Finite Group Databasebeta

← #30855526

Subgroups — #30855526

Order 17152.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864162C211yesmaximal, characteristic
2C2×#16863445C211yesmaximal, characteristic, complemented
3#22568726C211yesmaximal, characteristic
4C2×#16863395C211yesmaximal, characteristic
5#22560274C214yesmaximal, complemented
6#22560269C214yesmaximal, complemented
7#22559948C211yesmaximal, characteristic, complemented
8C2×#16863165C211yesmaximal, characteristic, complemented
9C2×#428×C67C211yesmaximal, characteristic, complemented
10C22×#11761628C2211yescharacteristic
11#16864162—24no—
12#16864162C418yescomplemented
13C2×#14524051—22no—
14C2×#14524049C2214yes—
15C22×#11761612C2211yescharacteristic
16C22×#11761612C2211yescharacteristic
17C22×#11761612C2211yescharacteristic
18C2×M16×D67—22no—
19#16863445—24no—
20C2×#14523979C2214yescomplemented
21#16863404—21no—
22#16863400—21no—
23#16863395—22no—
24#16863395C2212yes—
25#16863395C2214yes—
26#16863395C2212yes—
27#16863390—22no—
28C2×#14523966—21no—
29C2×#14523963—21no—
30#16863165C2212yescomplemented
31#16863165C2214yescomplemented
32#16863165C2212yescomplemented
33C22×M16×C67C2211yescharacteristic
34C2×#117×C67C2211yescharacteristic
35C22×C4⋊C4×C67C2211yescharacteristic
36#428×C67—22no—
37#428×C67—22no—
38C2×#107×C67C2212yescomplemented
39C2×#106×C67C2212yescomplemented
40C2×#11761628—21no—
41C2×#11761628—22no—
42C2×#11761628—21no—
43C2×#11761628—22no—
44C2×#11761628C2×C414yes—
45C2×#11761628D412yes—
46C2×#11761628Q814yes—
47C2×#11761628C2312yes—
48C22×Dic134D411yescharacteristic
49C22×D268D411yescharacteristic
50C22×C22⋊D67—21no—
51C22×C4×D67—21no—
52#14524051—216no—
53#14524049—28no—
54#14524049C2×C4116yescomplemented
55C2×#11761612—21no—
56C2×#11761612—22no—
57C2×#11761612—21no—
58C2×#11761612—21no—
59C2×#11761612—22no—
60C2×#11761612—22no—
61C2×#11761612—22no—
62C2×#11761612—22no—
63C2×#11761612C2×C414yes—
64C2×#11761612C2312yes—
65C2×#11761612C2312yes—
66C2×#11761612C2312yes—
67C2×#11761611—24no—
68M16×D67—216no—
69C22×C67⋊C8—21no—
70C22×C67⋊C8—21no—
71C22×C67⋊C8—21no—
72C2×C8×D67—24no—
73#14523979—28no—
74#14523966—24no—
75#14523966—22no—
76#14523966—22no—
77#14523963—24no—
78#14523963—22no—
79#14523963—22no—
80C23×C4×C67—21no—
81C23×C4×C67C2311yescharacteristic
82C2×M16×C67C2×C414yes—
83C2×M16×C67D414yes—
84C2×M16×C67D412yes—
85C2×M16×C67D412yes—
86C22×C8×C67C2312yes—
87#117×C67—21no—
88#117×C67—21no—
89#117×C67—21no—
90#117×C67—21no—
91C2×C4⋊C4×C67—21no—
92C2×C4⋊C4×C67—21no—
93C2×C4⋊C4×C67—21no—
94C2×C4⋊C4×C67—21no—
95C2×C4⋊C4×C67—21no—
96C2×C4⋊C4×C67C2312yes—
97C2×C4⋊C4×C67C2311yescharacteristic
98C2×C4⋊C4×C67C2311yescharacteristic
99C2×C22⋊C4×C67—21no—
100C2×C42×C67—21no—
101#107×C67—24no—
102#106×C67—24no—
103C23×D67—21no—
104#11761628—24no—
105#11761628—24no—
106#11761628—24no—
107#11761628—24no—
108#11761628—24no—
109#11761628—22no—
110#11761628—22no—
111#11761628C4⋊C418yes—
112#11761628C22×C414yes—
113#11761628C2×Q814yes—
114C2×Dic134—21no—
115C2×Dic134—22no—
116C2×Dic134C4⋊C414yes—
117C2×Dic134C2×D412yes—
118C2×D268—21no—
119C2×D268—22no—
120C2×D268C4⋊C414yes—
121C2×D268C2×D412yes—
122C2×C22⋊D67—24no—
