Finite Group Databasebeta

← #30825788

Subgroups — #30825788

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570227C211yesmaximal, characteristic
2#22569184C211yesmaximal, characteristic, complemented
3#22559087C211yesmaximal, characteristic, complemented
4#22559080C211yesmaximal, characteristic, complemented
5#22559073C211yesmaximal, characteristic, complemented
6#22559066C211yesmaximal, characteristic, complemented
7#22559031C211yesmaximal, characteristic, complemented
8#22559026C211yesmaximal, characteristic, complemented
9#22559021C211yesmaximal, characteristic, complemented
10#22559019C211yesmaximal, characteristic, complemented
11#22558987C211yesmaximal, characteristic, complemented
12#22558643C211yesmaximal, characteristic, complemented
13#22558615C211yesmaximal, characteristic, complemented
14#22558501C211yesmaximal, characteristic, complemented
15C8×#41×C67C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863737C411yescharacteristic
18#16863720C411yescharacteristic
19#16863703C411yescharacteristic
20#16863696C411yescharacteristic
21#16863618C411yescharacteristic
22#16863611C411yescharacteristic
23#16863558C411yescharacteristic
24#16863541C411yescharacteristic
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#16863449C2211yescharacteristic
38#16863449C2211yescharacteristic
39#16863350—21no—
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
56C4×#41×C67C2211yescharacteristic
57C8×Q8×C67C2211yescharacteristic
58C8×Q8×C67C2211yescharacteristic
59C8×D4×C67—21no—
60C8×D4×C67—21no—
61C8×D4×C67C2211yescharacteristic
62C8×D4×C67C2211yescharacteristic
63#431×C67—21no—
64#431×C67C2211yescharacteristic
65C2×C4×C8×C67—21no—
66C2×C4×C8×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761616C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761613C2×C411yescharacteristic
71C2×#11761611—21no—
72C2×C8×D67—21no—
73#14524037—21no—
74#14524035C2×C411yescharacteristic
75#14524031C2×C411yescharacteristic
76#14524026—21no—
77#14524022C2×C411yescharacteristic
78#14524017C2×C411yescharacteristic
79C4×Dic134C2311yescharacteristic
80C4×D268C2311yescharacteristic
81C4×C22⋊D67—21no—
82#14524000—21no—
83#14523976C2×C411yescharacteristic
84#14523976C2×C411yescharacteristic
85#14523974C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87C4⋊(C67⋊C8)—21no—
88C4⋊(C67⋊C8)C2×C411yescharacteristic
89C4⋊(C67⋊C8)C2×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—
103C22⋊(C67⋊C8)—21no—
104C22⋊(C67⋊C8)C2×C411yescharacteristic
105C22⋊(C67⋊C8)C2×C411yescharacteristic
106(C2×D67)⋊C8—21no—
107(C2×D67)⋊C8—21no—
108#14523954—21no—
109#14523953—21no—
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C4×C67⋊C8C2311yescharacteristic
114C8×Dic67—21no—
115C2×#41×C67C2×C411yescharacteristic
116C22×C8×C67—21no—
117C22×C8×C67—21no—
118C22×C8×C67D411yescharacteristic
119C22×C8×C67D411yescharacteristic
120C4×Q8×C67C2×C411yescharacteristic
121C4×Q8×C67C2311yescharacteristic
122C4×D4×C67—21no—
123C4×D4×C67—21no—
124C4×D4×C67C2×C411yescharacteristic
125C4×D4×C67C2311yescharacteristic
126#117×C67—21no—
127#117×C67C2×C411yescharacteristic
128C2×C42×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—
137C4×C8×C67—21no—
138C4×C8×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—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C8×D67—42no—
162C4⋊Dic67C22×C411yescharacteristic
163C4⋊Dic67C22×C411yescharacteristic
164Dic67⋊C4—21no—
165Dic67⋊C4—21no—
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168C22⋊Dic67—21no—
169(C2×D67)⋊C4—21no—
170(C2×D67)⋊C4—21no—
171(C2×D67)⋊C4—21no—
172(C2×D67)⋊C4—21no—
173C4×Dic67—21no—
174#41×C67—22no—
175#41×C67—22no—
176C2×Q8×C67C22×C411yescharacteristic
177C2×D4×C67—21no—
178C2×D4×C67C22×C411yescharacteristic
179C22×C4×C67—21no—
180C22×C4×C67—21no—
181C22×C4×C67—21no—
182C22×C4×C67C22×C411yescharacteristic
183C22×C4×C67C2×D411yescharacteristic
184C22×C4×C67#4111yescharacteristic
