Finite Group Databasebeta

← #29662552

Subgroups — #29662552

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456329C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic
3C2×#16839840C211yesmaximal, characteristic
4#21445340C212yesmaximal, complemented
5#21445330C212yesmaximal, complemented
6#21445320C212yesmaximal, complemented
7#21445316C212yesmaximal, complemented
8#21444565C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10#21444474C211yesmaximal, characteristic, complemented
11C61×#3101C211yesmaximal, characteristic
12C22×#11758194—21no—
13#16840319C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840165C2211yescharacteristic, complemented
16#16840161—21no—
17#16840161—21no—
18#16840158C2211yescharacteristic, complemented
19#16840158C2211yescharacteristic, complemented
20#16840092—21no—
21#16840052C2212yes—
22#16839967C2212yes—
23#16839913—21no—
24C4×#11758200—22no—
25#16839840—21no—
26#16839840—21no—
27#16839840C2212yes—
28#16839840C2212yes—
29#16839815C2211yescharacteristic
30#16839798—21no—
31#16839798—21no—
32C2×C4⋊#92418C2211yescharacteristic
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839512C2212yescomplemented
37#16839511C2212yescomplemented
38#16839510C2212yescomplemented
39#16839509C2212yescomplemented
40#16839508C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#483—21no—
44C61×#476C2212yes—
45C61×#461C2212yes—
46C61×#450—21no—
47C4×SD16×C61—22no—
48C61×#431C2211yescharacteristic
49C2×#11758200—22no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194—44no—
53C22×#92418—21no—
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—21no—
65#14519710—21no—
66#14519710—21no—
67#14519710—21no—
68#14519708—22no—
69#14519708—21no—
70#14519708—21no—
71#14519708—21no—
72#14519708—21no—
73C4⋊#92418—21no—
74C4⋊#92418C2312yes—
75C4⋊#92418C2312yes—
76#14519700—22no—
77#14519698—22no—
78C22⋊#92418—21no—
79C22⋊#92418—22no—
80C22⋊#92418—22no—
81C4×#92418—21no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—22no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C24×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—42no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×Dic122—21no—
125C2×Dic122—21no—
126SD16×C61—44no—
127C2×C8×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61C2×D412yes—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—41no—
145#11758200—44no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150#11758196—41no—
151#11758196—41no—
152C4×Dic61—41no—
153#11758194—44no—
154#11758194—44no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—42no—
159C2×#92418SD1614yescomplemented
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164C23×C61—22no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61#12111yescharacteristic
169Q8×C61—42no—
170Q8×C61—81no—
171D4×C61—22no—
172D4×C61—22no—
173D4×C61—42no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—44no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61C2×SD1612yes—
187C2×C4×C61#13712yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×Dic61—41no—
192C2×Dic61—41no—
193Dic122—42no—
194Dic122—81no—
195C8×C61—42no—
196#92418—21no—
197#92418—21no—
198#92418—42no—
199#92418—42no—
200#92418C2×SD1612yes—
201#92418C2×SD1612yes—
202#19137—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—84no—
209C22×C61#46112yes—
210C22×C61C2×#12111yescharacteristic
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—42no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61#46112yes—
218C4×C61C22×SD1611yescharacteristic
219C4×C61C2×#13711yescharacteristic
220C4×C61#57512yes—
221Dic61—81no—
222Dic61—81no—
223#3160—611no—
224#3101—611no—
225#3101D6111yescharacteristic
226C2×#436—611no—
227#1695—612no—
228#1693—612no—
229#1691—612no—
230#1691—612no—
231#1564—611no—
232#1542—611no—
233#1542—611no—
234C2×C61—22no—
235C2×C61—82no—
236C2×C61C2×#46111yescharacteristic
237C2×C61#305411yescharacteristic
238C2×C61#306711yescharacteristic
239C22×M16—1221no—
240#533—611no—
241#533C2×D6111yescharacteristic
242C2×C4×D4C2×D6111yescharacteristic
243#500—611no—
244#500—611no—
245#499—1221no—
246#499—1221no—
247#498—611no—
248#498—611no—
249#483—21no—
250#483—1221no—
251#476—612no—
252#476C2×D6112yes—
253#461—612no—
254#461C2×D6112yes—
255#450—21no—
256#450—1221no—
257C4×SD16—22no—
258C4×SD16—1222no—
