Finite Group Databasebeta

← #17013226

Subgroups — #17013226

Order 4352.

1032 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539722C211yesmaximal, characteristic, complemented
2#14538264C211yesmaximal, characteristic
3C2×#11763009C211yesmaximal, characteristic, complemented
4#14527887C212yesmaximal, complemented
5#14527880C212yesmaximal, complemented
6#14527862C212yesmaximal, complemented
7#14527856C212yesmaximal, complemented
8#14527474C211yesmaximal, characteristic, complemented
9#14527213C211yesmaximal, characteristic, complemented
10#14527171C211yesmaximal, characteristic, complemented
11Q8×#41×C17C211yesmaximal, characteristic, complemented
12C2×#10587635—21no—
13C22×#63230—21no—
14C2×#10587560C2211yescharacteristic
15#11763183C2212yes—
16#11763157C2212yes—
17#11763009—21no—
18#11763009—22no—
19#11763009—21no—
20#11763009—21no—
21#11763009C2212yes—
22#11763009C2212yes—
23#11763009C2212yes—
24#11763004—21no—
25#11763004C2212yes—
26#11762959—22no—
27#11762862C2211yescharacteristic
28#11762833C2211yescharacteristic
29C2×C4⋊#3622C2211yescharacteristic
30#11762813—21no—
31C2×#10587480—21no—
32#11762558C2212yescomplemented
33#11762555C2212yescomplemented
34Q8⋊#3622C2212yescomplemented
35Q8⋊#3622C2212yescomplemented
36Q8⋊#3622C2212yescomplemented
37D4⋊#3622C2212yescomplemented
38Q82×C17C2211yescharacteristic, complemented
39C17×#557—21no—
40C17×#557C2211yescharacteristic, complemented
41C17×#554—21no—
42C17×#554—21no—
43C17×#554—22no—
44C17×#554C2211yescharacteristic, complemented
45D4×Q8×C17—21no—
46D4×Q8×C17C2211yescharacteristic, complemented
47C17×#533—22no—
48C17×#533—21no—
49C17×#533—21no—
50C17×#533C2211yescharacteristic
51C4×#41×C17—21no—
52C4×#41×C17C2211yescharacteristic
53C2×C4×Q8×C17—21no—
54C2×C4×Q8×C17C2211yescharacteristic
55C2×#41×C17D411yescharacteristic
56C22×Q8×C17—21no—
57C22×Q8×C17D411yescharacteristic
58C22×D68—21no—
59C22×D68—21no—
60C22×D68—21no—
61C17×C4⋊Q8—22no—
62C17×C4⋊Q8—21no—
63C17×C4⋊Q8—22no—
64C17×C4⋊Q8—21no—
65C17×C4⋊Q8—21no—
66C17×C4⋊Q8—21no—
67C17×C4⋊Q8C2311yescharacteristic
68C17×C4⋊Q8C2311yescharacteristic
69C17×#125—22no—
70C17×#125—22no—
71C17×#125—22no—
72C17×#125—21no—
73C17×#125—21no—
74C17×#125—21no—
75C17×C22⋊Q8—22no—
76C17×C22⋊Q8—22no—
77C17×C22⋊Q8—22no—
78C17×C22⋊Q8—22no—
79C17×C22⋊Q8—22no—
80C17×C22⋊Q8—21no—
81C17×C22⋊Q8—22no—
82C17×C22⋊Q8—21no—
83C17×C22⋊Q8—21no—
84C17×C22⋊Q8—22no—
85C17×C22⋊Q8D412yes—
86C4×Q8×C17—22no—
87C4×Q8×C17—22no—
88C4×Q8×C17—22no—
89C4×Q8×C17—21no—
90C4×Q8×C17—21no—
91C4×Q8×C17—21no—
92C4×Q8×C17—21no—
93C4×Q8×C17—21no—
94C4×Q8×C17—21no—
95C4×Q8×C17—21no—
96C4×Q8×C17D411yescharacteristic
97C4×Q8×C17D411yescharacteristic
98C4×Q8×C17D411yescharacteristic
99C4×Q8×C17D412yes—
100C4×Q8×C17C2311yescharacteristic
101C4×Q8×C17C2311yescharacteristic
102C4×Q8×C17C2311yescharacteristic
103C4×D4×C17—22no—
104C4×D4×C17—22no—
105C4×D4×C17—21no—
106C4×D4×C17—21no—
107C4×D4×C17—21no—
108C4×D4×C17—21no—
