Finite Group Databasebeta

← #17097269

Subgroups — #17097269

Order 4352.

968 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763087C211yesmaximal, characteristic, complemented
2#14532628C211yesmaximal, characteristic, complemented
3D4×#63225C211yesmaximal, characteristic, complemented
4#14530310C211yesmaximal, characteristic, complemented
5#14530258C211yesmaximal, characteristic, complemented
6#14530170C211yesmaximal, characteristic, complemented
7#14530110C211yesmaximal, characteristic, complemented
8#14529517C211yesmaximal, characteristic, complemented
9#14529341C211yesmaximal, characteristic, complemented
10#14529134C211yesmaximal, characteristic
11#14529134C211yesmaximal, characteristic
12#14529033C211yesmaximal, characteristic, complemented
13#14529031C211yesmaximal, characteristic, complemented
14#14529031C211yesmaximal, characteristic, complemented
15C2×C4×D8×C17C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587610—21no—
18C2×D4×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20C22×#63225—21no—
21#11763087—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31#11762732—21no—
32#11762721—21no—
33#11762712C2211yescharacteristic
34#11762712C2211yescharacteristic
35#11762708C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762705C2211yescharacteristic
39#11762682—21no—
40#11762682—21no—
41#11762681—21no—
42#11762678C2211yescharacteristic
43#11762667—21no—
44C4×#63225C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×D8×C17C2211yescharacteristic
54C2×C4×D4×C17C2211yescharacteristic
55C2×C4×D4×C17C2211yescharacteristic
56C4×D8×C17—24no—
57C2×C17×#107C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C2×C17×#102C2211yescharacteristic
60C2×C4×C8×C17C2211yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C23×Dic17—21no—
66C23×Dic17—21no—
67C22×Dic34—21no—
68C2×D8×C17—22no—
69C2×D8×C17—22no—
70C2×D8×C17Q814yescomplemented
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C4×D4×C17—22no—
74C4×D4×C17—22no—
75C4×D4×C17—41no—
76C4×D4×C17—41no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×C22⋊Dic17—21no—
88D4×Dic17—22no—
89D4×Dic17—22no—
90D4×Dic17—22no—
91D4×Dic17—41no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97#10587610—44no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×#63225—21no—
104C2×#63225—44no—
105C2×#63225D411yescharacteristic
106C2×#63225D411yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C2×C4×Dic17C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111#10587574—22no—
112#10587574—41no—
113C17×#107—22no—
114C17×#102—22no—
115C17×#102D414yes—
116C4×C8×C17—22no—
117#10587503—22no—
118#10587503—22no—
119#10587503—22no—
120#10587503—41no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502D411yescharacteristic
126#10587502D411yescharacteristic
127#10587489—22no—
128#10587489—22no—
129#10587485—22no—
130#10587478—22no—
131#10587478—41no—
132C24×C17—21no—
133C24×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17C2×D412yes—
143C2×D4×C17C2×Q812yes—
144C2×D4×C17C2×Q812yes—
145C2×D4×C17#4112yes—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—42no—
149C22×C4×C17—42no—
150C22×C4×C17—42no—
151C22×C4×C17—42no—
152C22×C4×C17—81no—
153C22×C4×C17—81no—
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2411yescharacteristic
157C22×Dic17—24no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—42no—
163C22×Dic17—42no—
164C22×Dic17—42no—
165C22×Dic17—42no—
166C22×Dic17—81no—
167C22×Dic17C2×D411yescharacteristic
168C22×Dic17C2×D411yescharacteristic
169C22×Dic17C2×D411yescharacteristic
170C22×Dic17C2×D411yescharacteristic
171C2×Dic34—42no—
172C2×Dic34—42no—
173C2×Dic34—81no—
174D8×C17—28no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C2×D412yes—
180C2×C8×C17C2×Q812yes—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17C2×D412yes—
185C22⋊C4×C17—42no—
186C22⋊C4×C17—42no—
187C42×C17—21no—
188C42×C17—21no—
189C22⋊Dic17—24no—
190C22⋊Dic17—42no—
191C22⋊Dic17—42no—
192C22⋊Dic17—42no—
193C22⋊Dic17—42no—
194C22⋊Dic17—81no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—42no—
203C4⋊Dic17#4112yes—
204#63225—41no—
205#63225—42no—
206#63225—42no—
207#63225—42no—
