Finite Group Databasebeta

← #30827026

Subgroups — #30827026

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569369C211yesmaximal, characteristic
2#22569298C211yesmaximal, characteristic, complemented
3#22559106C211yesmaximal, characteristic, complemented
4#22559100C211yesmaximal, characteristic, complemented
5#22559088C211yesmaximal, characteristic, complemented
6#22559081C211yesmaximal, characteristic, complemented
7#22559081C211yesmaximal, characteristic, complemented
8#22559067C211yesmaximal, characteristic, complemented
9Q16×C67⋊C8C211yesmaximal, characteristic, complemented
10SD16×C67⋊C8C211yesmaximal, characteristic, complemented
11#22558994C211yesmaximal, characteristic, complemented
12#22558669C211yesmaximal, characteristic
13C2×Q8⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
14#22558493C211yesmaximal, characteristic, complemented
15C67×#3108C211yesmaximal, characteristic
16#16863527—21no—
17#16863527—21no—
18#16863527C2211yescharacteristic
19#16863527C2211yescharacteristic
20#16863527C2211yescharacteristic
21Q8×C67⋊C8C2211yescharacteristic
22Q8×C67⋊C8C2211yescharacteristic
23Q8×C67⋊C8C2211yescharacteristic
24#16863474C2211yescharacteristic
25#16863468—21no—
26D4×C67⋊C8—21no—
27D4×C67⋊C8C2211yescharacteristic
28#16863457—21no—
29#16863453—21no—
30#16863453C2211yescharacteristic
31#16863449—21no—
32#16863449C2211yescharacteristic
33#16863423—21no—
34C2×C4⋊(C67⋊C8)C2211yescharacteristic
35C2×C4×C67⋊C8C2211yescharacteristic
36#16863149C2211yescharacteristic
37#16863149C2211yescharacteristic
38#16863147C2211yescharacteristic
39#16863147C2211yescharacteristic
40Q8⋊(C67⋊C8)C2211yescharacteristic
41Q8⋊(C67⋊C8)C2211yescharacteristic
42Q8⋊(C67⋊C8)C2211yescharacteristic
43Q8⋊(C67⋊C8)C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46D4⋊(C67⋊C8)C2211yescharacteristic
47D4⋊(C67⋊C8)C2211yescharacteristic
48#16863118C2211yescharacteristic
49#16863118C2211yescharacteristic
50C8×C67⋊C8C2211yescharacteristic
51C8×C67⋊C8C2211yescharacteristic
52#529×C67C2211yescharacteristic
53C2×C4×Q8×C67C2211yescharacteristic
54#483×C67C411yescharacteristic
55#480×C67C411yescharacteristic
56#479×C67C411yescharacteristic
57#479×C67C411yescharacteristic
58#464×C67C411yescharacteristic
59#464×C67C411yescharacteristic
60#451×C67C411yescharacteristic
61#448×C67C411yescharacteristic
62C4×Q16×C67C2211yescharacteristic
63C4×Q16×C67C2211yescharacteristic
64C4×SD16×C67C2211yescharacteristic
65C4×SD16×C67C2211yescharacteristic
66#431×C67C2211yescharacteristic
67C2×#11761612—21no—
68C22×C67⋊C8—21no—
69C22×C67⋊C8—21no—
70C22×C67⋊C8—21no—
71C4⋊(C67⋊C8)—21no—
72C4⋊(C67⋊C8)—21no—
73C4⋊(C67⋊C8)—21no—
74C4⋊(C67⋊C8)—21no—
75C4⋊(C67⋊C8)—21no—
76C4⋊(C67⋊C8)—21no—
77C4⋊(C67⋊C8)—21no—
78C4⋊(C67⋊C8)D411yescharacteristic
79C4⋊(C67⋊C8)D411yescharacteristic
80C4⋊(C67⋊C8)C2311yescharacteristic
81C4⋊(C67⋊C8)C2311yescharacteristic
82C4⋊(C67⋊C8)C2311yescharacteristic
83C4⋊(C67⋊C8)C2311yescharacteristic
84C22⋊(C67⋊C8)—21no—
85C22⋊(C67⋊C8)—21no—
86C22⋊(C67⋊C8)—21no—
87C22⋊(C67⋊C8)D411yescharacteristic
88C22⋊(C67⋊C8)D411yescharacteristic
89#14523954—21no—
90#14523954—21no—
91#14523954—21no—
92#14523954—21no—
93C4×C67⋊C8—21no—
94C4×C67⋊C8—21no—
95C4×C67⋊C8—21no—
96C4×C67⋊C8—21no—
97C4×C67⋊C8C2311yescharacteristic
98C4×C67⋊C8C2311yescharacteristic
99C4×C67⋊C8C2311yescharacteristic
