Finite Group Databasebeta

← #29649805

Subgroups — #29649805

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454780C211yesmaximal, characteristic
2C2×#16839893C211yesmaximal, characteristic, complemented
3C4×#14519808C211yesmaximal, characteristic
4#21444725C212yesmaximal, complemented
5#21444724C212yesmaximal, complemented
6#21444713C212yesmaximal, complemented
7#21444706C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444207C211yesmaximal, characteristic
10#21444100C211yesmaximal, characteristic, complemented
11C61×#1479C211yesmaximal, characteristic, complemented
12C2×#14519808C411yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16839893—21no—
15#16839893—21no—
16#16839893C2212yes—
17#16839893C2212yes—
18#16839889C2211yescharacteristic
19#16839889C2211yescharacteristic
20#16839867C412yes—
21#16839860C412yes—
22C4×#11758202C2212yes—
23C4×#11758201C2212yes—
24#16839839C2211yescharacteristic
25#16839839C2211yescharacteristic
26#16839830—21no—
27#16839830—21no—
28#16839813—21no—
29#16839802C411yescharacteristic
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32#16839771C411yescharacteristic
33#16839761C411yescharacteristic
34C4×#11758194C2211yescharacteristic
35C4×#11758189—21no—
36C2×#14519688—21no—
37C2×C4×#92418C2211yescharacteristic
38#16839514C2212yes—
39#16839512C2212yes—
40#16839494C2212yes—
41#16839491C2212yes—
42#16839484C2212yes—
43C8×#92418C2212yes—
44C2×C4×Q8×C61C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×Q8⋊C8C2212yes—
48C61×Q8⋊C8C2212yes—
49C2×#11758222C2×C411yescharacteristic
50#14519808—24no—
51C2×#11758202C2×C412yes—
52C2×#11758201C2×C412yes—
53C2×#11758194C2×C411yescharacteristic
54C2×#11758194C2×C411yescharacteristic
55C2×#11758189—21no—
56C2×#11758189—21no—
57C22×#92418—21no—
58C22×#92418D411yescharacteristic
59C22×#92418D411yescharacteristic
60C4×Dic122C2×C411yescharacteristic
61C4×Dic122C2311yescharacteristic
62C4×D244C2×C411yescharacteristic
63C4×D244C2311yescharacteristic
64C4×C22⋊D61—21no—
65C4×C22⋊D61—21no—
66#14519736C2×C411yescharacteristic
67#14519734—21no—
68C42×D61—21no—
69#14519710C2×C412yes—
70#14519708C2×C412yes—
71C4⋊#92418—21no—
72C4⋊#92418—22no—
73C4⋊#92418—22no—
74C4⋊#92418C2312yes—
75#14519706—21no—
76#14519706—21no—
77#14519701C2×C412yes—
78#14519701C2×C412yes—
79#14519700C2×C412yes—
80#14519697C2×C412yes—
81#14519689—21no—
82#14519689—21no—
83#14519688—22no—
84#14519688—22no—
85#14519688C2×C412yes—
86#14519687—21no—
87#14519687—21no—
88C4×#92418D411yescharacteristic
89C4×#92418D411yescharacteristic
90C4×#92418C2312yes—
91C4×#92418C2311yescharacteristic
92C4×#92418C2311yescharacteristic
93C22×Q8×C61C2×C411yescharacteristic
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61C2×C412yes—
97C4×Q8×C61C2312yes—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C4⋊C8×C61C2311yescharacteristic
104C4⋊C8×C61C2311yescharacteristic
105C22⋊C8×C61—21no—
106C22⋊C8×C61—21no—
107C4×C8×C61—21no—
108C4×C8×C61C2312yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61C22×C412yes—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×C22⋊D61—21no—
119#11758222—22no—
120#11758222—22no—
121C2×D244C22×C411yescharacteristic
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×Dic122C22×C411yescharacteristic
126C2×C8×C61—21no—
127C2×C8×C61—21no—
128C2×C8×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61C22×C412yes—
133C42×C61—22no—
134C42×C61—22no—
135C42×C61C22×C411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61#4111yescharacteristic
