Finite Group Databasebeta

← C4×#16863833 #31009150

Subgroups — C4×#16863833 #31009150

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863833C211yesmaximal, characteristic
2C4×#14524066C211yesmaximal, characteristic, complemented
3C4×D8×D67C211yesmaximal, characteristic, complemented
4#22562084C211yesmaximal, characteristic, complemented
5#22562082C211yesmaximal, characteristic, complemented
6#22562076C211yesmaximal, characteristic, complemented
7#22562074C211yesmaximal, characteristic, complemented
8C4×#14523986C211yesmaximal, characteristic, complemented
9C4×D1072C211yesmaximal, characteristic, complemented
10C4×#14523984C211yesmaximal, characteristic, complemented
11#22561996C211yesmaximal, characteristic, complemented
12#22561923C211yesmaximal, characteristic, complemented
13#22561920C211yesmaximal, characteristic, complemented
14C4×#14523982C211yesmaximal, characteristic, complemented
15C4×SD32×C67C211yesmaximal, characteristic, complemented
16C2×#14524066C2211yescharacteristic
17C2×D8×D67C2211yescharacteristic
18C4×#11761629—21no—
19C4×D4×D67—21no—
20#16863833C418yesdirect product factor
21C2×#14523986C2211yescharacteristic
22C2×D1072C2211yescharacteristic
23C2×#14523984C2211yescharacteristic
24C2×#14523982C2211yescharacteristic
25C2×#14523982C2211yescharacteristic
26Q16×Dic67C2211yescharacteristic, complemented
27C4×#11761615—21no—
28#16863481—21no—
29C4×D536C2211yescharacteristic
30C4×D536C2211yescharacteristic, complemented
31D8×Dic67C2211yescharacteristic
32#16863478C2211yescharacteristic
33#16863478C2211yescharacteristic, complemented
34C4×#11761613—21no—
35#16863476—21no—
36#16863433C2211yescharacteristic
37#107×D67C2211yescharacteristic
38#16863391—21no—
39#102×D67—21no—
40C4×C8×D67C2211yescharacteristic
41#16863223C2211yescharacteristic, complemented
42#16863222C2211yescharacteristic, complemented
43#16863221C2211yescharacteristic, complemented
44#16863200C2211yescharacteristic, complemented
45#16863198C2211yescharacteristic
46#16863198C2211yescharacteristic, complemented
47#16863197C2211yescharacteristic
48#16863196C2211yescharacteristic
49#16863196C2211yescharacteristic, complemented
50#16863171C2211yescharacteristic, complemented
51#16863171C2211yescharacteristic, complemented
52C4×C67⋊C16C2211yescharacteristic, complemented
53C2×SD32×C67C2211yescharacteristic
54C4×Q16×C67C2211yescharacteristic, complemented
55C4×D8×C67C2211yescharacteristic
56#367×C67C2211yescharacteristic, complemented
57#358×C67C2211yescharacteristic, complemented
58#357×C67C2211yescharacteristic
59C4×C16×C67C2211yescharacteristic, complemented
60C2×#11761629—21no—
61C2×D4×D67—21no—
62C22×C4×D67—41no—
63#14524066—22no—
64#14524066C2×C414yescomplemented
65C2×#11761615—21no—
66C2×D536C2311yescharacteristic
67C2×D536C2311yescharacteristic
68C2×#11761613—21no—
