Finite Group Databasebeta

← #30827169

Subgroups — #30827169

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#22570218C211yesmaximal, characteristic
2#22569310C211yesmaximal, characteristic, complemented
3#22569243C211yesmaximal, characteristic
4#22559113C211yesmaximal, characteristic, complemented
5#22559101C211yesmaximal, characteristic, complemented
6#22559089C211yesmaximal, characteristic, complemented
7#22559068C211yesmaximal, characteristic, complemented
8#22559054C211yesmaximal, characteristic, complemented
9#22559047C211yesmaximal, characteristic, complemented
10#22559023C211yesmaximal, characteristic, complemented
11C8×#11761617C211yesmaximal, characteristic, complemented
12#22558989C211yesmaximal, characteristic, complemented
13#22558683C211yesmaximal, characteristic, complemented
14#22558601C211yesmaximal, characteristic
15C67×#1454C211yesmaximal, characteristic, complemented
16#16863953C2211yescharacteristic
17C2×C4×D268C2211yescharacteristic
18#16863727C411yescharacteristic
19#16863727C411yescharacteristic
20#16863687C411yescharacteristic
21#16863687C411yescharacteristic
22#16863602C411yescharacteristic
23#16863602C411yescharacteristic
24#16863549C411yescharacteristic
25#16863549C411yescharacteristic
26#16863526C2211yescharacteristic, complemented
27#16863526C2211yescharacteristic, complemented
28#16863523—21no—
29C8×Dic134C2211yescharacteristic, complemented
30C8×Dic134C2211yescharacteristic, complemented
31#16863520—21no—
32C4×#11761617C2211yescharacteristic
33C4×#11761617C2211yescharacteristic
34C4×#11761617C2211yescharacteristic
35C4×#11761617C2211yescharacteristic
36#16863475C2211yescharacteristic
37#16863475C2211yescharacteristic
38#16863466—21no—
39#16863466—21no—
40C8×D268C2211yescharacteristic
41C8×D268C2211yescharacteristic
42#16863454—21no—
43#16863454—21no—
44#16863453C2211yescharacteristic
45#16863449C2211yescharacteristic
46#16863358—21no—
47C22⋊C8×D67—21no—
48#16863149C2211yescharacteristic, complemented
49#16863149C2211yescharacteristic, complemented
50#16863147C2211yescharacteristic, complemented
51#16863147C2211yescharacteristic, complemented
52#16863128C2211yescharacteristic, complemented
53#16863128C2211yescharacteristic, complemented
54#16863128C2211yescharacteristic, complemented
55#16863128C2211yescharacteristic, complemented
56#16863125C2211yescharacteristic
57#16863125C2211yescharacteristic
58#16863125C2211yescharacteristic
59#16863125C2211yescharacteristic
60#431×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63#322×C67C2211yescharacteristic, complemented
64#322×C67C2211yescharacteristic, complemented
65#322×C67C2211yescharacteristic, complemented
66C82×C67C2211yescharacteristic, complemented
67C22×D268C2×C411yescharacteristic
68C22×C4×D67—21no—
69C2×#11761617C2×C411yescharacteristic
70C2×#11761617C2×C411yescharacteristic
71C2×#11761617C2×C411yescharacteristic
72C2×#11761617C2×C411yescharacteristic
73C2×#11761611—21no—
74C2×#11761611—21no—
75C2×C8×D67—21no—
76C2×C8×D67—21no—
77#14524043C2×C411yescharacteristic
78#14524043C2×C411yescharacteristic
79#14524035—21no—
80#14524022—21no—
81#14524022C2×C411yescharacteristic
82#14524022C2×C411yescharacteristic
83C4×Dic134—21no—
84C4×Dic134C2311yescharacteristic
85C4×Dic134C2311yescharacteristic
86C4×D268—21no—
87C4×D268—21no—