208#63225—42no—
209#63225—42no—
210#63225—81no—
211#63225D814yescomplemented
212C4×Dic17—21no—
213C4×Dic17—21no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222#26368—171nomaximal, Sylow
223C23×C17—22no—
224C23×C17—22no—
225C23×C17—22no—
226C23×C17—22no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17—41no—
231C23×C17—41no—
232C23×C17—41no—
233C23×C17—41no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17—41no—
237C23×C17C22×D411yescharacteristic
238D4×C17—22no—
239D4×C17—22no—
240D4×C17—22no—
241D4×C17—22no—
242D4×C17—22no—
243D4×C17—41no—
244D4×C17—41no—
245D4×C17C22⋊Q814yescomplemented
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—84no—
258C2×C4×C17—84no—
259C2×C4×C17C2×D812yes—
260C2×C4×C17C22×D411yescharacteristic
261C2×C4×C17C22×Q811yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17H(5,𝔽2)11yescharacteristic
265C2×C4×C172−1+411yescharacteristic
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—22no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—48no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—84no—
290C2×Dic17C2×D812yes—
291C2×Dic17C2×D812yes—
292Dic34—84no—
293C8×C17—22no—
294C8×C17—22no—
295#3622—41no—
296#3622—41no—
297C2×#474—171no—
298C2×C4×D8D1711yescharacteristic
299#2452—171no—
300D4×C4⋊C4—171no—
301#2093—171no—
302#2083—171no—
303#2067—171no—
304#2057—171no—
305#1942—171no—
306#1916—171no—
307#1879—171no—
308#1879—171no—
309#1862—171no—
310#1862—171no—
311#1862—171no—
312C22×C17—24no—
313C22×C17—42no—
314C22×C17—42no—
315C22×C17—42no—
316C22×C17—42no—
317C22×C17—42no—
318C22×C17—42no—
319C22×C17—42no—
320C22×C17—42no—
321C22×C17—42no—
322C22×C17—81no—
323C22×C17—81no—
324C22×C17—81no—
325C22×C17—81no—
326C22×C17#52511yescharacteristic
327C22×C17D4211yescharacteristic
328C22×C17D4×Q811yescharacteristic
329C22×C17C22×D811yescharacteristic
330C22×C17#57511yescharacteristic
331C22×C17#57611yescharacteristic
332C22×C17#57811yescharacteristic
333C4×C17—22no—
334C4×C17—41no—
335C4×C17—42no—
336C4×C17—81no—
337C4×C17C2×C22⋊Q811yescharacteristic
338C4×C17#53911yescharacteristic
339C4×C17#54711yescharacteristic
340C4×C17#55411yescharacteristic
341Dic17—22no—
342Dic17—22no—
343Dic17—42no—
344Dic17—41no—
345Dic17—41no—
346Dic17—81no—
347Dic17—81no—
348Dic17—81no—
349C22×D8C2×D1711yescharacteristic
350C2×C4⋊Q8—171no—
351C2×C22⋊Q8—341no—
352C2×C4×D4—171no—
353C2×C4×D4—171no—
354C2×C4×D4C2×D1711yescharacteristic
355C2×C4×D4C2×D1711yescharacteristic
356C22×C4⋊C4—341no—
357#474—344no—
358C4×D8—24no—
359C2×#107—171no—
360C2×#107—171no—
361C2×#107—171no—
362C2×#107—171no—
363C2×#107C2×D1711yescharacteristic
364C2×C4⋊C8—171no—
365C2×#102—171no—
366C2×#102—171no—
367C2×#102—171no—
368C2×#102—171no—
369C2×#102C2×D1711yescharacteristic
370C2×#102C2×D1711yescharacteristic
371C2×C4×C8C2×D1711yescharacteristic
372#396—341no—
373#393—341no—
374#390—171no—
375#390—171no—
376#389—171no—
377#389—171no—
378#389—171no—
379#389—171no—
380#385—341no—
381#385—341no—
382#385—341no—
383#384—171no—
384#380—341no—
385C4×C4⋊C4—171no—
386#340—171no—
387#340—171no—
388#340—171no—
389#340—171no—
390#336—171no—
391#336—171no—
392#336—171no—
393#336—171no—
394C2×C17—44no—
395C2×C17—82no—
396C2×C17#244911yescharacteristic
397C2×C17#310011yescharacteristic
398C2×C17#312411yescharacteristic
399C2×C17#326911yescharacteristic
400C2×C17D4×D811yescharacteristic
401C2×C17Q8×D811yescharacteristic
402C2×C17#338911yescharacteristic
403C22×Q8—341no—
404C22×D4C22×D1711yescharacteristic
405C22×D4C22×D1711yescharacteristic
406C23×C4—21no—
407C23×C4—21no—
408C23×C4—341no—
409C23×C4—341no—
410C2×D8—22no—
411C2×D8—22no—
412C2×D8Dic3414yescomplemented
413C22×C8—171no—
414C22×C8—171no—
415C22×C8C22×D1711yescharacteristic
416C22×C8C22×D1711yescharacteristic
417C4⋊Q8—342no—
418C4⋊Q8—681no—
419C22⋊Q8—684no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—41no—
423C4×D4—41no—
424C4×D4—342no—
425C4×D4—342no—
426C4×D4—342no—
427C4×D4—681no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4—684no—
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C22⋊C4—341no—
452C2×C42—171no—
453C2×C42—171no—
454C2×C42C22×D1711yescharacteristic
455#107—22no—
