Finite Group Databasebeta

← #32150767

Subgroups — #32150767

Order 18176.

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×#17986436C211yesmaximal, characteristic
2C4×#14556151C211yesmaximal, characteristic, complemented
3C4×D8×D71C211yesmaximal, characteristic, complemented
4#22590217C211yesmaximal, characteristic, complemented
5#22590214C211yesmaximal, characteristic, complemented
6#22590202C211yesmaximal, characteristic, complemented
7SD32×Dic71C211yesmaximal, characteristic, complemented
8#22590189C211yesmaximal, characteristic, complemented
9#22590186C211yesmaximal, characteristic, complemented
10#22590184C211yesmaximal, characteristic, complemented
11#22590125C211yesmaximal, characteristic, complemented
12#22590054C211yesmaximal, characteristic, complemented
13#22590051C211yesmaximal, characteristic, complemented
14#22589933C211yesmaximal, characteristic, complemented
15C71×#2175C211yesmaximal, characteristic, complemented
16C2×#14556151C2211yescharacteristic
17C2×D8×D71C2211yescharacteristic
18C4×#11765866—21no—
19C4×D4×D71—21no—
20#17986436C418yescomplemented
21C2×#14556071C2211yescharacteristic
22C2×D1136C2211yescharacteristic
23C2×#14556069C2211yescharacteristic
24C2×#14556067C2211yescharacteristic
25C2×#14556067C2211yescharacteristic
26Q16×Dic71C2211yescharacteristic, complemented
27C4×#11765852—21no—
28#17986084—21no—
29C4×D568C2211yescharacteristic
30C4×D568C2211yescharacteristic, complemented
31D8×Dic71C2211yescharacteristic
32#17986081C2211yescharacteristic
33#17986081C2211yescharacteristic, complemented
34C4×#11765850—21no—
35#17986079—21no—
36#17986036C2211yescharacteristic
37#107×D71C2211yescharacteristic
38#17985994—21no—
39#102×D71—21no—
40C4×C8×D71C2211yescharacteristic
41#17985827C2211yescharacteristic, complemented
42#17985826C2211yescharacteristic, complemented
43#17985824C2211yescharacteristic, complemented
44#17985803C2211yescharacteristic, complemented
45#17985801C2211yescharacteristic
46#17985801C2211yescharacteristic, complemented
47#17985800C2211yescharacteristic
48#17985799C2211yescharacteristic
49#17985799C2211yescharacteristic, complemented
50#17985775C2211yescharacteristic, complemented
51C16×Dic71C2211yescharacteristic, complemented
52C16×Dic71C2211yescharacteristic, complemented
53C2×SD32×C71C2211yescharacteristic
54C4×Q16×C71C2211yescharacteristic, complemented
55C4×D8×C71C2211yescharacteristic
56#366×C71C2211yescharacteristic, complemented
57#358×C71C2211yescharacteristic, complemented
58#357×C71C2211yescharacteristic
59#346×C71C2211yescharacteristic, complemented
60C2×#11765866—21no—
61C2×D4×D71—21no—
62C22×C4×D71—41no—
63#14556151—22no—
64#14556151C2×C414yescomplemented
65C2×#11765852—21no—
66C2×D568C2311yescharacteristic
67C2×D568C2311yescharacteristic
68C2×#11765850—21no—
