Finite Group Databasebeta

← #30873583

Subgroups — #30873583

Order 17152.

810 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
1C2×#16863439C211yesmaximal, characteristic
2#22563076C211yesmaximal, characteristic
3#22560944C211yesmaximal, characteristic, complemented
4#22560939C211yesmaximal, characteristic, complemented
5#22560798C211yesmaximal, characteristic, complemented
6#22560794C211yesmaximal, characteristic, complemented
7#22560787C211yesmaximal, characteristic, complemented
8#22560777C211yesmaximal, characteristic, complemented
9#22560304C211yesmaximal, characteristic
10#22560297C211yesmaximal, characteristic
11#22560240C211yesmaximal, characteristic
12#22560107C211yesmaximal, characteristic, complemented
13C4×#14523974C211yesmaximal, characteristic, complemented
14#22559993C211yesmaximal, characteristic
15C67×#2347C211yesmaximal, characteristic
16C22×#11761612C2211yescharacteristic
17C2×#14523999—21no—
18C22×C4⋊Dic67C2211yescharacteristic
19#16863439C418yescomplemented
20C2×#14523976C2211yescharacteristic
21C2×#14523976C2211yescharacteristic
22C2×#14523976C2211yescharacteristic
23C2×#14523974C2211yescharacteristic
24C2×#14523974C2211yescharacteristic
25C2×#14523974C2211yescharacteristic
26C2×C22⋊(C67⋊C8)C2211yescharacteristic
27C2×#14523954C2211yescharacteristic
28C2×C4×C67⋊C8C2211yescharacteristic
29#16863294C2211yescharacteristic
30#16863294C2211yescharacteristic
31#16863281—21no—
32#16863281—21no—
33#16863257—21no—
34#16863256C2211yescharacteristic
35C4×C4⋊Dic67C2211yescharacteristic
36C4×C4⋊Dic67C2211yescharacteristic
37#16863158C2211yescharacteristic
38#16863158C2211yescharacteristic
39#16863158C2211yescharacteristic
40#16863158C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863151C2211yescharacteristic
48#16863151C2211yescharacteristic
49C2×#117×C67C2211yescharacteristic
50#389×C67C2211yescharacteristic
51#389×C67C2211yescharacteristic
52#388×C67—21no—
53#388×C67—21no—
54#383×C67—21no—
55#380×C67C2211yescharacteristic
56C4×C4⋊C4×C67—21no—
57C4×C4⋊C4×C67C2211yescharacteristic
58C4×C4⋊C4×C67C2211yescharacteristic
59C4×C22⋊C4×C67C2211yescharacteristic
60C23×Dic67—21no—
61C2×#11761612—22no—
62C2×#11761612—22no—
63C2×#11761612C2×C414yes—
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C22×C67⋊C8C2311yescharacteristic
67C22×C67⋊C8C2311yescharacteristic
68#14523999—44no—
69C2×C4⋊Dic67—21no—
70C2×C4⋊Dic67—22no—
71C2×C4⋊Dic67C2×C414yes—
72C2×C4⋊Dic67C2311yescharacteristic
73C2×C4⋊Dic67C2311yescharacteristic
74C2×C4⋊Dic67C2311yescharacteristic
75C2×C4⋊Dic67C2311yescharacteristic
76C2×Dic67⋊C4—21no—
77C2×Dic67⋊C4—21no—
78C2×C22⋊Dic67—21no—
79C2×C4×Dic67—21no—
80C2×C4×Dic67—21no—
81#14523976—22no—
82#14523976C2×C414yes—
83#14523976C2×C414yes—
84#14523974—22no—
85#14523974C2×C414yes—
86#14523974C2×C414yes—
87C22⋊(C67⋊C8)—22no—
88#14523954—22no—
89C4×C67⋊C8—22no—
90C67⋊#95—21no—
91C67⋊#95—21no—
92C67⋊#95—21no—
93C67⋊#95—21no—
94C67⋊#95—21no—
95C23×C4×C67C2311yescharacteristic
96#117×C67—22no—
97#117×C67C2×C414yes—
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67C2311yescharacteristic
101C2×C4⋊C4×C67C2311yescharacteristic
102C2×C4⋊C4×C67C2311yescharacteristic
103C2×C4⋊C4×C67C2311yescharacteristic