259#436—612no—
260#436—612no—
261#436—1221no—
262#436—1221no—
263#431—611no—
264#431C2×D6111yescharacteristic
265#427—1221no—
266#427—1221no—
267C2×C4⋊C8—611no—
268C2×C22⋊C8—1221no—
269C2×C4×C8—611no—
270#335—612no—
271#334—612no—
272#333—612no—
273#333—612no—
274#332—612no—
275#332—612no—
276C61#1913711yescharacteristic, complemented, Sylow
277C22×D4C22⋊D6111yescharacteristic
278C23×C4—21no—
279C2×SD16—22no—
280C2×SD16—1222no—
281C2×M16—1222no—
282C2×M16—1222no—
283C2×M16—2444no—
284C22×C8—611no—
285C22×C8—611no—
286C22×C8—1221no—
287C4⋊Q8—611no—
288C4⋊Q8—611no—
289C4⋊Q8C22×D6111yescharacteristic
290C4⋊Q8C22×D6111yescharacteristic
291#125—21no—
292#125—1221no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—1221no—
296C22⋊Q8—1221no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—1221no—
300C4×Q8—1221no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C22⋊D6112yes—
304C4×D4C22×D6112yes—
305#117—21no—
306#117—1221no—
307C2×C4⋊C4C22⋊D6111yescharacteristic
308C2×C22⋊C4—21no—
309C2×C42C22×D6111yescharacteristic
310#107—1221no—
311#107—1221no—
312#107—1221no—
313#107—1221no—
314#106—22no—
315#106—1221no—
316#106—1221no—
317#106—1221no—
318#106—1221no—
319#106—1222no—
320C4⋊C8—612no—
321C4⋊C8—612no—
322C4⋊C8—1221no—
323C4⋊C8C22×D6112yes—
324#103—22no—
325#103—1222no—
326#102—22no—
327#102—1222no—
328C22⋊C8—21no—
329C22⋊C8—1222no—
330C22⋊C8—1222no—
331C22⋊C8—1221no—
332C4×C8—21no—
333C4×C8—611no—
334C4×C8—611no—
335C4×C8—611no—
336C4×C8—611no—
337C4×C8—1221no—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1221no—
342C2×Q8—1221no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4C2×C22⋊D6112yes—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4D4×D6111yescharacteristic
352C22×C4#1175822411yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354SD16—44no—
355SD16—2444no—
356M16—2444no—
357M16—2444no—
358C2×C8—22no—
359C2×C8—611no—
360C2×C8—611no—
361C2×C8—614no—
362C2×C8—611no—
363C2×C8—611no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—1222no—
367C2×C8—2442no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4C2×C22⋊D6112yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—41no—
385C22⋊C4—2441no—
386C42—41no—
387C42—2441no—
388C42D4×D6111yescharacteristic
389C42#1175822411yescharacteristic
390C42C2×C22⋊D6111yescharacteristic
391C42C23×D6111yescharacteristic
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23#1451974911yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—4881no—
400Q8—2442no—
401D4—22no—
402D4—22no—
403D4—42no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4#1451974911yescharacteristic
416C2×C4#1451974911yescharacteristic
417C2×C4#1451974911yescharacteristic
418C2×C4SD16×D6112yes—
419C2×C4#1451980612yes—
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C22×C22⋊D6111yescharacteristic
422C2×C4C2×#1175822411yescharacteristic
423C8—42no—
424C8—612no—
425C8—612no—
426C8—1221no—
427C8—1221no—
428C8—2442no—
429C8—2442no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—84no—
436C22#1683996512yes—
437C22C2×#1451974911yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—4881no—
445C4—4881no—
446C4#1683996512yes—
447C4C2×SD16×D6111yescharacteristic
448C4C2×#1451980611yescharacteristic
449C4#1684056012yes—
450C2—22no—
451C2—82no—
452C2C2×#1683996511yescharacteristic
453C2#2145542511yescharacteristic
454C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3101C2
socleC22×C61C2×#121
prime radicalC22×D4×C61D4
hypercenter#3101D61
hypocenterC61#19137
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #21456329C2semidirect
4. #21445340C2semidirect
5. #21445330C2semidirect
6. #21445320C2semidirect
7. #21445316C2semidirect
8. #21444565C2semidirect
9. #21444476C2semidirect
10. #21444474C2semidirect
14. #16840165C22semidirect
15. #16840165C22semidirect
18. #16840158C22semidirect
19. #16840158C22semidirect
35. #16839514C22semidirect
36. #16839512C22semidirect
37. #16839511C22semidirect
38. #16839510C22semidirect
39. #16839509C22semidirect
40. #16839508C22semidirect
159. C2×#92418SD16semidirect
276. C61#19137semidirect

Maximal subgroups

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

subgroupindex
#214563292
#214558272
C2×#168398402
#214453402
#214453302
#214453202
#214453162
#214445652
#214444762
#214444742
C61×#31012
#1913761