescharacteristic
500C42D4×D6711yescharacteristic
501C42#1176162711yescharacteristic
502C42C23×D6711yescharacteristic
503C23—1341no—
504C23—1341no—
505C23(C22⋊D67)⋊C411yescharacteristic
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—42no—
510Q8—5361no—
511Q8#1452404912yes—
512D4—2682no—
513D4—5361no—
514D4—5361no—
515D4—5361no—
516D4—1341no—
517D4—1341no—
518D4—1341no—
519D4—1341no—
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—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—2681no—
532C2×C4—2681no—
533C2×C4—2681no—
534C2×C4—2681no—
535C2×C4—1341no—
536C2×C4—1341no—
537C2×C4(C22⋊D67)⋊C411yescharacteristic
538C2×C4(C22⋊D67)⋊C411yescharacteristic
539C2×C4(C22⋊D67)⋊C411yescharacteristic
540C2×C4C22×C4×D6711yescharacteristic
541C2×C4C2×D4×D6711yescharacteristic
542C2×C4C2×#1176162711yescharacteristic
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—2681no—
548C8—2681no—
549C8—1341no—
550C8—1341no—
551C8—1341no—
552C8—1341no—
553C8—1341no—
554C8—1341no—
555C8—1341no—
556C8—1341no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—2681no—
561C22—2681no—
562C22—2681no—
563C22—2681no—
564C22—2681no—
565C22—2681no—
566C22C2×(C22⋊D67)⋊C411yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—2681no—
576C4—2681no—
577C4C2×#1452404911yescharacteristic
578C4#1686416211yescharacteristic
579C4#1686420511yescharacteristic
580C4#1686420511yescharacteristic
581C2—41no—
582C2—5361no—
583C2—5361no—
584C2#2256893011yescharacteristic
585C2#2256906411yescharacteristic
586C2#2256910811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1479C2
socleC22×C67C2×C4×D4
prime radical#16863946C22
cosocleC42×C67C24
hypercenter#1479D67
hypocenterC67#18030
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #22569298C2semidirect
4. #22559114C2semidirect
5. #22559096C2semidirect
6. #22559084C2semidirect
7. #22559069C2semidirect
8. #22559049C2semidirect
9. #22559048C2semidirect
10. #22559024C2semidirect
11. #22559006C2semidirect
12. #22558987C2semidirect
14. #22558509C2semidirect
15. C67×#1479C2semidirect
307. #1479D67semidirect
309. C67#18030semidirect

Maximal subgroups

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

subgroupindex
#225702402
#225692982
#225692752
#225591142
#225590962
#225590842
#225590692
#225590492
#225590482
#225590242
#225590062
#225589872
#225586792
#225585092
C67×#14792
#1803067