Finite Group Databasebeta

← #29662830

Subgroups — #29662830

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455823C211yesmaximal, characteristic
2C2×#16840039C211yesmaximal, characteristic
3M16×D244C211yesmaximal, characteristic
4#21445391C212yesmaximal, complemented
5#21445363C212yesmaximal, complemented
6#21445267C212yesmaximal, complemented
7#21445239C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444491C211yesmaximal, characteristic, complemented
11C61×#3139C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×C4×D244C2211yescharacteristic
14#16840090—21no—
15#16840049C2211yescharacteristic
16#16840049C2211yescharacteristic
17#16840039—21no—
18#16840039—21no—
19#16840039C2212yes—
20#16840039C2212yes—
21#16839964C2211yescharacteristic
22#16839964C2211yescharacteristic
23#16839912—21no—
24C4×#11758201—21no—
25C4×#11758201—21no—
26#16839841C2212yes—
27#16839835—22no—
28C8×D244C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519710—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519697—21no—
34#16839725—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839511C2212yescomplemented
38#16839508C2212yescomplemented
39#16839508C2212yescomplemented
40#16839508C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#491—22no—
44C61×#487C2212yescomplemented
45C61×#487C2212yescomplemented
46C61×#419—21no—
47C61×#419—21no—
48C4×M16×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758201—21no—
52C2×#11758201—21no—
53C2×#11758194—21no—
54C2×#11758189—22no—
55M16×D61—44no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244D411yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×D244C2311yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758198—21no—
66#14519710—22no—
67#14519710—22no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75C4⋊#92418C2312yes—
76#14519701—21no—
77#14519701—21no—
78#14519697—22no—
79#14519697—22no—
80#14519697—21no—
81#14519697—21no—
82C22⋊#92418—21no—
83#14519689—22no—
84#14519689—22no—
85C4×#92418—21no—
86C22×Q8×C61—21no—
87C2×M16×C61D411yescharacteristic
88C2×M16×C61D411yescharacteristic
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61C2312yes—
93C4⋊Q8×C61C2311yescharacteristic
94C4⋊Q8×C61C2311yescharacteristic
95#125×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100#117×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—22no—
105#103×C61—22no—
106#103×C61—22no—
107#103×C61—22no—
108#97×C61D412yes—
109C4×C8×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—41no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C22×C4×C61—41no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×Dic61—21no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—41no—
125C2×D244C2×D411yescharacteristic
126C2×D244C2×D411yescharacteristic
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133M16×C61—22no—
134M16×C61—22no—
135C2×C8×C61C2×D412yes—
136C2×C8×C61C2×D412yes—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61—41no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61#4111yescharacteristic
152C42×C61C2411yescharacteristic
153#11758201—42no—
154#11758201—42no—
155#11758198—21no—
156#11758198—21no—
157#11758198—21no—
158#11758198—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61C2×D411yescharacteristic
161C4⋊Dic61C2×D411yescharacteristic
162#11758194—42no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—42no—
168#11758189—44no—
169C8×D61—44no—
170C23×C61#12111yescharacteristic
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—81no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—81no—
181Q8×C61—82no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—82no—
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#13512yes—
193C2×C4×C61AGL(1,ℤ/8)12yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—42no—
200D244—21no—
201D244—21no—
202D244—21no—
203D244—42no—
204D244#13712yes—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—41no—
208C4×D61—41no—
209C4×D61—42no—
210C4×D61—42no—
211C8×C61—22no—
212C8×C61—22no—
213#92418—42no—
214#92418—41no—
215#92418—42no—
216#92418—41no—
217#19167—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61#46312yes—
221C22×C61C2×#12111yescharacteristic
222C2×D61—22no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—84no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—82no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61#46312yes—
236C4×C61C2×#13511yescharacteristic
237C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C61C2×#13711yescharacteristic
239C4×C61#57511yescharacteristic
240Dic61—41no—
241Dic61—41no—
242#3139D6111yescharacteristic, complemented
243#3101—611no—
244C2×#474—611no—
245D4×M16—611no—
246#1703—612no—
247#1699—612no—
248#1683—612no—
249#1679—612no—
250#1565—611no—
251#1550—611no—
252#1546—611no—
253C2×C61—22no—
254C2×C61C2×#46311yescharacteristic
