Finite Group Databasebeta

← #29650208

Subgroups — #29650208

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C211yesmaximal, characteristic
2#21454531C211yesmaximal, characteristic, complemented
3C2×#16839839C211yesmaximal, characteristic, complemented
4#21444701C214yesmaximal, complemented
5#21444645C214yesmaximal, complemented
6#21444593C211yesmaximal, characteristic, complemented
7#21444245C211yesmaximal, characteristic, complemented
8#21444103C211yesmaximal, characteristic, complemented
9C61×#1480C211yesmaximal, characteristic, complemented
10C2×#14519784—21no—
11C22×#11758189—21no—
12C2×C4×D244C2211yescharacteristic
13#16840073C412yes—
14#16840039C412yes—
15#16839954C412yes—
16#16839895C412yes—
17C4×#11758199C2214yes—
18#16839840—21no—
19#16839840C2212yescomplemented
20#16839839—21no—
21#16839839—21no—
22#16839839C2212yes—
23#16839839C2212yes—
24D4×#92418—21no—
25D4×#92418C2212yescomplemented
26#16839815C2211yescharacteristic
27#16839786C2211yescharacteristic
28#16839729—21no—
29C2×#14519689—21no—
30C2×C4×#92418C2211yescharacteristic
31#16839514C2214yescomplemented
32#16839491C2214yes—
33#16839484C2212yescomplemented
34C8×#92418C2212yescomplemented
35C61×#532C2211yescharacteristic
36C61×#431C2211yescharacteristic
37C2×C4⋊C8×C61C2211yescharacteristic
38C61×D4⋊C8C2214yescomplemented
39C22×D244C2×C411yescharacteristic
40C22×C4×D61—21no—
41C2×#11758199C2×C414yes—
42#14519784—44no—
43C2×#11758194—21no—
44C2×#11758194—22no—
45C2×#11758189—22no—
46C2×#11758189—22no—
47C2×#11758189—44no—
48C22×#92418—22no—
49C22×#92418D411yescharacteristic
50C22×#92418D411yescharacteristic
51C4×D244—21no—
52C4×D244—21no—
53C4×D244C2×C412yes—
54C4×D244C2312yes—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758198—21no—
57#14519710C2×C414yes—
58C4⋊#92418—21no—
59C4⋊#92418C2×C412yes—
60C4⋊#92418C2312yes—
61#14519698C2×C414yes—
62#14519697C2×C414yes—
63C22⋊#92418—22no—
64C22⋊#92418—22no—
65C22⋊#92418C2×C412yes—
66#14519689—22no—
67#14519689—22no—
68C4×#92418D411yescharacteristic
69C4×#92418D411yescharacteristic
70C4×#92418C2312yes—
71C4×#92418C2311yescharacteristic
72C4×#92418C2311yescharacteristic
73C2×#41×C61—21no—
74C22×C8×C61—21no—
75C4⋊Q8×C61C2×C411yescharacteristic
76#127×C61C2×C411yescharacteristic
77#124×C61—21no—
78#121×C61—21no—
79#121×C61C2×C412yes—
80C4×D4×C61—21no—
81C4×D4×C61C2312yes—
82C2×C42×C61C2311yescharacteristic
83C4⋊C8×C61—21no—
84C4⋊C8×C61—21no—
85C4⋊C8×C61C2312yes—
86C22⋊C8×C61—21no—
87C4×C8×C61C2312yes—
88C23×D61—21no—
89#41×C61—42no—
90#41×C61—42no—
91C2×Q8×C61—21no—
92C2×D4×C61—22no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61C22×C412yes—
96C22×C4×C61—22no—
97C22×C4×C61C22×C411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C22×Dic61—21no—
101C2×D244—21no—
102C2×D244—42no—
103C2×D244C22×C412yes—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—42no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61C22×C412yes—
113C22⋊C4×C61—22no—
114C22⋊C4×C61—22no—
115C42×C61C22×C411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758199—44no—
120#11758198—22no—
121#11758198—22no—
122C4⋊Dic61—21no—
123C4⋊Dic61C22×C412yes—
124#11758194—42no—
125#11758194—44no—
126C2×#92418—22no—
127C2×#92418—22no—
128C2×#92418—44no—
129C2×#92418D814yescomplemented
130C2×#92418C22×C414yes—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135#11758189—44no—
136#11758189—44no—
137C23×C61—22no—
138C23×C61C4×D411yescharacteristic
139C22×D61—22no—
140C22×D61—42no—
141C22×D61—42no—
142C22×D61—42no—
143Q8×C61—42no—
144D4×C61—24no—
145D4×C61—42no—
146D4×C61—44no—
147D4×C61—42no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61C2×D812yes—
157C2×C4×C61#13512yes—
158C2×C4×C61C23×C411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×Dic61—22no—
162C2×Dic61—42no—
163D244—22no—
164D244—22no—
165D244—42no—
166C4×D61—42no—
167C4×D61—42no—
168C4×D61—44no—
169C8×C61—42no—
170C8×C61—42no—
171#92418—24no—
172#92418—21no—
173#92418—21no—
174#92418—42no—
175#92418C2×D812yes—
176#92418C2×D812yes—
177#17928—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—42no—
181C22×C61—42no—
182C22×C61—42no—
183C22×C61C4×D812yes—
184C22×C61C2×C4×D411yescharacteristic
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—84no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61C4×D812yes—
194C4×C61#56812yes—
195C4×C61C22×D811yescharacteristic
196C4×C61C2×#13511yescharacteristic
197Dic61—42no—
198#3100—611no—
199#2931—611no—
200C2×#436—611no—
201#1589—614no—
202#1581—614no—
203#1571—611no—
204#1486—611no—
205#1480D6111yescharacteristic, complemented
206#1450—611no—
207C2×C61—22no—
208C2×C61—82no—