69D8×D71—22no—
70D8×D71C2×C414yescomplemented
71C2×C8×D71D411yescharacteristic
72C2×C8×D71C2311yescharacteristic
73Q8×Dic71—21no—
74C4×D284—21no—
75C4×D284—21no—
76C4×D284—41no—
77D4×Dic71—21no—
78#14556090—21no—
79#14556090—21no—
80#14556090—41no—
81C4×C22⋊D71—41no—
82(C22⋊D71)⋊C4—41no—
83#14556086—21no—
84#14556086—41no—
85C4⋊C4×D71—21no—
86C22⋊C4×D71—41no—
87C42×D71—41no—
88C42×D71D411yescharacteristic
89#14556071C2×C414yescomplemented
90D1136C2×C414yescomplemented
91#14556069C2×C414yescomplemented
92#14556067—22no—
93#14556067C2×C414yescomplemented
94C2×C71⋊C16C2311yescharacteristic
95C2×C71⋊C16C2311yescharacteristic
96#14556062C2311yescharacteristic
97#14556062C2311yescharacteristic
98#14556061—21no—
99#14556059—21no—
100#14556052—21no—
101#14556050—21no—
102#14556050—21no—
103#14556049—21no—
104#14556048—21no—
105#14556048—21no—
106C4×C71⋊C8D411yescharacteristic
107C8×Dic71D411yescharacteristic
108C8×Dic71C2311yescharacteristic
109C2×Q16×C71C2311yescharacteristic
110C2×D8×C71C2311yescharacteristic
111C4×Q8×C71—21no—
112C4×D4×C71—21no—
113SD32×C71C2×C414yescomplemented
114C2×C16×C71C2311yescharacteristic
115C2×C16×C71C2311yescharacteristic
116#107×C71C2311yescharacteristic
117#107×C71C2311yescharacteristic
118#103×C71—21no—
119#102×C71—21no—
120C4×C8×C71C2311yescharacteristic
121C23×D71—41no—
122#11765866—24no—
123#11765866—42no—
124C2×D284—21no—
125C2×D284—21no—
126C2×D284—41no—
127C2×C22⋊D71—41no—
128D4×D71—24no—
129D4×D71—42no—
130C22×Dic71—41no—
131C2×C4×D71—41no—
132C2×C4×D71—41no—
133C2×C4×D71—41no—
134C2×C4×D71—41no—
135C2×C4×D71—41no—
136C2×C4×D71—41no—
137C2×C4×D71—84no—
138C2×C4×D71D811yescharacteristic
139C2×C4×D71D811yescharacteristic
140C2×C4×D71C2×D411yescharacteristic
141#11765852—24no—
142D568—21no—
143D568—21no—
144D568C22×C412yes—
145D568C22×C412yescomplemented
146#11765850—24no—
147C2×C71⋊C8C2×D411yescharacteristic
148C2×C71⋊C8C2×D411yescharacteristic
149C8×D71—22no—
150C8×D71C22×C412yes—
151C8×D71#4112yes—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—41no—
155Dic71⋊C4—41no—
156C22⋊Dic71—41no—
157(C2×D71)⋊C4—41no—
158(C2×D71)⋊C4—41no—
159(C2×D71)⋊C4—41no—
160(C2×D71)⋊C4—41no—
161(C2×D71)⋊C4—41no—
162C4×Dic71—41no—
163C4×Dic71—41no—
164C4×Dic71D811yescharacteristic
165C4×Dic71D811yescharacteristic
166C4×Dic71C2×D411yescharacteristic
167C71⋊C16—21no—
168C71⋊C16C22×C412yescomplemented
169C2×Q8×C71—21no—
170C2×D4×C71—21no—
171C22×C4×C71—41no—
172Q16×C71—21no—
173Q16×C71C22×C412yescomplemented
174D8×C71—21no—
175D8×C71C22×C412yes—
176C2×C8×C71C2×D411yescharacteristic
177C2×C8×C71C2411yescharacteristic
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—41no—
181C22⋊C4×C71—41no—
182C42×C71—41no—
183C42×C71C2×D411yescharacteristic
184C16×C71—21no—
185C16×C71C22×C412yescomplemented
186C23×C71—41no—
