Finite Group Databasebeta

← #17097350

Subgroups — #17097350

Order 4352.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763094C211yesmaximal, characteristic, complemented
2C2×C4×#63230C211yesmaximal, characteristic
3#14532634C211yesmaximal, characteristic
4#14530318C211yesmaximal, characteristic
5#14530261C211yesmaximal, characteristic, complemented
6#14530155C211yesmaximal, characteristic
7#14530117C211yesmaximal, characteristic, complemented
8#14529509C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529139C211yesmaximal, characteristic
11#14529139C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14529052C211yesmaximal, characteristic, complemented
14#14529044C211yesmaximal, characteristic
15Q8×C4⋊C4×C17C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587582—21no—
19C2×C4×D68C2211yescharacteristic
20#11763094—24no—
21C4×#63230—24no—
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
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×Q8×C17C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C17×#389—21no—
54C17×#389—21no—
55C17×#389C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#384—21no—
58C17×#384C2211yescharacteristic
59C4×C4⋊C4×C17—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×Dic34—21no—
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C4×Q8×C17—41no—
70C4×Q8×C17—41no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C42×C17—21no—
84C2×C42×C17—21no—
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63230—22no—
90C2×#63230—22no—
91C2×#63230Q814yescomplemented
92C2×#63227—21no—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×#63227—21no—
97C2×C4⋊Dic17—21no—
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×#63225—21no—
102C2×#63225—21no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107#10587582—44no—
108C4×D68—22no—
109C4×D68—41no—
110#10587555—22no—
111#10587555—41no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587506—22no—
117#10587506—22no—
118#10587506D414yes—
119#10587502—21no—
120#10587502—21no—
121#10587480—22no—
122#10587480—22no—
123#10587480—41no—
124#10587479—22no—
125#10587479—22no—
126#10587479—22no—
127#10587479—22no—
128#10587479—41no—
129C4⋊#3622—22no—
130C4×#3622—22no—
131C23×D17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17C2×D412yes—
136C2×Q8×C17C2×Q812yes—
137C2×Q8×C17#4112yes—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68—21no—
157C2×D68C2×Q812yes—
158C2×C4×D17—42no—
159C2×C4×D17—42no—
160C2×C4×D17—81no—
161C2×Dic34—42no—
162C2×Dic34—42no—
163C2×Dic34—81no—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—41no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17—42no—
174C4⋊C4×C17—81no—
175C4⋊C4×C17—81no—
176C4⋊C4×C17—81no—
177C4⋊C4×C17SD1614yescomplemented
178C4⋊C4×C17#4112yes—
179C42×C17—24no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—42no—
187C42×C17—42no—
188#63230—28no—
189#63227—42no—
190#63227—42no—
191#63227—42no—
192#63227—42no—
193#63227—81no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200C4⋊Dic17C2×D412yes—
201#63225—42no—
202#63225—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#26373—171nomaximal, Sylow
216C23×C17C22×D411yescharacteristic
217C22×D17—22no—
218C22×D17—41no—
219C22×D17—41no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—41no—
223C22×D17—41no—
224Q8×C17—22no—
225Q8×C17—22no—
226Q8×C17—22no—
227Q8×C17C22⋊Q814yescomplemented
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
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—41no—
254C2×C4×C17—41no—
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—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273D68—22no—
274D68—22no—
275D68—41no—
276D68—41no—
277C4×D17—84no—
278Dic34—84no—
279#3622—22no—
280#3622—22no—
281#3622—41no—
282#3622—41no—
283C2×#475—171no—
284C2×C4×SD16—171no—
285#2452—171no—
286Q8×C4⋊C4D1711yescharacteristic, complemented
287#2094—171no—
288#2084—171no—
289#2065—171no—
290#2058—171no—
291#1941—171no—
292#1916—171no—
293#1880—171no—
294#1880—171no—
295#1865—171no—
296#1865—171no—
297#1864—171no—
298C22×C17#52511yescharacteristic
299C22×C17D4211yescharacteristic
300C22×C17D4×Q811yescharacteristic
301C22×C17C22×SD1611yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57711yescharacteristic
304C22×C17#57711yescharacteristic
305C2×D17—42no—
306C2×D17—42no—
307C2×D17—42no—
308C2×D17—81no—
309C2×D17—81no—
310C2×D17—81no—
311C2×D17—81no—
312C4×C17—22no—
313C4×C17—24no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—42no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—81no—
321C4×C17—81no—
322C4×C17C2×C22⋊Q811yescharacteristic
323C4×C17#53911yescharacteristic
324C4×C17#54711yescharacteristic
325C4×C17#55411yescharacteristic
326Dic17—42no—
327Dic17—81no—
328Dic17—81no—
329C22×SD16—171no—
330C2×C4⋊Q8—171no—
331C2×C22⋊Q8—341no—
332C2×C4×Q8C2×D1711yescharacteristic
333C2×C4×Q8C2×D1711yescharacteristic
334C2×C4×Q8C2×D1711yescharacteristic
335C2×C4×D4—171no—
336#475—344no—
337C4×SD16—344no—
338C2×#106—171no—
339C2×#106—171no—
340C2×#106—171no—
341C2×#106—171no—
342C2×#106—171no—