109C4×D4×C17D411yescharacteristic
110C4×D4×C17C2311yescharacteristic
111C17×#117—22no—
112C17×#117—21no—
113C17×#117—21no—
114C17×#117D411yescharacteristic
115C2×C4⋊C4×C17—22no—
116C2×C4⋊C4×C17—21no—
117C2×C4⋊C4×C17—21no—
118C2×C4⋊C4×C17D411yescharacteristic
119C2×C42×C17—22no—
120C2×C42×C17—21no—
121C2×C42×C17—21no—
122C2×C42×C17C2311yescharacteristic
123#10587635—44no—
124C2×#63230—22no—
125C2×#63230—22no—
126C2×#63230—22no—
127C2×#63230—24no—
128C2×#63230—44no—
129C2×#63228—22no—
130C2×#63223—21no—
131C22×#3622—21no—
132#10587560—21no—
133#10587560—21no—
134#10587560—21no—
135#10587560C2312yes—
136C22⋊#3622—21no—
137#10587480—24no—
138#10587480—22no—
139#10587480—22no—
140#10587480—22no—
141#10587480—22no—
142C4⋊#3622—21no—
143C4⋊#3622—21no—
144C4⋊#3622—21no—
145C4⋊#3622C2312yes—
146C4⋊#3622C2312yes—
147C4×#3622C2312yes—
148C23×D17—41no—
149#41×C17—22no—
150#41×C17SD1614yescomplemented
151C2×Q8×C17—22no—
152C2×Q8×C17—22no—
153C2×Q8×C17—21no—
154C2×Q8×C17—21no—
155C2×Q8×C17—22no—
156C2×Q8×C17—41no—
157C2×Q8×C17—42no—
158C2×Q8×C17—42no—
159C2×Q8×C17C2×D411yescharacteristic
160C2×Q8×C17C2×D411yescharacteristic
161C2×Q8×C17C2×D411yescharacteristic
162C2×Q8×C17C2×D412yes—
163C2×D4×C17—21no—
164C2×D4×C17C2×D411yescharacteristic
165C22×C4×C17—21no—
166C22×C4×C17—22no—
167C22×C4×C17—21no—
168C22×C4×C17—22no—
169C22×C4×C17—21no—
170C22×C4×C17—21no—
171C22×C4×C17—21no—
172C22×C4×C17C2×D411yescharacteristic
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17C2×D411yescharacteristic
175C2×D68—22no—
176C2×D68—22no—
177C2×D68—22no—
178C2×D68—42no—
179C2×D68—42no—
180C2×D68—42no—
181C2×D68—42no—
182C2×D68—41no—
183C2×D68—41no—
184C2×D68—41no—
185C2×D68—82no—
186C4⋊C4×C17—22no—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—22no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17—22no—
191C4⋊C4×C17—22no—
192C4⋊C4×C17—22no—
193C4⋊C4×C17—22no—
194C4⋊C4×C17—22no—
195C4⋊C4×C17—21no—
196C4⋊C4×C17—21no—
197C4⋊C4×C17—22no—
198C4⋊C4×C17—21no—
199C4⋊C4×C17—21no—
200C4⋊C4×C17—21no—
201C4⋊C4×C17—21no—
202C4⋊C4×C17—21no—
203C4⋊C4×C17—21no—
204C4⋊C4×C17—21no—
205C4⋊C4×C17—21no—
206C4⋊C4×C17C2×D412yes—
207C4⋊C4×C17C2×D411yescharacteristic
208C4⋊C4×C17C2×D411yescharacteristic
209C4⋊C4×C17C2×D411yescharacteristic
210C4⋊C4×C17C2×D411yescharacteristic
211C22⋊C4×C17—22no—
212C22⋊C4×C17—22no—
213C22⋊C4×C17—21no—
214C22⋊C4×C17—22no—
215C22⋊C4×C17—21no—
216C22⋊C4×C17—21no—
217C42×C17—22no—
218C42×C17—22no—
219C42×C17—21no—
220C42×C17—21no—
221C42×C17—22no—
222C42×C17—21no—
223C42×C17—21no—
224C42×C17—21no—
225C42×C17—21no—
226C42×C17—21no—
227C42×C17C2×D411yescharacteristic
228C42×C17C2×D411yescharacteristic
229C42×C17C2×D411yescharacteristic
230C42×C17C2411yescharacteristic
231#63230—44no—
232#63230—44no—
233#63230—44no—
234#63230—84no—
235#63228—44no—
236#63223—42no—
237C2×#3622—22no—
238C2×#3622—22no—
239C2×#3622—22no—
240C2×#3622—42no—
241#18788—171nomaximal, Sylow