185C2×C8×C67—21no—
186C2×C8×C67—21no—
187C2×C8×C67—41no—
188C2×C8×C67—41no—
189C2×C8×C67—42no—
190C2×C8×C67—42no—
191C2×C8×C67C2×D411yescharacteristic
192C2×C8×C67C2×D411yescharacteristic
193C2×C8×C67C2×D411yescharacteristic
194C2×C8×C67C2×D411yescharacteristic
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67C22×C411yescharacteristic
198C4⋊C4×C67C22×C411yescharacteristic
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67C4×D411yescharacteristic
211C22×D67—21no—
212Q8×C67—21no—
213Q8×C67#13112yes—
214D4×C67—24no—
215D4×C67—21no—
216D4×C67#13112yes—
217C2×C4×C67—24no—
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—41no—
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
235C22⋊D67—42no—
236C2×Dic67—21no—
237C2×Dic67—21no—
238C2×Dic67—21no—
239D268—21no—
240D268—21no—
241C4×D67—41no—
242C4×D67—41no—
243Dic134—21no—
244Dic134—21no—
245C8×C67—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—41no—
250C8×C67—41no—
251C67⋊C8—21no—
252C67⋊C8—21no—
253C67⋊C8—21no—
254C67⋊C8—21no—
255C67⋊C8—41no—
256C67⋊C8—41no—
257C67⋊C8—41no—
258C67⋊C8—41no—
259C22×C67—21no—
260C22×C67—22no—
261C22×C67—21no—
262C22×C67—21no—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67C2×C4×D411yescharacteristic
266C2×D67—41no—
267C2×D67—41no—
268C2×D67—41no—
269C4×C67—21no—
270C4×C67—22no—
271C4×C67—21no—
272C4×C67—21no—
273C4×C67—21no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67C2×#13111yescharacteristic
277C4×C67C2×#13111yescharacteristic
278C4×C67#57511yescharacteristic
279C4×C67#57511yescharacteristic
280Dic67—41no—
281Dic67—41no—
282Dic67—41no—
283#17909—671nomaximal, Sylow
284C2×C67—41no—
285C2×C67—42no—
286C2×C67#292911yescharacteristic
287C2×C67#294311yescharacteristic
288C2×C67#295511yescharacteristic
289D67—81no—
290#3102—671no—
291#2966—671no—
292C8×#41D6711yescharacteristic, complemented
293#1588—671no—
294#1587—671no—
295#1586—671no—
296#1585—671no—
297#1580—671no—
298#1579—671no—
299SD16⋊C8—671no—
300SD16⋊C8—671no—
301#1572—671no—
302#1488—671no—
303#1482—671no—
304#1452—671no—
305C67#1790911yescharacteristic, complemented, Sylow
306#529—671no—
307C4×#41C2×D6711yescharacteristic
308#485—671no—
309#482—671no—
310#479—671no—
311#478—671no—
312#464—671no—
313#463—671no—
314#452—671no—
315#449—671no—
316#446—671no—
317C8×Q8—671no—
318C8×Q8C2×D6711yescharacteristic
319C8×Q8C2×D6711yescharacteristic
320C4×Q16—671no—
321C4×SD16—671no—
322C4×SD16—671no—
323C4×D8—671no—
324#436—671no—
325#435—1341no—
326C8×D4—21no—
327C8×D4—21no—
328C8×D4—671no—
329C8×D4—1341no—
330C8×D4C2×D6711yescharacteristic
331C8×D4C2×D6711yescharacteristic
332#433—1341no—
333#432—671no—
334#431—21no—
335#431—671no—
336#431—671no—
337#431C2×D6711yescharacteristic
338#405—1341no—
339C2×C4×C8—21no—
340C2×C4×C8C2×D6711yescharacteristic
341#335—671no—
342#335—671no—
343#334—671no—
344#334—671no—
345Q8⋊C8—671no—
346Q8⋊C8—671no—
347Q8⋊C8—671no—
348Q8⋊C8—671no—
349D4⋊C8—671no—
350D4⋊C8—671no—
351D4⋊C8—671no—
352D4⋊C8—671no—
353#322—671no—
354#322—671no—
355#322—671no—
356#322—671no—
357C2×#41C4×D6711yescharacteristic
358C2×Q16—671no—
359C2×SD16—671no—
360C2×SD16—671no—
361C2×D8—671no—
362C2×M16—1341no—
363C22×C8—21no—
364C22×C8—21no—
365C22×C8—1341no—
366C22×C8C22⋊D6711yescharacteristic
367C22×C8C22⋊D6711yescharacteristic
368#126—1341no—
369#125—671no—
370#124—671no—
371#123—1341no—
372C22⋊Q8—671no—
373#121—671no—
374C4×Q8—671no—
375C4×Q8C4×D6711yescharacteristic
376C4×Q8C22×D6711yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—671no—
380C4×D4—1341no—
381C4×D4C4×D6711yescharacteristic
382C4×D4C22×D6711yescharacteristic
383#117—21no—
384#117—1341no—
385#117C4×D6711yescharacteristic
386C2×C42—21no—
387C2×C42C22×D6711yescharacteristic
388#107—671no—
389#107—671no—
390#106—671no—
391#106—671no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—671no—
395C4⋊C8—671no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—1341no—