138C42×C61C2411yescharacteristic
139C22⋊Dic61—21no—
140#11758202—24no—
141#11758201—24no—
142#11758198—21no—
143#11758198—21no—
144C4⋊Dic61C22×C411yescharacteristic
145C4⋊Dic61C22×C411yescharacteristic
146#11758196—21no—
147#11758196—21no—
148C4×Dic61—21no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151#11758194—22no—
152#11758194M1614yescomplemented
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—41no—
156C2×#92418—42no—
157C2×#92418C22×C412yes—
158C2×#92418C22×C412yes—
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163#11758189—42no—
164#11758189—42no—
165C23×C61C4×D411yescharacteristic
166C22×D61—21no—
167Q8×C61—22no—
168Q8×C61—22no—
169Q8×C61—42no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61C4×D411yescharacteristic
178C2×C4×C61C4×D411yescharacteristic
179C2×C4×C61C4×D411yescharacteristic
180C2×C4×C61C2×M1612yes—
181C2×C4×C61#13112yes—
182C2×C4×C61C23×C411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C22⋊D61—42no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189D244—21no—
190D244—21no—
191C4×D61—42no—
192C4×D61—41no—
193C4×D61—41no—
194C4×D61—42no—
195Dic122—21no—
196Dic122—21no—
197C8×C61—42no—
198C8×C61—41no—
199C8×C61—41no—
200#92418—22no—
201#92418—21no—
202#92418—21no—
203#92418—21no—
204#92418—42no—
205#92418C2×M1614yes—
206#92418#13712yescomplemented
207#17892—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61#43612yes—
211C22×C61C2×C4×D411yescharacteristic
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—41no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61#43612yes—
221C4×C61C22×M1611yescharacteristic
222C4×C61C2×#13111yescharacteristic
223C4×C61C2×#13711yescharacteristic
224C4×C61#57511yescharacteristic
225Dic61—41no—
226Dic61—41no—
227Dic61—41no—
228#2965—611no—
229C2×#446—611no—
230C4×#137—611no—
231#1593—612no—
232#1592—612no—
233#1591—612no—
234#1590—612no—
235#1565—611no—
236#1479D6111yescharacteristic, complemented
237#1477—611no—
238C8×M16—611no—
239C2×C61—22no—
240C2×C61C2×#43611yescharacteristic
241C2×C61C4×#13711yescharacteristic
242C2×C61#295511yescharacteristic
243D61—81no—
244C2×#137—611no—
245C4×#41—611no—
246C2×C4×Q8C2×D6111yescharacteristic
247#446—612no—
248#446—612no—
249#446—611no—
250#446—611no—
251#446—1221no—
252#446—1221no—
253#441—612no—
254#440—612no—
255C4×Q16—612no—
256C4×SD16—612no—
257#436—611no—
258#436—611no—
259#435—1221no—
260#435—1221no—
261#431—1221no—
262#431C2×D6111yescharacteristic
263#431C2×D6111yescharacteristic
264#428—611no—
265C2×C4⋊C8—611no—
266C2×C4⋊C8—1221no—
267#419—611no—
268#417—611no—
269C4×M16—611no—
270C4×M16—1221no—
271C2×#97—1221no—
272C2×C4×C8—611no—
273#335—612no—
274#334—612no—
275Q8⋊C8—612no—
276Q8⋊C8C2×D6112yes—
277Q8⋊C8C2×D6112yes—
278D4⋊C8—612no—
279#322—612no—
280C82—612no—
281C61#1789211yescharacteristic, complemented, Sylow
282C2×#41—611no—
283C22×Q8C4×D6111yescharacteristic
284#137—1224no—
285C2×Q16—612no—
286C2×SD16—612no—
287C2×M16—611no—
288C2×M16—611no—
289C2×M16—1221no—
290C2×M16—1221no—
291C22×C8—611no—
292C22×C8—611no—
293C22×C8—1221no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—611no—
297C4×Q8—611no—
298C4×Q8C4×D6112yes—
299C4×Q8C22×D6112yes—
300C4×D4—611no—
301C4×D4—611no—
302C4×D4—1221no—
303C4×D4—1221no—
304#117—611no—
305#117—1221no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C4×D6111yescharacteristic
308C2×C42—21no—
309C2×C42—1221no—
