Finite Group Databasebeta

← #30827171

Subgroups — #30827171

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#22570213C211yesmaximal, characteristic
2#22569311C211yesmaximal, characteristic, complemented
3#22569246C211yesmaximal, 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#22558990C211yesmaximal, characteristic, complemented
13#22558601C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic, complemented
15C67×#1497C211yesmaximal, characteristic, complemented
16C2×C4×D268C2211yescharacteristic
17#16863724C411yescharacteristic
18#16863724C411yescharacteristic
19#16863683C411yescharacteristic
20#16863683C411yescharacteristic
21#16863598C411yescharacteristic
22#16863598C411yescharacteristic
23#16863546C411yescharacteristic
24#16863546C411yescharacteristic
25#16863527—21no—
26#16863527C2211yescharacteristic, complemented
27#16863527C2211yescharacteristic, complemented
28Q8×C67⋊C8—21no—
29Q8×C67⋊C8C2211yescharacteristic, complemented
30Q8×C67⋊C8C2211yescharacteristic, complemented
31C4×#11761615C2211yescharacteristic
32C4×#11761615C2211yescharacteristic
33C4×#11761615C2211yescharacteristic
34C4×#11761615C2211yescharacteristic
35#16863473C2211yescharacteristic
36#16863473C2211yescharacteristic
37#16863469—21no—
38#16863469—21no—
39#16863461C2211yescharacteristic
40#16863461C2211yescharacteristic
41#16863457—21no—
42#16863457—21no—
43#16863449C2211yescharacteristic
44#16863449C2211yescharacteristic
45#16863449C2211yescharacteristic
46#16863358—21no—
47C22⋊C8×D67—21no—
48#16863148C2211yescharacteristic, complemented
49#16863148C2211yescharacteristic, complemented
50#16863146C2211yescharacteristic, complemented
51#16863146C2211yescharacteristic, complemented
52#16863125C2211yescharacteristic
53#16863125C2211yescharacteristic
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863118C2211yescharacteristic, complemented
57#16863118C2211yescharacteristic, complemented
58#16863117C2211yescharacteristic, complemented
59C8×C67⋊C8C2211yescharacteristic, complemented
60#533×C67C2211yescharacteristic
61#432×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic, complemented
64Q8⋊C8×C67C2211yescharacteristic, complemented
65Q8⋊C8×C67C2211yescharacteristic, complemented
66Q8⋊C8×C67C2211yescharacteristic, complemented
67C22×D268C2×C411yescharacteristic
68C22×C4×D67—21no—
69C2×#11761615C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761615C2×C411yescharacteristic
72C2×#11761615C2×C411yescharacteristic
73C2×#11761611—21no—
74C2×#11761611—21no—
75C2×C8×D67—21no—
76C2×C8×D67—21no—
77C4×D268—21no—
78C4×D268—21no—
79C4×D268C2×C411yescharacteristic
80C4×D268C2×C411yescharacteristic
81C4×D268C2311yescharacteristic
82C4×D268C2311yescharacteristic
83C2×C4⋊Dic67C2×C411yescharacteristic
84C2×(C2×D67)⋊C4—21no—
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—
118C4×C67⋊C8—21no—
119C4×C67⋊C8—21no—
120C4×C67⋊C8—21no—
121C4×C67⋊C8C2311yescharacteristic
122C4×C67⋊C8C2311yescharacteristic
123C4×C67⋊C8C2311yescharacteristic
124C4×C67⋊C8C2311yescharacteristic
125C4⋊Q8×C67C2×C411yescharacteristic
126C4⋊Q8×C67C2×C411yescharacteristic
127#125×C67—21no—
128C22⋊Q8×C67—21no—
129C22⋊Q8×C67C2×C411yescharacteristic
130C22⋊Q8×C67C2×C411yescharacteristic
131C4×Q8×C67—21no—
132C4×Q8×C67C2311yescharacteristic
133C4×Q8×C67C2311yescharacteristic
134#117×C67—21no—
135C2×C42×C67C2311yescharacteristic
136C4⋊C8×C67—21no—
137C4⋊C8×C67C2311yescharacteristic
138C4⋊C8×C67C2311yescharacteristic
139C22⋊C8×C67—21no—
140C22⋊C8×C67—21no—
141#97×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C23×D67—21no—