255C2×C61#305611yescharacteristic
256C2×C61#306911yescharacteristic
257D61—42no—
258D61—81no—
259C22×M16—1221no—
260#533C2×D6111yescharacteristic
261C2×C4⋊Q8C2×D6111yescharacteristic
262C2×C4×D4—611no—
263#491—22no—
264#487C2×D6112yescomplemented
265#487C2×D6112yescomplemented
266#483—1221no—
267#476—611no—
268#476—611no—
269#474—612no—
270#474—612no—
271#474—1221no—
272#474—1221no—
273#461—611no—
274#461—611no—
275#450—1221no—
276C4×SD16—1221no—
277C4×SD16—1221no—
278#436—612no—
279#435—1222no—
280C8×D4—612no—
281#431—611no—
282C2×#107—1221no—
283#423—611no—
284C2×C4⋊C8—611no—
285#419—21no—
286#419—21no—
287C2×#102—1221no—
288#407—1221no—
289C4×M16C2×D6111yescharacteristic
290#335—612no—
291#334—612no—
292#333—612no—
293#332—612no—
294#332—612no—
295#332—612no—
296C61#1916711yescharacteristic, complemented, Sylow
297C22×Q8—21no—
298C22×D4—611no—
299C23×C4—1221no—
300C2×SD16—1221no—
301C2×SD16—1221no—
302C2×M16—1221no—
303C2×M16—1222no—
304C2×M16—2444no—
305C2×M16C22⋊D6111yescharacteristic
306C2×M16C22⋊D6111yescharacteristic
307C22×C8—1221no—
308C22×C8—1222no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8C22×D6112yes—
313C4⋊Q8C22×D6111yescharacteristic
314C4⋊Q8C22×D6111yescharacteristic
315#125—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—611no—
323C4×D4—611no—
324C4×D4—1221no—
325C4×D4—1221no—
326#117—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—611no—
330C2×C22⋊C4—1221no—
331C2×C42C22×D6111yescharacteristic
332#107—1222no—
333#107—1222no—
334#106—1221no—
335#106—1221no—
336C4⋊C8—612no—
337C4⋊C8—611no—
338C4⋊C8—611no—
339C4⋊C8—612no—
340C4⋊C8—1221no—
341C4⋊C8—1221no—
342#103—22no—
343#103—22no—
344#103—22no—
345#103—22no—
346#103—1221no—
347#103—1221no—
348#102—1222no—
349#102—1222no—
350#102—1221no—
351#102—1221no—
352C22⋊C8—1221no—
353C22⋊C8—1222no—
354C22⋊C8—1222no—
355#97C22⋊D6112yes—
356C4×C8—1221no—
357C4×C8C22×D6112yes—
358C24—1221no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×D4—611no—
366C2×D4—611no—
367C2×D4—1222no—
368C2×D4—1221no—
369C2×D4—2441no—
370C22×C4—41no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4D4×D6111yescharacteristic
379C22×C4#1175822411yescharacteristic
380C22×C4C2×C22⋊D6111yescharacteristic
381SD16—2442no—
382SD16—2442no—
383M16—22no—
384M16—22no—
385M16—2444no—
386M16—2442no—
387C2×C8—1221no—
388C2×C8—1221no—
389C2×C8—1222no—
390C2×C8—1222no—
391C2×C8—2444no—
392C2×C8—2442no—
393C2×C8C2×C22⋊D6112yes—
394C2×C8C2×C22⋊D6112yes—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—611no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C22⋊C4—41no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C42—41no—
414C42D4×D6111yescharacteristic
415C42#1175822411yescharacteristic
416C42C2×C22⋊D6111yescharacteristic
417C42C23×D6111yescharacteristic
418C23—1221no—
419C23—1221no—
420C23—4881no—
421C23—2442no—
422C23—2442no—
423C23#1451974811yescharacteristic
424Q8—42no—
425Q8—42no—
426Q8—41no—
427Q8—41no—
428Q8—81no—
429Q8—82no—
430D4—612no—
431D4—1221no—
432D4—1221no—
433D4—1221no—
434D4—2442no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4#1451974811yescharacteristic
452C2×C4#1451974811yescharacteristic
453C2×C4#1451974811yescharacteristic
454C2×C4#1451979712yes—
455C2×C4#1451980212yes—
456C2×C4C2×D4×D6111yescharacteristic
457C2×C4C22×C22⋊D6111yescharacteristic
458C2×C4C2×#1175822411yescharacteristic
459C8—22no—
460C8—22no—
461C8—2442no—
462C8—2441no—
463C8—2442no—
464C8—2441no—
465C22—21no—
466C22—21no—
467C22—1222no—
468C22—4884no—
469C22—2441no—
470C22—2441no—
471C22—2441no—
472C22—2441no—
473C22—2441no—
474C22#1683997412yes—
475C22C2×#1451974811yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4—2441no—
483C4—2441no—
484C4#1683997412yes—
485C4C2×#1451979711yescharacteristic
486C4C2×#1451980211yescharacteristic
487C4#137×D6111yescharacteristic
488C4#1684056111yescharacteristic
489C2—22no—
490C2—4881no—
491C2—2442no—
492C2C2×#1683997411yescharacteristic
493C2#2145544311yescharacteristic
494C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3139D61
hypocenterC61#19167
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
4. #21445391C2semidirect
5. #21445363C2semidirect
6. #21445267C2semidirect
7. #21445239C2semidirect
8. #21444571C2semidirect
9. #21444509C2semidirect
10. #21444491C2semidirect
11. C61×#3139C2semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839511C22semidirect
38. #16839508C22semidirect
39. #16839508C22semidirect
40. #16839508C22semidirect
44. C61×#487C22semidirect
45. C61×#487C22semidirect
242. #3139D61semidirect
264. #487C2×D61semidirect
265. #487C2×D61semidirect
296. C61#19167semidirect

Maximal subgroups

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

subgroupindex
#214558232
C2×#168400392
M16×D2442
#214453912
#214453632
#214452672
#214452392
#214445712
#214445092
#214444912
C61×#31392
#1916761