69D8×D67—22no—
70D8×D67C2×C414yescomplemented
71C2×C8×D67D411yescharacteristic
72C2×C8×D67C2311yescharacteristic
73Q8×Dic67—21no—
74C4×D268—21no—
75C4×D268—21no—
76C4×D268—41no—
77D4×Dic67—21no—
78#14524005—21no—
79#14524005—21no—
80#14524005—41no—
81C4×C22⋊D67—41no—
82(C22⋊D67)⋊C4—41no—
83#14524001—21no—
84#14524001—41no—
85C4⋊C4×D67—21no—
86C22⋊C4×D67—41no—
87C42×D67—41no—
88C42×D67D411yescharacteristic
89#14523986C2×C414yescomplemented
90D1072C2×C414yescomplemented
91#14523984C2×C414yescomplemented
92#14523982—22no—
93#14523982C2×C414yescomplemented
94C2×C67⋊C16C2311yescharacteristic
95C2×C67⋊C16C2311yescharacteristic
96#14523977C2311yescharacteristic
97#14523977C2311yescharacteristic
98#14523976—21no—
99#14523974—21no—
100#14523967—21no—
101#14523965—21no—
102#14523965—21no—
103#14523964—21no—
104#14523963—21no—
105#14523963—21no—
106C4×C67⋊C8D411yescharacteristic
107C8×Dic67D411yescharacteristic
108C8×Dic67C2311yescharacteristic
109C2×Q16×C67C2311yescharacteristic
110C2×D8×C67C2311yescharacteristic
111C4×Q8×C67—21no—
112C4×D4×C67—21no—
113SD32×C67C2×C414yescomplemented
114C2×C16×C67C2311yescharacteristic
115C2×C16×C67C2311yescharacteristic
116#107×C67C2311yescharacteristic
117#107×C67C2311yescharacteristic
118#103×C67—21no—
119#102×C67—21no—
120C4×C8×C67C2311yescharacteristic
121C23×D67—41no—
122#11761629—24no—
123#11761629—42no—
124C2×D268—21no—
125C2×D268—21no—
126C2×D268—41no—
127C2×C22⋊D67—41no—
128D4×D67—24no—
129D4×D67—42no—
130C22×Dic67—41no—
131C2×C4×D67—41no—
132C2×C4×D67—41no—
133C2×C4×D67—41no—
134C2×C4×D67—41no—
135C2×C4×D67—41no—
136C2×C4×D67—41no—
137C2×C4×D67—84no—
138C2×C4×D67D811yescharacteristic
139C2×C4×D67D811yescharacteristic
140C2×C4×D67C2×D411yescharacteristic
141#11761615—24no—
142D536—21no—
143D536—21no—
144D536C22×C412yes—
145D536C22×C412yescomplemented
146#11761613—24no—
147C2×C67⋊C8C2×D411yescharacteristic
148C2×C67⋊C8C2×D411yescharacteristic
149C8×D67—22no—
150C8×D67C22×C412yes—
151C8×D67#4112yes—
152C4⋊Dic67—21no—
153C4⋊Dic67—21no—
154C4⋊Dic67—41no—
155Dic67⋊C4—41no—
156C22⋊Dic67—41no—
157(C2×D67)⋊C4—41no—
158(C2×D67)⋊C4—41no—
159(C2×D67)⋊C4—41no—
160(C2×D67)⋊C4—41no—
161(C2×D67)⋊C4—41no—
162C4×Dic67—41no—
163C4×Dic67—41no—
164C4×Dic67D811yescharacteristic
165C4×Dic67D811yescharacteristic
166C4×Dic67C2×D411yescharacteristic
167C67⋊C16—21no—
168C67⋊C16C22×C412yescomplemented
169C2×Q8×C67—21no—
170C2×D4×C67—21no—
171C22×C4×C67—41no—
172Q16×C67—21no—
173Q16×C67C22×C412yescomplemented
174D8×C67—21no—
175D8×C67C22×C412yes—
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2411yescharacteristic
178C4⋊C4×C67—21no—
179C4⋊C4×C67—21no—
180C4⋊C4×C67—41no—
181C22⋊C4×C67—41no—
182C42×C67—41no—
183C42×C67C2×D411yescharacteristic
184C16×C67—21no—
185C16×C67C22×C412yescomplemented
186C23×C67—41no—
187C22×D67—41no—
188C22×D67—41no—
