Finite Group Databasebeta

← #29650131

Subgroups — #29650131

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455820C211yesmaximal, characteristic
2#21454897C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21444682C212yesmaximal, complemented
5#21444647C212yesmaximal, complemented
6#21444642C212yesmaximal, complemented
7#21444613C212yesmaximal, complemented
8#21444261C211yesmaximal, characteristic, complemented
9#21444235C211yesmaximal, characteristic, complemented
10#21444114C211yesmaximal, characteristic, complemented
11C61×#1572C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840076C412yes—
15#16840043C412yes—
16#16839958C412yes—
17#16839898C412yes—
18C4×D488C2212yes—
19C4×D488C2212yes—
20#16839839C2212yescomplemented
21#16839835—22no—
22C8×D244—22no—
23#16839827C2212yes—
24#16839827C2212yes—
25#16839827C2212yescomplemented
26#16839815C2211yescharacteristic
27#16839727—22no—
28C22⋊C8×D61—22no—
29C4×#11758194C2211yescharacteristic
30#16839489C2212yes—
31#16839489C2212yes—
32#16839489C2212yescomplemented
33#16839489C2212yescomplemented
34#16839484C2211yescharacteristic, complemented
35#16839484C2211yescharacteristic, complemented
36#16839483C2211yescharacteristic, complemented
37C8×#92418C2211yescharacteristic, complemented
38C61×#432C2211yescharacteristic
39C61×#431C2211yescharacteristic
40C61×#431C2211yescharacteristic
41C61×#335C2212yescomplemented
42C61×#334C2212yescomplemented
43C2×#11758222—21no—
44C22×D244C2×C411yescharacteristic
45C22×C4×D61—21no—
46C2×D488C2×C412yes—
47C2×D488C2×C412yes—
48C2×#11758194—21no—
49C2×#11758189—22no—
50C2×C8×D61—22no—
51C2×C8×D61—22no—
52C2×C8×D61—22no—
53#14519777C2×C411yescharacteristic
54#14519774C2×C411yescharacteristic
55#14519765—21no—
56#14519751—21no—
57#14519751C2×C412yes—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244C2×C412yes—
62C4×D244C2312yes—
63C4×D244C2312yes—
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×#11758198—21no—
66#14519711C2×C412yes—
67#14519711C2×C412yes—
68C4⋊#92418D412yes—
69#14519699C2×C412yes—
70#14519699C2×C412yes—
71#14519699C2×C412yes—
72#14519699C2×C412yes—
73C22⋊#92418D412yes—
74#14519689—22no—
75#14519689—22no—
76#14519689—22no—
77#14519689—22no—
78#14519688—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82C4×#92418C2311yescharacteristic
83C2×C42×C61C2311yescharacteristic
84C4⋊C8×C61C2×C412yes—
85C4⋊C8×C61C2312yes—
86C4⋊C8×C61C2312yes—
87C22⋊C8×C61—21no—
88C22⋊C8×C61—21no—
89C22⋊C8×C61C2×C412yes—
90#97×C61—21no—
91C4×C8×C61—21no—
92C4×C8×C61C2311yescharacteristic
93C4×C8×C61C2311yescharacteristic
94C23×D61—21no—
95C22×C4×C61C22×C411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—22no—
99C22×Dic61—21no—
100#11758222—82no—
101C2×D244—21no—
102C2×D244—24no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C22×C412yes—
106C2×D244C22×C412yes—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—82no—
111C2×Dic122—21no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61C22×C412yes—
115C2×C8×C61C22×C412yes—
116C42×C61C22×C411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C42×C61C2411yescharacteristic
120#11758198—22no—
121#11758198—22no—
122#11758198—22no—
123#11758198—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61C22×C412yes—
127C4⋊Dic61C22×C412yes—
128#11758194—42no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418C2×D412yes—
132C2×#92418C2×D412yes—
133D488—24no—
134D488—24no—
135#11758189—82no—
136C8×D61—82no—
137C8×D61—82no—
138C8×D61—82no—
139C23×C61C4×D411yescharacteristic
140C22×D61—22no—
141C22×D61—24no—
142C22×D61—22no—
143C22×D61—81no—
144C22×D61—81no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—41no—
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61C23×C411yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C22⋊D61—82no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—81no—
158D244—24no—
159D244—22no—
160D244—22no—
161D244—22no—
162D244—81no—
163D244—81no—
164D244#13114yes—
165C4×D61—42no—
166C4×D61—42no—
167C4×D61—42no—
168C4×D61—42no—
169C4×D61—82no—
170Dic122—81no—
171C8×C61—22no—
172C8×C61—22no—
173C8×C61—42no—
174C8×C61—42no—
175#92418—22no—
176#92418—22no—
177#92418—22no—
178#92418—41no—
179#17922—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61C2×C4×D411yescharacteristic
184C2×D61—24no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—84no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61C2×#13112yes—
196C4×C61#57612yes—
197Dic61—42no—
198Dic61—42no—
199#3100—611no—
200#2981—611no—
201#2972—611no—
202#1586—612no—
203#1581—612no—
204#1580—612no—
205C8×D8—612no—
206#1572D6111yescharacteristic, complemented
207#1490—611no—
208#1484—611no—
209#1453—611no—
210C2×C61—41no—
211C2×C61#292911yescharacteristic
212C2×C61#294811yescharacteristic