187C22×D71—41no—
188C22×D71—41no—
189C22×D71—44no—
190C22×D71—82no—
191C22×D71—82no—
192C22×D71C2×D811yescharacteristic
193Q8×C71—22no—
194Q8×C71—41no—
195D4×C71—22no—
196D4×C71—41no—
197C2×C4×C71—41no—
198C2×C4×C71—41no—
199C2×C4×C71—41no—
200C2×C4×C71—82no—
201C2×C4×C71C2×D811yescharacteristic
202C2×C4×C71C2×D811yescharacteristic
203C2×C4×C71C22×D411yescharacteristic
204C22⋊D71—44no—
205C2×Dic71—41no—
206C2×Dic71—41no—
207C2×Dic71—82no—
208C2×Dic71C2×D811yescharacteristic
209C2×Dic71C2×D811yescharacteristic
210C2×Dic71C2×D811yescharacteristic
211D284—22no—
212D284—22no—
213D284—44no—
214D284—41no—
215D284—41no—
216C4×D71—22no—
217C4×D71—41no—
218C4×D71—44no—
219C4×D71—82no—
220C4×D71—82no—
221C4×D71—82no—
222C4×D71—82no—
223C4×D71—82no—
224C4×D71C4×D412yes—
225C4×D71#13512yes—
226C8×C71—21no—
227C8×C71C23×C411yescharacteristic
228C8×C71C2×#4111yescharacteristic
229C71⋊C8—21no—
230C71⋊C8C4×D412yes—
231C22×C71—42no—
232C22×C71—81no—
233C22×C71—81no—
234C22×C71C22×D811yescharacteristic
235C2×D71—21no—
236C2×D71—21no—
237C2×D71—42no—
238C2×D71—42no—
239C2×D71—88no—
240C2×D71—81no—
241C2×D71—81no—
242C2×D71—81no—
243C2×D71—81no—
244C2×D71C4×D812yes—
245C4×C71—21no—
246C4×C71—21no—
247C4×C71—42no—
248C4×C71—81no—
249C4×C71—81no—
250C4×C71C2×C4×D411yescharacteristic
251C4×C71C2×#13511yescharacteristic
252Dic71—21no—
253Dic71—81no—
254Dic71—81no—
255Dic71C4×D812yes—
256Dic71#50912yescomplemented
257C4×#509—711nomaximal, Sylow
258C2×C71—82no—
259C2×C71C2×C4×D811yescharacteristic
260C2×C71C2×#50911yescharacteristic
261C2×C71#341411yescharacteristic
262D71—22no—
263D71—82no—
264D71—82no—
265C2×#509—711no—
266C4×#135—711no—
267C2×C4×D8—711no—
268#2177—711no—
269#2177—711no—
270#2175—711no—
271#2175D7111yescharacteristic, complemented
272C4×SD32—711no—
273C4×SD32—711no—
274C4×D16—711no—
275C4×D16—711no—
276#2159—711no—
277#2140—711no—
278#2139—711no—
279C4×M32—711no—
280C71C4×#50911yescharacteristic, complemented, Sylow
281C2×#135—711no—
282C22×D8—711no—
283C4×#41—1421no—
284C2×C4×D4—1421no—
285#509—718no—
286C2×SD32—711no—
287C2×SD32C2×D7111yescharacteristic
288C2×D16—711no—
289C2×D16—711no—
290C2×M32—711no—
291C2×M32—711no—
292C4×Q16—711no—
293C4×Q16C2×D7111yescharacteristic, complemented
294C4×SD16—1421no—
295C4×SD16—1421no—
296C4×D8—711no—
297C4×D8—711no—
298C4×D8—711no—
299C4×D8—711no—
300C4×D8—711no—
301C4×D8—1421no—
302C4×D8—1421no—
303C4×D8C2×D7111yescharacteristic
304#427—711no—
305C2×#107—711no—
306#416—1421no—
307C2×#102—1421no—
308C2×C4×C8—711no—
309#367—711no—
310#367—711no—
311#366—711no—
312#366C2×D7111yescharacteristic, complemented
313#358—711no—
314#358C2×D7111yescharacteristic, complemented
315#357—711no—
316#357—711no—
317#357—711no—
318#357—711no—
319#357—711no—
320#357C2×D7111yescharacteristic
321#346—711no—
