Finite Group Databasebeta

← #17097395

Subgroups — #17097395

Order 4352.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763099C211yesmaximal, characteristic
2C2×C4×#63229C211yesmaximal, characteristic
3#14532623C211yesmaximal, characteristic, complemented
4#14530328C211yesmaximal, characteristic, complemented
5#14530261C211yesmaximal, characteristic, complemented
6#14530162C211yesmaximal, characteristic, complemented
7#14530117C211yesmaximal, characteristic, complemented
8#14529509C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529118C211yesmaximal, characteristic
11#14529118C211yesmaximal, characteristic
12#14529051C211yesmaximal, characteristic, complemented
13#14529045C211yesmaximal, characteristic
14#14529045C211yesmaximal, characteristic
15D4×C4⋊C4×C17C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19#11763099—24no—
20C4×#63229—24no—
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762743—21no—
35#11762729—21no—
36#11762715C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762713—21no—
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762695—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×Dic34—21no—
66C22×Dic34C2311yescharacteristic
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17—41no—
72C4×D4×C17—41no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—44no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63229—22no—
90C2×#63229—22no—
91C2×#63229Q814yescomplemented
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587555—22no—
105#10587555—41no—
106C4×Dic34—22no—
107C4×Dic34—41no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95D411yescharacteristic
111C17×#95D411yescharacteristic
112#10587503—22no—
113#10587503—22no—
114#10587503D414yes—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587481—22no—
122#10587481—22no—
123#10587481—41no—
124#10587479—22no—
125#10587479—22no—
126#10587479—22no—
127#10587479—22no—
128#10587479—41no—
129C4⋊#3622—22no—
130C4×#3622—22no—
131C24×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17C2×D412yes—
136C2×D4×C17C2×Q812yes—
137C2×D4×C17#4112yes—
138C22×C4×C17—24no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—42no—
148C22×C4×C17—81no—
149C22×C4×C17—81no—
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2411yescharacteristic
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—42no—
167C2×Dic34—42no—
168C2×Dic34—81no—
169C2×Dic34C2×Q812yes—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—41no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—42no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17—81no—
180C4⋊C4×C17SD1614yescomplemented
181C4⋊C4×C17#4112yes—
182C22⋊C4×C17—24no—
183C22⋊C4×C17—42no—
184C22⋊C4×C17—42no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—21no—
191#63229—28no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—42no—
198C4⋊Dic17C2×D412yes—
199#63225—42no—
200#63225—81no—
201#63225—81no—
202C4×Dic17—42no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622C2×D412yes—
214C2×#3622C2×Q812yes—
215#26376—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—22no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17C22×D411yescharacteristic
224D4×C17—22no—
225D4×C17—22no—
226D4×C17—22no—
227D4×C17C22⋊Q814yes—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—48no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—84no—
254C2×C4×C17—84no—
255C2×C4×C17C2×SD1612yes—
256C2×C4×C17C2×SD1612yes—
257C2×C4×C17C2×SD1612yes—
258C2×C4×C17C22×D411yescharacteristic
259C2×C4×C17C22×Q811yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17H(5,𝔽2)11yescharacteristic
263C2×C4×C172−1+411yescharacteristic
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276Dic34—22no—
277Dic34—22no—
278Dic34—41no—
279Dic34—41no—
280Dic34—84no—
281#3622—22no—
282#3622—22no—
283#3622—41no—
284#3622—41no—
285C2×#476—171no—
286C2×C4×SD16—171no—
287#2451—171no—
288D4×C4⋊C4D1711yescharacteristic
289#2096—171no—
290#2084—171no—
291#2066—171no—
292#2058—171no—
293#1941—171no—
294#1916—171no—
295#1877—171no—
296#1877—171no—
297#1865—171no—
298#1864—171no—
299#1864—171no—
300C22×C17—24no—
301C22×C17—42no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17#52511yescharacteristic
308C22×C17D4211yescharacteristic
309C22×C17D4×Q811yescharacteristic
310C22×C17C22×SD1611yescharacteristic
311C22×C17#57511yescharacteristic
312C22×C17#57711yescharacteristic
313C22×C17#57711yescharacteristic
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—22no—
317C4×C17—42no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17C2×C22⋊Q811yescharacteristic
324C4×C17#53911yescharacteristic
325C4×C17#54711yescharacteristic
326C4×C17#55411yescharacteristic
327Dic17—42no—
328Dic17—42no—
329Dic17—81no—
330Dic17—81no—
331C22×SD16—171no—
332C2×C4⋊Q8—171no—
333C2×C4×Q8—171no—
334C2×C4×D4C2×D1711yescharacteristic
335C2×C4×D4C2×D1711yescharacteristic
336C2×C4×D4C2×D1711yescharacteristic
337C22×C4⋊C4—21no—
338#476—344no—
339C4×SD16—344no—
340C2×#106—171no—
341C2×#106—171no—
342C2×#106—171no—
