Finite Group Databasebeta

← #30826402

Subgroups — #30826402

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570773C211yesmaximal, characteristic
2#22569184C211yesmaximal, characteristic, complemented
3#22559121C211yesmaximal, characteristic, complemented
4#22559116C211yesmaximal, characteristic, complemented
5#22559111C211yesmaximal, characteristic, complemented
6#22559109C211yesmaximal, characteristic, complemented
7#22559031C211yesmaximal, characteristic, complemented
8#22559026C211yesmaximal, characteristic, complemented
9#22559021C211yesmaximal, characteristic, complemented
10#22559019C211yesmaximal, characteristic, complemented
11#22558970C211yesmaximal, characteristic, complemented
12#22558669C211yesmaximal, characteristic, complemented
13#22558667C211yesmaximal, characteristic, complemented
14C2×#16863118C211yesmaximal, characteristic, complemented
15C67×#2966C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863795C411yescharacteristic
18#16863778C411yescharacteristic
19#16863753C411yescharacteristic
20#16863750C411yescharacteristic
21#16863635C411yescharacteristic
22#16863630C411yescharacteristic
23#16863627C411yescharacteristic
24#16863623C411yescharacteristic
25#16863526C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C4×#11761618C2211yescharacteristic
28C4×#11761615C2211yescharacteristic
29C4×#11761616C2211yescharacteristic
30C4×#11761613C2211yescharacteristic
31#16863475C2211yescharacteristic
32#16863467—21no—
33C8×D268C2211yescharacteristic
34C8×C22⋊D67—21no—
35#16863455—21no—
36#16863453C2211yescharacteristic
37#16863350—21no—
38C2×C4×C67⋊C8C2211yescharacteristic
39C2×C4×C67⋊C8C2211yescharacteristic
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46#16863126C2211yescharacteristic
47#16863126C2211yescharacteristic
48D4⋊(C67⋊C8)C2211yescharacteristic
49D4⋊(C67⋊C8)C2211yescharacteristic
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863118C2211yescharacteristic
55#16863118C2211yescharacteristic
56#529×C67C2211yescharacteristic
57#446×C67C2211yescharacteristic
58C8×Q8×C67C2211yescharacteristic
59#436×C67C2211yescharacteristic
60#435×C67—21no—
61C8×D4×C67—21no—
62C8×D4×C67C2211yescharacteristic
63#433×C67—21no—
64#432×C67C2211yescharacteristic
65#405×C67—21no—
66C2×C4×C8×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761616C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761613C2×C411yescharacteristic
71C2×#11761611—21no—
72C22×C67⋊C8—21no—
73C22×C67⋊C8C2×C411yescharacteristic
74C22×C67⋊C8C2×C411yescharacteristic
75C2×C8×D67—21no—
76#14524037—21no—
77#14524035C2×C411yescharacteristic
78#14524031C2×C411yescharacteristic
79#14524026—21no—
80#14524022C2×C411yescharacteristic
81#14524017C2×C411yescharacteristic
82C4×Dic134C2311yescharacteristic
83C4×D268C2311yescharacteristic
84C4×C22⋊D67—21no—
85#14524000—21no—
86#14523976C2×C411yescharacteristic
87#14523976C2×C411yescharacteristic
88#14523974C2×C411yescharacteristic
89#14523974C2×C411yescharacteristic
90C4⋊(C67⋊C8)C2311yescharacteristic
91C4⋊(C67⋊C8)C2311yescharacteristic
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94#14523967C2×C411yescharacteristic
95#14523967C2×C411yescharacteristic
96#14523966C2×C411yescharacteristic
97#14523966C2×C411yescharacteristic
98#14523964C2×C411yescharacteristic
99#14523964C2×C411yescharacteristic
