Finite Group Databasebeta

← #17005759

Subgroups — #17005759

Order 4352.

716 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#63218C211yesmaximal, characteristic, complemented
2C4×SD16×D17C211yesmaximal, characteristic
3#14537520C211yesmaximal, characteristic
4#14527677C211yesmaximal, characteristic, complemented
5#14527672C211yesmaximal, characteristic, complemented
6#14527628C211yesmaximal, characteristic, complemented
7#14527622C211yesmaximal, characteristic, complemented
8#14527619C211yesmaximal, characteristic, complemented
9C8×#63230C211yesmaximal, characteristic, complemented
10#14527581C211yesmaximal, characteristic, complemented
11#14527538C211yesmaximal, characteristic, complemented
12#14527211C211yesmaximal, characteristic, complemented
13#14527207C211yesmaximal, characteristic
14#14527063C211yesmaximal, characteristic, complemented
15C17×#1583C211yesmaximal, characteristic, complemented
16C2×SD16×D17C411yescharacteristic
17C2×M16×D17—21no—
18C4×Q8×D17C2211yescharacteristic
19C4×D4×D17C2211yescharacteristic
20#11762939—21no—
21#11762939C2211yescharacteristic, complemented
22#11762936—21no—
23#11762936C2211yescharacteristic, complemented
24Q8×#3622C2211yescharacteristic, complemented
25#11762900C411yescharacteristic
26C4×#63219C2211yescharacteristic
27SD16×Dic17C411yescharacteristic
28C4×#63230C2211yescharacteristic
29#11762896C411yescharacteristic
30C4×#63229C2211yescharacteristic
31#11762894C411yescharacteristic
32#11762887C2211yescharacteristic
33#11762883—21no—
34#11762878—21no—
35C8×D68C2211yescharacteristic
36#11762874C2211yescharacteristic
37#11762860C2211yescharacteristic
38#106×D17C411yescharacteristic
39#11762831—21no—
40#11762831C2211yescharacteristic
41#11762831C2211yescharacteristic
42#103×D17C411yescharacteristic
43#102×D17C411yescharacteristic
44#11762771—21no—
45C4×#63218—21no—
46C4×#63218C2211yescharacteristic
47C4×C8×D17C2211yescharacteristic
48#11762560C2211yescharacteristic, complemented
49#11762559C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51#11762541C2211yescharacteristic, complemented
52Q8⋊#3622C2211yescharacteristic, complemented
53#11762539C2211yescharacteristic, complemented
54#11762538C2211yescharacteristic
55D4⋊#3622C2211yescharacteristic
56#11762536C2211yescharacteristic
57#11762531C2211yescharacteristic, complemented
58C8×#3622C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C8×Q8×C17C2211yescharacteristic, complemented
61C4×SD16×C17C2211yescharacteristic
62C17×#436C2211yescharacteristic
63C17×#334C2211yescharacteristic, complemented
64C17×Q8⋊C8C2211yescharacteristic, complemented
65C17×D4⋊C8C2211yescharacteristic
66C17×#322C2211yescharacteristic, complemented
67C2×Q8×D17C2×C411yescharacteristic
68C2×D4×D17C2×C411yescharacteristic
69C22×C4×D17—21no—
70C2×SD16×C17C2×C411yescharacteristic
71C2×M16×C17—21no—
72C4×Q8×C17C2311yescharacteristic
73C4×D4×C17C2311yescharacteristic
74Q8×Dic17C2×C411yescharacteristic
75C2×#63230C2×C411yescharacteristic
76D4×Dic17C2×C411yescharacteristic
77C2×#63229C2×C411yescharacteristic
78C4×C22⋊D17—21no—
79C2×#63223—21no—