310C2×C42C22×D6111yescharacteristic
311#107—612no—
312#106—612no—
313C4⋊C8—21no—
314C4⋊C8—612no—
315C4⋊C8—1222no—
316C4⋊C8—1222no—
317C4⋊C8—1221no—
318C4⋊C8—1221no—
319C4⋊C8—1221no—
320C4⋊C8C22×D6111yescharacteristic
321C4⋊C8C22×D6111yescharacteristic
322#103—612no—
323#103—612no—
324#103—612no—
325#102—612no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—1221no—
329C22⋊C8—1221no—
330#97—612no—
331#97—1222no—
332#97—1222no—
333#97—1221no—
334#97—1221no—
335C4×C8—21no—
336C4×C8—611no—
337C4×C8—611no—
338C4×C8—611no—
339C4×C8—612no—
340C4×C8—611no—
341C4×C8C22×D6112yes—
342#41—1222no—
343#41—1222no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—611no—
348C2×Q8C2×C4×D6112yes—
349C2×D4—611no—
350C2×D4—1221no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4C2×C4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4#1175822411yescharacteristic
359Q16—1224no—
360SD16—1224no—
361M16—614no—
362M16—1222no—
363M16—2442no—
364M16—2442no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—612no—
369C2×C8—611no—
370C2×C8—612no—
371C2×C8—611no—
372C2×C8—611no—
373C2×C8—611no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8—2441no—
377C2×C8—2442no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—611no—
382C4⋊C4—611no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4C2×C4×D6112yes—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C22⋊C4—1221no—
389C42—22no—
390C42—22no—
391C42—1221no—
392C42—1221no—
393C42—1221no—
394C42C2×C4×D6111yescharacteristic
395C42D4×D6111yescharacteristic
396C42#1175822411yescharacteristic
397C42C23×D6111yescharacteristic
398C23—1221no—
399C23#1451973711yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—42no—
403Q8—1221no—
404Q8—1221no—
405D4—1221no—
406D4—1221no—
407D4—2442no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4#1451973711yescharacteristic
423C2×C4#1451973711yescharacteristic
424C2×C4#1451973711yescharacteristic
425C2×C4M16×D6112yes—
426C2×C4#1451978312yes—
427C2×C4C22×C4×D6111yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C2×#1175822411yescharacteristic
430C8—42no—
431C8—41no—
432C8—41no—
433C8—614no—
434C8—612no—
435C8—1222no—
436C8—1221no—
437C8—1221no—
438C8—1221no—
439C8—2442no—
440C22—21no—
441C22—21no—
442C22—2441no—
443C22—2441no—
444C22—2441no—
445C22#1683983712yes—
446C22C2×#1451973711yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—42no—
452C4—2441no—
453C4—2441no—
454C4—2441no—
455C4#1683983712yes—
456C4C2×M16×D6111yescharacteristic
457C4C2×#1451978311yescharacteristic
458C4#137×D6111yescharacteristic
459C4#1684056111yescharacteristic
460C2—22no—
461C2—4881no—
462C2C2×#1683983711yescharacteristic
463C2#2145463411yescharacteristic
464C2#2145471811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1479C2
socleC22×C61C2×C4×D4
prime radicalC2×#11758222C2×C4
cosocleC42×C61C24
hypercenter#1479D61
hypocenterC61#17892
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. C2×#16839893C2semidirect
4. #21444725C2semidirect
5. #21444724C2semidirect
6. #21444713C2semidirect
7. #21444706C2semidirect
8. #21444572C2semidirect
10. #21444100C2semidirect
11. C61×#1479C2semidirect
152. #11758194M16semidirect
206. #92418#137semidirect
236. #1479D61semidirect
281. C61#17892semidirect

Maximal subgroups

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

subgroupindex
#214547802
C2×#168398932
C4×#145198082
#214447252
#214447242
#214447132
#214447062
#214445722
#214442072
#214441002
C61×#14792
#1789261