100C4×C67⋊C8C2311yescharacteristic
101C22×Q8×C67C2×C411yescharacteristic
102C2×Q16×C67C2×C411yescharacteristic
103C2×Q16×C67C2×C411yescharacteristic
104C2×SD16×C67C2×C411yescharacteristic
105C2×SD16×C67C2×C411yescharacteristic
106#126×C67—21no—
107#125×C67C2×C411yescharacteristic
108#124×C67C2×C411yescharacteristic
109#123×C67—21no—
110C22⋊Q8×C67C2×C411yescharacteristic
111#121×C67C2×C411yescharacteristic
112C4×Q8×C67—21no—
113C4×Q8×C67—21no—
114C4×Q8×C67C2×C411yescharacteristic
115C4×Q8×C67C2×C411yescharacteristic
116C4×Q8×C67C2311yescharacteristic
117C4×Q8×C67C2311yescharacteristic
118C4×Q8×C67C2311yescharacteristic
119C4×D4×C67—21no—
120C4×D4×C67C2311yescharacteristic
121#117×C67—21no—
122C2×C4⋊C4×C67—21no—
123C2×C4⋊C4×C67C2×C411yescharacteristic
124C2×C42×C67—21no—
125C2×C42×C67C2311yescharacteristic
126#107×C67C2×C411yescharacteristic
127#107×C67C2×C411yescharacteristic
128#106×C67C2×C411yescharacteristic
129#106×C67C2×C411yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67C2×C411yescharacteristic
132#103×C67C2×C411yescharacteristic
133#103×C67C2×C411yescharacteristic
134#103×C67C2×C411yescharacteristic
135#103×C67C2×C411yescharacteristic
136#103×C67C2×C411yescharacteristic
137#103×C67C2×C411yescharacteristic
138#102×C67C2×C411yescharacteristic
139#102×C67C2×C411yescharacteristic
140C22⋊C8×C67C2×C411yescharacteristic
141C22⋊C8×C67C2×C411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#11761612—81no—
145C2×C67⋊C8—24no—
146C2×C67⋊C8—21no—
147C2×C67⋊C8—21no—
148C2×C67⋊C8—21no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—81no—
153C2×C67⋊C8—81no—
154C2×C67⋊C8C2×D411yescharacteristic
155C2×C67⋊C8C2×D411yescharacteristic
156C2×C67⋊C8C2×D411yescharacteristic
157C2×C67⋊C8C2×D411yescharacteristic
158C2×Q8×C67—21no—
159C2×Q8×C67—22no—
160C2×Q8×C67—41no—
161C2×Q8×C67C22×C411yescharacteristic
162C2×Q8×C67C22×C411yescharacteristic
163C2×Q8×C67C22×C411yescharacteristic
164C2×D4×C67—21no—
165C2×D4×C67C22×C411yescharacteristic
166C22×C4×C67—21no—
167C22×C4×C67—21no—
168C22×C4×C67—21no—
169C22×C4×C67C22×C411yescharacteristic
170C22×C4×C67C2×D411yescharacteristic
171C22×C4×C67#4111yescharacteristic
172Q16×C67—22no—
173Q16×C67—41no—
174SD16×C67—22no—
175SD16×C67—41no—
176C2×C8×C67C22×C411yescharacteristic
177C2×C8×C67C22×C411yescharacteristic
178C2×C8×C67C22×C411yescharacteristic
179C2×C8×C67C22×C411yescharacteristic
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67C22×C411yescharacteristic
190C4⋊C4×C67C22×C411yescharacteristic
191C4⋊C4×C67C22×C411yescharacteristic
192C4⋊C4×C67C22×C411yescharacteristic
193C22⋊C4×C67—21no—
194C22⋊C4×C67—21no—
195C22⋊C4×C67—21no—
196C22⋊C4×C67—21no—
197C22⋊C4×C67—21no—
198C42×C67—21no—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67—21no—
208C23×C67C4×D411yescharacteristic
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—42no—
215Q8×C67#13112yes—
216D4×C67—21no—
217D4×C67—21no—
218D4×C67—42no—
219C2×C4×C67—21no—
220C2×C4×C67—22no—
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—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67C4×D411yescharacteristic
236C2×C4×C67C4×D411yescharacteristic
237C2×C4×C67C4×D411yescharacteristic
238C2×C4×C67C23×C411yescharacteristic
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C2×#4111yescharacteristic