242C23×C17—21no—
243C23×C17#12011yescharacteristic
244C22×D17—42no—
245C22×D17—82no—
246C22×D17—82no—
247C22×D17—82no—
248Q8×C17—21no—
249Q8×C17—21no—
250Q8×C17—21no—
251Q8×C17—21no—
252Q8×C17—42no—
253Q8×C17—42no—
254Q8×C17—44no—
255Q8×C17—41no—
256Q8×C17—41no—
257Q8×C17—82no—
258Q8×C17C2×SD1612yescomplemented
259Q8×C17AGL(1,ℤ/8)12yescomplemented
260D4×C17—24no—
261D4×C17—21no—
262D4×C17C2×SD1612yescomplemented
263C2×C4×C17—21no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—21no—
267C2×C4×C17—24no—
268C2×C4×C17—21no—
269C2×C4×C17—21no—
270C2×C4×C17—22no—
271C2×C4×C17—22no—
272C2×C4×C17—21no—
273C2×C4×C17—21no—
274C2×C4×C17—21no—
275C2×C4×C17—41no—
276C2×C4×C17—42no—
277C2×C4×C17—44no—
278C2×C4×C17—41no—
279C2×C4×C17—41no—
280C2×C4×C17—81no—
281C2×C4×C17—81no—
282C2×C4×C17—82no—
283C2×C4×C17#12011yescharacteristic
284C2×C4×C17#12011yescharacteristic
285C2×C4×C17#12011yescharacteristic
286C2×C4×C17C2×SD1612yes—
287C2×C4×C17#13712yes—
288C2×C4×C17C22×D411yescharacteristic
289C2×C4×C17C22×D411yescharacteristic
290C2×C4×C17C22×D411yescharacteristic
291D68—44no—
292D68—42no—
293D68—42no—
294D68—42no—
295D68—42no—
296D68—82no—
297D68—82no—
298D68—82no—
299#3622—42no—
300#3622—42no—
301#3622—42no—
302Q8×#41D1711yescharacteristic, complemented
303#3208—171no—
304#3038—171no—
305C2×#460—171no—
306#1627—172no—
307#1626—172no—
308#1623—172no—
309#1622—172no—
310#1550—171no—
311#1487—171no—
312#1477—171no—
313C22×C17—21no—
314C22×C17—21no—
315C22×C17—22no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17#44812yes—
319C22×C17C2×#12011yescharacteristic
320C2×D17—82no—
321C2×D17—82no—
322C2×D17—82no—
323C2×D17—164no—
324C4×C17—22no—
325C4×C17—21no—
326C4×C17—22no—
327C4×C17—21no—
328C4×C17—21no—
329C4×C17—41no—
330C4×C17—42no—
331C4×C17—41no—
332C4×C17—41no—
333C4×C17—42no—
334C4×C17—41no—
335C4×C17—41no—
336C4×C17#44812yes—
337C4×C17C22×SD1611yescharacteristic
338C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
339C4×C17C2×#13711yescharacteristic
340C4×C17#57511yescharacteristic
341C2×AGL(1,ℤ/8)—341no—
342C22×SD16—341no—
343Q82C2×D1711yescharacteristic, complemented
344#557—21no—
345#557C2×D1711yescharacteristic, complemented
346#554—22no—
347#554—21no—
348#554—21no—
349#554C2×D1711yescharacteristic, complemented
350D4×Q8—21no—
351D4×Q8C2×D1711yescharacteristic, complemented
352#533—22no—
353#533—21no—
354#533—21no—
355#533C2×D1711yescharacteristic
356C2×#127—171no—
357C4×#41—21no—
358C4×#41C2×D1711yescharacteristic
359C2×C4×Q8—21no—
360C2×C4×Q8C2×D1711yescharacteristic
361#492—172no—
362#486—172no—
363#460—172no—
364#460—172no—
365#460—172no—
366#460—342no—
367#460—341no—
368#460—341no—
369#460—341no—
370#459—172no—
371#459—341no—
372#450—342no—
373#431—171no—
374#423—171no—
375C2×C4⋊C8—171no—
376#418—341no—
377C2×#102—341no—
378#333—172no—
379#332—172no—
380Q8⋊C8—172no—
381Q8⋊C8—172no—
382Q8⋊C8—172no—
383D4⋊C8—172no—
384C2×C17—22no—
385C2×C17—42no—