123C2×C22⋊D67—22no—
124C2×C22⋊D67—24no—
125C2×C22⋊D67—22no—
126C22×Dic67—21no—
127C2×C4×D67—24no—
128C2×C4×D67—22no—
129C2×C4×D67—24no—
130C2×C4×D67—22no—
131#11761612—24no—
132#11761612—22no—
133#11761612—22no—
134#11761612—22no—
135#11761612—22no—
136#11761612—22no—
137#11761612—42no—
138#11761612—42no—
139#11761612C22×C414yes—
140#11761611—216no—
141C2×C67⋊C8—24no—
142C2×C67⋊C8—22no—
143C2×C67⋊C8—22no—
144C2×C67⋊C8—22no—
145C2×C67⋊C8—42no—
146C2×C67⋊C8—42no—
147C8×D67—216no—
148C24×C67#4111yescharacteristic
149C22×C4×C67—21no—
150C22×C4×C67—21no—
151C22×C4×C67—22no—
152C22×C4×C67—41no—
153C22×C4×C67—41no—
154C22×C4×C67—82no—
155C22×C4×C67C22×C412yes—
156C22×C4×C67C2×D411yescharacteristic
157C22×C4×C67C2×D411yescharacteristic
158C22×C4×C67C2×D411yescharacteristic
159C22×C4×C67C2×D411yescharacteristic
160C22×C4×C67C2×D412yes—
161C22×C4×C67C2×Q812yes—
162C22×C4×C67#4111yescharacteristic
163C22×C4×C67#4112yes—
164C22×C4×C67C2411yescharacteristic
165M16×C67C22⋊C418yes—
166M16×C67C22⋊C418yes—
167C2×C8×C67C22×C414yes—
168C2×C8×C67C2×D414yes—
169C2×C8×C67C2×D414yes—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—22no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—22no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—41no—
177C4⋊C4×C67—41no—
178C22⋊C4×C67—41no—
179C22⋊C4×C67—41no—
180C42×C67—41no—
181C42×C67—41no—
182C23×C67—41no—
183C23×C67—41no—
184C23×C67#11712yes—
185C23×C67#12111yescharacteristic
186C23×C67#12111yescharacteristic
187C23×C67C22⋊Q812yes—
188C23×C67C2×#4111yescharacteristic
189C22×D67—22no—
190C22×D67—24no—
191C22×D67—81no—
192C22×D67—81no—
193C2×C4×C67—41no—
194C2×C4×C67—41no—
195C2×C4×C67—42no—
196C2×C4×C67—82no—
197C2×C4×C67—82no—
198C2×C4×C67—81no—
199C2×C4×C67—81no—
200C2×C4×C67—82no—
201C2×C4×C67C2×C22⋊C412yes—
202C2×C4×C67C2×C22⋊C412yes—
203C2×C4×C67C2×C4⋊C412yes—
204C2×C4×C67C2×C4⋊C412yes—
205C2×C4×C67#11712yes—
206C2×C4×C67#12111yescharacteristic
207C2×C4×C67#12111yescharacteristic
208C2×C4×C67#12112yes—
209C2×C4×C67#12112yes—
210C2×C4×C67C22⋊Q812yes—
211C2×C4×C67C22⋊Q812yes—
212C2×C4×C67C22⋊Q812yes—
213C2×C4×C67C23×C411yescharacteristic
214C2×C4×C67C22×D411yescharacteristic
215C2×C4×C67C22×D411yescharacteristic
216C2×C4×C67C22×D411yescharacteristic
217C2×C4×C67C22×Q811yescharacteristic
218C2×C4×C67C2×#4111yescharacteristic
219C22⋊D67—24no—
220C22⋊D67—24no—
221C22⋊D67—28no—
222C2×Dic67—22no—
223C2×Dic67—24no—
224D268—22no—
225D268—24no—
226D268C2×C4⋊C414yes—
227C4×D67—24no—
228C4×D67—24no—
229C4×D67—28no—
230Dic134—22no—
231Dic134—24no—
232Dic134C2×C4⋊C414yes—
233C8×C67C2×C22⋊C418yes—
234C67⋊C8—24no—
235C67⋊C8—42no—
236C67⋊C8—42no—
237C22×C67—21no—
238C22×C67—21no—
239C22×C67—21no—
240C22×C67—21no—
241C22×C67—21no—
242C22×C67—21no—
243C22×C67—42no—
244C22×C67—42no—
245C22×C67#38012yes—
246C22×C67#38012yes—
247C22×C67C2×#11711yescharacteristic
248C22×C67C2×#12111yescharacteristic
249C22×C67C2×C22⋊Q811yescharacteristic
250C2×D67—24no—
251C2×D67—42no—
252C2×D67—42no—
253C2×D67—84no—
254C4×C67—81no—
255C4×C67—82no—
256C4×C67—81no—
257C4×C67#38012yes—
258C4×C67#38012yes—
259C4×C67C22×C22⋊C411yescharacteristic
260C4×C67C22×C4⋊C411yescharacteristic