209C2×C61#293011yescharacteristic
210C2×C61C2×C4×D811yescharacteristic
211C2×C61#295511yescharacteristic
212D61—82no—
213C2×#131—1221no—
214C22×M16—1221no—
215#532C2×D6111yescharacteristic
216C2×C4×D4—611no—
217#480—612no—
218#474—612no—
219#459—612no—
220#447—612no—
221C4×D8—614no—
222#436—612no—
223#436—612no—
224#436—612no—
225#436—1221no—
226#436—1221no—
227#436—1221no—
228C8×D4—612no—
229C8×D4—1221no—
230#431—611no—
231#431C2×D6111yescharacteristic
232#423—611no—
233C2×C4⋊C8C2×D6111yescharacteristic
234#408—1221no—
235C2×C22⋊C8—1221no—
236C2×C4×C8—611no—
237#335—614no—
238D4⋊C8—614no—
239D4⋊C8C2×D6114yescomplemented
240#322—612no—
241C82—612no—
242C61#1792811yescharacteristic, complemented, Sylow
243C2×#41—21no—
244C22×D4—611no—
245C23×C4—1221no—
246C2×D8—614no—
247#131—2444no—
248C2×M16—1222no—
249C2×M16—1222no—
250C2×M16—1222no—
251C2×M16—1221no—
252C2×M16—2444no—
253C22×C8—21no—
254C22×C8—611no—
255C22×C8—611no—
256C22×C8—1222no—
257C4⋊Q8C4×D6111yescharacteristic
258#127C4×D6111yescharacteristic
259#124—21no—
260#121—21no—
261#121C4×D6112yes—
262C4×D4—21no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—1221no—
266C4×D4—1221no—
267C4×D4C22×D6112yes—
268C2×C4⋊C4—611no—
269C2×C22⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—614no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—612no—
275C4⋊C8—612no—
276C4⋊C8—1221no—
277C4⋊C8C22×D6112yes—
278#102—614no—
279#102—614no—
280C22⋊C8—21no—
281C22⋊C8—612no—
282C22⋊C8—1222no—
283C22⋊C8—1222no—
284C22⋊C8—1222no—
285C22⋊C8—1222no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—612no—
290C4×C8—611no—
291C4×C8C22×D6112yes—
292C24—1221no—
293#41—42no—
294#41—42no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—612no—
300C2×D4—1221no—
301C2×D4—2442no—
302C2×D4C2×C4×D6112yes—
303C22×C4—22no—
304C22×C4—1221no—
305C22×C4—1222no—
306C22×C4—1222no—
307C22×C4—2442no—
308C22×C4—2442no—
309C22×C4C2×C4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4#1175822411yescharacteristic
312D8—2444no—
313M16—2444no—
314M16—2444no—
315M16—2444no—
316M16—2442no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—611no—
321C2×C8—611no—
322C2×C8—614no—
323C2×C8—614no—
324C2×C8—611no—
325C2×C8—611no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8—2444no—
329C4⋊C4—21no—
330C4⋊C4—612no—
331C4⋊C4—1221no—
332C4⋊C4C2×C4×D6112yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—1222no—
336C22⋊C4—1222no—
337C42C2×C4×D6111yescharacteristic
338C42D4×D6111yescharacteristic
339C42#1175822411yescharacteristic
340C42C23×D6111yescharacteristic
341C23—22no—
342C23—1222no—
343C23—2442no—
344C23—2442no—
345C23—2442no—
346C23#1451973711yescharacteristic
347Q8—42no—
348D4—24no—
349D4—42no—
350D4—44no—
351D4—42no—
352D4—1222no—
353D4—1222no—
354D4—2442no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—1222no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2444no—
365C2×C4#1451973711yescharacteristic
366C2×C4#1451973711yescharacteristic
367C2×C4#1451973711yescharacteristic
368C2×C4D8×D6112yes—
369C2×C4#1451979812yes—
370C2×C4C22×C4×D6111yescharacteristic
371C2×C4C2×D4×D6111yescharacteristic
372C2×C4C2×#1175822411yescharacteristic
373C8—42no—
374C8—42no—
375C8—612no—
376C8—612no—
377C8—1224no—
378C8—1221no—
379C8—1221no—
380C8—2442no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—4884no—
387C22—2442no—
388C22—2442no—
389C22—2442no—
390C22#1683984212yes—
391C22C2×#1451973711yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—2442no—
397C4#1683984212yes—
398C4#1684052712yes—
399C4C2×D8×D6111yescharacteristic
400C4C2×#1451979811yescharacteristic
401C2—22no—
402C2—82no—
403C2—4882no—
404C2#2145452511yescharacteristic
405C2C2×#1683984211yescharacteristic
406C2#2145471811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1480C2
socleC22×C61C2×C4×D4
prime radical#21455815C2
cosocleC42×C61C24
hypercenter#1480D61
hypocenterC61#17928
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #21454531C2semidirect
3. C2×#16839839C2semidirect
4. #21444701C2semidirect
5. #21444645C2semidirect
6. #21444593C2semidirect
7. #21444245C2semidirect
8. #21444103C2semidirect
9. C61×#1480C2semidirect
19. #16839840C22semidirect
25. D4×#92418C22semidirect
31. #16839514C22semidirect
33. #16839484C22semidirect
34. C8×#92418C22semidirect
38. C61×D4⋊C8C22semidirect
129. C2×#92418D8semidirect
205. #1480D61semidirect
239. D4⋊C8C2×D61semidirect
242. C61#17928semidirect

Maximal subgroups

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

subgroupindex
#214558152
#214545312
C2×#168398392
#214447012
#214446452
#214445932
#214442452
#214441032
C61×#14802
#1792861