88C4×D268C2×C411yescharacteristic
89C4×D268C2×C411yescharacteristic
90C4×D268C2311yescharacteristic
91C4×D268C2311yescharacteristic
92#14524002—21no—
93C2×C4⋊Dic67C2×C411yescharacteristic
94C2×(C2×D67)⋊C4—21no—
95#14523975C2×C411yescharacteristic
96#14523975C2×C411yescharacteristic
97#14523975C2×C411yescharacteristic
98#14523975C2×C411yescharacteristic
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—
107#14523968C2×C411yescharacteristic
108#14523968C2×C411yescharacteristic
109#14523968C2×C411yescharacteristic
110#14523968C2×C411yescharacteristic
111#14523965C2×C411yescharacteristic
112#14523965C2×C411yescharacteristic
113#14523965C2×C411yescharacteristic
114#14523965C2×C411yescharacteristic
115C22⋊(C67⋊C8)—21no—
116C22⋊(C67⋊C8)—21no—
117(C2×D67)⋊C8—21no—
118(C2×D67)⋊C8—21no—
119(C2×D67)⋊C8—21no—
120(C2×D67)⋊C8—21no—
121#14523954—21no—
122#14523953—21no—
123#14523953—21no—
124C4×C67⋊C8—21no—
125C8×Dic67—21no—
126C8×Dic67—21no—
127C2×C42×C67C2311yescharacteristic
128C4⋊C8×C67—21no—
129C4⋊C8×C67C2×C411yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67D411yescharacteristic
133C22⋊C8×C67—21no—
134C22⋊C8×C67C2×C411yescharacteristic
135C22⋊C8×C67C2×C411yescharacteristic
136C22⋊C8×C67D411yescharacteristic
137C22⋊C8×C67D411yescharacteristic
138C4×C8×C67C2311yescharacteristic
139C4×C8×C67C2311yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C23×D67—21no—
145C2×Dic134—21no—
146C2×Dic134C22×C411yescharacteristic
147C2×Dic134C22×C411yescharacteristic
148C2×D268—22no—
149C2×D268—21no—
150C2×D268—41no—
151C2×D268C22×C411yescharacteristic
152C2×D268C22×C411yescharacteristic
153C22×Dic67—21no—
154C2×C4×D67—21no—
155C2×C4×D67—21no—
156C2×C4×D67—21no—
157C2×C4×D67—21no—
158C2×C4×D67—81no—
159C2×C4×D67—81no—
160#11761617—22no—
161#11761617—22no—
162#11761617—41no—
163#11761617—41no—
164#11761611—81no—
165#11761611—81no—
166C2×C67⋊C8—21no—
167C2×C67⋊C8—21no—
168C2×C67⋊C8—21no—
169C2×C67⋊C8—21no—
170C8×D67—81no—
171C8×D67—81no—
172C4⋊Dic67—21no—
173C4⋊Dic67—21no—
174C4⋊Dic67—21no—
175C4⋊Dic67—21no—
176C4⋊Dic67C22×C411yescharacteristic
177C4⋊Dic67C22×C411yescharacteristic
178C4⋊Dic67C22×C411yescharacteristic
179C4⋊Dic67C22×C411yescharacteristic
180Dic67⋊C4—21no—
181Dic67⋊C4—21no—
182Dic67⋊C4—21no—
183Dic67⋊C4—21no—
184C22⋊Dic67—21no—
185C22⋊Dic67—21no—
186(C2×D67)⋊C4—21no—
187(C2×D67)⋊C4—21no—
188(C2×D67)⋊C4—21no—
189(C2×D67)⋊C4—21no—
190C4×Dic67—21no—
191C4×Dic67—21no—
192C22×C4×C67C22×C411yescharacteristic
193C22×C4×C67C2×D411yescharacteristic
194C22×C4×C67#4111yescharacteristic
195C2×C8×C67—21no—
196C2×C8×C67—21no—
197C2×C8×C67C22×C411yescharacteristic
198C2×C8×C67C22×C411yescharacteristic
199C2×C8×C67C22×C411yescharacteristic
200C2×C8×C67C22×C411yescharacteristic
201C2×C8×C67C2×D411yescharacteristic
202C2×C8×C67C2×D411yescharacteristic
203C2×C8×C67C2×D411yescharacteristic
204C2×C8×C67C2×D411yescharacteristic
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—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—41no—
220C2×C4×C67C4×D411yescharacteristic
221C2×C4×C67C4×D411yescharacteristic
222C2×C4×C67C4×D411yescharacteristic
223C2×C4×C67C23×C411yescharacteristic
224C2×C4×C67C22×D411yescharacteristic
225C2×C4×C67C2×#4111yescharacteristic
226C2×Dic67—21no—
227C2×Dic67—21no—
228C2×Dic67—21no—
229C2×Dic67—21no—
230C2×Dic67—21no—
