Finite Group Databasebeta

← #17097275

Subgroups — #17097275

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×#11763089C211yesmaximal, characteristic, complemented
2C2×C4×#63228C211yesmaximal, characteristic
3#14532634C211yesmaximal, characteristic, complemented
4#14530313C211yesmaximal, characteristic, complemented
5#14530256C211yesmaximal, characteristic, complemented
6#14530169C211yesmaximal, characteristic, complemented
7#14530112C211yesmaximal, characteristic, complemented
8#14529514C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529132C211yesmaximal, characteristic
11#14529132C211yesmaximal, characteristic
12#14529031C211yesmaximal, characteristic, complemented
13#14529031C211yesmaximal, characteristic, complemented
14#14529030C211yesmaximal, characteristic, complemented
15D4×C4⋊C4×C17C211yesmaximal, characteristic, complemented
16C22×#63228C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587582—21no—
19C2×C4×D68C2211yescharacteristic
20#11763089—24no—
21C4×#63228—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587480C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×C4×#3622C2211yescharacteristic
35#11762731—21no—
36#11762722—21no—
37#11762711C2211yescharacteristic
38#11762711C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762683—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C22×C4⋊C4×C17—21no—
54C17×#389C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#385—21no—
57C17×#385—21no—
58C17×#384C2211yescharacteristic
59C17×#380—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C22×D68C2311yescharacteristic
66C22×C4×D17—21no—
67C22×Dic34—21no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17—22no—
72C4×D4×C17—41no—
73C4×D4×C17—41no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—44no—
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17D411yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C42×C17C2311yescharacteristic
90C2×#63228—22no—
91C2×#63228—22no—
92C2×#63228Q814yescomplemented
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×#63227—21no—
97C2×#63227—21no—
98C2×C4⋊Dic17—21no—
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×#63225—21no—
103C2×#63225—21no—
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587582—44no—
109C4×D68—22no—
110C4×D68—41no—
111#10587555—22no—
112#10587555—41no—
113C17×#95—21no—
114C17×#95—21no—
115C17×#95D411yescharacteristic
116C17×#95D411yescharacteristic
117#10587503—22no—
118#10587503—22no—
119#10587503D414yes—
120#10587502—21no—
121#10587502—21no—
122#10587480—22no—
123#10587480—22no—
124#10587480—41no—
125#10587478—22no—
126#10587478—22no—
127#10587478—22no—
128#10587478—22no—
129#10587478—41no—
130C4⋊#3622—22no—
131C4×#3622—22no—
132C24×C17—21no—
133C23×D17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C2×D412yes—
138C2×D4×C17C2×Q812yes—
139C2×D4×C17#4112yes—
140C22×C4×C17—24no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—42no—
148C22×C4×C17—42no—
149C22×C4×C17—42no—
150C22×C4×C17—81no—
151C22×C4×C17—81no—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2411yescharacteristic
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C2×D68—21no—
163C2×D68—21no—
164C2×D68—21no—
165C2×D68—21no—
166C2×D68—21no—
167C2×D68C2×Q812yes—
168C2×C4×D17—42no—
169C2×C4×D17—42no—
170C2×C4×D17—81no—
171C2×Dic34—42no—
172C2×Dic34—42no—
173C2×Dic34—81no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—41no—
180C4⋊C4×C17—42no—
181C4⋊C4×C17—42no—
182C4⋊C4×C17—42no—
183C4⋊C4×C17—81no—
184C4⋊C4×C17D814yescomplemented
185C4⋊C4×C17#4112yes—
186C22⋊C4×C17—24no—
187C22⋊C4×C17—42no—
188C22⋊C4×C17—42no—
189C42×C17—21no—
190C42×C17—21no—
191C42×C17—21no—
192C42×C17—21no—
193C42×C17—21no—
194C42×C17—21no—
195#63228—28no—
196#63227—42no—
197#63227—42no—
198#63227—42no—
199#63227—42no—
200#63227—81no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—42no—
207C4⋊Dic17C2×D412yes—
208#63225—42no—
209#63225—42no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622C2×D412yes—
221C2×#3622C2×Q812yes—