80SD16×D17—24no—
81M16×D17—44no—
82C2×#63219C2×C411yescharacteristic
83C2×#63218—21no—
84C2×#63218—21no—
85C2×#63218D411yescharacteristic
86C2×#63218D411yescharacteristic
87C2×C8×D17—21no—
88C2×C8×D17—21no—
89C2×C8×D17C2×C411yescharacteristic
90C2×C8×D17C2×C411yescharacteristic
91#10587579C2×C411yescharacteristic
92C4⋊C4×D17—21no—
93C4⋊C4×D17C2×C411yescharacteristic
94C4⋊C4×D17C2×C411yescharacteristic
95#10587573—21no—
96#10587573C2×C411yescharacteristic
97#10587567—21no—
98C22⋊C4×D17—21no—
99C4×D68C2311yescharacteristic
100C42×D17—21no—
101C42×D17C2311yescharacteristic
102C4×Dic34—21no—
103C4×Dic34C2311yescharacteristic
104C17×#106C2×C411yescharacteristic
105C17×C4⋊C8—21no—
106C17×C4⋊C8C2311yescharacteristic
107C17×C4⋊C8C2311yescharacteristic
108C17×#103C2×C411yescharacteristic
109C17×#102C2×C411yescharacteristic
110C17×C22⋊C8—21no—
111C4×C8×C17—21no—
112C4×C8×C17C2311yescharacteristic
113C4×C8×C17C2311yescharacteristic
114C4×C8×C17C2311yescharacteristic
115#10587506C2×C411yescharacteristic
116#10587503C2×C411yescharacteristic
117C22⋊#3622—21no—
118#10587492C2×C411yescharacteristic
119#10587491—21no—
120#10587491—21no—
121#10587491—21no—
122#10587488C2×C411yescharacteristic
123#10587487C2×C411yescharacteristic
124#10587486—21no—
125#10587486—21no—
126#10587486D411yescharacteristic
127#10587486D411yescharacteristic
128#10587485—21no—
129#10587485—21no—
130#10587485—21no—
131#10587485—21no—
132#10587485—21no—
133C8×Dic17C2×C411yescharacteristic
134C8×Dic17C2×C411yescharacteristic
135#10587481C2×C411yescharacteristic
136#10587480C2×C411yescharacteristic
137#10587479C2×C411yescharacteristic
138#10587479C2×C411yescharacteristic
139C4⋊#3622—21no—
140C4⋊#3622C2311yescharacteristic
141C4⋊#3622C2311yescharacteristic
142C4×#3622—21no—
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C4×#3622C2311yescharacteristic
146C23×D17—21no—
147C2×Q8×C17C22×C411yescharacteristic
148C2×D4×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C2×C22⋊D17—21no—
151C22×Dic17—21no—
152Q8×D17—22no—
153Q8×D17—41no—
154D4×D17—22no—
155D4×D17—41no—
156C2×D68C22×C411yescharacteristic
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—42no—
163C2×C4×D17—42no—
164C2×C4×D17C22×C411yescharacteristic
165C2×C4×D17C2×D411yescharacteristic
166C2×C4×D17#4111yescharacteristic
167C2×Dic34—21no—
168C2×Dic34C22×C411yescharacteristic
169SD16×C17—22no—
170M16×C17—42no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17C22×C411yescharacteristic
175C2×C8×C17C22×C411yescharacteristic
176C2×C8×C17C2×D411yescharacteristic
177C2×C8×C17C2×D411yescharacteristic
178C4⋊C4×C17—21no—
179C4⋊C4×C17C22×C411yescharacteristic
180C4⋊C4×C17C22×C411yescharacteristic
181C22⋊C4×C17—21no—
182C42×C17—21no—
183C42×C17C2411yescharacteristic
184C22⋊Dic17—21no—
185#63230M1614yescomplemented
186#63229—22no—
187#63227—21no—
188#63227—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17C22×C411yescharacteristic
191C4⋊Dic17C22×C411yescharacteristic
192#63225—21no—
193#63225—21no—
194#63225—21no—
195#63225—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17C22×C411yescharacteristic
