Finite Group Databasebeta

← #29683563

Subgroups — #29683563

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D61C211yesmaximal, characteristic
2#21454406C212yesmaximal
3C2×#16839799C211yesmaximal, characteristic
4#405×D61C211yesmaximal, characteristic
5#21446605C211yesmaximal, characteristic, complemented
6#21446604C211yesmaximal, characteristic, complemented
7#21446602C211yesmaximal, characteristic, complemented
8#21446598C212yesmaximal, complemented
9#21446596C212yesmaximal, complemented
10#21445556C212yesmaximal
11C61×#1952C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C2×M16×D61C2211yescharacteristic
14C22×C8×D61C2211yescharacteristic
15C2×#14519735C2212yes—
16#117×D61C2211yescharacteristic
17C2×#14519713C2211yescharacteristic
18C2×#14519712C2212yes—
19#108×D61C418yescomplemented
20#16839805C2212yes—
21#16839804C2212yes—
22#16839803—22no—
23#16839802C2212yes—
24#16839801—22no—
25#16839799—21no—
26#16839799—21no—
27#16839799—21no—
28#16839799—21no—
29#16839799—22no—
30#16839797—22no—
31C2×#14519711C2211yescharacteristic
32C2×#14519708C2212yes—
33#16839720C2211yescharacteristic
34#16839719—21no—
35#16839718C2211yescharacteristic
36#16839717C2211yescharacteristic
37#16839716—21no—
38#97×D61—21no—
39C4×C8×D61—21no—
40#16839531C2212yes—
41#16839530C2212yes—
42#16839530C2212yes—
43C2×#108×C61C2211yescharacteristic
44C61×#428C2212yes—
45C2×#107×C61C2211yescharacteristic
46C61×#405C2211yescharacteristic
47C61×#343C2212yes—
48C22×C4×D61C2311yescharacteristic
49C2×#11758194C2312yes—
50C2×#11758194C2311yescharacteristic
51C2×#11758189—21no—
52C2×#11758189—22no—
53M16×D61—24no—
54M16×D61—22no—
55M16×D61C2×C418yes—
56M16×D61Q814yes—
57C22×#92418C2311yescharacteristic
58C2×C8×D61—21no—
59C2×C8×D61—22no—
60C2×C8×D61C2×C414yes—
61C2×C8×D61D411yescharacteristic
62C2×C8×D61D411yescharacteristic
63C2×C8×D61D411yescharacteristic
64C2×C8×D61D411yescharacteristic
65C2×C8×D61Q814yes—
66#14519736C2311yescharacteristic
67#14519735—22no—
68#14519735—22no—
69#14519735—22no—
70#14519735—22no—
71#14519735D411yescharacteristic
72#14519735D411yescharacteristic
73#14519734—21no—
74#14519733—21no—
75C2×C4⋊Dic61C2312yes—
76C4⋊C4×D61D411yescharacteristic
77C4⋊C4×D61D411yescharacteristic
78C2×#11758198—22no—
79C22⋊C4×D61—21no—
80C2×C4×Dic61—22no—
81C42×D61—21no—
82#14519713C2×C414yescomplemented
83#14519712C2×C418yescomplemented
84#14519711—21no—
85#14519711—21no—
86#14519711—22no—
87#14519710—22no—
88#14519710—22no—
89#14519709—22no—
90#14519708—22no—
91#14519708—22no—
92#14519708—22no—
93#14519708—22no—
94#14519688—21no—
95#14519687—21no—
96#14519687—21no—
97C4×#92418—21no—
98C8×Dic61—21no—
99C8×Dic61—21no—
100C2×M16×C61C2312yes—
101C2×M16×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103#117×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105#108×C61C2×C414yescomplemented
106#107×C61—21no—
107#107×C61—21no—
108#107×C61—22no—
109#106×C61—22no—
110#97×C61—21no—
111C4×C8×C61—21no—
112C23×D61#4111yescharacteristic
113C22×C4×C61—22no—
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61#4111yescharacteristic
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61C22×C412yes—
120C2×C4×D61C2×D411yescharacteristic
121C2×C4×D61C2×D411yescharacteristic
122C2×C4×D61C2×D411yescharacteristic
123C2×C4×D61C2×D411yescharacteristic
124C2×C4×D61C2×Q812yes—
125C2×C4×D61C2×Q812yes—
126C2×C4×D61#4112yes—
127M16×C61—22no—
128M16×C61—21no—
129M16×C61C22×C414yes—
130M16×C61C2×Q812yes—
131C2×C8×C61—21no—
132C2×C8×C61—22no—
133C2×C8×C61C22×C412yes—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×Q812yes—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61C2×D411yescharacteristic
140C4⋊C4×C61C2×D411yescharacteristic
141C22⋊C4×C61—21no—
142C42×C61—21no—
143C22⋊Dic61—21no—
144#11758198—21no—
145#11758198—21no—