104C2×C22⋊C4×C67—21no—
105C2×C22⋊C4×C67—21no—
106C2×C22⋊C4×C67D411yescharacteristic
107C2×C22⋊C4×C67D411yescharacteristic
108C2×C42×C67—21no—
109C2×C42×C67—21no—
110C2×C42×C67—21no—
111C2×C42×C67C2311yescharacteristic
112C2×C42×C67C2311yescharacteristic
113#95×C67—21no—
114#95×C67—21no—
115#95×C67—21no—
116#95×C67—21no—
117#95×C67—21no—
118#95×C67D411yescharacteristic
119#95×C67D411yescharacteristic
120C22×Dic67—24no—
121C22×Dic67—21no—
122C22×Dic67—21no—
123C22×Dic67—21no—
124C22×Dic67—21no—
125C22×Dic67—42no—
126C22×Dic67—42no—
127#11761612—28no—
128C2×C67⋊C8—21no—
129C2×C67⋊C8—21no—
130C2×C67⋊C8—21no—
131C2×C67⋊C8—21no—
132C2×C67⋊C8—21no—
133C2×C67⋊C8—21no—
134C2×C67⋊C8—21no—
135C2×C67⋊C8—21no—
136C2×C67⋊C8C22×C412yes—
137C2×C67⋊C8C22×C412yes—
138C2×C67⋊C8#4112yes—
139C2×C67⋊C8#4112yes—
140C4⋊Dic67—24no—
141C4⋊Dic67—21no—
142C4⋊Dic67—21no—
143C4⋊Dic67—21no—
144C4⋊Dic67—21no—
145C4⋊Dic67—21no—
146C4⋊Dic67—21no—
147C4⋊Dic67C22×C412yes—
148C4⋊Dic67C22×C412yes—
149Dic67⋊C4—42no—
150Dic67⋊C4—42no—
151C22⋊Dic67—42no—
152C4×Dic67—42no—
153C4×Dic67—42no—
154C24×C67C2×D411yescharacteristic
155C22×C4×C67—21no—
156C22×C4×C67—21no—
157C22×C4×C67—21no—
158C22×C4×C67—21no—
159C22×C4×C67—21no—
160C22×C4×C67—21no—
161C22×C4×C67—21no—
162C22×C4×C67—21no—
163C22×C4×C67C22×C412yes—
164C22×C4×C67C2×D411yescharacteristic
165C22×C4×C67C2×D411yescharacteristic
166C22×C4×C67C2×D411yescharacteristic
167C22×C4×C67C2×D411yescharacteristic
168C22×C4×C67C2×D411yescharacteristic
169C22×C4×C67#4112yes—
170C22×C4×C67C2411yescharacteristic
171C4⋊C4×C67—21no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—21no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—21no—
177C4⋊C4×C67—42no—
178C4⋊C4×C67—42no—
179C4⋊C4×C67C22×C412yes—
180C4⋊C4×C67C22×C412yes—
181C22⋊C4×C67—24no—
182C22⋊C4×C67—22no—
183C22⋊C4×C67—41no—
184C22⋊C4×C67—81no—
185C42×C67—24no—
186C42×C67—21no—
187C42×C67—21no—
188C42×C67—21no—
189C42×C67—21no—
190C42×C67—81no—
191C42×C67—81no—
192C23×C67—21no—
193C23×C67—21no—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67—21no—
197C23×C67—21no—
198C23×C67C4×D412yes—
199C23×C67C22×D411yescharacteristic
200C2×C4×C67—21no—
201C2×C4×C67—21no—
202C2×C4×C67—21no—
203C2×C4×C67—21no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—22no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67C4×D412yes—
230C2×C4×C67AGL(1,ℤ/8)12yes—
231C2×C4×C67C23×C411yescharacteristic
232C2×C4×C67C2×#4111yescharacteristic
233C2×C4×C67C2×#4111yescharacteristic
234C2×C4×C67C2×#4111yescharacteristic
235C2×C4×C672−1+411yescharacteristic
236C2×C4×C672−1+411yescharacteristic
237C2×Dic67—22no—
238C2×Dic67—22no—
239C2×Dic67—48no—
240C2×Dic67—41no—
241C2×Dic67—41no—
242C2×Dic67—41no—
243C2×Dic67—41no—
244C2×Dic67—41no—
245C2×Dic67—41no—
246C2×Dic67—41no—
247C2×Dic67—41no—
248C2×Dic67—41no—
249C2×Dic67—41no—
250C67⋊C8—22no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8#11714yescomplemented
254C22×C67—22no—
255C22×C67—22no—
256C22×C67—22no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67C2×C4×D411yescharacteristic
262C22×C67#54711yescharacteristic
263C22×C67#54711yescharacteristic
264C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C67#57511yescharacteristic
