Finite Group Databasebeta

← #29662797

Subgroups — #29662797

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455823C211yesmaximal, characteristic
2C2×#16839964C211yesmaximal, characteristic
3M16×D244C211yesmaximal, characteristic
4#21445329C212yesmaximal, complemented
5#21445315C212yesmaximal, complemented
6#21445205C212yesmaximal, complemented
7#21445191C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444504C211yesmaximal, characteristic, complemented
11C61×#3165C211yesmaximal, characteristic, complemented
12C22×#11758201—21no—
13C2×M16×D61—21no—
14C2×C4×D244C2211yescharacteristic
15#16840090—21no—
16#16840049C2211yescharacteristic
17#16840049C2211yescharacteristic
18#16839964—21no—
19#16839964—21no—
20#16839964C2212yes—
21#16839964C2211yescharacteristic
22#16839964C2212yes—
23#16839964C2211yescharacteristic
24#16839912—21no—
25C4×#11758201—21no—
26C4×#11758201—21no—
27#16839841C2212yes—
28#16839835—22no—
29C8×D244C2212yes—
30#16839815C2211yescharacteristic
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519701—21no—
34#16839725—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839511C2212yescomplemented
38#16839508C2212yescomplemented
39#16839508C2212yescomplemented
40#16839508C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#501—22no—
44C61×#500C2212yescomplemented
45C61×#498C2212yescomplemented
46C61×#428—21no—
47C61×#428—21no—
48C4×M16×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C2×#11758201—21no—
54C2×#11758201—21no—
55C2×#11758201—44no—
56C2×#11758194—21no—
57C2×#11758189—22no—
58M16×D61—44no—
59C22×#92418—21no—
60C2×C8×D61—22no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D411yescharacteristic
64C4×D244D411yescharacteristic
65C4×D244C2311yescharacteristic
66C4×D244C2311yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758198—21no—
69#14519708—21no—
70#14519708—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519701—22no—
78#14519701—22no—
79#14519701—21no—
80#14519701—21no—
81#14519697—21no—
82#14519697—21no—
83C22⋊#92418—21no—
84#14519689—22no—
85#14519689—22no—
86C4×#92418—21no—
87C22×Q8×C61—21no—
88C2×M16×C61D411yescharacteristic
89C2×M16×C61D411yescharacteristic
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4⋊Q8×C61C2311yescharacteristic
95C4⋊Q8×C61C2311yescharacteristic
96#125×C61—21no—
97C22⋊Q8×C61—21no—
98C22⋊Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#107×C61—22no—
106#107×C61—22no—
107#106×C61—22no—
108#106×C61—22no—
109#97×C61D412yes—
110C4×C8×C61C2312yes—
111C23×D61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—21no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C2×Q8×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×Dic61—21no—
123C2×D244—22no—
124C2×D244—21no—
125C2×D244—41no—
126C2×D244C2×D411yescharacteristic
127C2×D244C2×D411yescharacteristic
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134M16×C61—22no—
135M16×C61SD1614yescomplemented
136C2×C8×C61C2×D412yes—
137C2×C8×C61C2×D412yes—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758201—42no—
155#11758201—44no—
156#11758201—42no—
157#11758201—44no—
158#11758198—21no—
159#11758198—21no—
160#11758198—21no—
161#11758198—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61C2×D411yescharacteristic
164C4⋊Dic61C2×D411yescharacteristic
165#11758194—42no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—42no—
171#11758189—44no—
172C8×D61—44no—
173C23×C61#12111yescharacteristic
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—81no—
179Q8×C61—42no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—42no—
183Q8×C61—82no—
184Q8×C61—81no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—82no—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61C2×SD1612yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—42no—
203D244—21no—
204D244—21no—
205D244—21no—
206D244—42no—
207D244AGL(1,ℤ/8)12yes—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—41no—
212C4×D61—42no—
213C4×D61—42no—
214C8×C61—22no—
215C8×C61C2×SD1614yescomplemented
216#92418—42no—
217#92418—42no—
218#92418—41no—
219#92418—41no—
220#19164—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61#46112yes—
224C22×C61C2×#12111yescharacteristic
225C2×D61—22no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61—84no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—82no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61#46112yes—
239C4×C61C22×SD1611yescharacteristic
240C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C61C2×#13711yescharacteristic
242C4×C61#57511yescharacteristic
243Dic61—41no—
244Dic61—41no—
245#3165D6111yescharacteristic, complemented
246#3101—611no—
247C2×#461—611no—
248D4×M16—611no—
249#1693—612no—
250#1691—612no—
251#1673—612no—
252#1671—612no—
253#1565—611no—
254#1550—611no—
255#1549—611no—
256C2×C61—22no—
257C2×C61C2×#46111yescharacteristic
258C2×C61#305511yescharacteristic
259C2×C61#306911yescharacteristic
260D61—42no—
261D61—81no—
262C22×SD16—1221no—
263C22×M16—1221no—