146#11758198—42no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61C2×D411yescharacteristic
151C4⋊Dic61C2×D411yescharacteristic
152#11758196—21no—
153#11758196—21no—
154C4×Dic61—21no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C4×Dic61—42no—
158C4×Dic61—42no—
159#11758194—22no—
160#11758194—21no—
161#11758194C22×C414yes—
162#11758194C2×Q812yes—
163C2×#92418—21no—
164C2×#92418—22no—
165C2×#92418C22×C412yes—
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×Q812yes—
169#11758189—28no—
170#11758189—24no—
171C8×D61—28no—
172C8×D61—24no—
173C8×D61C4⋊C414yes—
174C8×D61C4⋊C414yes—
175C8×D61C4⋊C414yes—
176C8×D61C4⋊C414yes—
177C23×C61C2×#4111yescharacteristic
178C22×D61—41no—
179C22×D61—41no—
180C22×D61C4×Q812yes—
181C22×D61#12111yescharacteristic
182C22×D61#12111yescharacteristic
183C22×D61#12512yes—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—42no—
187C2×C4×C61—82no—
188C2×C4×C61C23×C411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×Q811yescharacteristic
192C2×C4×C61C22×Q811yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—42no—
197C2×Dic61—82no—
198C2×Dic61C4×Q812yes—
199C2×Dic61#12111yescharacteristic
200C2×Dic61#12111yescharacteristic
201C2×Dic61#12512yes—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61C2×C4⋊C412yes—
205C4×D61C2×C4⋊C412yes—
206C4×D61C4×D412yes—
207C4×D61C4×D412yes—
208C4×D61C22⋊Q812yes—
209C4×D61C22⋊Q812yes—
210C4×D61C4⋊Q812yes—
211C4×D61C4⋊Q812yes—
212C8×C61—24no—
213C8×C61—22no—
214C8×C61C2×C4⋊C412yes—
215C8×C61C2×C4⋊C412yes—
216#92418—24no—
217#92418—22no—
218#92418C2×C4⋊C412yes—
219#92418C2×C4⋊C412yes—
220#21181—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61C2×C4×Q811yescharacteristic
224C22×C61C2×#12111yescharacteristic
225C22×C61C2×#12511yescharacteristic
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—42no—
231C2×D61—42no—
232C2×D61#38912yes—
233C2×D61#38912yes—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61—82no—
237C4×C61C22×C4⋊C411yescharacteristic
238C4×C61C2×C4×D411yescharacteristic
239C4×C61C2×C22⋊Q811yescharacteristic
240C4×C61C2×C4⋊Q811yescharacteristic
241Dic61—41no—
242Dic61—41no—
243Dic61—82no—
244Dic61#38912yes—
245Dic61#38912yes—
246C22×#108—611no—
247#2911—612no—
248C2×#427—611no—
249C2×#405—611no—
250#1952—611no—
251#1952—611no—
252#1952—611no—
253#1952D6111yescharacteristic, complemented
254#1951—612no—
255#1951—612no—
256#1742—612no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61C2×#38911yescharacteristic
260C2×C61#306711yescharacteristic
261C2×C61#306711yescharacteristic
262D61—22no—
263D61—22no—
264C22×M16—611no—
265C22×M16—612no—
266C23×C8—611no—
267C2×#117—611no—
268C2×#117—612no—
269C2×#108—612no—
270C2×#108—611no—
271C2×#108—618no—
272C2×#108C2×D6111yescharacteristic
273#428—612no—
274#428—612no—
275#428—612no—
276#428—1222no—
277#428—1222no—
278#428C2×D6112yes—
279#427—1222no—
280#427—1222no—
281#427—1221no—
282#427—1221no—
283#427—1221no—
284#427—1221no—
285C2×#107—611no—
286C2×#107C2×D6111yescharacteristic
287C2×#106—612no—
288#405—611no—
289#405—611no—
290#405—611no—
291#405—1221no—
292#405—1221no—
293#405C2×D6111yescharacteristic
294C2×#97—1221no—
295C2×C4×C8—1221no—
296#343—612no—
297#343—612no—
298#343—612no—
299#343C2×D6112yes—
300C61#2118111yescharacteristic, complemented, Sylow
301C23×C4—611no—
302C2×M16—611no—
303C2×M16—612no—
304C2×M16—618no—
305C2×M16—614no—
306C2×M16—1221no—
307C2×M16—1222no—
308C2×M16—1222no—
309C2×M16—1224no—
310C2×M16C22×D6112yes—
311C2×M16C22×D6111yescharacteristic
312C22×C8—611no—
313C22×C8—614no—
314C22×C8—611no—
315C22×C8—611no—
316C22×C8—611no—
317C22×C8—611no—
318C22×C8—614no—
319C22×C8—1221no—
320C22×C8—1222no—
321C22×C8C22×D6111yescharacteristic
322#117—611no—
323#117—611no—
324#117—611no—
325#117—1221no—
326#117—1221no—
327#117—1222no—
328#117—1222no—