322#346C2×D7111yescharacteristic, complemented
323C4×C16—711no—
324C4×C16—711no—
325C2×#41—1421no—
326C22×D4—1421no—
327C23×C4—2841no—
328#135—714no—
329#135—1422no—
330C2×Q16C22×D7111yescharacteristic
331C2×SD16—1421no—
332C2×D8—714no—
333C2×D8—711no—
334C2×D8—711no—
335C2×D8—1422no—
336C2×D8—1421no—
337C2×D8C22×D7111yescharacteristic
338C22×C8—711no—
339C22×C8—711no—
340C4×Q8—21no—
341C4×Q8—1421no—
342C4×D4—21no—
343C4×D4—2841no—
344C4×D4—2841no—
345C4×D4—2841no—
346C4×D4—2841no—
347C4×D4—1421no—
348C4×D4—1421no—
349C4×D4—1421no—
350C4×D4—1421no—
351C4×D4—1421no—
352#117—2841no—
353#117—1421no—
354C2×C4⋊C4—1421no—
355C2×C22⋊C4—2841no—
356C2×C42—2841no—
357C2×C42—711no—
358SD32—714no—
359SD32C4×D7114yescomplemented
360D16—714no—
361D16—714no—
362M32—714no—
363M32—1422no—
364C2×C16—711no—
365C2×C16—711no—
366C2×C16C22×D7111yescharacteristic
367C2×C16C22×D7111yescharacteristic
368#107—711no—
369#107—711no—
370#107—1421no—
371#107C22×D7111yescharacteristic
372#107C22×D7111yescharacteristic
373#106—1421no—
374#103—21no—
375#103—1421no—
376#102—21no—
377#102—1421no—
378#102—1421no—
379#102—1421no—
380#102—1421no—
381#102—1421no—
382C4×C8—711no—
383C4×C8—711no—
384C4×C8—711no—
385C4×C8C22×D7111yescharacteristic
386C24—2841no—
387#41—2842no—
388#41—1424no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—2841no—
392C2×D4—2842no—
393C2×D4—2841no—
394C2×D4—1424no—
395C2×D4—1421no—
396C2×D4—1421no—
397C22×C4—41no—
398C22×C4—2841no—
399C22×C4—2841no—
400C22×C4—2841no—
401C22×C4—2841no—
402C22×C4—2841no—
403C22×C4—2841no—
404C22×C4—2841no—
405C22×C4—5684no—
406C22×C4—711no—
407C22×C4—711no—
408C22×C4—711no—
409Q16—21no—
410Q16C2×C4×D7112yescomplemented
411SD16—1424no—
412D8—21no—
413D8—712no—
414D8—712no—
415D8—1424no—
416D8—1421no—
417D8—1421no—
418D8C2×C4×D7112yes—
419C2×C8—711no—
420C2×C8—712no—
421C2×C8—712no—
422C2×C8—711no—
423C2×C8—1422no—
424C2×C8D4×D7111yescharacteristic
425C2×C8C23×D7111yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—2841no—
430C4⋊C4—2841no—
431C4⋊C4—1421no—
432C4⋊C4—1421no—
433C22⋊C4—41no—
434C22⋊C4—2841no—
435C22⋊C4—2841no—
436C22⋊C4—2841no—
437C22⋊C4—2841no—
438C22⋊C4—2841no—
439C22⋊C4—2841no—
440C42—41no—
441C42—2841no—
442C42—2841no—
443C42—711no—
444C42—711no—
445C42—711no—
446C42D4×D7111yescharacteristic
447C16—21no—
448C16—712no—
449C16—1421no—
450C16C2×C4×D7112yescomplemented
451C23—41no—
452C23—2844no—
453C23—2841no—
454C23—2841no—
455C23—5682no—
456C23—5682no—
457C23—711no—
458Q8—22no—
459Q8—41no—
460D4—22no—
461D4—41no—
462D4—2844no—
463D4—2844no—
464D4—2841no—
465D4—2841no—
466D4—1422no—
467D4—1422no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—2841no—
473C2×C4—2844no—
474C2×C4—2841no—
475C2×C4—2841no—
476C2×C4—5682no—
477C2×C4—5682no—
478C2×C4—5682no—
479C2×C4—5682no—
