Finite Group Databasebeta

← #30855434

Subgroups — #30855434

Order 17152.

1438 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524051C211yesmaximal, characteristic
2C2×#16863438C211yesmaximal, characteristic, complemented
3#22562802C211yesmaximal, characteristic
4#22562797C211yesmaximal, characteristic
5#22560243C212yesmaximal, complemented
6#22560239C212yesmaximal, complemented
7#22560235C212yesmaximal, complemented
8#22560231C212yesmaximal, complemented
9#22559925C211yesmaximal, characteristic, complemented
10C2×#16863158C211yesmaximal, characteristic, complemented
11C2×#428×C67C211yesmaximal, characteristic, complemented
12C22×#11761628C2211yescharacteristic
13C2×#14524051—24no—
14C2×#14524051—24no—
15C2×#14524051C418yescomplemented
16C22×#11761611C2212yes—
17C23×C67⋊C8C2211yescharacteristic
18C22×C8×D67C2212yes—
19C2×#14524002C2211yescharacteristic
20C22×C4⋊Dic67C2211yescharacteristic
21#16863438—24no—
22C2×#14523977C2212yes—
23C2×#14523975C2212yes—
24#16863302—21no—
25#16863301—21no—
26#16863298—21no—
27#16863297—21no—
28#16863275—22no—
29#16863273—22no—
30#16863253C2212yes—
31#16863253C2212yes—
32#16863253C2212yes—
33#16863253C2212yes—
34#16863158C2212yes—
35#16863158C2212yes—
36#16863158C2211yescharacteristic
37#16863158C2211yescharacteristic
38#16863158C2211yescharacteristic
39#16863158C2211yescharacteristic
40C22×M16×C67C2211yescharacteristic
41C2×#117×C67C2211yescharacteristic
42C22×C4⋊C4×C67C2211yescharacteristic
43#428×C67—24no—
44C2×#107×C67C2212yes—
45C2×#106×C67C2212yes—
46C2×#11761628—22no—
47C2×#11761628—22no—
48C2×#11761628—22no—
49C2×#11761628—22no—
50C2×#11761628C2×C414yes—
51C2×#11761628Q814yes—
52C22×Dic134D411yescharacteristic
53C22×D268D411yescharacteristic
54C22×C22⋊D67D412yes—
55C23×Dic67—21no—
56C22×C4×D67C2312yes—
57#14524051—232no—
58C2×#11761611—24no—
59C2×#11761611—24no—
60C2×#11761611C2×C418yescomplemented
61C22×C67⋊C8—22no—
62C22×C67⋊C8—22no—
63C22×C67⋊C8C2×C414yes—
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C2×C8×D67—24no—
67C2×C8×D67—24no—
68C2×C8×D67C2×C418yescomplemented
69#14524002—22no—
70#14524002—22no—
71C2×C4⋊Dic67—22no—
72C2×C4⋊Dic67—21no—
73C2×C4⋊Dic67—22no—
74C2×C4⋊Dic67C2312yes—
75C2×C4⋊Dic67C2311yescharacteristic
76C2×C4⋊Dic67C2311yescharacteristic
77C2×C22⋊Dic67—21no—
78C2×(C2×D67)⋊C4—22no—
79C2×(C2×D67)⋊C4—22no—
80C2×(C2×D67)⋊C4—22no—
81C2×(C2×D67)⋊C4—22no—
82C2×C4×Dic67—21no—
83#14523977—24no—
84#14523975—24no—
85C67⋊#95—22no—
86C67⋊#95—22no—
87C67⋊#95—22no—
88C67⋊#95—22no—
89C23×C4×C67—21no—
90C23×C4×C67C2311yescharacteristic
91C2×M16×C67C2×C414yes—
92C2×M16×C67D414yes—
93C2×M16×C67D414yes—
94C22×C8×C67C2312yes—