231C2×Dic67—21no—
232C2×Dic67—81no—
233D268—21no—
234D268—21no—
235D268—21no—
236D268—42no—
237D268#13112yes—
238C4×D67—41no—
239C4×D67—41no—
240C4×D67—41no—
241C4×D67—41no—
242C4×D67—81no—
243C4×D67—81no—
244Dic134—21no—
245Dic134—21no—
246Dic134—21no—
247Dic134—21no—
248Dic134—42no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—21no—
254C8×C67—21no—
255C8×C67—21no—
256C8×C67—21no—
257C8×C67—41no—
258C8×C67—41no—
259C67⋊C8—41no—
260C67⋊C8—41no—
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×C67C2×#13111yescharacteristic
279C4×C67#56811yescharacteristic
280C4×C67#57711yescharacteristic
281C4×C67#57711yescharacteristic
282Dic67—41no—
283Dic67—41no—
284Dic67—41no—
285Dic67—41no—
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#1497—671no—
308#1478—671no—
309#1454D6711yescharacteristic, complemented
310C67#1802811yescharacteristic, complemented, Sylow
311#533—671no—
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#432—671no—
340#431—671no—
341#431C2×D6711yescharacteristic
342#431C2×D6711yescharacteristic
343#431C2×D6711yescharacteristic
344#407—1341no—
345C2×C22⋊C8—1341no—
346#335—671no—
347#335—671no—
348#334—671no—
349#334—671no—
350Q8⋊C8—671no—
351Q8⋊C8—671no—
352Q8⋊C8—671no—
353Q8⋊C8—671no—
354D4⋊C8—671no—
355D4⋊C8—671no—
356D4⋊C8—671no—
357D4⋊C8—671no—
358#322C2×D6711yescharacteristic, complemented
359#322C2×D6711yescharacteristic, complemented
360#322C2×D6711yescharacteristic, complemented
361C82C2×D6711yescharacteristic, complemented
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⋊Q8—671no—
373C4⋊Q8—671no—
374#125—1341no—
375C22⋊Q8—671no—
376C22⋊Q8—671no—
377C22⋊Q8—1341no—
378C4×Q8—671no—
379C4×Q8—671no—
380C4×Q8—1341no—
381C4×D4—671no—
382C4×D4—671no—
383C4×D4—671no—
384C4×D4—671no—
385C4×D4—1341no—
386C4×D4—1341no—
387#117—1341no—
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—1341no—
401C4⋊C8—1341no—
402C4⋊C8—1341no—
403C4⋊C8—1341no—
404C4⋊C8C4×D6711yescharacteristic
405C4⋊C8C4×D6711yescharacteristic
406C4⋊C8D26811yescharacteristic
407C4⋊C8D26811yescharacteristic
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—1341no—
418C22⋊C8—1341no—
419C22⋊C8—1341no—
420C22⋊C8—1341no—
421C22⋊C8—1341no—
422C22⋊C8—1341no—
423C22⋊C8C4×D6711yescharacteristic
424C22⋊C8C4×D6711yescharacteristic
425C22⋊C8D26811yescharacteristic
426C22⋊C8D26811yescharacteristic
427#97—1341no—
428#97—1341no—
429#97—1341no—
430C4×C8—1341no—
431C4×C8—1341no—
432C4×C8—1341no—
433C4×C8C22×D6711yescharacteristic
434C4×C8C22×D6711yescharacteristic
435C4×C8C22×D6711yescharacteristic
436C4×C8C22×D6711yescharacteristic
437C4×C8C22×D6711yescharacteristic
438C4×C8C22×D6711yescharacteristic
439C24—1341no—
440C2×Q8—671no—
441C2×Q8—671no—
442C2×Q8—1341no—
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×C4C2×D26811yescharacteristic
457C22×C4#1176162811yescharacteristic
458SD16—2681no—
459SD16—2681no—
460SD16—1342no—
461SD16—1342no—
462M16—5361no—
463M16—5361no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—5361no—
467C2×C8—5361no—
468C2×C8—1341no—
469C2×C8—1341no—
470C2×C8—1341no—
471C2×C8—1341no—
472C2×C8C2×C4×D6711yescharacteristic
473C2×C8C2×C4×D6711yescharacteristic
474C2×C8C2×C4×D6711yescharacteristic
475C2×C8C2×C4×D6711yescharacteristic
476C2×C8C2×D26811yescharacteristic
477C2×C8C2×D26811yescharacteristic