261C4×C67C2×#12111yescharacteristic
262C4×C67C2×C22⋊Q811yescharacteristic
263Dic67—24no—
264#20679—671nomaximal, Sylow
265C2×C67—22no—
266C2×C67—21no—
267C2×C67—21no—
268C2×C67—22no—
269C2×C67C2×#38011yescharacteristic
270C2×C67#306811yescharacteristic
271C2×C67#306911yescharacteristic
272D67—44no—
273C2×#568—671no—
274C2×#430—671no—
275C2×#428D6711yescharacteristic, complemented
276#2896—671no—
277C2×#417—671no—
278#1816—674no—
279#1815—674no—
280#1742—671no—
281C2×#343—671no—
282C67#2067911yescharacteristic, complemented, Sylow
283C22×#41—671no—
284#568—678no—
285#568—1344no—
286C2×#131—674no—
287C2×#131—1342no—
288C22×M16—671no—
289C22×M16—671no—
290C22×M16—671no—
291C22×M16—1342no—
292C22×M16C2×D6711yescharacteristic
293C2×#117C2×D6711yescharacteristic
294C22×C4⋊C4C2×D6711yescharacteristic
295#430—1344no—
296C2×#108—674no—
297#428—22no—
298#428—22no—
299C2×#107C2×D6712yescomplemented
300C2×#106C2×D6712yescomplemented
301#419—1341no—
302#418—1341no—
303#417—672no—
304#417—674no—
305#417—672no—
306#417—1342no—
307#416—1342no—
308C2×#103—1341no—
309C2×#102—1341no—
310#343—672no—
311#343—674no—
312#343—672no—
313C2×#41—674no—
314C2×#41—672no—
315C2×#41—674no—
316C2×#41—672no—
317C2×#41—1341no—
318C2×#41—1342no—
319C2×#41—1341no—
320C2×#41—1342no—
321C22×Q8—671no—
322C22×D4—671no—
323C22×D4—1341no—
324C23×C4—21no—
325C23×C4—1341no—
326C23×C4C22×D6711yescharacteristic
327#131—6716no—
328#131—13416no—
329#131—1348no—
330C2×M16—674no—
331C2×M16—672no—
332C2×M16—672no—
333C2×M16—672no—
334C2×M16—1342no—
335C2×M16—1342no—
336C2×M16—1342no—
337C2×M16—1342no—
338C2×M16—1341no—
339C2×M16—1341no—
340C2×M16—1341no—
341C2×M16—1342no—
342C2×M16—13416no—
343C2×M16—1344no—
344C2×M16C4×D6714yes—
345C2×M16D26814yes—
346C2×M16C22⋊D6712yes—
347C2×M16C22⋊D6712yes—
348C22×C8—1341no—
349C22×C8—1341no—
350C22×C8—1341no—
351C22×C8—1344no—
352C22×C8C22×D6712yes—
353#117—21no—
354#117—21no—
355#117—21no—
356#117—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4C22×D6712yes—
363C2×C4⋊C4C22×D6711yescharacteristic
364C2×C4⋊C4C22×D6711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C42—21no—
367#108—1348no—
368#107—24no—
369#106—24no—
370#103—1344no—
371#103—1342no—
372#103—1342no—
373#102—1344no—
374#102—1342no—
375#102—1342no—
376C24—1341no—
377C24#1176162711yescharacteristic
378#41—678no—
379#41—674no—
380#41—674no—
381#41—1344no—
382#41—1344no—
383#41—1344no—
384#41—1344no—
385#41—1344no—
386#41—1342no—
387#41—1342no—
388C2×Q8—674no—
389C2×Q8—672no—
390C2×Q8—1341no—
391C2×Q8—1342no—
392C2×D4—674no—
393C2×D4—672no—
394C2×D4—1344no—
395C2×D4—1342no—
396C2×D4—1344no—
397C2×D4—1342no—
398C2×D4—1341no—
399C2×D4—1342no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—82no—
406C22×C4—1341no—
407C22×C4—1344no—
408C22×C4—1342no—
409C22×C4—1344no—
410C22×C4—1342no—
411C22×C4C2×C4×D6712yes—
412C22×C4D4×D6711yescharacteristic
413C22×C4D4×D6711yescharacteristic
414C22×C4C2×C22⋊D6711yescharacteristic
415C22×C4C2×C22⋊D6711yescharacteristic
416C22×C4C2×D26812yes—
417C22×C4Q8×D6712yes—
418C22×C4#1176162711yescharacteristic
419C22×C4#1176162912yes—
420C22×C4C23×D6711yescharacteristic
421M16—2682no—