189C22×D67—44no—
190C22×D67—82no—
191C22×D67—82no—
192C22×D67C2×D811yescharacteristic
193Q8×C67—22no—
194Q8×C67—41no—
195D4×C67—22no—
196D4×C67—41no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—41no—
200C2×C4×C67—82no—
201C2×C4×C67C2×D811yescharacteristic
202C2×C4×C67C2×D811yescharacteristic
203C2×C4×C67C22×D411yescharacteristic
204C22⋊D67—44no—
205C2×Dic67—41no—
206C2×Dic67—41no—
207C2×Dic67—82no—
208C2×Dic67C2×D811yescharacteristic
209C2×Dic67C2×D811yescharacteristic
210C2×Dic67C2×D811yescharacteristic
211D268—22no—
212D268—22no—
213D268—44no—
214D268—41no—
215D268—41no—
216C4×D67—22no—
217C4×D67—41no—
218C4×D67—44no—
219C4×D67—82no—
220C4×D67—82no—
221C4×D67—82no—
222C4×D67—82no—
223C4×D67—82no—
224C4×D67C4×D412yes—
225C4×D67#13512yes—
226C8×C67—21no—
227C8×C67C23×C411yescharacteristic
228C8×C67C2×#4111yescharacteristic
229C67⋊C8—21no—
230C67⋊C8C4×D412yes—
231C22×C67—42no—
232C22×C67—81no—
233C22×C67—81no—
234C22×C67C22×D811yescharacteristic
235C2×D67—21no—
236C2×D67—21no—
237C2×D67—42no—
238C2×D67—42no—
239C2×D67—88no—
240C2×D67—81no—
241C2×D67—81no—
242C2×D67—81no—
243C2×D67—81no—
244C2×D67C4×D812yes—
245C4×C67—21no—
246C4×C67—42no—
247C4×C67—81no—
248C4×C67—81no—
249C4×C67#50912yescomplemented
250C4×C67C2×C4×D411yescharacteristic
251C4×C67C2×#13511yescharacteristic
252Dic67—21no—
253Dic67—21no—
254Dic67—81no—
255Dic67—81no—
256Dic67C4×D812yes—
257C4×#509—671nomaximal, Sylow
258C2×C67—82no—
259C2×C67C2×C4×D811yescharacteristic
260C2×C67C2×#50911yescharacteristic
261C2×C67#341411yescharacteristic
262D67—22no—
263D67—82no—
264D67—82no—
265C2×#509—671no—
266C4×#135—671no—
267C2×C4×D8—671no—
268#2177—671no—
269#2177—671no—
270#2175—671no—
271#2175—671no—
272C4×SD32—671no—
273C4×SD32D6711yescharacteristic, complemented
274C4×D16—671no—
275C4×D16—671no—
276#2159—671no—
277#2140—671no—
278#2139—671no—
279C4×M32—671no—
280C67C4×#50911yescharacteristic, complemented, Sylow
281C2×#135—671no—
282C22×D8—671no—
283C4×#41—1341no—
284C2×C4×D4—1341no—
285#509—678no—
286C2×SD32—671no—
287C2×SD32C2×D6711yescharacteristic
288C2×D16—671no—
289C2×D16—671no—
290C2×M32—671no—
291C2×M32—671no—
292C4×Q16—671no—
293C4×Q16C2×D6711yescharacteristic, complemented
294C4×SD16—1341no—
295C4×SD16—1341no—
296C4×D8—671no—
297C4×D8—671no—
298C4×D8—671no—
299C4×D8—671no—
300C4×D8—671no—
301C4×D8—1341no—
302C4×D8—1341no—
303C4×D8C2×D6711yescharacteristic
304#427—671no—
305C2×#107—671no—
306#416—1341no—
307C2×#102—1341no—
308C2×C4×C8—671no—
309#367—671no—
310#367C2×D6711yescharacteristic, complemented
311#366—671no—
312#366—671no—
313#358—671no—
314#358C2×D6711yescharacteristic, complemented
315#357—671no—
316#357—671no—
317#357—671no—
318#357—671no—
319#357—671no—
320#357C2×D6711yescharacteristic
321#346—671no—
322#346—671no—
323C4×C16—671no—