100#14523963C2×C411yescharacteristic
101#14523963C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103(C2×D67)⋊C8—21no—
104(C2×D67)⋊C8—21no—
105#14523954—21no—
106#14523953—21no—
107C4×C67⋊C8—21no—
108C4×C67⋊C8C2×C411yescharacteristic
109C4×C67⋊C8C2×C411yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C4×C67⋊C8C2311yescharacteristic
114C8×Dic67—21no—
115C2×M16×C67—21no—
116C22×C8×C67—21no—
117C22×C8×C67D411yescharacteristic
118C22×C8×C67D411yescharacteristic
119#126×C67—21no—
120#125×C67C2×C411yescharacteristic
121#124×C67C2×C411yescharacteristic
122#123×C67—21no—
123C22⋊Q8×C67C2×C411yescharacteristic
124#121×C67C2×C411yescharacteristic
125C4×Q8×C67C2311yescharacteristic
126C4×D4×C67—21no—
127C4×D4×C67C2311yescharacteristic
128#117×C67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67—21no—
137#97×C67—21no—
138#97×C67—21no—
139C4×C8×C67—21no—
140C4×C8×C67D411yescharacteristic
141C4×C8×C67D411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134C22×C411yescharacteristic
145C2×D268C22×C411yescharacteristic
146C2×C22⋊D67—21no—
147C2×C4×D67—21no—
148#11761618—22no—
149#11761616—22no—
150#11761615—22no—
151#11761613—22no—
152#11761611—42no—
153C2×C67⋊C8—22no—
154C2×C67⋊C8—22no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—42no—
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C8×D67—42no—
165C4⋊Dic67C22×C411yescharacteristic
166C4⋊Dic67C22×C411yescharacteristic
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171C22⋊Dic67—21no—
172(C2×D67)⋊C4—21no—
173(C2×D67)⋊C4—21no—
174(C2×D67)⋊C4—21no—
175(C2×D67)⋊C4—21no—
176C4×Dic67—21no—
177C2×Q8×C67C22×C411yescharacteristic
178C2×D4×C67—21no—
179C2×D4×C67C22×C411yescharacteristic
180C22×C4×C67—21no—
181C22×C4×C67C22×C411yescharacteristic
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67#4111yescharacteristic
184M16×C67—42no—
185C2×C8×C67—21no—
186C2×C8×C67—21no—
187C2×C8×C67—41no—
188C2×C8×C67—41no—
189C2×C8×C67—42no—
190C2×C8×C67C2×D411yescharacteristic
191C2×C8×C67C2×D411yescharacteristic
192C2×C8×C67C2×D411yescharacteristic
193C2×C8×C67C2×D411yescharacteristic
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67C22×C411yescharacteristic
199C4⋊C4×C67C22×C411yescharacteristic
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C22⋊C4×C67—21no—
204C22⋊C4×C67—21no—
205C42×C67—21no—
206C42×C67C22×C411yescharacteristic
207C42×C67C2×D411yescharacteristic
208C42×C67#4111yescharacteristic
209C42×C67C2411yescharacteristic
210C23×C67—21no—
211C23×C67C4×D411yescharacteristic
212C22×D67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215D4×C67—21no—
216D4×C67—21no—
217D4×C67—42no—
218C2×C4×C67—22no—
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—21no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
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—42no—
235C2×Dic67—21no—
236C2×Dic67—21no—
237C2×Dic67—21no—
238D268—21no—
239D268—21no—
240C4×D67—41no—
241C4×D67—41no—
242Dic134—21no—
243Dic134—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—41no—
249C8×C67—41no—
250C67⋊C8—21no—
251C67⋊C8—21no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C67⋊C8—41no—
256C67⋊C8#13112yes—
257C67⋊C8#13112yes—
258C22×C67—21no—
259C22×C67—21no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67#44412yes—
264C22×C67C2×C4×D411yescharacteristic
265C2×D67—41no—
266C2×D67—41no—