480C2×C4—5682no—
481C2×C4—5682no—
482C2×C4—711no—
483C2×C4—711no—
484C2×C4—712no—
485C2×C4—712no—
486C2×C4—711no—
487C2×C4—1422no—
488C2×C4D8×D7111yescharacteristic
489C2×C4D8×D7111yescharacteristic
490C2×C4C2×D4×D7111yescharacteristic
491C8—21no—
492C8—712no—
493C8—1421no—
494C8C22×C4×D7111yescharacteristic
495C8#41×D7111yescharacteristic
496C22—42no—
497C22—81no—
498C22—81no—
499C22—2842no—
500C22—2842no—
501C22—5688no—
502C22—5681no—
503C22—5681no—
504C22—5681no—
505C22—5681no—
506C22—712no—
507C22—1421no—
508C22—1421no—
509C22C2×D8×D7111yescharacteristic
510C4—21no—
511C4—21no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—5681no—
516C4—5681no—
517C4—712no—
518C4—712no—
519C4—1421no—
520C4C4×D4×D7111yescharacteristic
521C4#135×D7111yescharacteristic
522C2—82no—
523C2—5682no—
524C2—5682no—
525C2—1422no—
526C2C4×D8×D7111yescharacteristic
527C2#509×D7111yescharacteristic
528C2#2260147511yescharacteristic
subgroupG/H
centerC22C2×D8×D71
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2175C2
socleC22×C71C22×D8
prime radicalC2×#17986436C2
cosocleC2×C8×C71C24
hypercenter#2175D71
hypocenterC71C4×#509
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. C4×#14556151C2semidirect
3. C4×D8×D71C2semidirect
4. #22590217C2semidirect
5. #22590214C2semidirect
6. #22590202C2semidirect
7. SD32×Dic71C2semidirect
8. #22590189C2semidirect
9. #22590186C2semidirect
10. #22590184C2semidirect
11. #22590125C2semidirect
12. #22590054C2semidirect
13. #22590051C2semidirect
14. #22589933C2semidirect
15. C71×#2175C2semidirect
20. #17986436C4semidirect
26. Q16×Dic71C22semidirect
30. C4×D568C22semidirect
33. #17986081C22semidirect
41. #17985827C22semidirect
42. #17985826C22semidirect
43. #17985824C22semidirect
44. #17985803C22semidirect
46. #17985801C22semidirect
49. #17985799C22semidirect
50. #17985775C22semidirect
51. C16×Dic71C22semidirect
52. C16×Dic71C22semidirect
54. C4×Q16×C71C22semidirect
56. #366×C71C22semidirect
57. #358×C71C22semidirect
59. #346×C71C22semidirect
64. #14556151C2×C4semidirect
70. D8×D71C2×C4semidirect
89. #14556071C2×C4semidirect
90. D1136C2×C4semidirect
91. #14556069C2×C4semidirect
93. #14556067C2×C4semidirect
113. SD32×C71C2×C4semidirect
145. D568C22×C4semidirect
168. C71⋊C16C22×C4semidirect
173. Q16×C71C22×C4semidirect
185. C16×C71C22×C4semidirect
256. Dic71#509semidirect
271. #2175D71semidirect
280. C71C4×#509semidirect
293. C4×Q16C2×D71semidirect
312. #366C2×D71semidirect
314. #358C2×D71semidirect
322. #346C2×D71semidirect
359. SD32C4×D71semidirect
410. Q16C2×C4×D71semidirect
450. C16C2×C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864362
C4×#145561512
C4×D8×D712
#225902172
#225902142
#225902022
SD32×Dic712
#225901892
#225901862
#225901842
#225901252
#225900542
#225900512
#225899332
C71×#21752
C4×#50971