324C4×C16C2×D6711yescharacteristic, complemented
325C2×#41—1341no—
326C22×D4—1341no—
327C23×C4—2681no—
328#135—674no—
329#135—1342no—
330C2×Q16C22×D6711yescharacteristic
331C2×SD16—1341no—
332C2×D8—674no—
333C2×D8—671no—
334C2×D8—671no—
335C2×D8—1342no—
336C2×D8—1341no—
337C2×D8C22×D6711yescharacteristic
338C22×C8—671no—
339C22×C8—671no—
340C4×Q8—21no—
341C4×Q8—1341no—
342C4×D4—21no—
343C4×D4—2681no—
344C4×D4—2681no—
345C4×D4—2681no—
346C4×D4—2681no—
347C4×D4—1341no—
348C4×D4—1341no—
349C4×D4—1341no—
350C4×D4—1341no—
351C4×D4—1341no—
352#117—2681no—
353#117—1341no—
354C2×C4⋊C4—1341no—
355C2×C22⋊C4—2681no—
356C2×C42—2681no—
357C2×C42—671no—
358SD32—674no—
359SD32C4×D6714yescomplemented
360D16—674no—
361D16—674no—
362M32—674no—
363M32—1342no—
364C2×C16—671no—
365C2×C16—671no—
366C2×C16C22×D6711yescharacteristic
367C2×C16C22×D6711yescharacteristic
368#107—671no—
369#107—671no—
370#107—1341no—
371#107C22×D6711yescharacteristic
372#107C22×D6711yescharacteristic
373#106—1341no—
374#103—21no—
375#103—1341no—
376#102—21no—
377#102—1341no—
378#102—1341no—
379#102—1341no—
380#102—1341no—
381#102—1341no—
382C4×C8—671no—
383C4×C8—671no—
384C4×C8—671no—
385C4×C8C22×D6711yescharacteristic
386C24—2681no—
387#41—2682no—
388#41—1344no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—2681no—
392C2×D4—2682no—
393C2×D4—2681no—
394C2×D4—1344no—
395C2×D4—1341no—
396C2×D4—1341no—
397C22×C4—41no—
398C22×C4—2681no—
399C22×C4—2681no—
400C22×C4—2681no—
401C22×C4—2681no—
402C22×C4—2681no—
403C22×C4—2681no—
404C22×C4—2681no—
405C22×C4—5364no—
406C22×C4—671no—
407C22×C4—671no—
408C22×C4—671no—
409Q16—21no—
410Q16C2×C4×D6712yescomplemented
411SD16—1344no—
412D8—21no—
413D8—672no—
414D8—672no—
415D8—1344no—
416D8—1341no—
417D8—1341no—
418D8C2×C4×D6712yes—
419C2×C8—671no—
420C2×C8—672no—
421C2×C8—672no—
422C2×C8—671no—
423C2×C8—1342no—
424C2×C8D4×D6711yescharacteristic
425C2×C8C23×D6711yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—2681no—
430C4⋊C4—2681no—
431C4⋊C4—1341no—
432C4⋊C4—1341no—
433C22⋊C4—41no—
434C22⋊C4—2681no—
435C22⋊C4—2681no—
436C22⋊C4—2681no—
437C22⋊C4—2681no—
438C22⋊C4—2681no—
439C22⋊C4—2681no—
440C42—41no—
441C42—2681no—
442C42—2681no—
443C42—671no—
444C42—671no—
445C42—671no—
446C42D4×D6711yescharacteristic
447C16—21no—
448C16—672no—
449C16—1341no—
450C16C2×C4×D6712yescomplemented
451C23—41no—
452C23—2684no—
453C23—2681no—
454C23—2681no—
455C23—5362no—
456C23—5362no—
457C23—671no—
458Q8—22no—
459Q8—41no—
460D4—22no—
461D4—41no—
462D4—2684no—
463D4—2684no—
464D4—2681no—
465D4—2681no—
466D4—1342no—
467D4—1342no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—2681no—
473C2×C4—2684no—
474C2×C4—2681no—
475C2×C4—2681no—
476C2×C4—5362no—
477C2×C4—5362no—
478C2×C4—5362no—
479C2×C4—5362no—
480C2×C4—5362no—
481C2×C4—5362no—
482C2×C4—671no—
483C2×C4—672no—
484C2×C4—672no—
