Finite Group Databasebeta

← #29665250

Subgroups — #29665250

Order 15616.

728 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#21456942C211yesmaximal, characteristic, complemented
2#21455285C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445257C214yesmaximal, complemented
6#21445209C214yesmaximal, complemented
7#21444492C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C61×#1571C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11C22×#11758190—21no—
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840167C2212yescomplemented
16#16840109C2212yescomplemented
17#16840076—22no—
18#16840043—22no—
19#16839978—22no—
20#16839968—22no—
21#16839968C2212yes—
22#16839968C2214yes—
23#16839958C2212yes—
24C8×D244C2214yes—
25#16839827—22no—
26#16839815C2211yescharacteristic
27#16839764—21no—
28C2×#14519702—21no—
29C22⋊C8×D61—22no—
30#16839509C2212yescomplemented
31#16839509C2214yescomplemented
32#16839505C2212yescomplemented
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#335C2214yescomplemented
37C2×#11758222—21no—
38C22×Dic122—21no—
39C22×D244D411yescharacteristic
40C22×C4×D61—21no—
41#14519805—44no—
42C2×#11758190—24no—
43C2×#11758190—22no—
44C2×#11758190—44no—
45C2×D488—22no—
46C2×C8×D61—24no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777C2312yes—
51#14519777C2311yescharacteristic
52#14519774C2311yescharacteristic
53#14519765—21no—
54#14519751—21no—
55#14519751—21no—
56C4×D244—21no—
57C4×D244—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244D412yes—
61C4×D244C2312yes—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758198—21no—
66#14519711—24no—
67C4⋊#92418C2312yes—
68#14519702—24no—
69#14519702—22no—
70#14519699—24no—
71#14519699—22no—
72C22⋊#92418—21no—
73#14519689—24no—
74C4×#92418—21no—
75C2×M16×C61—21no—
76C22×C8×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C4⋊C8×C61—21no—
80C4⋊C8×C61D412yes—
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61D412yes—
84C4×C8×C61C2312yes—
85C23×D61—21no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C2×C22⋊D61—41no—
90C22×Dic61—21no—
91C22×Dic61—41no—
92#11758222—42no—
93#11758222—42no—
94C2×D244—21no—
95C2×D244—22no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244D814yes—
100C2×D244C2×D412yes—
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61—41no—
104C2×C4×D61—82no—
105C2×Dic122—22no—
106C2×Dic122—21no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—41no—
110M16×C61—42no—
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61—42no—
114C2×C8×C61C2×D414yes—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758198—22no—
120#11758198—22no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61C2×D412yes—
131C2×#92418—22no—
132D488—44no—
133#11758190—48no—
134#11758190—44no—
135C8×D61—84no—
136C23×C61#12111yescharacteristic
137C22×D61—22no—
138C22×D61—24no—
139C22×D61—41no—
140C22×D61—81no—
141C22×D61—81no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61C2×D812yes—
148C2×C4×C61#13512yes—
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C2×#4111yescharacteristic
152C22⋊D61—82no—
153C2×Dic61—22no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—81no—
157C2×Dic61—82no—
158D244—22no—
159D244—24no—
160D244—41no—
161D244—41no—
162D244—42no—
163D244—81no—
164D244C2×D814yes—
165C4×D61—44no—
166C4×D61—82no—
167C4×D61—82no—
168Dic122—42no—
169Dic122—41no—
170Dic122—42no—
171Dic122—41no—
172Dic122—82no—
173C8×C61—24no—
174C8×C61—42no—
175C8×C61—42no—
176#92418—42no—
177#19392—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61#46612yes—
181C22×C61C2×#12111yescharacteristic
182C2×D61—24no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—84no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61#46612yes—
193C4×C61C22×D811yescharacteristic
194C4×C61C2×#13511yescharacteristic
195C4×C61#57712yes—
196Dic61—42no—
197Dic61—82no—
198Dic61—81no—
199#3226—611no—
200#3036—611no—
201C2×#461—611no—
202#2972—611no—
203#1681—614no—
204#1673—614no—
205#1571D6111yescharacteristic, complemented
206#1546—611no—
207#1540—611no—
208C2×C61—22no—
209C2×C61#304211yescharacteristic
210C2×C61C2×#46611yescharacteristic
211C2×C61#306911yescharacteristic
212D61—44no—
213D61—161no—
214C2×AGL(1,ℤ/8)—1221no—
215C22×SD16—1221no—
216#532—611no—
217C2×C4⋊Q8—611no—
218C2×C4×D4—611no—
219#500—612no—
220#486—612no—
221#480—1222no—
222#474—1222no—
223#463—1222no—