200C4×Dic17C2×D411yescharacteristic
201C4×Dic17#4111yescharacteristic
202#63223—42no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622C22×C411yescharacteristic
207C2×#3622C22×C411yescharacteristic
208C2×#3622C2×D411yescharacteristic
209C2×#3622C2×D411yescharacteristic
210#63219—22no—
211#63218—22no—
212#63218—42no—
213#63218—42no—
214#63218SD1614yescomplemented
215C8×D17—22no—
216C8×D17—41no—
217C8×D17—42no—
218C8×D17—42no—
219M16×SD16—171nomaximal, Sylow
220C23×C17—21no—
221C22×D17—21no—
222C22×D17—42no—
223C22×D17—42no—
224C22×D17—81no—
225C22×D17C4×D411yescharacteristic
226Q8×C17—21no—
227Q8×C17C2×M1612yescomplemented
228D4×C17—21no—
229D4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17C23×C411yescharacteristic
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C22⋊D17—42no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—42no—
243C2×Dic17C4×D411yescharacteristic
244C2×Dic17C4×D411yescharacteristic
245C2×Dic17C4×D411yescharacteristic
246D68—21no—
247D68C2×M1612yes—
248C4×D17—21no—
249C4×D17—21no—
250C4×D17—41no—
251C4×D17—42no—
252C4×D17—42no—
253C4×D17—41no—
254C4×D17—41no—
255C4×D17—42no—
256C4×D17—81no—
257C4×D17C2×SD1612yes—
258C4×D17#13512yes—
259Dic34—21no—
260Dic34—21no—
261Dic34—42no—
262C8×C17—21no—
263C8×C17—21no—
264C8×C17—21no—
265C8×C17—41no—
266C8×C17—41no—
267C8×C17—41no—
268C8×C17C2×SD1612yescomplemented
269#3622—21no—
270#3622—21no—
271#3622—41no—
272#3622—41no—
273#3622—41no—
274#3622C2×M1612yescomplemented
275#3622C2×SD1612yescomplemented
276Q8×M16—171no—
277C2×C4×SD16—171no—
278D4×M16—171no—
279#1590—171no—
280#1590—171no—
281#1583—171no—
282#1583D1711yescharacteristic, complemented
283#1582—171no—
284#1582—171no—
285C8×SD16—171no—
286C8×SD16—171no—
287#1566—171no—
288#1487—171no—
289#1486—171no—
290C8×M16—171no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17C2×C4×D411yescharacteristic
295C2×D17—21no—
296C2×D17—21no—
297C2×D17—22no—
298C2×D17—41no—
299C2×D17—41no—
300C2×D17—84no—
301C2×D17C4×SD1612yes—
302C4×C17—22no—
303C4×C17—21no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17C22×M1611yescharacteristic
307C4×C17#56811yescharacteristic
308C4×C17C22×SD1611yescharacteristic
309C4×C17C2×#13511yescharacteristic
310Dic17—21no—
311Dic17—21no—
312Dic17—41no—
313Dic17—41no—
314Dic17—41no—
315Dic17C4×SD1612yes—
316C22×SD16—171no—
317C22×M16—341no—
318C2×C4×Q8—171no—
319C2×C4×D4—171no—
320#446—171no—
321#446—171no—
322#446—341no—
323#446—341no—
324C8×Q8—171no—
325C8×Q8C2×D1711yescharacteristic, complemented
326C4×SD16—171no—
327C4×SD16—171no—
328C4×SD16—171no—
329C4×SD16—171no—
330C4×SD16—171no—
331C4×SD16—171no—
332C4×SD16—171no—
333C4×SD16C2×D1711yescharacteristic
334#436—171no—
335#436C2×D1711yescharacteristic
336#435—341no—
337#435—341no—
338C8×D4—171no—
339C8×D4—171no—
340#431—171no—
341C2×#106—171no—
342#423—171no—
343#423—171no—
344#423—341no—
345C2×#103—171no—
346C2×#102—171no—
347#407—341no—
348C4×M16—171no—
349C4×M16—341no—
350C2×C4×C8—171no—
351#334—171no—
352#334—171no—
353#334—171no—
354#334C2×D1711yescharacteristic, complemented
355Q8⋊C8—171no—