95#117×C67—22no—
96#117×C67—22no—
97C2×C4⋊C4×C67—22no—
98C2×C4⋊C4×C67—22no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67C2312yes—
101C2×C4⋊C4×C67C2311yescharacteristic
102C2×C4⋊C4×C67C2311yescharacteristic
103C2×C22⋊C4×C67—21no—
104C2×C42×C67—21no—
105#107×C67—24no—
106#106×C67—24no—
107C23×D67C2×D412yes—
108#11761628—28no—
109#11761628—28no—
110#11761628—28no—
111#11761628—28no—
112C2×Dic134—22no—
113C2×Dic134—22no—
114C2×Dic134C4⋊C414yes—
115C2×D268—22no—
116C2×D268—22no—
117C2×D268C4⋊C414yes—
118C2×C22⋊D67—24no—
119C2×C22⋊D67—24no—
120C2×C22⋊D67C4⋊C418yes—
121C22×Dic67—21no—
122C22×Dic67—21no—
123C22×Dic67—22no—
124C22×Dic67—42no—
125C22×Dic67—81no—
126C22×Dic67—81no—
127C22×Dic67C2×D412yes—
128C2×C4×D67—22no—
129C2×C4×D67—22no—
130C2×C4×D67—22no—
131C2×C4×D67—22no—
132C2×C4×D67C22×C414yes—
133C2×C4×D67C2×Q814yes—
134#11761611—216no—
135C2×C67⋊C8—28no—
136C2×C67⋊C8—21no—
137C2×C67⋊C8—21no—
138C2×C67⋊C8—21no—
139C2×C67⋊C8—21no—
140C2×C67⋊C8C22×C412yes—
141C2×C67⋊C8C22×C412yes—
142C8×D67—216no—
143C4⋊Dic67—21no—
144C4⋊Dic67—22no—
145C4⋊Dic67—21no—
146C4⋊Dic67—22no—
147C4⋊Dic67—21no—
148C4⋊Dic67—21no—
149C4⋊Dic67—42no—
150C22⋊Dic67—42no—
151(C2×D67)⋊C4—82no—
152(C2×D67)⋊C4—82no—
153(C2×D67)⋊C4—82no—
154(C2×D67)⋊C4—82no—
155C4×Dic67—42no—
156C24×C67C2×D411yescharacteristic
157C22×C4×C67—21no—
158C22×C4×C67—21no—
159C22×C4×C67—22no—
160C22×C4×C67—42no—
161C22×C4×C67—81no—
162C22×C4×C67—81no—
163C22×C4×C67C22×C412yes—
164C22×C4×C67C2×D411yescharacteristic
165C22×C4×C67C2×D412yes—
166C22×C4×C67C2×D412yes—
167C22×C4×C67C2×Q812yes—
168C22×C4×C67#4112yes—
169C22×C4×C67#4112yes—
170C22×C4×C67C2411yescharacteristic
171M16×C67C22⋊C4116yescomplemented
172C2×C8×C67C22×C414yes—
173C2×C8×C67C2×D414yes—
174C2×C8×C67C2×D414yes—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—22no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—22no—
179C4⋊C4×C67—21no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—42no—
182C22⋊C4×C67—42no—
183C42×C67—42no—
184C23×C67—21no—
185C23×C67—21no—
186C23×C67—21no—
187C23×C67—21no—
188C23×C67C2×C4⋊C412yes—
189C23×C67#12112yes—
190C23×C67#12112yes—
191C23×C67C22×D411yescharacteristic
192C22×D67—22no—
193C22×D67—22no—
194C22×D67—22no—
195C22×D67—22no—
196C22×D67—22no—
197C22×D67—22no—
198C22×D67C2×C4⋊C414yes—
199C2×C4×C67—41no—
200C2×C4×C67—41no—
201C2×C4×C67—42no—
202C2×C4×C67—42no—
203C2×C4×C67—41no—
204C2×C4×C67—41no—
205C2×C4×C67—42no—
206C2×C4×C67—41no—
207C2×C4×C67—41no—
208C2×C4×C67—84no—
209C2×C4×C67C2×C22⋊C414yes—
210C2×C4×C67C2×C4⋊C412yes—
211C2×C4×C67#11714yes—
212C2×C4×C67#12112yes—
213C2×C4×C67#12112yes—
214C2×C4×C67C22⋊Q814yes—