478C2×C8C2×D26811yescharacteristic
479C2×C8C2×D26811yescharacteristic
480C4⋊C4—671no—
481C4⋊C4—671no—
482C4⋊C4—671no—
483C4⋊C4—671no—
484C4⋊C4—1341no—
485C4⋊C4—1341no—
486C4⋊C4—1341no—
487C4⋊C4—1341no—
488C4⋊C4—1341no—
489C4⋊C4—1341no—
490C4⋊C4—1341no—
491C4⋊C4—1341no—
492C22⋊C4—1341no—
493C22⋊C4—1341no—
494C22⋊C4—1341no—
495C22⋊C4—1341no—
496C22⋊C4—1341no—
497C22⋊C4—1341no—
498C42—1341no—
499C42—1341no—
500C42C2×C4×D6711yescharacteristic
501C42C2×D26811yescharacteristic
502C42#1176162811yescharacteristic
503C42C23×D6711yescharacteristic
504C23—2682no—
505C23—2682no—
506C23—5361no—
507C23—1341no—
508C23—1341no—
509C23C4×D26811yescharacteristic
510Q8—2682no—
511Q8—1341no—
512Q8—1341no—
513Q8—1341no—
514Q8—1341no—
515D4—2682no—
516D4—672no—
517D4—1341no—
518D4—1341no—
519D4—1341no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—2681no—
526C2×C4—2681no—
527C2×C4—2681no—
528C2×C4—2681no—
529C2×C4—5361no—
530C2×C4—5361no—
531C2×C4—5361no—
532C2×C4—1341no—
533C2×C4—1341no—
534C2×C4—1341no—
535C2×C4—1341no—
536C2×C4—1341no—
537C2×C4—1341no—
538C2×C4C4×D26811yescharacteristic
539C2×C4C4×D26811yescharacteristic
540C2×C4C4×D26811yescharacteristic
541C2×C4C22×C4×D6711yescharacteristic
542C2×C4C22×D26811yescharacteristic
543C2×C4C2×#1176162811yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—2681no—
555C8—2681no—
556C8—2681no—
557C8—2681no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—2681no—
562C22—2681no—
563C22—2681no—
564C22—2681no—
565C22—2681no—
566C22—5364no—
567C22—1342no—
568C22C2×C4×D26811yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—2681no—
574C4—2681no—
575C4—2681no—
576C4—2681no—
577C4—2681no—
578C4—2681no—
579C4#131×D6711yescharacteristic
580C4#1686416111yescharacteristic
581C4#1686420811yescharacteristic
582C4#1686420811yescharacteristic
583C2—41no—
584C2—2682no—
585C2—5361no—
586C2#2256893711yescharacteristic
587C2#2256907011yescharacteristic
588C2#2256911311yescharacteristic
subgroupG/H
centerC2×C4C22×D268
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1454C2
socleC22×C67C2×C4×D4
prime radicalC22×D268C2×C4
cosocleC42×C67C24
hypercenter#1454D67
hypocenterC67#18028
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #22569310C2semidirect
4. #22559113C2semidirect
5. #22559101C2semidirect
6. #22559089C2semidirect
7. #22559068C2semidirect
8. #22559054C2semidirect
9. #22559047C2semidirect
10. #22559023C2semidirect
11. C8×#11761617C2semidirect
12. #22558989C2semidirect
13. #22558683C2semidirect
15. C67×#1454C2semidirect
26. #16863526C22semidirect
27. #16863526C22semidirect
29. C8×Dic134C22semidirect
30. C8×Dic134C22semidirect
48. #16863149C22semidirect
49. #16863149C22semidirect
50. #16863147C22semidirect
51. #16863147C22semidirect
52. #16863128C22semidirect
53. #16863128C22semidirect
54. #16863128C22semidirect
55. #16863128C22semidirect
63. #322×C67C22semidirect
64. #322×C67C22semidirect
65. #322×C67C22semidirect
66. C82×C67C22semidirect
309. #1454D67semidirect
310. C67#18028semidirect
358. #322C2×D67semidirect
359. #322C2×D67semidirect
360. #322C2×D67semidirect
361. C82C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225702182
#225693102
#225692432
#225591132
#225591012
#225590892
#225590682
#225590542
#225590472
#225590232
C8×#117616172
#225589892
#225586832
#225586012
C67×#14542
#1802867