267C2×D67—41no—
268C4×C67—21no—
269C4×C67—21no—
270C4×C67—21no—
271C4×C67—21no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67C2×#13111yescharacteristic
276C4×C67C2×#13111yescharacteristic
277C4×C67#57511yescharacteristic
278C4×C67#57511yescharacteristic
279Dic67—41no—
280Dic67—41no—
281Dic67—41no—
282#17963—671nomaximal, Sylow
283C2×C67—22no—
284C2×C67—81no—
285C2×C67#293111yescharacteristic
286C2×C67#294311yescharacteristic
287C2×C67C2×#44411yescharacteristic
288D67—81no—
289#3168—671no—
290#2966—671no—
291#2966D6711yescharacteristic, complemented
292#1594—671no—
293#1593—671no—
294#1592—671no—
295#1592—671no—
296#1580—671no—
297#1579—671no—
298SD16⋊C8—671no—
299SD16⋊C8—671no—
300#1568—671no—
301#1494—671no—
302#1494—671no—
303C2×#322—671no—
304C67#1796311yescharacteristic, complemented, Sylow
305#529—671no—
306#529C2×D6711yescharacteristic
307#499—671no—
308#495—671no—
309#488—671no—
310#488—671no—
311#467—671no—
312#466—671no—
313#465—671no—
314#465—671no—
315#446—671no—
316#446C2×D6711yescharacteristic
317C8×Q8—671no—
318C8×Q8C2×D6711yescharacteristic
319C4×Q16—671no—
320C4×SD16—671no—
321C4×SD16—671no—
322C4×D8—671no—
323#436—671no—
324#436C2×D6711yescharacteristic
325#435—21no—
326#435—1341no—
327C8×D4—21no—
328C8×D4—671no—
329C8×D4—1341no—
330C8×D4C2×D6711yescharacteristic
331#433—21no—
332#433—1341no—
333#432—671no—
334#432C2×D6711yescharacteristic
335#405—21no—
336#405—1341no—
337C2×C4×C8—671no—
338C2×C4×C8—671no—
339C2×C4×C8C2×D6711yescharacteristic
340#335—671no—
341#335—671no—
342#334—671no—
343#334—671no—
344Q8⋊C8—671no—
345Q8⋊C8—671no—
346Q8⋊C8—671no—
347Q8⋊C8—671no—
348D4⋊C8—671no—
349D4⋊C8—671no—
350D4⋊C8—671no—
351D4⋊C8—671no—
352#322—671no—
353#322—671no—
354#322—671no—
355#322—671no—
356C2×Q16—671no—
357C2×SD16—671no—
358C2×SD16—671no—
359C2×D8—671no—
360C2×M16—21no—
361C2×M16—1341no—
362C22×C8—21no—
363C22×C8—671no—
364C22×C8—671no—
365C22×C8—1341no—
366C22×C8—1341no—
367C22×C8C22⋊D6711yescharacteristic
368C22×C8C22⋊D6711yescharacteristic
369#126—21no—
370#126—1341no—
371#125—671no—
372#125C4×D6711yescharacteristic
373#124—671no—
374#124C4×D6711yescharacteristic
375#123—21no—
376#123—1341no—
377C22⋊Q8—671no—
378C22⋊Q8C4×D6711yescharacteristic
379#121—671no—
380#121C4×D6711yescharacteristic
381C4×Q8—671no—
382C4×Q8C22×D6711yescharacteristic
383C4×D4—21no—
384C4×D4—671no—
385C4×D4—1341no—
386C4×D4C22×D6711yescharacteristic
387#117—21no—
388#117—1341no—
389C2×C42C22×D6711yescharacteristic
390#107—671no—
391#107—671no—
392#106—671no—
393#106—671no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—1341no—
399C4⋊C8—1341no—
400C4⋊C8C22×D6711yescharacteristic
401C4⋊C8C22×D6711yescharacteristic
402#103—671no—
403#103—671no—
404#103—671no—
405#103—671no—
406#102—671no—
407#102—671no—
408#102—671no—
409#102—671no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—1341no—
414C22⋊C8—1341no—
415C22⋊C8—1341no—
416#97—21no—
417#97—21no—
418#97—1341no—
419#97—1341no—
420C4×C8—21no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—671no—
424C4×C8—671no—
425C4×C8—671no—
426C4×C8—671no—
427C4×C8—1341no—
428C4×C8—1341no—
429C4×C8C22⋊D6711yescharacteristic
430C4×C8C22⋊D6711yescharacteristic
431C4×C8C22×D6711yescharacteristic
432C4×C8C22×D6711yescharacteristic
433C2×Q8—671no—
434C2×Q8C2×C4×D6711yescharacteristic
435C2×D4—21no—