224#461—612no—
225#461—614no—
226#461—1222no—
227#459—612no—
228C8×D4—614no—
229C8×D4—1222no—
230#431—611no—
231#431C2×D6111yescharacteristic
232#423C2×D6111yescharacteristic
233C2×C4⋊C8C2×D6111yescharacteristic
234#417—1221no—
235C2×#103—1221no—
236C2×C22⋊C8—1222no—
237#335C2×D6114yescomplemented
238#332—612no—
239#332—614no—
240#331—612no—
241C61#1939211yescharacteristic, complemented, Sylow
242C2×#41—1221no—
243C22×Q8—1221no—
244C22×D4—611no—
245C23×C4—1221no—
246AGL(1,ℤ/8)—2444no—
247C2×SD16—1224no—
248C2×SD16—1222no—
249C2×SD16—2444no—
250C2×D8—1222no—
251C2×M16—21no—
252C22×C8—21no—
253C22×C8—1224no—
254C4⋊Q8—612no—
255C4⋊Q8—611no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258C4⋊Q8—1221no—
259#127—611no—
260#124—1221no—
261#121—1221no—
262#121—1221no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—1221no—
266C4×D4—1221no—
267C4×D4—1221no—
268C4×D4—1221no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C42C22×D6111yescharacteristic
274#107—1224no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—612no—
278C4⋊C8D24412yes—
279C4⋊C8C22×D6112yes—
280C4⋊C8C22×D6112yes—
281#103—1224no—
282#103—1222no—
283#102—1222no—
284#102—1224no—
285C22⋊C8—1221no—
286C22⋊C8—1224no—
287C22⋊C8D24412yes—
288C4×C8—1221no—
289C4×C8C22×D6112yes—
290C24—1221no—
291#41—2442no—
292#41—2442no—
293C2×Q8—1221no—
294C2×Q8—1222no—
295C2×Q8—2442no—
296C2×Q8—2442no—
297C2×Q8—2441no—
298C2×D4—612no—
299C2×D4—614no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1222no—
305C2×D4—2441no—
306C22×C4—1221no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—4882no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4C2×D24411yescharacteristic
313C22×C4D4×D6111yescharacteristic
314C22×C4#1175822611yescharacteristic
315SD16—2448no—
316SD16—2444no—
317D8—2444no—
318M16—42no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—42no—
322C2×C8—1222no—
323C2×C8—4884no—
324C2×C8C2×D24414yes—
325C4⋊C4—612no—
326C4⋊C4—1221no—
327C4⋊C4—1222no—
328C4⋊C4—1221no—
329C4⋊C4—1222no—
330C4⋊C4—1221no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C22⋊C4—1222no—
334C22⋊C4—1222no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C42C2×D24411yescharacteristic
338C42D4×D6111yescharacteristic
339C42#1175822611yescharacteristic
340C42C23×D6111yescharacteristic
341C23—1224no—
342C23—1222no—
343C23—4881no—
344C23—4881no—
345C23—2441no—
346C23#1451975211yescharacteristic
347Q8—4882no—
348Q8—2442no—
349Q8—2441no—
350Q8—2442no—
351Q8—2441no—
352D4—614no—
353D4—1224no—
354D4—1222no—
355D4—4881no—
356D4—4882no—
357D4—2441no—
358D4—2441no—
359D4—2442no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—1222no—
363C2×C4—4882no—
364C2×C4—4882no—
365C2×C4—4882no—
366C2×C4—4881no—
367C2×C4—2441no—
368C2×C4—2442no—
369C2×C4—2444no—
370C2×C4#1451975211yescharacteristic
371C2×C4#1451975211yescharacteristic
372C2×C4#1451975211yescharacteristic
373C2×C4D8×D6112yes—
374C2×C4#1451980012yes—
375C2×C4C2×D4×D6111yescharacteristic
376C2×C4C22×D24411yescharacteristic
377C2×C4C2×#1175822611yescharacteristic
378C8—24no—
379C8—42no—
380C8—42no—
381C8—2442no—
382C22—21no—
383C22—21no—
384C22—1224no—
385C22—4884no—
386C22—4881no—
387C22—4881no—
388C22—4881no—
389C22—2442no—
390C22—2442no—
391C22#1683999912yes—
392C22C2×#1451975211yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—4881no—
397C4—4882no—
398C4—2442no—
399C4#1683999912yes—
400C4C2×D8×D6111yescharacteristic
401C4C2×#1451980011yescharacteristic
402C4#1684057412yes—
403C2—22no—
404C2—9761no—
405C2—2444no—
406C2#2145533211yescharacteristic
407C2C2×#1683999911yescharacteristic
408C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1571C2
socleC22×C61C2×#121
prime radical#21455285C2
hypercenter#1571D61
hypocenterC61#19392
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
1. #21456942C2semidirect
3. C2×#16839968C2semidirect
4. #21454842C2semidirect
5. #21445257C2semidirect
6. #21445209C2semidirect
7. #21444492C2semidirect
8. #21444469C2semidirect
9. C61×#1571C2semidirect
15. #16840167C22semidirect
16. #16840109C22semidirect
30. #16839509C22semidirect
31. #16839509C22semidirect
32. #16839505C22semidirect
36. C61×#335C22semidirect
205. #1571D61semidirect
237. #335C2×D61semidirect
241. C61#19392semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214552852
C2×#168399682
#214548422
#214452572
#214452092
#214444922
#214444692
C61×#15712
#1939261