222#26368—171nomaximal, Sylow
223C23×C17—22no—
224C23×C17—22no—
225C23×C17—22no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17C22×D411yescharacteristic
231C22×D17—22no—
232C22×D17—41no—
233C22×D17—41no—
234C22×D17—41no—
235C22×D17—41no—
236C22×D17—41no—
237C22×D17—41no—
238D4×C17—22no—
239D4×C17—22no—
240D4×C17—22no—
241D4×C17C22⋊Q814yescomplemented
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
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—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—48no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—84no—
268C2×C4×C17—84no—
269C2×C4×C17C2×D812yes—
270C2×C4×C17C2×D812yes—
271C2×C4×C17C2×D812yes—
272C2×C4×C17C22×D411yescharacteristic
273C2×C4×C17C22×Q811yescharacteristic
274C2×C4×C17C2×#4111yescharacteristic
275C2×C4×C17C2×#4111yescharacteristic
276C2×C4×C17H(5,𝔽2)11yescharacteristic
277C2×C4×C172−1+411yescharacteristic
278C2×Dic17—22no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287D68—22no—
288D68—22no—
289D68—41no—
290D68—41no—
291C4×D17—84no—
292Dic34—84no—
293#3622—22no—
294#3622—22no—
295#3622—41no—
296#3622—41no—
297C2×#474—171no—
298C2×C4×D8—171no—
299#2452—171no—
300D4×C4⋊C4D1711yescharacteristic, complemented
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#52511yescharacteristic
320C22×C17D4211yescharacteristic
321C22×C17D4×Q811yescharacteristic
322C22×C17C22×D811yescharacteristic
323C22×C17#57511yescharacteristic
324C22×C17#57611yescharacteristic
325C22×C17#57811yescharacteristic
326C2×D17—42no—
327C2×D17—42no—
328C2×D17—42no—
329C2×D17—81no—
330C2×D17—81no—
331C2×D17—81no—
332C2×D17—81no—
333C4×C17—22no—
334C4×C17—22no—
335C4×C17—22no—
336C4×C17—42no—
337C4×C17—41no—
338C4×C17—41no—
339C4×C17—41no—
340C4×C17—81no—
341C4×C17—81no—
342C4×C17C2×C22⋊Q811yescharacteristic
343C4×C17#53911yescharacteristic
344C4×C17#54711yescharacteristic
345C4×C17#55411yescharacteristic
346Dic17—42no—
347Dic17—81no—
348Dic17—81no—
349C22×D8—171no—
350C2×C4⋊Q8—171no—
351C2×C22⋊Q8—341no—
352C2×C4×D4—171no—
353C2×C4×D4C2×D1711yescharacteristic
354C2×C4×D4C2×D1711yescharacteristic
355C2×C4×D4C2×D1711yescharacteristic
356C22×C4⋊C4—21no—
357#474—344no—
358C4×D8—344no—
359C2×#107—171no—
360C2×#107—171no—
361C2×#107—171no—
362C2×#107—171no—
363C2×#107—171no—
364C2×C4⋊C8—171no—
365C2×#102—171no—
366C2×#102—171no—
367C2×#102—171no—
368C2×#102—171no—
369C2×#102—171no—
370C2×#102—171no—
371C2×C4×C8—171no—
372#396—341no—
373#393—341no—
374#390—171no—
375#390—171no—
376#389—171no—
377#389—171no—
378#389C2×D1711yescharacteristic
379#389C2×D1711yescharacteristic
380#385—21no—
381#385—21no—
382#385—341no—
383#384C2×D1711yescharacteristic
384#380—21no—
385C4×C4⋊C4C2×D1711yescharacteristic
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#244911yescharacteristic
396C2×C17#310011yescharacteristic
397C2×C17#312411yescharacteristic
398C2×C17#326911yescharacteristic
399C2×C17D4×D811yescharacteristic
400C2×C17Q8×D811yescharacteristic
401C2×C17#338911yescharacteristic
402D17—82no—
403C22×Q8—341no—
404C22×D4—171no—
405C22×D4C22×D1711yescharacteristic
406C23×C4—21no—
407C23×C4—21no—
408C23×C4—21no—
409C23×C4—341no—
410C2×D8—174no—
411C2×D8—342no—
412C2×D8—342no—
413C22×C8—171no—
414C22×C8—171no—
415C22×C8—171no—
416C22×C8—171no—
417C4⋊Q8—342no—
418C4⋊Q8—681no—
419C22⋊Q8—684no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—41no—
425C4×D4—41no—
426C4×D4—342no—
427C4×D4—681no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—44no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4—171no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4C22⋊D1711yescharacteristic
440C2×C4⋊C4C22⋊D1711yescharacteristic
441C2×C4⋊C4C22×D1711yescharacteristic
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C22⋊C4—341no—
452C2×C42C22×D1711yescharacteristic
453C2×C42C22×D1711yescharacteristic
454C2×C42C22×D1711yescharacteristic
455#107—342no—
456#107—342no—
457#107—342no—
458#107—342no—
459#107—681no—
460C4⋊C8—342no—
461#102—174no—
462#102—342no—
463#102—342no—
464#102—342no—
465#102—342no—