386C2×C17C2×#44811yescharacteristic
387C2×C17#300611yescharacteristic
388C2×C17#302111yescharacteristic
389D17—162no—
390C2×#41C22⋊D1711yescharacteristic
391C22×Q8—21no—
392C22×Q8C22⋊D1711yescharacteristic
393C22×D4—341no—
394C22×D4—341no—
395C22×D4—341no—
396AGL(1,ℤ/8)—684no—
397C2×SD16—342no—
398C2×SD16—342no—
399C2×SD16—342no—
400C2×SD16—344no—
401C2×SD16—684no—
402C2×D8—342no—
403C2×M16—341no—
404C22×C8—341no—
405C4⋊Q8—22no—
406C4⋊Q8—21no—
407C4⋊Q8—22no—
408C4⋊Q8—21no—
409C4⋊Q8—21no—
410C4⋊Q8—21no—
411C4⋊Q8C22×D1711yescharacteristic
412C4⋊Q8C22×D1711yescharacteristic
413#127—172no—
414#127—341no—
415#127—341no—
416#127—341no—
417#125—22no—
418#125—22no—
419#125—22no—
420#125—21no—
421#125—21no—
422#125—21no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—22no—
427C22⋊Q8—22no—
428C22⋊Q8—21no—
429C22⋊Q8—22no—
430C22⋊Q8—21no—
431C22⋊Q8—21no—
432C22⋊Q8—22no—
433C22⋊Q8C22⋊D1712yes—
434C4×Q8—22no—
435C4×Q8—22no—
436C4×Q8—22no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8—21no—
442C4×Q8—21no—
443C4×Q8—21no—
444C4×Q8C22⋊D1711yescharacteristic
445C4×Q8C22⋊D1711yescharacteristic
446C4×Q8C22⋊D1711yescharacteristic
447C4×Q8C22⋊D1712yes—
448C4×Q8C22×D1711yescharacteristic
449C4×Q8C22×D1711yescharacteristic
450C4×Q8C22×D1711yescharacteristic
451C4×D4—22no—
452C4×D4—22no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4—21no—
457C4×D4C22⋊D1711yescharacteristic
458C4×D4C22×D1711yescharacteristic
459#117—22no—
460#117—21no—
461#117—21no—
462#117C22⋊D1711yescharacteristic
463C2×C4⋊C4—22no—
464C2×C4⋊C4—21no—
465C2×C4⋊C4—21no—
466C2×C4⋊C4C22⋊D1711yescharacteristic
467C2×C42—22no—
468C2×C42—21no—
469C2×C42—21no—
470C2×C42C22×D1711yescharacteristic
471C4⋊C8—172no—
472C4⋊C8—172no—
473C4⋊C8—341no—
474C4⋊C8—341no—
475C4⋊C8—341no—
476#102—342no—
477#102—342no—
478#102—342no—
479#102—342no—
480#102—344no—
481C22⋊C8—341no—
482C4×C8—172no—
483C17#1878811yescharacteristic, complemented, Sylow
484C24—681no—
485#41—22no—
486#41#6323014yescomplemented
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—22no—
492C2×Q8—41no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8C2×C22⋊D1711yescharacteristic
496C2×Q8C2×C22⋊D1711yescharacteristic
497C2×Q8C2×C22⋊D1711yescharacteristic
498C2×Q8C2×C22⋊D1712yes—
499C2×D4—21no—
500C2×D4—342no—
501C2×D4—342no—
502C2×D4—342no—
503C2×D4—682no—
504C2×D4—682no—
505C2×D4—682no—
506C2×D4—682no—
507C2×D4—681no—
508C2×D4—681no—
509C2×D4—681no—
510C2×D4—1362no—
511C2×D4C2×C22⋊D1711yescharacteristic
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4C2×C22⋊D1711yescharacteristic
520C22×C4C2×C22⋊D1711yescharacteristic
521C22×C4C2×C22⋊D1711yescharacteristic
522SD16—684no—
523SD16—684no—
524SD16—684no—
525SD16—1364no—
526D8—684no—
527M16—682no—
528C2×C8—342no—
529C2×C8—342no—
530C2×C8—342no—
531C2×C8—682no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4C2×C22⋊D1711yescharacteristic
553C4⋊C4C2×C22⋊D1711yescharacteristic
554C4⋊C4C2×C22⋊D1711yescharacteristic