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C67⋊C8—22no—
246C67⋊C8—22no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67#44312yes—
261C22×C67C2×C4×D411yescharacteristic
262C4×C67—21no—
263C4×C67—22no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67C2×#13111yescharacteristic
274C4×C67#56811yescharacteristic
275C4×C67#57711yescharacteristic
276C4×C67#57811yescharacteristic
277#18017—671nomaximal, Sylow
278C2×C67—22no—
279C2×C67—81no—
280C2×C67#293311yescharacteristic
281C2×C67#294711yescharacteristic
282C2×C67C2×#44311yescharacteristic
283#3108D6711yescharacteristic
284#2994—671no—
285#2984—671no—
286#1591—671no—
287#1590—671no—
288#1588—671no—
289#1587—671no—
290#1587—671no—
291#1585—671no—
292C8×Q16—671no—
293C8×SD16—671no—
294#1573—671no—
295#1494—671no—
296C2×Q8⋊C8—671no—
297#1450—671no—
298C67#1801711yescharacteristic, complemented, Sylow
299#529C2×D6711yescharacteristic
300C2×C4×Q8C2×D6711yescharacteristic
301#483Dic6711yescharacteristic
302#480Dic6711yescharacteristic
303#479Dic6711yescharacteristic
304#479Dic6711yescharacteristic
305#464Dic6711yescharacteristic
306#464Dic6711yescharacteristic
307#451Dic6711yescharacteristic
308#448Dic6711yescharacteristic
309#446—671no—
310#446—671no—
311#446—671no—
312#446—1341no—
313#446—1341no—
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8—671no—
317C4×Q16C2×D6711yescharacteristic
318C4×Q16C2×D6711yescharacteristic
319C4×SD16C2×D6711yescharacteristic
320C4×SD16C2×D6711yescharacteristic
321#436—671no—
322#435—1341no—
323C8×D4—671no—
324C8×D4—1341no—
325#433—1341no—
326#432—671no—
327#432—1341no—
328#431—671no—
329#431—1341no—
330#431C2×D6711yescharacteristic
331#424—1341no—
332C2×C4⋊C8—671no—
333C2×C4×C8—671no—
334#335—671no—
335#335—671no—
336#334—671no—
337#334—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344D4⋊C8—671no—
345D4⋊C8—671no—
346#322—671no—
347#322—671no—
348C82—671no—
349C82—671no—
350C22×Q8C2×Dic6711yescharacteristic
351C2×Q16C2×Dic6711yescharacteristic
352C2×Q16C2×Dic6711yescharacteristic
353C2×SD16C2×Dic6711yescharacteristic
354C2×SD16C2×Dic6711yescharacteristic
355C2×M16—1341no—
356C22×C8—1341no—
357C22×C8—1341no—
358C22×C8—1341no—
359#126—21no—
360#125C2×Dic6711yescharacteristic
361#124C2×Dic6711yescharacteristic
362#123—21no—
363C22⋊Q8C2×Dic6711yescharacteristic
364#121C2×Dic6711yescharacteristic
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C2×Dic6711yescharacteristic
368C4×Q8C2×Dic6711yescharacteristic
369C4×Q8C22×D6711yescharacteristic
370C4×Q8C22×D6711yescharacteristic
371C4×Q8C22×D6711yescharacteristic
372C4×D4—21no—
373C4×D4C22×D6711yescharacteristic
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C2×Dic6711yescharacteristic
377C2×C42—21no—
378C2×C42C22×D6711yescharacteristic
379#107C2×Dic6711yescharacteristic
380#107C2×Dic6711yescharacteristic
381#106C2×Dic6711yescharacteristic
382#106C2×Dic6711yescharacteristic
383C4⋊C8—671no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8C2×Dic6711yescharacteristic
397C4⋊C8C2×Dic6711yescharacteristic
398#103C2×Dic6711yescharacteristic
399#103C2×Dic6711yescharacteristic
400#103C2×Dic6711yescharacteristic
401#103C2×Dic6711yescharacteristic
402#103C2×Dic6711yescharacteristic
403#103C2×Dic6711yescharacteristic
404#102C2×Dic6711yescharacteristic
405#102C2×Dic6711yescharacteristic
406C22⋊C8—671no—