466#102—681no—
467C4×C8—342no—
468#95—21no—
469#95—21no—
470#95—341no—
471#95—341no—
472#95C22⋊D1711yescharacteristic
473#95C22⋊D1711yescharacteristic
474C17#2636811yescharacteristic, complemented, Sylow
475C24—21no—
476C24—341no—
477C2×Q8—682no—
478C2×Q8—682no—
479C2×Q8—1361no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—172no—
484C2×D4—341no—
485C2×D4—341no—
486C2×D4—341no—
487C2×D4—341no—
488C2×D4—341no—
489C2×D4#6325212yes—
490C2×D4D4×D1712yes—
491C2×D4Q8×D1712yes—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—682no—
508C22×C4—682no—
509C22×C4—1361no—
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C2×C22⋊D1711yescharacteristic
512C22×C4C2×C22⋊D1711yescharacteristic
513C22×C4C2×C22⋊D1711yescharacteristic
514C22×C4C2×C22⋊D1711yescharacteristic
515C22×C4C2×C22⋊D1711yescharacteristic
516C22×C4C23×D1711yescharacteristic
517D8—348no—
518C2×C8—172no—
519C2×C8—172no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C2×C8—341no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—41no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—172no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—682no—
547C4⋊C4—682no—
548C4⋊C4—682no—
549C4⋊C4#6322814yescomplemented
550C4⋊C4#6325212yes—
551C22⋊C4—24no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—682no—
555C22⋊C4—682no—
556C22⋊C4—682no—
557C22⋊C4—682no—
558C22⋊C4—1361no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—342no—
573C23—681no—
574C23—681no—
575C23—681no—
576C23—681no—
577C23—681no—
578C23—681no—
579C23C22×C22⋊D1711yescharacteristic
580Q8—1364no—
581D4—22no—
582D4—22no—
583D4—22no—
584D4—342no—
585D4—342no—
586D4—681no—
587D4—681no—
588D4#1058758214yescomplemented
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—342no—
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—1364no—
626C2×C4C2×#6322812yes—
627C2×C4C2×#6322812yes—
628C2×C4C2×#6322812yes—
629C2×C4C2×#6325211yescharacteristic
630C2×C4C2×#6325211yescharacteristic
631C2×C4C2×D4×D1711yescharacteristic
632C2×C4#1058769111yescharacteristic
633C2×C4C2×Q8×D1711yescharacteristic
634C2×C4#1058769411yescharacteristic
635C8—342no—
636C8—342no—
637C8—681no—
638C8—681no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—682no—
647C22—682no—
648C22—682no—
649C22—1361no—
650C22—1361no—
651C22—1361no—
652C22—1361no—
653C22#1176332011yescharacteristic
654C22D4×C22⋊D1711yescharacteristic
655C22Q8×C22⋊D1711yescharacteristic
656C22C22×#6322811yescharacteristic
657C22#1176360711yescharacteristic
658C22#1176361211yescharacteristic
659C22#1176362411yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—81no—
668C4—81no—
669C4—682no—
670C4—1361no—
671C4—1361no—
672C4C2×#1058758211yescharacteristic
673C4#1176340611yescharacteristic
674C4#1176346011yescharacteristic
675C4#1176351411yescharacteristic
676C2—44no—
677C2—1362no—
678C2#1453259311yescharacteristic
679C2#1453878411yescharacteristic
680C2#1453904611yescharacteristic
681C2#1454033511yescharacteristic
682C2D4×#6322811yescharacteristic
683C2Q8×#6322811yescharacteristic
684C2#1454173811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingD4×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×#63228C22
hypercenterD4×C4⋊C4D17
hypocenterC17#26368
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11763089C2semidirect
3. #14532634C2semidirect
4. #14530313C2semidirect
5. #14530256C2semidirect
6. #14530169C2semidirect
7. #14530112C2semidirect
8. #14529514C2semidirect
9. #14529343C2semidirect
12. #14529031C2semidirect
13. #14529031C2semidirect
14. #14529030C2semidirect
15. D4×C4⋊C4×C17C2semidirect
92. C2×#63228Q8semidirect
184. C4⋊C4×C17D8semidirect
241. D4×C17C22⋊Q8semidirect
300. D4×C4⋊C4D17semidirect
474. C17#26368semidirect
549. C4⋊C4#63228semidirect
588. D4#10587582semidirect

Maximal subgroups

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

subgroupindex
C2×#117630892
C2×C4×#632282
#145326342
#145303132
#145302562
#145301692
#145301122
#145295142
#145293432
#145291322
#145290312
#145290302
D4×C4⋊C4×C172
#2636817