329#117—1222no—
330#117—1222no—
331#117C22×D6111yescharacteristic
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4C22×D6112yes—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1222no—
338C2×C42—1221no—
339C2×C42—1222no—
340#108—618no—
341#108—614no—
342#108C4×D6114yescomplemented
343#107—21no—
344#107—21no—
345#107—22no—
346#107—1222no—
347#107—1222no—
348#107—1221no—
349#107—1221no—
350#107—1222no—
351#106—22no—
352#106—1222no—
353#106—1222no—
354#106—1222no—
355#106—1222no—
356#106—1222no—
357#97—21no—
358#97—1221no—
359#97—1221no—
360#97—1221no—
361C4×C8—21no—
362C4×C8—1221no—
363C4×C8—1221no—
364C4×C8—1221no—
365C24—611no—
366C22×C4—22no—
367C22×C4—611no—
368C22×C4—612no—
369C22×C4—611no—
370C22×C4—611no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—612no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1222no—
379C22×C4C23×D6111yescharacteristic
380M16—22no—
381M16—21no—
382M16—614no—
383M16—612no—
384M16—1228no—
385M16—1224no—
386M16—1222no—
387M16—1221no—
388M16C2×C4×D6114yes—
389M16Q8×D6112yes—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—612no—
393C2×C8—611no—
394C2×C8—611no—
395C2×C8—612no—
396C2×C8—614no—
397C2×C8—614no—
398C2×C8—614no—
399C2×C8—614no—
400C2×C8—1228no—
401C2×C8—1224no—
402C2×C8—1221no—
403C2×C8—1222no—
404C2×C8C2×C4×D6112yes—
405C2×C8D4×D6111yescharacteristic
406C2×C8D4×D6111yescharacteristic
407C2×C8Q8×D6112yes—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—611no—
411C4⋊C4—611no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4D4×D6111yescharacteristic
417C4⋊C4D4×D6111yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C42—21no—
425C42—1221no—
426C42—1221no—
427C42—1221no—
428C42—2442no—
429C42—2442no—
430C23—611no—
431C23—611no—
432C23—612no—
433C23—612no—
434C23—2441no—
435C23—2441no—
436C23#41×D6111yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—612no—
442C2×C4—611no—
443C2×C4—611no—
444C2×C4—612no—
445C2×C4—612no—
446C2×C4—612no—
447C2×C4—612no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4—612no—
451C2×C4—612no—
452C2×C4—612no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—4882no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4C22×C4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×Q8×D6111yescharacteristic
463C2×C4C2×Q8×D6111yescharacteristic
464C2×C4#41×D6111yescharacteristic
465C8—24no—
466C8—22no—
467C8—612no—
468C8—612no—
469C8—1224no—
470C8—1222no—
471C8C4⋊C4×D6112yes—
472C8C4⋊C4×D6112yes—
473C22—21no—
474C22—21no—
475C22—612no—
476C22—612no—
477C22—1221no—
478C22—1221no—
479C22—1221no—
480C22—1221no—
481C22—2442no—
482C22—2442no—
483C22C4×Q8×D6111yescharacteristic
484C22#121×D6111yescharacteristic
485C22#125×D6111yescharacteristic
486C4—41no—
487C4—41no—
488C4—82no—
489C4—612no—
490C4—612no—
491C4—4882no—
492C4—2441no—
493C4—2441no—
494C4C2×C4⋊C4×D6111yescharacteristic
495C4C4×D4×D6111yescharacteristic
496C4C22⋊Q8×D6111yescharacteristic
497C4C4⋊Q8×D6111yescharacteristic
498C2—21no—
499C2—21no—
500C2—1222no—
501C2—1222no—
502C2#389×D6111yescharacteristic
503C2#2145553711yescharacteristic
504C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1952C2
socleC22×C61C2×#121
prime radicalC23×D61#41
cosocleC22×C4×C61C24
hypercenter#1952D61
hypocenterC61#21181
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
5. #21446605C2semidirect
6. #21446604C2semidirect
7. #21446602C2semidirect
8. #21446598C2semidirect
9. #21446596C2semidirect
11. C61×#1952C2semidirect
19. #108×D61C4semidirect
82. #14519713C2×C4semidirect
83. #14519712C2×C4semidirect
105. #108×C61C2×C4semidirect
253. #1952D61semidirect
300. C61#21181semidirect
342. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#108×D612
#214544062
C2×#168397992
#405×D612
#214466052
#214466042
#214466022
#214465982
#214465962
#214455562
C61×#19522
#2118161