407C22⋊C8—671no—
408C22⋊C8—1341no—
409C22⋊C8—1341no—
410C22⋊C8—1341no—
411C22⋊C8C2×Dic6711yescharacteristic
412C22⋊C8C2×Dic6711yescharacteristic
413#97—1341no—
414#97—1341no—
415#97—1341no—
416#97—1341no—
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8C22×D6711yescharacteristic
426C4×C8C22×D6711yescharacteristic
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8C22×Dic6711yescharacteristic
431C2×Q8C22×Dic6711yescharacteristic
432C2×Q8C22×Dic6711yescharacteristic
433C2×D4—21no—
434C2×D4C22×Dic6711yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4C22×Dic6711yescharacteristic
439C22×C4D4×D6711yescharacteristic
440C22×C4#1176162711yescharacteristic
441Q16—22no—
442Q16—41no—
443SD16—22no—
444SD16—41no—
445M16—5361no—
446C2×C8—5361no—
447C2×C8—5361no—
448C2×C8—671no—
449C2×C8—671no—
450C2×C8—671no—
451C2×C8—671no—
452C2×C8—1341no—
453C2×C8—1341no—
454C2×C8—1341no—
455C2×C8—1341no—
456C2×C8—1344no—
457C2×C8—1341no—
458C2×C8—1341no—
459C2×C8C22×Dic6711yescharacteristic
460C2×C8C22×Dic6711yescharacteristic
461C2×C8C22×Dic6711yescharacteristic
462C2×C8C22×Dic6711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4C22×Dic6711yescharacteristic
473C4⋊C4C22×Dic6711yescharacteristic
474C4⋊C4C22×Dic6711yescharacteristic
475C4⋊C4C22×Dic6711yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42C22×Dic6711yescharacteristic
487C42D4×D6711yescharacteristic
488C42#1176162711yescharacteristic
489C42C23×D6711yescharacteristic
490C23—21no—
491C23D4×Dic6711yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8#1452405012yes—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4D4×Dic6711yescharacteristic
519C2×C4D4×Dic6711yescharacteristic
520C2×C4D4×Dic6711yescharacteristic
521C2×C4C23×Dic6711yescharacteristic
522C2×C4C2×D4×D6711yescharacteristic
523C2×C4C2×#1176162711yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—2681no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8—1342no—
533C8—1342no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8—1341no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1686351312yes—
544C22C2×D4×Dic6711yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4C2×#1452405011yescharacteristic
557C4#1686416311yescharacteristic
558C4#1686420511yescharacteristic
559C4#1686421211yescharacteristic
560C2—22no—
561C2—81no—
562C2#2256893611yescharacteristic
563C2#2256905311yescharacteristic
564C2C2×#1686351311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#3108C2
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
cosocleC42×C67C24
hypercenter#3108D67
hypocenterC67#18017
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3108

Complements

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

NH ≅ G/Ntype
2. #22569298C2semidirect
3. #22559106C2semidirect
4. #22559100C2semidirect
5. #22559088C2semidirect
6. #22559081C2semidirect
7. #22559081C2semidirect
8. #22559067C2semidirect
9. Q16×C67⋊C8C2semidirect
10. SD16×C67⋊C8C2semidirect
11. #22558994C2semidirect
13. C2×Q8⋊(C67⋊C8)C2semidirect
14. #22558493C2semidirect
298. C67#18017semidirect

Maximal subgroups

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

subgroupindex
#225693692
#225692982
#225591062
#225591002
#225590882
#225590812
#225590672
Q16×C67⋊C82
SD16×C67⋊C82
#225589942
#225586692
C2×Q8⋊(C67⋊C8)2
#225584932
C67×#31082
#1801767