399C4⋊C8—1341no—
400C4⋊C8—1341no—
401C4⋊C8C22×D6711yescharacteristic
402C4⋊C8C22×D6711yescharacteristic
403#103—671no—
404#103—671no—
405#103—671no—
406#103—671no—
407#102—671no—
408#102—671no—
409#102—671no—
410#102—671no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—671no—
415C22⋊C8—671no—
416C22⋊C8—1341no—
417C22⋊C8—1341no—
418C22⋊C8—1341no—
419C22⋊C8—1341no—
420#97—1341no—
421#97—1341no—
422C4×C8—21no—
423C4×C8—21no—
424C4×C8—21no—
425C4×C8—671no—
426C4×C8—671no—
427C4×C8—671no—
428C4×C8—671no—
429C4×C8—1341no—
430C4×C8C22⋊D6711yescharacteristic
431C4×C8C22⋊D6711yescharacteristic
432C4×C8C22×D6711yescharacteristic
433C4×C8C22×D6711yescharacteristic
434#41—22no—
435#41—22no—
436C2×Q8—671no—
437C2×Q8C2×C4×D6711yescharacteristic
438C2×D4—21no—
439C2×D4—671no—
440C2×D4—1341no—
441C2×D4C2×C4×D6711yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1341no—
446C22×C4C2×C4×D6711yescharacteristic
447C22×C4C2×C22⋊D6711yescharacteristic
448C22×C4#1176162811yescharacteristic
449Q16—1342no—
450SD16—1342no—
451SD16—1342no—
452D8—1342no—
453M16—2682no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—2682no—
461C2×C8—671no—
462C2×C8—671no—
463C2×C8—671no—
464C2×C8—671no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8C2×C22⋊D6711yescharacteristic
470C2×C8C2×C22⋊D6711yescharacteristic
471C2×C8C2×C22⋊D6711yescharacteristic
472C2×C8C2×C22⋊D6711yescharacteristic
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—1341no—
487C22⋊C4—1341no—
488C22⋊C4—1341no—
489C22⋊C4—1341no—
490C22⋊C4—1341no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—1341no—
495C42C2×C4×D6711yescharacteristic
496C42C2×C22⋊D6711yescharacteristic
497C42#1176162811yescharacteristic
498C42C23×D6711yescharacteristic
499C23—21no—
500C23—1341no—
501C23C4×C22⋊D6711yescharacteristic
502Q8—21no—
503Q8—1341no—
504Q8—1341no—
505Q8#1452404912yes—
506D4—24no—
507D4—21no—
508D4—2682no—
509D4—1341no—
510D4—1341no—
511D4#1452404912yes—
512C2×C4—24no—
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—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—2681no—
525C2×C4—2681no—
526C2×C4—1341no—
527C2×C4—1341no—
528C2×C4—1341no—
529C2×C4C4×C22⋊D6711yescharacteristic
530C2×C4C4×C22⋊D6711yescharacteristic
531C2×C4C4×C22⋊D6711yescharacteristic
532C2×C4C22×C4×D6711yescharacteristic
533C2×C4C22×C22⋊D6711yescharacteristic
534C2×C4C2×#1176162811yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—2681no—
542C8—2681no—
543C8—2681no—
544C8—2681no—
545C8—1341no—
546C8—1341no—
547C8—1341no—
548C8—1341no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—2681no—
556C22—2681no—
557C22—2681no—
558C22C2×C4×C22⋊D6711yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—2681no—
567C4—2681no—
568C4—2681no—
569C4C2×#1452404911yescharacteristic
570C4C2×#1452404911yescharacteristic
571C4#1686419411yescharacteristic
572C4#1686419411yescharacteristic
573C2—41no—
574C2—42no—
575C2—5361no—
576C2#2256890311yescharacteristic
577C2#2256900711yescharacteristic
578C2#2256911011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC8×#41×C67C2
socleC22×C67C2×C4×D4
prime radical#16863541C4
cosocleC42×C67C24
hypercenterC8×#41D67
hypocenterC67#17909
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #22569184C2semidirect
3. #22559087C2semidirect
4. #22559080C2semidirect
5. #22559073C2semidirect
6. #22559066C2semidirect
7. #22559031C2semidirect
8. #22559026C2semidirect
9. #22559021C2semidirect
10. #22559019C2semidirect
11. #22558987C2semidirect
12. #22558643C2semidirect
13. #22558615C2semidirect
14. #22558501C2semidirect
15. C8×#41×C67C2semidirect
292. C8×#41D67semidirect
305. C67#17909semidirect

Maximal subgroups

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

subgroupindex
#225702272
#225691842
#225590872
#225590802
#225590732
#225590662
#225590312
#225590262
#225590212
#225590192
#225589872
#225586432
#225586152
#225585012
C8×#41×C672
#1790967