266C22×C67#57711yescharacteristic
267C22×C67#57811yescharacteristic
268C4×C67—22no—
269C4×C67—22no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—42no—
274C4×C67—42no—
275C4×C67—81no—
276C4×C67—81no—
277C4×C67C2×#11711yescharacteristic
278C4×C67#51911yescharacteristic
279C4×C67#55211yescharacteristic
280C4×C67#55211yescharacteristic
281Dic67—42no—
282Dic67—42no—
283Dic67—81no—
284Dic67—81no—
285#22622—671nomaximal, Sylow
286C2×C67—42no—
287C2×C67#234411yescharacteristic
288C2×C67C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C67#294711yescharacteristic
290C2×C67#334511yescharacteristic
291C2×C67#334911yescharacteristic
292C2×C67#335111yescharacteristic
293C2×C67#335311yescharacteristic
294C2×#428—671no—
295#2347D6711yescharacteristic
296#2344—671no—
297#1943—671no—
298#1942—671no—
299#1921—671no—
300#1920—671no—
301#1919—671no—
302#1918—671no—
303#1825—671no—
304#1824—671no—
305#1809—671no—
306#1776—671no—
307C4×#106—671no—
308#1753—671no—
309C67#2262211yescharacteristic, complemented, Sylow
310C22×M16—671no—
311C2×#117—1341no—
312C2×#117C2×D6711yescharacteristic
313C22×C4⋊C4—671no—
314#428—678no—
315C2×#107—671no—
316C2×#107—671no—
317C2×#107—671no—
318C2×#106—671no—
319C2×#106—671no—
320C2×#106—671no—
321C2×C22⋊C8—671no—
322C2×#97—671no—
323C2×C4×C8—671no—
324#389—671no—
325#389—671no—
326#389C2×D6711yescharacteristic
327#389C2×D6711yescharacteristic
328#388—21no—
329#388—21no—
330#387—1341no—
331#387—1341no—
332#383—21no—
333#381—1341no—
334#380—671no—
335#380C2×D6711yescharacteristic
336C4×C4⋊C4—21no—
337C4×C4⋊C4—671no—
338C4×C4⋊C4—671no—
339C4×C4⋊C4C2×D6711yescharacteristic
340C4×C4⋊C4C2×D6711yescharacteristic
341C4×C22⋊C4C2×D6711yescharacteristic
342#340—671no—
343#340—671no—
344#340—671no—
345#340—671no—
346#340—671no—
347#340—671no—
348#340—671no—
349#340—671no—
350#340—671no—
351#340—671no—
352#336—671no—
353#336—671no—
354C23×C4—1341no—
355C23×C4C22×D6711yescharacteristic
356C2×M16—674no—
357C2×M16—1342no—
358C2×M16—1342no—
359C22×C8—671no—
360C22×C8—671no—
361C22×C8—671no—
362C22×C8—671no—
363#117—22no—
364#117—2684no—
365#117C4×D6714yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—674no—
369C2×C4⋊C4—671no—
370C2×C4⋊C4—671no—
371C2×C4⋊C4—671no—
372C2×C4⋊C4—671no—
373C2×C4⋊C4—1341no—
374C2×C4⋊C4—1341no—
375C2×C4⋊C4—1341no—
376C2×C4⋊C4—1342no—
377C2×C4⋊C4C22×D6711yescharacteristic
378C2×C4⋊C4C22×D6711yescharacteristic
379C2×C4⋊C4C22×D6711yescharacteristic
380C2×C4⋊C4C22×D6711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1341no—
384C2×C22⋊C4C22⋊D6711yescharacteristic
385C2×C22⋊C4C22⋊D6711yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—1341no—
390C2×C42—1341no—
391C2×C42C22×D6711yescharacteristic
392C2×C42C22×D6711yescharacteristic
393#107—674no—
394#107—674no—
395#107—1342no—
396#106—674no—
397#106—674no—
398#106—1342no—
399C22⋊C8—1342no—
400#97—1342no—
401C4×C8—1342no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1341no—
408#95—1341no—
409#95—1341no—
410#95—1341no—
411#95—1341no—
412#95C22⋊D6711yescharacteristic
413#95C22⋊D6711yescharacteristic
414C24C2×C22⋊D6711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—2682no—
424C22×C4—2682no—
425C22×C4—1341no—
426C22×C4—1341no—
427C22×C4—1344no—