215C2×C4×C67C22⋊Q814yes—
216C2×C4×C67C23×C411yescharacteristic
217C2×C4×C67C22×D411yescharacteristic
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C22×Q811yescharacteristic
220C2×C4×C67C2×#4111yescharacteristic
221C2×C4×C67C2×#4111yescharacteristic
222C22⋊D67—216no—
223C2×Dic67—22no—
224C2×Dic67—22no—
225C2×Dic67—41no—
226C2×Dic67—41no—
227C2×Dic67—42no—
228C2×Dic67—41no—
229C2×Dic67—42no—
230C2×Dic67—41no—
231C2×Dic67—42no—
232C2×Dic67—41no—
233C2×Dic67—41no—
234C2×Dic67—84no—
235C2×Dic67C2×C4⋊C414yes—
236D268—28no—
237C4×D67—24no—
238C4×D67—24no—
239C4×D67—24no—
240C4×D67—24no—
241Dic134—28no—
242C8×C67C2×C22⋊C418yes—
243C67⋊C8—22no—
244C67⋊C8—22no—
245C22×C67—22no—
246C22×C67—22no—
247C22×C67—22no—
248C22×C67—22no—
249C22×C67—22no—
250C22×C67—22no—
251C22×C67#38014yes—
252C22×C67C22×C4⋊C411yescharacteristic
253C22×C67C2×#12111yescharacteristic
254C22×C67C2×#12111yescharacteristic
255C22×C67#57611yescharacteristic
256C22×C67#57711yescharacteristic
257C22×C67#57711yescharacteristic
258C22×C67#57811yescharacteristic
259C2×D67—24no—
260C2×D67—24no—
261C2×D67—24no—
262C2×D67—42no—
263C2×D67—42no—
264C2×D67—42no—
265C2×D67—42no—
266C4×C67—81no—
267C4×C67—82no—
268C4×C67—81no—
269C4×C67#38014yes—
270C4×C67C22×C22⋊C411yescharacteristic
271C4×C67C2×#11711yescharacteristic
272C4×C67C2×C22⋊Q811yescharacteristic
273C4×C67C2×C22⋊Q811yescharacteristic
274Dic67—24no—
275Dic67—81no—
276Dic67—82no—
277Dic67—81no—
278#20664—671nomaximal, Sylow
279C2×C67—24no—
280C2×C67C2×#38011yescharacteristic
281C2×C67#291211yescharacteristic
282C2×C67#291311yescharacteristic
283C2×C67#306211yescharacteristic
284C2×C67#306211yescharacteristic
285C2×C67#306311yescharacteristic
286C2×C67#306311yescharacteristic
287D67—44no—
288C22×#131—671no—
289C2×#428—671no—
290C2×#428D6711yescharacteristic, complemented
291#2311—671no—
292#2311—671no—
293#1809—672no—
294#1809—672no—
295#1808—672no—
296#1807—672no—
297#1735—671no—
298C2×#340—671no—
299C67#2066411yescharacteristic, complemented, Sylow
300C22×#41—671no—
301C2×#131—678no—
302C2×#131—1344no—
303C2×#131—1344no—
304C22×M16—672no—
305C22×M16C2×D6711yescharacteristic
306C23×C8—671no—
307C23×C8—672no—
308C2×#117—671no—
309C2×#117C2×D6711yescharacteristic
310C22×C4⋊C4—671no—
311C22×C4⋊C4C2×D6711yescharacteristic
312#428—24no—
313#428—1344no—
314C2×#107—672no—
315C2×#107C2×D6712yes—
316C2×#106—672no—
317C2×#106C2×D6712yes—
318#391—1341no—
319#391—1341no—
320#390—1341no—
321#390—1341no—
322#386—1342no—
323#386—1342no—
324#380—672no—
325#380—672no—
326#380—672no—
327#380—672no—
328#340—671no—
329#340—672no—
330#340—671no—
331#340—672no—
332#340—671no—
333#340—671no—
334C2×#41—674no—