436C2×D4—671no—
437C2×D4—1341no—
438C2×D4C2×C4×D6711yescharacteristic
439C22×C4—21no—
440C22×C4—1341no—
441C22×C4C2×C4×D6711yescharacteristic
442C22×C4C2×C22⋊D6711yescharacteristic
443C22×C4#1176162811yescharacteristic
444Q16—1342no—
445SD16—1342no—
446SD16—1342no—
447D8—1342no—
448M16—42no—
449M16—2682no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—2682no—
456C2×C8—2682no—
457C2×C8—671no—
458C2×C8—671no—
459C2×C8—671no—
460C2×C8—671no—
461C2×C8—1341no—
462C2×C8—1341no—
463C2×C8—1341no—
464C2×C8—1341no—
465C2×C8—1342no—
466C2×C8—1342no—
467C2×C8C2×C22⋊D6711yescharacteristic
468C2×C8C2×C22⋊D6711yescharacteristic
469C2×C8C2×C22⋊D6711yescharacteristic
470C2×C8C2×C22⋊D6711yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—671no—
476C4⋊C4—671no—
477C4⋊C4—1341no—
478C4⋊C4—1341no—
479C4⋊C4—1341no—
480C4⋊C4—1341no—
481C4⋊C4C2×C4×D6711yescharacteristic
482C4⋊C4C2×C4×D6711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1341no—
489C22⋊C4—1341no—
490C22⋊C4—1341no—
491C22⋊C4—1341no—
492C22⋊C4—1341no—
493C42—21no—
494C42—1341no—
495C42C2×C4×D6711yescharacteristic
496C42C2×C22⋊D6711yescharacteristic
497C42#1176162811yescharacteristic
498C42C23×D6711yescharacteristic
499C23—21no—
500C23—1341no—
501C23C4×C22⋊D6711yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—1341no—
505Q8—1341no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—2682no—
510D4—1341no—
511D4—1341no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2681no—
523C2×C4—2681no—
524C2×C4—1341no—
525C2×C4—1341no—
526C2×C4—1341no—
527C2×C4C4×C22⋊D6711yescharacteristic
528C2×C4C4×C22⋊D6711yescharacteristic
529C2×C4C4×C22⋊D6711yescharacteristic
530C2×C4C22×C4×D6711yescharacteristic
531C2×C4C22×C22⋊D6711yescharacteristic
532C2×C4C2×#1176162811yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—41no—
538C8—41no—
539C8—2681no—
540C8—2681no—
541C8—2681no—
542C8—2681no—
543C8—672no—
544C8—672no—
545C8—1341no—
546C8—1341no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—2681no—
553C22—2681no—
554C22—2681no—
555C22#1686351612yes—
556C22C2×C4×C22⋊D6711yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—2681no—
565C4—2681no—
566C4—2681no—
567C4#131×D6711yescharacteristic
568C4#131×D6711yescharacteristic
569C4#1686419411yescharacteristic
570C4#1686419411yescharacteristic
571C2—22no—
572C2—81no—
573C2—5361no—
574C2#2256891911yescharacteristic
575C2#2256900711yescharacteristic
576C2C2×#1686351611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2966C2
socleC22×C67C2×C4×D4
prime radical#16863630C4
cosocleC42×C67C24
hypercenter#2966D67
hypocenterC67#17963
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #22569184C2semidirect
3. #22559121C2semidirect
4. #22559116C2semidirect
5. #22559111C2semidirect
6. #22559109C2semidirect
7. #22559031C2semidirect
8. #22559026C2semidirect
9. #22559021C2semidirect
10. #22559019C2semidirect
11. #22558970C2semidirect
12. #22558669C2semidirect
13. #22558667C2semidirect
14. C2×#16863118C2semidirect
15. C67×#2966C2semidirect
291. #2966D67semidirect
304. C67#17963semidirect

Maximal subgroups

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

subgroupindex
#225707732
#225691842
#225591212
#225591162
#225591112
#225591092
#225590312
#225590262
#225590212
#225590192
#225589702
#225586692
#225586672
C2×#168631182
C67×#29662
#1796367