343C2×C4⋊C8—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×#102—171no—
350C2×C4×C8—171no—
351#396—341no—
352#393—341no—
353#390—171no—
354#390—171no—
355#389—21no—
356#389—21no—
357#389—171no—
358#389—171no—
359#389C2×D1711yescharacteristic
360#389C2×D1711yescharacteristic
361#385—341no—
362#384—21no—
363#384C2×D1711yescharacteristic
364C4×C4⋊C4—21no—
365C4×C4⋊C4C2×D1711yescharacteristic
366#340—171no—
367#340—171no—
368#340—171no—
369#340—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374C2×C17#244911yescharacteristic
375C2×C17#310311yescharacteristic
376C2×C17#312511yescharacteristic
377C2×C17#327111yescharacteristic
378C2×C17D4×SD1611yescharacteristic
379C2×C17Q8×SD1611yescharacteristic
380C2×C17#338811yescharacteristic
381D17—82no—
382C22×Q8—341no—
383C22×Q8C22×D1711yescharacteristic
384C22×D4—171no—
385C23×C4—341no—
386C2×SD16—174no—
387C2×SD16—342no—
388C2×SD16—342no—
389C22×C8—171no—
390C22×C8—171no—
391C22×C8—171no—
392C22×C8—171no—
393C4⋊Q8—342no—
394C4⋊Q8—681no—
395C22⋊Q8—684no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—41no—
402C4×D4—342no—
403C4×D4—681no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4C22⋊D1711yescharacteristic
418C2×C4⋊C4C22⋊D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42C22×D1711yescharacteristic
431C2×C42C22×D1711yescharacteristic
432C2×C42C22×D1711yescharacteristic
433#106—342no—
434#106—342no—
435#106—342no—
436#106—342no—
437#106—681no—
438C4⋊C8—342no—
439#103—174no—
440#103—342no—
441#103—342no—
442#102—342no—
443#102—342no—
444#102—681no—
445C4×C8—342no—
446#95—21no—
447#95—21no—
448#95—341no—
449#95—341no—
450#95C22⋊D1711yescharacteristic
451#95C22⋊D1711yescharacteristic
452C17#2637311yescharacteristic, complemented, Sylow
453C24—341no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—682no—
458C2×Q8—682no—
459C2×Q8—1361no—
460C2×Q8#6325212yes—
461C2×Q8D4×D1712yes—
462C2×Q8Q8×D1712yes—
463C2×D4—172no—
464C2×D4—341no—
465C2×D4—341no—
466C2×D4—341no—
467C2×D4—341no—
468C2×D4—341no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—682no—
477C22×C4—682no—
478C22×C4—1361no—
479C22×C4C2×C22⋊D1711yescharacteristic
480C22×C4C2×C22⋊D1711yescharacteristic
481C22×C4C2×C22⋊D1711yescharacteristic
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C23×D1711yescharacteristic
486SD16—348no—
487C2×C8—172no—
488C2×C8—172no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—172no—
513C4⋊C4—341no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—682no—
519C4⋊C4—682no—
520C4⋊C4—682no—
521C4⋊C4#6323014yescomplemented
522C4⋊C4#6325212yes—
523C22⋊C4—682no—
524C22⋊C4—682no—
525C22⋊C4—682no—
526C22⋊C4—682no—
527C22⋊C4—1361no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—42no—
537C23—342no—
538C23—681no—
539C23—681no—
540C23—681no—
541C23—681no—
542C23—681no—
543C23—681no—
544C23C22×C22⋊D1711yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—1364no—
549Q8#1058758214yescomplemented
550D4—342no—
551D4—342no—
552D4—681no—
553D4—681no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
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—342no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—1364no—
591C2×C4C2×#6323012yes—
592C2×C4C2×#6323012yes—
593C2×C4C2×#6323012yes—
594C2×C4C2×#6325211yescharacteristic
595C2×C4C2×#6325211yescharacteristic
596C2×C4C2×D4×D1711yescharacteristic
597C2×C4#1058769111yescharacteristic
598C2×C4C2×Q8×D1711yescharacteristic
599C2×C4#1058769411yescharacteristic
600C8—342no—
601C8—342no—
602C8—681no—
603C8—681no—
604C22—682no—
605C22—682no—
606C22—682no—
607C22—1361no—
608C22—1361no—
609C22—1361no—
610C22—1361no—
611C22#1176332011yescharacteristic
612C22D4×C22⋊D1711yescharacteristic
613C22Q8×C22⋊D1711yescharacteristic
614C22C22×#6323011yescharacteristic
615C22#1176360711yescharacteristic
616C22#1176361711yescharacteristic
617C22#1176362011yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—682no—
629C4—1361no—
630C4—1361no—
631C4C2×#1058758211yescharacteristic
632C4#1176340611yescharacteristic
633C4#1176346011yescharacteristic
634C4#1176351411yescharacteristic
635C2—1362no—
636C2#1453259311yescharacteristic
637C2#1453881511yescharacteristic
638C2#1453906111yescharacteristic
639C2#1454036511yescharacteristic
640C2D4×#6323011yescharacteristic
641C2Q8×#6323011yescharacteristic
642C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingQ8×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenterQ8×C4⋊C4D17
hypocenterC17#26373
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11763094C2semidirect
5. #14530261C2semidirect
7. #14530117C2semidirect
8. #14529509C2semidirect
9. #14529343C2semidirect
12. #14529052C2semidirect
13. #14529052C2semidirect
15. Q8×C4⋊C4×C17C2semidirect
91. C2×#63230Q8semidirect
177. C4⋊C4×C17SD16semidirect
227. Q8×C17C22⋊Q8semidirect
286. Q8×C4⋊C4D17semidirect
452. C17#26373semidirect
521. C4⋊C4#63230semidirect
549. Q8#10587582semidirect

Maximal subgroups

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

subgroupindex
C2×#117630942
C2×C4×#632302
#145326342
#145303182
#145302612
#145301552
#145301172
#145295092
#145293432
#145291392
#145290522
#145290442
Q8×C4⋊C4×C172
#2637317