555C4⋊C4C2×C22⋊D1711yescharacteristic
556C4⋊C4C2×C22⋊D1712yes—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42C2×C22⋊D1711yescharacteristic
574C42C2×C22⋊D1711yescharacteristic
575C42C2×C22⋊D1711yescharacteristic
576C42C23×D1711yescharacteristic
577C23—21no—
578C23—682no—
579C23—1362no—
580C23—1362no—
581C23—1362no—
582C23#1058763911yescharacteristic
583Q8—21no—
584Q8—21no—
585Q8—21no—
586Q8—21no—
587Q8—42no—
588Q8—42no—
589Q8—44no—
590Q8—41no—
591Q8—41no—
592Q8—82no—
593Q8C2×#6323012yescomplemented
594Q8#1058763512yescomplemented
595D4—24no—
596D4—21no—
597D4—684no—
598D4—682no—
599D4—682no—
600D4—682no—
601D4—682no—
602D4—1362no—
603D4—1362no—
604D4—1362no—
605D4C2×#6323012yescomplemented
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4C2×#6323012yes—
627C2×C4#1058762812yes—
628C2×C4#1058763911yescharacteristic
629C2×C4#1058763911yescharacteristic
630C2×C4#1058763911yescharacteristic
631C2×C4C22×C22⋊D1711yescharacteristic
632C2×C4C22×C22⋊D1711yescharacteristic
633C2×C4C22×C22⋊D1711yescharacteristic
634C8—682no—
635C8—682no—
636C8—682no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—2724no—
643C22—1362no—
644C22—1362no—
645C22—1362no—
646C22#1176294812yes—
647C22C2×#1058763911yescharacteristic
648C4—21no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4#1176294812yes—
661C4C22×#6323011yescharacteristic
662C4C2×#1058763511yescharacteristic
663C4C2×#1058762811yescharacteristic
664C4#1176360711yescharacteristic
665C2—22no—
666C2—42no—
667C2—2722no—
668C2C2×#1176294811yescharacteristic
669C2#1453803611yescharacteristic
670C2#1453815611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingQ8×#41×C17C2
socleC22×C17C2×#120
prime radical#14538264C2
hypercenterQ8×#41D17
hypocenterC17#18788
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. #14539722C2semidirect
3. C2×#11763009C2semidirect
4. #14527887C2semidirect
5. #14527880C2semidirect
6. #14527862C2semidirect
7. #14527856C2semidirect
8. #14527474C2semidirect
9. #14527213C2semidirect
10. #14527171C2semidirect
11. Q8×#41×C17C2semidirect
32. #11762558C22semidirect
33. #11762555C22semidirect
34. Q8⋊#3622C22semidirect
35. Q8⋊#3622C22semidirect
36. Q8⋊#3622C22semidirect
37. D4⋊#3622C22semidirect
38. Q82×C17C22semidirect
40. C17×#557C22semidirect
44. C17×#554C22semidirect
46. D4×Q8×C17C22semidirect
150. #41×C17SD16semidirect
258. Q8×C17C2×SD16semidirect
259. Q8×C17AGL(1,ℤ/8)semidirect
262. D4×C17C2×SD16semidirect
302. Q8×#41D17semidirect
343. Q82C2×D17semidirect
345. #557C2×D17semidirect
349. #554C2×D17semidirect
351. D4×Q8C2×D17semidirect
483. C17#18788semidirect
486. #41#63230semidirect
593. Q8C2×#63230semidirect
594. Q8#10587635semidirect
605. D4C2×#63230semidirect

Maximal subgroups

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

subgroupindex
#145397222
#145382642
C2×#117630092
#145278872
#145278802
#145278622
#145278562
#145274742
#145272132
#145271712
Q8×#41×C172
#1878817