356Q8⋊C8—171no—
357Q8⋊C8—171no—
358Q8⋊C8C2×D1711yescharacteristic, complemented
359D4⋊C8—171no—
360D4⋊C8—171no—
361D4⋊C8—171no—
362D4⋊C8C2×D1711yescharacteristic
363#322—171no—
364#322C2×D1711yescharacteristic, complemented
365C82—171no—
366C82—171no—
367C2×C17—81no—
368C2×C17D4×M1611yescharacteristic
369C2×C17C2×C4×SD1611yescharacteristic
370C2×C17#295711yescharacteristic
371D17—22no—
372D17—42no—
373C22×Q8—171no—
374C22×D4—171no—
375C23×C4—341no—
376C2×SD16—171no—
377C2×SD16—171no—
378C2×SD16—171no—
379C2×SD16—344no—
380C2×SD16C4×D1711yescharacteristic
381C2×M16—21no—
382C2×M16—171no—
383C2×M16—171no—
384C2×M16—341no—
385C2×M16—341no—
386C2×M16—341no—
387C2×M16—684no—
388C22×C8—171no—
389C22×C8—171no—
390C22×C8—341no—
391C22×C8—341no—
392C4×Q8—171no—
393C4×Q8—171no—
394C4×Q8—171no—
395C4×Q8—341no—
396C4×Q8—341no—
397C4×Q8C22×D1711yescharacteristic
398C4×D4—171no—
399C4×D4—171no—
400C4×D4—171no—
401C4×D4—341no—
402C4×D4—341no—
403C4×D4C22×D1711yescharacteristic
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—341no—
407C2×C22⋊C4—341no—
408C2×C42—171no—
409C2×C42—341no—
410#106—171no—
411#106—171no—
412#106—171no—
413#106C4×D1711yescharacteristic
414C4⋊C8—21no—
415C4⋊C8—171no—
416C4⋊C8—171no—
417C4⋊C8—341no—
418C4⋊C8—341no—
419C4⋊C8—341no—
420C4⋊C8—341no—
421C4⋊C8—341no—
422C4⋊C8—341no—
423C4⋊C8C22×D1711yescharacteristic
424C4⋊C8C22×D1711yescharacteristic
425#103—171no—
426#103—171no—
427#103—171no—
428#103C4×D1711yescharacteristic
429#102—171no—
430#102—171no—
431#102—171no—
432#102C4×D1711yescharacteristic
433C22⋊C8—21no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437C22⋊C8—341no—
438#97—171no—
439#97—171no—
440#97—341no—
441#97—341no—
442C4×C8—21no—
443C4×C8—171no—
444C4×C8—171no—
445C4×C8—171no—
446C4×C8—171no—
447C4×C8—171no—
448C4×C8—341no—
449C4×C8C22×D1711yescharacteristic
450C4×C8C22×D1711yescharacteristic
451C4×C8C22×D1711yescharacteristic
452C17M16×SD1611yescharacteristic, complemented, Sylow
453C24—341no—
454C2×Q8—171no—
455C2×Q8—341no—
456C2×Q8—342no—
457C2×Q8—681no—
458C2×Q8C2×C4×D1711yescharacteristic
459C2×D4—171no—
460C2×D4—341no—
461C2×D4—342no—
462C2×D4—681no—
463C2×D4C2×C4×D1711yescharacteristic
464C22×C4—21no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—171no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—682no—
475C22×C4—682no—
476SD16—22no—
477SD16—174no—
478SD16—342no—
479SD16—342no—
480M16—42no—
481M16—174no—
482M16—342no—
483M16—682no—
484M16—682no—
485M16—682no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—171no—
490C2×C8—171no—
491C2×C8—171no—
492C2×C8—171no—
493C2×C8—342no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—681no—
498C2×C8—682no—
499C2×C8—682no—
500C2×C8C2×C4×D1711yescharacteristic
501C2×C8C2×C4×D1711yescharacteristic
502C2×C8D4×D1711yescharacteristic
503C2×C8D4×D1711yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—171no—
506C4⋊C4—171no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4C2×C4×D1711yescharacteristic
513C4⋊C4C2×C4×D1711yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—341no—
516C22⋊C4—341no—