428C22×C4—1341no—
429C22×C4—1341no—
430C22×C4C2×C4×D6712yes—
431C22×C4C2×C22⋊D6711yescharacteristic
432C22×C4C2×C22⋊D6711yescharacteristic
433C22×C4C2×C22⋊D6711yescharacteristic
434C22×C4C2×C22⋊D6711yescharacteristic
435C22×C4C2×C22⋊D6711yescharacteristic
436C22×C4#1176162812yes—
437C22×C4C23×D6711yescharacteristic
438M16—1348no—
439C2×C8—672no—
440C2×C8—672no—
441C2×C8—672no—
442C2×C8—672no—
443C2×C8—1341no—
444C2×C8—1341no—
445C2×C8—1341no—
446C2×C8—1341no—
447C2×C8—1341no—
448C2×C8—1341no—
449C2×C8—1341no—
450C2×C8—1341no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—2682no—
460C4⋊C4—2682no—
461C4⋊C4—672no—
462C4⋊C4—672no—
463C4⋊C4—1341no—
464C4⋊C4—1341no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1344no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4C2×C4×D6712yes—
471C4⋊C4C2×C4×D6712yes—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C22⋊C4—81no—
476C22⋊C4—2682no—
477C42—24no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—2682no—
485C42—2682no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D6712yes—
493C23C22×C22⋊D6711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—2688no—
524C2×C4—2681no—
525C2×C4—2681no—
526C2×C4—2681no—
527C2×C4—2681no—
528C2×C4—2681no—
529C2×C4—2681no—
530C2×C4—2681no—
531C2×C4—2681no—
532C2×C4—2681no—
533C2×C4—2681no—
534C2×C4—1342no—
535C2×C4—1342no—
536C2×C4C4×C22⋊D6712yes—
537C2×C4#1452406812yes—
538C2×C4C22×C4×D6711yescharacteristic
539C2×C4#41×D6711yescharacteristic
540C2×C4#41×D6711yescharacteristic
541C2×C4C2×#1176162811yescharacteristic
542C2×C4#1452409011yescharacteristic
543C2×C4#1452409011yescharacteristic
544C8—2681no—
545C8—2681no—
546C8—674no—
547C8—1342no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D6711yescharacteristic
556C22#1686404511yescharacteristic
557C22#1686404511yescharacteristic
558C22C2×#1452406811yescharacteristic
559C22#1686419411yescharacteristic
560C22#1686420711yescharacteristic
561C22#1686421111yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—2682no—
572C4—2682no—
573C4—5361no—
574C4—5361no—
575C4#117×D6711yescharacteristic
576C4#1686387311yescharacteristic
577C4#1686407911yescharacteristic
578C4#1686407911yescharacteristic
579C2—42no—
580C2#2256307911yescharacteristic
581C2C4×#1452406811yescharacteristic
582C2#2256904811yescharacteristic
583C2#2257267311yescharacteristic
584C2#2257273311yescharacteristic
585C2#2257276311yescharacteristic
586C2#2257278711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2347C2
socleC23×C67C22×D4
prime radicalC24×C67C2×D4
cosocleC22×C4×C67C24
hypercenter#2347D67
hypocenterC67#22622
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
3. #22560944C2semidirect
4. #22560939C2semidirect
5. #22560798C2semidirect
6. #22560794C2semidirect
7. #22560787C2semidirect
8. #22560777C2semidirect
12. #22560107C2semidirect
13. C4×#14523974C2semidirect
19. #16863439C4semidirect
253. C67⋊C8#117semidirect
309. C67#22622semidirect

Maximal subgroups

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

subgroupindex
C2×#168634392
#225630762
#225609442
#225609392
#225607982
#225607942
#225607872
#225607772
#225603042
#225602972
#225602402
#225601072
C4×#145239742
#225599932
C67×#23472
#2262267