145C2×D268—22no—
146C2×D268—21no—
147C2×D268—41no—
148C2×D268C22×C411yescharacteristic
149C2×D268C22×C411yescharacteristic
150C22×Dic67—21no—
151C2×C4×D67—21no—
152C2×C4×D67—21no—
153C2×C4×D67—21no—
154C2×C4×D67—21no—
155C2×C4×D67—81no—
156C2×C4×D67—81no—
157#11761615—22no—
158#11761615—22no—
159#11761615—41no—
160#11761615—41no—
161#11761611—81no—
162#11761611—81no—
163C2×C67⋊C8—21no—
164C2×C67⋊C8—21no—
165C2×C67⋊C8C22×C411yescharacteristic
166C2×C67⋊C8C22×C411yescharacteristic
167C2×C67⋊C8C22×C411yescharacteristic
168C2×C67⋊C8C22×C411yescharacteristic
169C2×C67⋊C8C2×D411yescharacteristic
170C2×C67⋊C8C2×D411yescharacteristic
171C2×C67⋊C8C2×D411yescharacteristic
172C2×C67⋊C8C2×D411yescharacteristic
173C8×D67—81no—
174C8×D67—81no—
175C4⋊Dic67—21no—
176C4⋊Dic67C22×C411yescharacteristic
177C4⋊Dic67C22×C411yescharacteristic
178(C2×D67)⋊C4—21no—
179(C2×D67)⋊C4—21no—
180(C2×D67)⋊C4—21no—
181(C2×D67)⋊C4—21no—
182C2×Q8×C67—21no—
183C2×Q8×C67C22×C411yescharacteristic
184C2×Q8×C67C22×C411yescharacteristic
185C22×C4×C67C22×C411yescharacteristic
186C22×C4×C67C2×D411yescharacteristic
187C22×C4×C67#4111yescharacteristic
188C2×C8×C67—21no—
189C2×C8×C67—21no—
190C2×C8×C67—21no—
191C2×C8×C67—21no—
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—
211C22×D67—21no—
212C22×D67—42no—
213C22×D67—42no—
214C22×D67—81no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—21no—
219Q8×C67—42no—
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—41no—
229C2×C4×C67C4×D411yescharacteristic
230C2×C4×C67C4×D411yescharacteristic
231C2×C4×C67C4×D411yescharacteristic
232C2×C4×C67C23×C411yescharacteristic
233C2×C4×C67C22×D411yescharacteristic
234C2×C4×C67C2×#4111yescharacteristic
235C2×Dic67—21no—
236C2×Dic67—21no—
237C2×Dic67—81no—
238D268—21no—
239D268—21no—
240D268—21no—
241D268—42no—
242D268#13112yes—
243C4×D67—41no—
244C4×D67—41no—
245C4×D67—41no—
246C4×D67—41no—
247C4×D67—81no—
248C4×D67—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—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×C67—41no—
279C4×C67—41no—
280C4×C67—41no—
281C4×C67—41no—
282C4×C67C2×#13111yescharacteristic
283C4×C67#56811yescharacteristic
284C4×C67#57711yescharacteristic
285C4×C67#57711yescharacteristic
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#1572—671no—
307#1497D6711yescharacteristic, complemented
308#1478—671no—
309#1454—671no—
310C67#1802811yescharacteristic, complemented, Sylow
311#533C2×D6711yescharacteristic
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#431—671no—
343#431C2×D6711yescharacteristic
344#407—1341no—
345C2×C22⋊C8—1341no—
346#335—671no—
347#335—671no—
348#334—671no—
349#334—671no—
350Q8⋊C8C2×D6711yescharacteristic, complemented
351Q8⋊C8C2×D6711yescharacteristic, complemented
352Q8⋊C8C2×D6711yescharacteristic, complemented
353Q8⋊C8C2×D6711yescharacteristic, complemented
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⋊Q8C4×D6711yescharacteristic
373C4⋊Q8C4×D6711yescharacteristic
374#125—21no—
375C22⋊Q8—21no—
376C22⋊Q8C4×D6711yescharacteristic
377C22⋊Q8C4×D6711yescharacteristic
378C4×Q8—21no—
379C4×Q8C22×D6711yescharacteristic
380C4×Q8C22×D6711yescharacteristic
381C4×D4—671no—
382C4×D4—671no—
383C4×D4—671no—
384C4×D4—671no—
385C4×D4—1341no—
386C4×D4—1341no—
387#117—21no—
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—21no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—671no—
399C4⋊C8—671no—
400C4⋊C8—671no—
401C4⋊C8—671no—
402C4⋊C8—1341no—
403C4⋊C8—1341no—
404C4⋊C8—1341no—
405C4⋊C8—1341no—
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—671no—
421C22⋊C8—671no—