335C2×#41—674no—
336C2×#41—1342no—
337C2×#41—1342no—
338C2×#41—1342no—
339C2×#41—1342no—
340C22×Q8—671no—
341C22×D4—672no—
342C22×D4—671no—
343C23×C4—21no—
344C23×C4—672no—
345C23×C4—1341no—
346C23×C4C22×D6711yescharacteristic
347#131—13432no—
348C2×M16—678no—
349C2×M16—1344no—
350C2×M16—1344no—
351C2×M16C4×D6714yes—
352C2×M16D26814yes—
353C2×M16C22⋊D6714yes—
354C22×C8—674no—
355C22×C8—671no—
356C22×C8—671no—
357C22×C8—678no—
358C22×C8—1342no—
359C22×C8—1342no—
360C22×C8—1344no—
361C22×C8—1344no—
362C22×C8C22×D6712yes—
363#117—22no—
364#117—22no—
365#117—1342no—
366#117—1342no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—672no—
371C2×C4⋊C4—671no—
372C2×C4⋊C4—671no—
373C2×C4⋊C4—1341no—
374C2×C4⋊C4—1342no—
375C2×C4⋊C4—1342no—
376C2×C4⋊C4C22×D6712yes—
377C2×C4⋊C4C22×D6711yescharacteristic
378C2×C4⋊C4C22×D6711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1341no—
381C2×C22⋊C4—1342no—
382C2×C22⋊C4—1342no—
383C2×C22⋊C4—1342no—
384C2×C22⋊C4—1342no—
385C2×C42—21no—
386C2×C42—1341no—
387#107—24no—
388#107—1344no—
389#106—24no—
390#106—1344no—
391#95—1342no—
392#95—1342no—
393#95—1342no—
394#95—1342no—
395C24—672no—
396C24D4×D6711yescharacteristic
397#41—1348no—
398#41—1348no—
399#41—1348no—
400#41—1348no—
401C2×Q8—674no—
402C2×Q8—1342no—
403C2×Q8—1342no—
404C2×D4—678no—
405C2×D4—674no—
406C2×D4—1344no—
407C2×D4—1344no—
408C2×D4—1342no—
409C2×D4—1342no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—2682no—
417C22×C4—5361no—
418C22×C4—5361no—
419C22×C4—672no—
420C22×C4—674no—
421C22×C4—674no—
422C22×C4—1341no—
423C22×C4—1341no—
424C22×C4—1342no—
425C22×C4—1342no—
426C22×C4—1342no—
427C22×C4—1342no—
428C22×C4—1342no—
429C22×C4C2×C4×D6712yes—
430C22×C4D4×D6711yescharacteristic
431C22×C4C2×C22⋊D6712yes—
432C22×C4C2×D26812yes—
433C22×C4Q8×D6712yes—
434C22×C4#1176162712yes—
435C22×C4#1176162912yes—
436C22×C4C23×D6711yescharacteristic
437M16—13416no—
438M16(C2×D67)⋊C4116yescomplemented
439C2×C8—672no—
440C2×C8—672no—
441C2×C8—1348no—
442C2×C8—1341no—
443C2×C8—1341no—
444C2×C8—1341no—
445C2×C8—1341no—
446C2×C8—13416no—
447C2×C8C2×C4×D6714yes—
448C2×C8C2×C22⋊D6714yes—
449C2×C8C2×D26814yes—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—2682no—
458C4⋊C4—1341no—
459C4⋊C4—1342no—
460C4⋊C4—1341no—
461C4⋊C4—1342no—
462C4⋊C4—1341no—
463C4⋊C4—1341no—
464C22⋊C4—42no—
465C22⋊C4—2682no—
466C22⋊C4—5362no—
467C22⋊C4—5362no—
468C22⋊C4—5362no—
469C22⋊C4—5362no—
470C42—42no—
471C42—2682no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—674no—
477C23—1342no—
478C23—1342no—
479C23—1342no—
480C23—1342no—
481C23—1342no—
482C23—1342no—
483C23C4⋊C4×D6712yes—
484C23#1452401412yes—