264#533C2×D6111yescharacteristic
265C2×C4⋊Q8C2×D6111yescharacteristic
266C2×C4×D4—611no—
267#501—22no—
268#500C2×D6112yescomplemented
269#498C2×D6112yescomplemented
270#483—1221no—
271#476—611no—
272#476—611no—
273#461—611no—
274#461—611no—
275#461—612no—
276#461—612no—
277#461—1221no—
278#461—1221no—
279#450—1221no—
280C4×SD16—1221no—
281C4×SD16—1221no—
282#436—612no—
283#435—1222no—
284C8×D4—612no—
285#431—611no—
286#428—21no—
287#428—21no—
288#423—611no—
289C2×C4⋊C8—611no—
290C2×#103—1221no—
291#407—1221no—
292C4×M16C2×D6111yescharacteristic
293#335—612no—
294#334—612no—
295#333—612no—
296#332—612no—
297#332—612no—
298#332—612no—
299C61#1916411yescharacteristic, complemented, Sylow
300C22×Q8—21no—
301C22×D4—611no—
302C23×C4—1221no—
303C2×SD16—1222no—
304C2×SD16—1222no—
305C2×SD16—1221no—
306C2×SD16—1221no—
307C2×SD16—2444no—
308C2×M16—1221no—
309C2×M16—1222no—
310C2×M16—2444no—
311C2×M16C22⋊D6111yescharacteristic
312C2×M16C22⋊D6111yescharacteristic
313C22×C8—1221no—
314C22×C8—1222no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8C22×D6112yes—
319C4⋊Q8C22×D6111yescharacteristic
320C4⋊Q8C22×D6111yescharacteristic
321#125—21no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—611no—
327C4×D4—611no—
328C4×D4—611no—
329C4×D4—611no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C22⋊C4—1221no—
337C2×C42C22×D6111yescharacteristic
338#107—22no—
339#107—22no—
340#106—22no—
341#106—22no—
342#106—1221no—
343#106—1221no—
344C4⋊C8—612no—
345C4⋊C8—611no—
346C4⋊C8—611no—
347C4⋊C8—612no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350#103—1222no—
351#103—1222no—
352#103—1221no—
353#103—1221no—
354#102—1221no—
355#102—1221no—
356C22⋊C8—1221no—
357C22⋊C8—1222no—
358C22⋊C8—1222no—
359#97C22⋊D6112yes—
360C4×C8—1221no—
361C4×C8C22×D6112yes—
362C24—1221no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—41no—
369C2×D4—611no—
370C2×D4—611no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—2441no—
374C22×C4—41no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—2442no—
381C22×C4—2442no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822411yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385SD16—2442no—
386SD16—2444no—
387SD16—2442no—
388SD16—2444no—
389M16—22no—
390M16—2442no—
391M16—2444no—
392M16#1175820114yescomplemented
393C2×C8—1221no—
394C2×C8—1221no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—2442no—
398C2×C8—2444no—
399C2×C8C2×C22⋊D6112yes—
400C2×C8C2×C22⋊D6112yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—611no—
412C4⋊C4—611no—
413C4⋊C4—1221no—
414C22⋊C4—41no—
415C22⋊C4—1221no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C42—41no—
420C42D4×D6111yescharacteristic
421C42#1175822411yescharacteristic
422C42C2×C22⋊D6111yescharacteristic
423C42C23×D6111yescharacteristic
424C23—1221no—
425C23—1221no—
426C23—4881no—
427C23—2442no—
428C23—2442no—
429C23#1451974811yescharacteristic
430Q8—42no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434Q8—81no—
435Q8—82no—
436D4—612no—
437D4—1221no—
438D4—1221no—
439D4—1221no—
440D4—2442no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4#1451974811yescharacteristic
458C2×C4#1451974811yescharacteristic
459C2×C4#1451974811yescharacteristic
460C2×C4C2×#1175820112yes—
461C2×C4#1451980812yes—
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C22×C22⋊D6111yescharacteristic
464C2×C4C2×#1175822411yescharacteristic
465C8—22no—
466C8—2442no—
467C8—2442no—
468C8—2441no—
469C8—2441no—
470C8C2×#1175820114yescomplemented
471C22—21no—
472C22—21no—
473C22—1222no—
474C22—4884no—
475C22—2441no—
476C22—2441no—
477C22—2441no—
478C22—2441no—
479C22—2441no—
480C22#1683996412yes—
481C22C2×#1451974811yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—81no—
487C4—81no—
488C4—2441no—
489C4—2441no—
490C4#1683996412yes—
491C4C22×#1175820111yescharacteristic
492C4AGL(1,ℤ/8)×D6111yescharacteristic
493C4C2×#1451980811yescharacteristic
494C4#1684055811yescharacteristic
495C2—22no—
496C2—4881no—
497C2—2442no—
498C2C2×#1683996411yescharacteristic
499C2#2145543211yescharacteristic
500C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3165D61
hypocenterC61#19164
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #21445329C2semidirect
5. #21445315C2semidirect
6. #21445205C2semidirect
7. #21445191C2semidirect
8. #21444571C2semidirect
9. #21444509C2semidirect
10. #21444504C2semidirect
11. C61×#3165C2semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839511C22semidirect
38. #16839508C22semidirect
39. #16839508C22semidirect
40. #16839508C22semidirect
44. C61×#500C22semidirect
45. C61×#498C22semidirect
135. M16×C61SD16semidirect
215. C8×C61C2×SD16semidirect
245. #3165D61semidirect
268. #500C2×D61semidirect
269. #498C2×D61semidirect
299. C61#19164semidirect
392. M16#11758201semidirect
470. C8C2×#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558232
C2×#168399642
M16×D2442
#214453292
#214453152
#214452052
#214451912
#214445712
#214445092
#214445042
C61×#31652
#1916461