422C22⋊C8—1341no—
423C22⋊C8—1341no—
424C22⋊C8—1341no—
425C22⋊C8—1341no—
426C22⋊C8—1341no—
427#97—21no—
428#97—1341no—
429#97—1341no—
430C4×C8—671no—
431C4×C8—671no—
432C4×C8—671no—
433C4×C8—671no—
434C4×C8—1341no—
435C4×C8—1341no—
436C4×C8—1341no—
437C4×C8C22×D6711yescharacteristic
438C4×C8C22×D6711yescharacteristic
439C24—1341no—
440C2×Q8—21no—
441C2×Q8C2×C4×D6711yescharacteristic
442C2×Q8C2×C4×D6711yescharacteristic
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#1176162711yescharacteristic
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—671no—
475C2×C8—671no—
476C2×C8—671no—
477C2×C8—671no—
478C2×C8—1341no—
479C2×C8—1341no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—671no—
488C4⋊C4—671no—
489C4⋊C4—1341no—
490C4⋊C4C2×C4×D6711yescharacteristic
491C4⋊C4C2×C4×D6711yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—1341no—
495C22⋊C4—1341no—
496C22⋊C4—1341no—
497C22⋊C4—1341no—
498C42—21no—
499C42—21no—
500C42C2×C4×D6711yescharacteristic
501C42D4×D6711yescharacteristic
502C42#1176162711yescharacteristic
503C42C23×D6711yescharacteristic
504C23—2682no—
505C23—2682no—
506C23—5361no—
507C23—1341no—
508C23—1341no—
509C23(C22⋊D67)⋊C411yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—42no—
515D4—2682no—
516D4—672no—
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—2681no—
530C2×C4—2681no—
531C2×C4—2681no—
532C2×C4—2681no—
533C2×C4—5361no—
534C2×C4—5361no—
535C2×C4—5361no—
536C2×C4—1341no—
537C2×C4—1341no—
538C2×C4(C22⋊D67)⋊C411yescharacteristic
539C2×C4(C22⋊D67)⋊C411yescharacteristic
540C2×C4(C22⋊D67)⋊C411yescharacteristic
541C2×C4C22×C4×D6711yescharacteristic
542C2×C4C2×D4×D6711yescharacteristic
543C2×C4C2×#1176162711yescharacteristic
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—2681no—
549C8—2681no—
550C8—1341no—
551C8—1341no—
552C8—1341no—
553C8—1341no—
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×(C22⋊D67)⋊C411yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—2681no—
578C4—2681no—
579C4#131×D6711yescharacteristic
580C4#1686416111yescharacteristic
581C4#1686420511yescharacteristic
582C4#1686420511yescharacteristic
583C2—41no—
584C2—2682no—
585C2—5361no—
586C2#2256893911yescharacteristic
587C2#2256906411yescharacteristic
588C2#2256910811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1497C2
socleC22×C67C2×C4×D4
prime radicalC22×D268C2×C4
cosocleC42×C67C24
hypercenter#1497D67
hypocenterC67#18028
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
2. #22569311C2semidirect
4. #22559114C2semidirect
5. #22559096C2semidirect
6. #22559084C2semidirect
7. #22559069C2semidirect
8. #22559049C2semidirect
9. #22559048C2semidirect
10. #22559024C2semidirect
11. #22559006C2semidirect
12. #22558990C2semidirect
14. #22558509C2semidirect
15. C67×#1497C2semidirect
26. #16863527C22semidirect
27. #16863527C22semidirect
29. Q8×C67⋊C8C22semidirect
30. Q8×C67⋊C8C22semidirect
48. #16863148C22semidirect
49. #16863148C22semidirect
50. #16863146C22semidirect
51. #16863146C22semidirect
56. #16863118C22semidirect
57. #16863118C22semidirect
58. #16863117C22semidirect
59. C8×C67⋊C8C22semidirect
63. Q8⋊C8×C67C22semidirect
64. Q8⋊C8×C67C22semidirect
65. Q8⋊C8×C67C22semidirect
66. Q8⋊C8×C67C22semidirect
307. #1497D67semidirect
310. C67#18028semidirect
350. Q8⋊C8C2×D67semidirect
351. Q8⋊C8C2×D67semidirect
352. Q8⋊C8C2×D67semidirect
353. Q8⋊C8C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225702132
#225693112
#225692462
#225591142
#225590962
#225590842
#225590692
#225590492
#225590482
#225590242
#225590062
#225589902
#225586012
#225585092
C67×#14972
#1802867