485C23#1452401812yes—
486C23C2×D4×D6711yescharacteristic
487Q8—1348no—
488D4—13416no—
489D4—1348no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4—2681no—
501C2×C4—2681no—
502C2×C4—2682no—
503C2×C4—2681no—
504C2×C4—2682no—
505C2×C4—2682no—
506C2×C4—2681no—
507C2×C4—2681no—
508C2×C4—2681no—
509C2×C4—5364no—
510C2×C4—674no—
511C2×C4—1342no—
512C2×C4—1342no—
513C2×C4—1344no—
514C2×C4—1344no—
515C2×C4—1344no—
516C2×C4—1344no—
517C2×C4C2×(C2×D67)⋊C414yes—
518C2×C4C4⋊C4×D6712yes—
519C2×C4#1452400114yes—
520C2×C4#1452401412yes—
521C2×C4#1452401812yes—
522C2×C4#1452401914yes—
523C2×C4#1452402314yes—
524C2×C4C22×C4×D6711yescharacteristic
525C2×C4C22×C22⋊D6711yescharacteristic
526C2×C4C22×D26811yescharacteristic
527C2×C4C2×Q8×D6711yescharacteristic
528C2×C4C2×#1176162711yescharacteristic
529C2×C4C2×#1176162911yescharacteristic
530C8—1342no—
531C8—1342no—
532C8C2×(C2×D67)⋊C418yes—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—2682no—
540C22—2682no—
541C22—2682no—
542C22—2682no—
543C22—1344no—
544C22—1344no—
545C22—1344no—
546C22#1686325314yes—
547C22C2×C4⋊C4×D6711yescharacteristic
548C22C2×#1452401411yescharacteristic
549C22C2×#1452401811yescharacteristic
550C22#1686420011yescharacteristic
551C22#1686420511yescharacteristic
552C22#1686420611yescharacteristic
553C22#1686421211yescharacteristic
554C4—82no—
555C4—81no—
556C4—81no—
557C4—5362no—
558C4—5361no—
559C4—5361no—
560C4—1344no—
561C4#1686325314yes—
562C4C22×(C2×D67)⋊C411yescharacteristic
563C4C2×#1452400111yescharacteristic
564C4C2×#1452401911yescharacteristic
565C4C2×#1452402311yescharacteristic
566C2—24no—
567C2—2684no—
568C2C2×#1686325311yescharacteristic
569C2#2256880311yescharacteristic
570C2#2256880811yescharacteristic
571C2#2256988311yescharacteristic
572C2#2256989011yescharacteristic
573C2#2256989411yescharacteristic
574C2#2256990111yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#428×C67C2
socleC23×C67C22×D4
prime radicalC22×#11761628C22
cosocleC22×C4×C67C24
hypercenterC2×#428D67
hypocenterC67#20664
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16863438C2semidirect
5. #22560243C2semidirect
6. #22560239C2semidirect
7. #22560235C2semidirect
8. #22560231C2semidirect
9. #22559925C2semidirect
10. C2×#16863158C2semidirect
11. C2×#428×C67C2semidirect
15. C2×#14524051C4semidirect
60. C2×#11761611C2×C4semidirect
68. C2×C8×D67C2×C4semidirect
171. M16×C67C22⋊C4semidirect
290. C2×#428D67semidirect
299. C67#20664semidirect
438. M16(C2×D67)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145240512
C2×#168634382
#225628022
#225627972
#225602432
#225602392
#225602352
#225602312
#225599252
C2×#168631582
C2×#428×C672
#2066467