456#107—342no—
457#107—342no—
458#107—342no—
459#107—681no—
460C4⋊C8—342no—
461#102—22no—
462#102—342no—
463#102—342no—
464#102—342no—
465#102—681no—
466#102C22⋊D1714yes—
467C4×C8—22no—
468#95—171no—
469#95—171no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474C17#2636811yescharacteristic, complemented, Sylow
475C24—21no—
476C24—21no—
477C2×Q8—682no—
478C2×Q8—682no—
479C2×Q8—1361no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×Dic3412yes—
489C2×D4C2×Dic3412yes—
490C2×D4#6325212yes—
491C2×D4C2×C22⋊D1712yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—171no—
501C22×C4—171no—
502C22×C4—171no—
503C22×C4—171no—
504C22×C4—344no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—682no—
510C22×C4—682no—
511C22×C4—682no—
512C22×C4—682no—
513C22×C4—1361no—
514C22×C4D4×D1711yescharacteristic
515C22×C4D4×D1711yescharacteristic
516C22×C4C23×D1711yescharacteristic
517D8—28no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8C2×Dic3412yes—
529C2×C8C2×C22⋊D1712yes—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—174no—
534C4⋊C4—172no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—341no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—681no—
543C4⋊C4—682no—
544C4⋊C4—682no—
545C4⋊C4—682no—
546C4⋊C4—682no—
547C4⋊C4—682no—
548C4⋊C4—682no—
549C4⋊C4—1361no—
550C4⋊C4C2×C22⋊D1712yes—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—344no—
554C22⋊C4—682no—
555C22⋊C4—682no—
556C22⋊C4—682no—
557C22⋊C4—682no—
558C22⋊C4—1361no—
559C42—21no—
560C42—21no—
561C42—341no—
562C42—341no—
563C42—341no—
564C42—341no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C2×D4×D1711yescharacteristic
580Q8—1364no—
581D4—22no—
582D4—22no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588D4#1058761014yescomplemented
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—172no—
603C2×C4—172no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—341no—
608C2×C4—341no—
609C2×C4—342no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—341no—
613C2×C4—341no—
614C2×C4—341no—
615C2×C4—688no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—1364no—
628C2×C4D8×D1712yes—
629C2×C4C22×Dic3411yescharacteristic
630C2×C4C2×#6325211yescharacteristic
631C2×C4#1058769111yescharacteristic
632C2×C4#41×D1711yescharacteristic
633C2×C4#1058769711yescharacteristic
634C2×C4C22×C22⋊D1711yescharacteristic
635C8—22no—
636C8—22no—
637C8—681no—
638C8—681no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22#1176331811yescharacteristic
654C22D4×C22⋊D1711yescharacteristic
655C22D4×Dic3411yescharacteristic
656C22C2×D8×D1711yescharacteristic
657C22#1176360611yescharacteristic
658C22#1176361011yescharacteristic
659C22#1176362311yescharacteristic
660C4—22no—
661C4—41no—
662C4—42no—
663C4—81no—
664C4—342no—
665C4—342no—
666C4—682no—
667C4—681no—
668C4—681no—
669C4—1361no—
670C4—1361no—
671C4—1361no—
672C4C2×#1058761011yescharacteristic
673C4#1176340111yescharacteristic
674C4#1176345811yescharacteristic
675C4#1176351811yescharacteristic
676C2—44no—
677C2—82no—
678C2#1453258711yescharacteristic
679C2#1453878211yescharacteristic
680C2#1453904211yescharacteristic
681C2#1454033911yescharacteristic
682C2D8×C22⋊D1711yescharacteristic
683C2D8×Dic3411yescharacteristic
684C2#1454174011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C4×D8×C17C2
socleC23×C17C22×D4
prime radicalC22×D8×C17C22
hypercenterC2×C4×D8D17
hypocenterC17#26368
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#11763087C2semidirect
2. #14532628C2semidirect
3. D4×#63225C2semidirect
4. #14530310C2semidirect
5. #14530258C2semidirect
6. #14530170C2semidirect
7. #14530110C2semidirect
8. #14529517C2semidirect
9. #14529341C2semidirect
12. #14529033C2semidirect
13. #14529031C2semidirect
14. #14529031C2semidirect
70. C2×D8×C17Q8semidirect
211. #63225D8semidirect
245. D4×C17C22⋊Q8semidirect
412. C2×D8Dic34semidirect
474. C17#26368semidirect
588. D4#10587610semidirect

Maximal subgroups

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

subgroupindex
C2×#117630872
#145326282
D4×#632252
#145303102
#145302582
#145301702
#145301102
#145295172
#145293412
#145291342
#145290332
#145290312
C2×C4×D8×C172
#2636817