485C2×C4—671no—
486C2×C4—671no—
487C2×C4—1342no—
488C2×C4D8×D6711yescharacteristic
489C2×C4D8×D6711yescharacteristic
490C2×C4C2×D4×D6711yescharacteristic
491C8—21no—
492C8—672no—
493C8—1341no—
494C8C22×C4×D6711yescharacteristic
495C8#41×D6711yescharacteristic
496C22—42no—
497C22—81no—
498C22—81no—
499C22—2682no—
500C22—2682no—
501C22—5368no—
502C22—5361no—
503C22—5361no—
504C22—5361no—
505C22—5361no—
506C22—672no—
507C22—1341no—
508C22—1341no—
509C22C2×D8×D6711yescharacteristic
510C4—21no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—5361no—
515C4—5361no—
516C4—672no—
517C4—1341no—
518C4—1341no—
519C4#1686383312yesdirect product factor
520C4C4×D4×D6711yescharacteristic
521C4#135×D6711yescharacteristic
522C2—82no—
523C2—5362no—
524C2—5362no—
525C2—1342no—
526C2C4×D8×D6711yescharacteristic
527C2C2×#1686383311yescharacteristic
528C2#2257334211yescharacteristic
subgroupG/H
centerC2×C4D8×D67
commutatorC8×C67C23×C4
FrattiniC2×C8C23×D67
FittingC4×SD32×C67C2
socleC22×C67C22×D8
prime radicalC2×#16863833C2
cosocleC2×C8×C67C24
hypercenterC4×SD32D67
hypocenterC67C4×#509
LCSG ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. C4×#14524066C2semidirect
3. C4×D8×D67C2semidirect
4. #22562084C2semidirect
5. #22562082C2semidirect
6. #22562076C2semidirect
7. #22562074C2semidirect
8. C4×#14523986C2semidirect
9. C4×D1072C2semidirect
10. C4×#14523984C2semidirect
11. #22561996C2semidirect
12. #22561923C2semidirect
13. #22561920C2semidirect
14. C4×#14523982C2semidirect
15. C4×SD32×C67C2semidirect
20. #16863833C4direct (N × H)
26. Q16×Dic67C22semidirect
30. C4×D536C22semidirect
33. #16863478C22semidirect
41. #16863223C22semidirect
42. #16863222C22semidirect
43. #16863221C22semidirect
44. #16863200C22semidirect
46. #16863198C22semidirect
49. #16863196C22semidirect
50. #16863171C22semidirect
51. #16863171C22semidirect
52. C4×C67⋊C16C22semidirect
54. C4×Q16×C67C22semidirect
56. #367×C67C22semidirect
57. #358×C67C22semidirect
59. C4×C16×C67C22semidirect
64. #14524066C2×C4semidirect
70. D8×D67C2×C4semidirect
89. #14523986C2×C4semidirect
90. D1072C2×C4semidirect
91. #14523984C2×C4semidirect
93. #14523982C2×C4semidirect
113. SD32×C67C2×C4semidirect
145. D536C22×C4semidirect
168. C67⋊C16C22×C4semidirect
173. Q16×C67C22×C4semidirect
185. C16×C67C22×C4semidirect
249. C4×C67#509semidirect
273. C4×SD32D67semidirect
280. C67C4×#509semidirect
293. C4×Q16C2×D67semidirect
310. #367C2×D67semidirect
314. #358C2×D67semidirect
324. C4×C16C2×D67semidirect
359. SD32C4×D67semidirect
410. Q16C2×C4×D67semidirect
450. C16C2×C4×D67semidirect
519. C4#16863833direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168638332
C4×#145240662
C4×D8×D672
#225620842
#225620822
#225620762
#225620742
C4×#145239862
C4×D10722
C4×#145239842
#225619962
#225619232
#225619202
C4×#145239822
C4×SD32×C672
C4×#50967