422M16—2682no—
423M16—674no—
424M16—1342no—
425M16—1342no—
426M16—1342no—
427M16—1342no—
428M16—1344no—
429M16—1342no—
430M16—13416no—
431M16(C2×D67)⋊C418yes—
432M16(C2×D67)⋊C418yes—
433C2×C8—2682no—
434C2×C8—2682no—
435C2×C8—1342no—
436C2×C8—1342no—
437C2×C8—1344no—
438C2×C8—1342no—
439C2×C8—13416no—
440C2×C8C2×C4×D6714yes—
441C2×C8C2×C22⋊D6714yes—
442C2×C8C2×D26814yes—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—41no—
454C42—41no—
455C23—41no—
456C23—41no—
457C23—5361no—
458C23—5361no—
459C23—1342no—
460C23—1344no—
461C23#1452400112yes—
462C23#1452401411yescharacteristic
463C23#1452401411yescharacteristic
464C23#1452402312yes—
465C23C2×#1176162711yescharacteristic
466Q8—674no—
467Q8—1344no—
468Q8—1342no—
469D4—674no—
470D4—1348no—
471D4—1344no—
472D4—1344no—
473D4—1344no—
474D4—1342no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—1344no—
484C2×C4—1342no—
485C2×C4—1348no—
486C2×C4—1344no—
487C2×C4—1344no—
488C2×C4C2×(C2×D67)⋊C412yes—
489C2×C4C2×(C2×D67)⋊C412yes—
490C2×C4C4⋊C4×D6712yes—
491C2×C4C4⋊C4×D6712yes—
492C2×C4#1452400112yes—
493C2×C4#1452401411yescharacteristic
494C2×C4#1452401411yescharacteristic
495C2×C4#1452401812yes—
496C2×C4#1452401812yes—
497C2×C4#1452401912yes—
498C2×C4#1452401912yes—
499C2×C4#1452402312yes—
500C2×C4C22×C4×D6711yescharacteristic
501C2×C4C2×D4×D6711yescharacteristic
502C2×C4C22×C22⋊D6711yescharacteristic
503C2×C4C22×D26811yescharacteristic
504C2×C4C2×Q8×D6711yescharacteristic
505C2×C4C2×#1176162911yescharacteristic
506C8—2682no—
507C8—2682no—
508C8—1344no—
509C8C2×(C2×D67)⋊C418yes—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—2682no—
519C22—2682no—
520C22—5364no—
521C22—1344no—
522C22#1686325312yes—
523C22#1686325312yes—
524C22C2×#1452400111yescharacteristic
525C22C2×#1452401411yescharacteristic
526C22C2×#1452402311yescharacteristic
527C4—82no—
528C4—81no—
529C4—81no—
530C4—1344no—
531C4#1686325312yes—
532C4#1686325312yes—
533C4C22×(C2×D67)⋊C411yescharacteristic
534C4C2×C4⋊C4×D6711yescharacteristic
535C4C2×#1452401811yescharacteristic
536C4C2×#1452401911yescharacteristic
537C2—22no—
538C2—22no—
539C2—21no—
540C2—21no—
541C2—2684no—
542C2C2×#1686325311yescharacteristic
543C2#2256993311yescharacteristic
544C2#2256994211yescharacteristic
subgroupG/H
centerC22C2×#14524014
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#428×C67C2
socleC22×C67C2×#121
prime radicalC22×#11761628C22
cosocleC22×C4×C67C24
hypercenterC2×#428D67
hypocenterC67#20679
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16863445C2semidirect
5. #22560274C2semidirect
6. #22560269C2semidirect
7. #22559948C2semidirect
8. C2×#16863165C2semidirect
9. C2×#428×C67C2semidirect
12. #16864162C4semidirect
20. C2×#14523979C22semidirect
30. #16863165C22semidirect
31. #16863165C22semidirect
32. #16863165C22semidirect
38. C2×#107×C67C22semidirect
39. C2×#106×C67C22semidirect
54. #14524049C2×C4semidirect
275. C2×#428D67semidirect
282. C67#20679semidirect
299. C2×#107C2×D67semidirect
300. C2×#106C2×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168641622
C2×#168634452
#225687262
C2×#168633952
#225602742
#225602692
#225599482
C2×#168631652
C2×#428×C672
#2067967