517C22⋊C4—341no—
518C42—21no—
519C42—171no—
520C42—171no—
521C42—171no—
522C42—341no—
523C42—341no—
524C42—341no—
525C42C23×D1711yescharacteristic
526C23—21no—
527C23—171no—
528C23—341no—
529C23—682no—
530C23—682no—
531C23—1361no—
532Q8—21no—
533Q8—341no—
534Q8—341no—
535Q8—682no—
536Q8M16×D1712yescomplemented
537D4—21no—
538D4—21no—
539D4—172no—
540D4—341no—
541D4—682no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—171no—
548C2×C4—172no—
549C2×C4—172no—
550C2×C4—171no—
551C2×C4—171no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—681no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—1361no—
565C2×C4C22×C4×D1711yescharacteristic
566C2×C4C2×D4×D1711yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C8—21no—
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—172no—
575C8—172no—
576C8—341no—
577C8—341no—
578C8—681no—
579C8—681no—
580C8—681no—
581C8SD16×D1712yescomplemented
582C22—41no—
583C22—41no—
584C22—41no—
585C22—172no—
586C22—341no—
587C22—341no—
588C22—342no—
589C22—681no—
590C22—681no—
591C22—1364no—
592C22C4×D4×D1711yescharacteristic
593C4—22no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—172no—
598C4—341no—
599C4—341no—
600C4—681no—
601C4—681no—
602C4—681no—
603C4C2×M16×D1711yescharacteristic
604C4#1176357411yescharacteristic
605C4C2×SD16×D1711yescharacteristic
606C4#135×D1711yescharacteristic
607C2—81no—
608C2—342no—
609C2—682no—
610C2#1453751711yescharacteristic
611C2C4×SD16×D1711yescharacteristic
612C2#1453769811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1583C2
socleC22×C17C2×C4×D4
prime radicalC2×D4×D17C2×C4
cosocleC42×C17C24
hypercenter#1583D17
hypocenterC17M16×SD16
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
1. Q8×#63218C2semidirect
4. #14527677C2semidirect
5. #14527672C2semidirect
6. #14527628C2semidirect
7. #14527622C2semidirect
8. #14527619C2semidirect
9. C8×#63230C2semidirect
10. #14527581C2semidirect
11. #14527538C2semidirect
12. #14527211C2semidirect
14. #14527063C2semidirect
15. C17×#1583C2semidirect
21. #11762939C22semidirect
23. #11762936C22semidirect
24. Q8×#3622C22semidirect
48. #11762560C22semidirect
49. #11762559C22semidirect
50. #11762559C22semidirect
51. #11762541C22semidirect
52. Q8⋊#3622C22semidirect
53. #11762539C22semidirect
57. #11762531C22semidirect
58. C8×#3622C22semidirect
59. C8×#3622C22semidirect
60. C8×Q8×C17C22semidirect
63. C17×#334C22semidirect
64. C17×Q8⋊C8C22semidirect
66. C17×#322C22semidirect
185. #63230M16semidirect
214. #63218SD16semidirect
227. Q8×C17C2×M16semidirect
268. C8×C17C2×SD16semidirect
274. #3622C2×M16semidirect
275. #3622C2×SD16semidirect
282. #1583D17semidirect
325. C8×Q8C2×D17semidirect
354. #334C2×D17semidirect
358. Q8⋊C8C2×D17semidirect
364. #322C2×D17semidirect
452. C17M16×SD16semidirect
536. Q8M16×D17semidirect
581. C8SD16×D17semidirect

Maximal subgroups

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

subgroupindex
Q8×#632182
C4×SD16×D172
#145375202
#145276772
#145276722
#145276282
#145276222
#145276192
C8×#632302
#145275812
#145275382
#145272112
#145272072
#145270632
C17×#15832
M16×SD1617