213C2×C61#295611yescharacteristic
214D61—44no—
215D61—82no—
216#532—611no—
217C2×C4×D4—611no—
218#480—612no—
219#474—612no—
220#459—612no—
221#447—612no—
222C4×D8—612no—
223C4×D8—612no—
224#436—612no—
225#435—1222no—
226C8×D4—612no—
227C8×D4—612no—
228C8×D4—612no—
229C8×D4—1222no—
230#432C2×D6111yescharacteristic
231#431—611no—
232#431C2×D6111yescharacteristic
233#431C2×D6111yescharacteristic
234#408—1222no—
235C2×C22⋊C8—1222no—
236C4×M16—611no—
237#335C2×D6112yescomplemented
238#334C2×D6112yescomplemented
239D4⋊C8—612no—
240D4⋊C8—612no—
241D4⋊C8—612no—
242D4⋊C8—612no—
243#322—611no—
244#322—611no—
245#322—611no—
246C82—611no—
247C61#1792211yescharacteristic, complemented, Sylow
248C2×#41—1221no—
249C22×D4—611no—
250C23×C4—1221no—
251C2×D8—612no—
252C2×D8—612no—
253C2×M16—1222no—
254C2×M16—1221no—
255C22×C8—1222no—
256C22×C8—1222no—
257C22×C8—1222no—
258C4⋊Q8—611no—
259#127—611no—
260#124—1221no—
261#121—612no—
262#121—1221no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—612no—
266C4×D4—1221no—
267C4×D4—1221no—
268C4×D4—1221no—
269C2×C4⋊C4—611no—
270C2×C22⋊C4—1221no—
271C2×C42C22×D6111yescharacteristic
272#107—612no—
273#107—612no—
274C4⋊C8—612no—
275C4⋊C8C4×D6112yes—
276C4⋊C8C22×D6112yes—
277C4⋊C8C22×D6112yes—
278#102—612no—
279#102—612no—
280#102—612no—
281#102—612no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—612no—
285C22⋊C8—1222no—
286C22⋊C8—1222no—
287C22⋊C8—1222no—
288C22⋊C8—1222no—
289C22⋊C8C4×D6112yes—
290#97—21no—
291#97—1221no—
292C4×C8—21no—
293C4×C8—611no—
294C4×C8—611no—
295C4×C8—611no—
296C4×C8—611no—
297C4×C8C22×D6111yescharacteristic
298C4×C8C22×D6111yescharacteristic
299C24—1221no—
300#41—4882no—
301C2×Q8—1221no—
302C2×D4—612no—
303C2×D4—612no—
304C2×D4—1224no—
305C2×D4—1222no—
306C2×D4—1221no—
307C2×D4—1221no—
308C2×D4—1221no—
309C22×C4—1221no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—1222no—
313C22×C4—4882no—
314C22×C4C2×C4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4#1175822611yescharacteristic
317D8—1224no—
318D8—1224no—
319M16—4882no—
320M16—2442no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—612no—
324C2×C8—612no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—4882no—
328C2×C8—4882no—
329C2×C8—4882no—
330C2×C8C2×C4×D6112yes—
331C2×C8C2×C4×D6112yes—
332C4⋊C4—612no—
333C4⋊C4—612no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C22⋊C4—1222no—
337C22⋊C4—1222no—
338C22⋊C4—1222no—
339C22⋊C4—1222no—
340C42C2×C4×D6111yescharacteristic
341C42D4×D6111yescharacteristic
342C42#1175822611yescharacteristic
343C42C23×D6111yescharacteristic
344C23—1224no—
345C23—1222no—
346C23—1222no—
347C23—4881no—
348C23—4881no—
349C23#1451973911yescharacteristic
350Q8—4881no—
351D4—614no—
352D4—1224no—
353D4—1222no—
354D4—1222no—
355D4—1222no—
356D4—4881no—
357D4—4881no—
358D4—4882no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—1222no—
363C2×C4—1222no—
364C2×C4—4882no—
365C2×C4—4881no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4#1451973911yescharacteristic
371C2×C4#1451973911yescharacteristic
372C2×C4#1451973911yescharacteristic
373C2×C4C22×C4×D6111yescharacteristic
374C2×C4C2×D4×D6111yescharacteristic
375C2×C4C2×#1175822611yescharacteristic
376C8—22no—
377C8—22no—
378C8—42no—
379C8—42no—
380C8—1222no—
381C8—1222no—
382C8—1222no—
383C8—2441no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—1224no—
388C22—4884no—
389C22—2442no—
390C22—2442no—
391C22—2442no—
392C22—2442no—
393C22—2442no—
394C22C2×#1451973911yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—2442no—
400C4—2442no—
401C4#131×D6112yes—
402C4#1684056412yes—
403C2—41no—
404C2—4882no—
405C2—2444no—
406C2#2145451211yescharacteristic
407C2#2145467011yescharacteristic
408C2#2145472811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1572C2
socleC22×C61C2×C4×D4
prime radical#21455820C2
cosocleC42×C61C24
hypercenter#1572D61
hypocenterC61#17922
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #21454897C2semidirect
3. #21454842C2semidirect
4. #21444682C2semidirect
5. #21444647C2semidirect
6. #21444642C2semidirect
7. #21444613C2semidirect
8. #21444261C2semidirect
9. #21444235C2semidirect
10. #21444114C2semidirect
11. C61×#1572C2semidirect
20. #16839839C22semidirect
25. #16839827C22semidirect
32. #16839489C22semidirect
33. #16839489C22semidirect
34. #16839484C22semidirect
35. #16839484C22semidirect
36. #16839483C22semidirect
37. C8×#92418C22semidirect
41. C61×#335C22semidirect
42. C61×#334C22semidirect
206. #1572D61semidirect
237. #335C2×D61semidirect
238. #334C2×D61semidirect
247. C61#17922semidirect

Maximal subgroups

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

subgroupindex
#214558202
#214548972
#214548422
#214446822
#214446472
#214446422
#214446132
#214442612
#214442352
#214441142
C61×#15722
#1792261