343C2×#106—171no—
344C2×#106—171no—
345C2×C4⋊C8—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#102—171no—
350C2×#102—171no—
351C2×#102—171no—
352C2×C4×C8—171no—
353#398—341no—
354#395—341no—
355#391—171no—
356#391—171no—
357#391—341no—
358#389—171no—
359#389—171no—
360#389C2×D1711yescharacteristic
361#389C2×D1711yescharacteristic
362#387—341no—
363#385—21no—
364#385—21no—
365#384C2×D1711yescharacteristic
366#380—21no—
367C4×C4⋊C4C2×D1711yescharacteristic
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#336—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376C2×C17—44no—
377C2×C17#244911yescharacteristic
378C2×C17#310111yescharacteristic
379C2×C17#312511yescharacteristic
380C2×C17#326711yescharacteristic
381C2×C17D4×SD1611yescharacteristic
382C2×C17Q8×SD1611yescharacteristic
383C2×C17#338811yescharacteristic
384C22×Q8—171no—
385C22×Q8—341no—
386C22×D4C22×D1711yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—21no—
390C2×SD16—174no—
391C2×SD16—342no—
392C2×SD16—342no—
393C22×C8—171no—
394C22×C8—171no—
395C22×C8—171no—
396C22×C8—171no—
397C4⋊Q8—342no—
398C4⋊Q8—681no—
399C4×Q8—342no—
400C4×Q8—681no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—41no—
406C4×D4—41no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—44no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4C22⋊D1711yescharacteristic
420C2×C4⋊C4C22⋊D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C4⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C42—341no—
428C2×C42C22×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431#106—342no—
432#106—342no—
433#106—342no—
434#106—342no—
435#106—681no—
436C4⋊C8—342no—
437#103—342no—
438#103—342no—
439#103—681no—
440#102—174no—
441#102—342no—
442#102—342no—
443C4×C8—342no—
444#95—21no—
445#95—21no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95C22⋊D1711yescharacteristic
453#95C22⋊D1711yescharacteristic
454C17#2637611yescharacteristic, complemented, Sylow
455C24—21no—
456C2×Q8—172no—
457C2×Q8—341no—
458C2×Q8—341no—
459C2×Q8—341no—
460C2×Q8—341no—
461C2×Q8—341no—
462C2×Q8—682no—
463C2×Q8—682no—
464C2×Q8—1361no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4#6325212yes—
469C2×D4D4×D1712yes—
470C2×D4Q8×D1712yes—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4C2×C22⋊D1711yescharacteristic
488C22×C4C2×C22⋊D1711yescharacteristic
489C22×C4C2×C22⋊D1711yescharacteristic
490C22×C4C2×C22⋊D1711yescharacteristic
491C22×C4C2×C22⋊D1711yescharacteristic
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C23×D1711yescharacteristic
494SD16—348no—
495C2×C8—172no—
496C2×C8—172no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—172no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—682no—
524C4⋊C4—682no—
525C4⋊C4—1361no—
526C4⋊C4—1361no—
527C4⋊C4#6322914yescomplemented
528C4⋊C4#6325212yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—682no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D1711yescharacteristic
547Q8—342no—
548Q8—342no—
549Q8—681no—
550Q8—681no—
551Q8—1364no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#1058758214yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—342no—
584C2×C4—342no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4C2×#6322912yes—
596C2×C4C2×#6322912yes—
597C2×C4C2×#6322912yes—
598C2×C4C2×#6325211yescharacteristic
599C2×C4C2×#6325211yescharacteristic
600C2×C4C2×D4×D1711yescharacteristic
601C2×C4#1058769111yescharacteristic
602C2×C4C2×Q8×D1711yescharacteristic
603C2×C4#1058769411yescharacteristic
604C8—342no—
605C8—342no—
606C8—681no—
607C8—681no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1176332011yescharacteristic
616C22D4×C22⋊D1711yescharacteristic
617C22Q8×C22⋊D1711yescharacteristic
618C22C22×#6322911yescharacteristic
619C22#1176360711yescharacteristic
620C22#1176361711yescharacteristic
621C22#1176362011yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—682no—
632C4—682no—
633C4—1361no—
634C4—1361no—
635C4C2×#1058758211yescharacteristic
636C4#1176340611yescharacteristic
637C4#1176346011yescharacteristic
638C4#1176351411yescharacteristic
639C2—44no—
640C2#1453259311yescharacteristic
641C2#1453879211yescharacteristic
642C2#1453906111yescharacteristic
643C2#1454031011yescharacteristic
644C2D4×#6322911yescharacteristic
645C2Q8×#6322911yescharacteristic
646C2#1454172911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingD4×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterD4×C4⋊C4D17
hypocenterC17#26376
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
3. #14532623C2semidirect
4. #14530328C2semidirect
5. #14530261C2semidirect
6. #14530162C2semidirect
7. #14530117C2semidirect
8. #14529509C2semidirect
9. #14529343C2semidirect
12. #14529051C2semidirect
91. C2×#63229Q8semidirect
180. C4⋊C4×C17SD16semidirect
454. C17#26376semidirect
527. C4⋊C4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117630992
C2×C4×#632292
#145326232
#145303282
#145302612
#145301622
#145301172
#145295092
#145293432
#145291182
#145290512
#145290452
D4×C4⋊C4×C172
#2637617