Finite Group Databasebeta

← #29665249

Subgroups — #29665249

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#21455283C211yesmaximal, characteristic
3C2×#16839967C211yesmaximal, characteristic, complemented
4#21445256C214yesmaximal, complemented
5#21445208C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444490C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic, complemented
9C61×#2972C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11C22×#11758200—21no—
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840167C2212yescomplemented
15#16840109C2212yescomplemented
16#16840076—22no—
17#16840043—22no—
18#16839977—22no—
19#16839967—22no—
20#16839967C2212yes—
21#16839967C2214yes—
22#16839957C2212yes—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519700—21no—
27#16839515C2214yescomplemented
28#16839507C2212yescomplemented
29#16839507C2214yescomplemented
30#16839505C2212yescomplemented
31C2×C4×D4×C61C2211yescharacteristic
32C8×D4×C61—22no—
33C8×D4×C61C2214yes—
34C61×#431C2211yescharacteristic
35C61×#431C2211yescharacteristic
36C2×C22⋊C8×C61—22no—
37C2×#11758222—21no—
38C22×Dic122—21no—
39#14519803—44no—
40C2×#11758200—24no—
41C2×#11758200—22no—
42C2×#11758200—44no—
43C2×#11758199—22no—
44C2×#11758194—21no—
45C22×#92418—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777C2312yes—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59#14519711—24no—
60C4⋊#92418—21no—
61C4⋊#92418—21no—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519700—24no—
65#14519700—22no—
66#14519699—24no—
67#14519697—22no—
68C22×D4×C61D411yescharacteristic
69C23×C4×C61—21no—
70C22×C8×C61—24no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61D412yes—
74C4×D4×C61C2312yes—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61D412yes—
79C4⋊C8×C61C2312yes—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—24no—
82C22⋊C8×C61D412yes—
83C4×C8×C61—21no—
84C4×C8×C61C2312yes—
85C24×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—22no—
88C2×D4×C61D814yes—
89C2×D4×C61C2×D412yes—
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—82no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C2×C22⋊D61—41no—
98C22×Dic61—41no—
99#11758222—42no—
100#11758222—42no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244—21no—
104C2×C4×D61—41no—
105C2×Dic122—22no—
106C2×Dic122—21no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—41no—
110C2×C8×C61—22no—
111C2×C8×C61—84no—
112C2×C8×C61C2×D414yes—
113C4⋊C4×C61—21no—
114C4⋊C4×C61C2×D412yes—
115C22⋊C4×C61—22no—
116C22⋊C4×C61—22no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61C2411yescharacteristic
121#11758200—48no—
122#11758200—44no—
123#11758199—44no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—22no—
127C4⋊Dic61—21no—
128C4⋊Dic61—22no—
129C4⋊Dic61—21no—
130C4⋊Dic61—41no—
131C4⋊Dic61—41no—
132#11758194—42no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—42no—
136C23×C61—22no—
137C23×C61—24no—
138C23×C61—81no—
139C23×C61—81no—
140C23×C61#12111yescharacteristic
141C22×D61—41no—
142D4×C61—22no—
143D4×C61—24no—
144D4×C61C2×D814yes—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—41no—
148C2×C4×C61—44no—
149C2×C4×C61—82no—
150C2×C4×C61—81no—
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61C2×D812yes—
155C2×C4×C61#13512yes—
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C22⋊D61—82no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—82no—
163D244—41no—
164D244—41no—
165D244—42no—
166D244—81no—
167C4×D61—82no—
168Dic122—42no—
169Dic122—41no—
170Dic122—42no—
171Dic122—41no—
172Dic122—82no—
173C8×C61—24no—
174C8×C61—42no—
175#92418—42no—
176#92418—42no—
177#19392—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—24no—
181C22×C61—42no—
182C22×C61—42no—
183C22×C61—84no—
184C22×C61#46612yes—
185C22×C61C2×#12111yescharacteristic
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61#46612yes—
194C4×C61C22×D811yescharacteristic
195C4×C61C2×#13511yescharacteristic
196C4×C61#57712yes—
197Dic61—82no—
198Dic61—81no—
199#3226—611no—
200#3036—611no—
201C2×#461—611no—
202#2972D6111yescharacteristic, complemented
203#1681—614no—
204#1673—614no—
205#1571—611no—
206#1546—611no—
207#1540—611no—
208C2×C61—22no—
209C2×C61—44no—
210C2×C61#304211yescharacteristic
211C2×C61C2×#46611yescharacteristic
212C2×C61#306911yescharacteristic
213D61—161no—
214C2×AGL(1,ℤ/8)—1221no—
215C22×SD16—1221no—
216#532—611no—
217C2×C4⋊Q8—611no—
218C2×C4×D4C2×D6111yescharacteristic
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—22no—
229C8×D4C2×D6114yes—
230#431C2×D6111yescharacteristic
231#431C2×D6111yescharacteristic
232#423—611no—
233C2×C4⋊C8—611no—
234#417—1221no—
235C2×#103—1221no—
236C2×C22⋊C8—22no—
237#335—614no—
238#332—612no—
239#332—614no—
240#331—612no—
241C61#1939211yescharacteristic, complemented, Sylow
242C2×#41—1221no—
243C22×Q8—1221no—
244C22×D4D24411yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—2444no—
247C2×SD16—1224no—
248C2×SD16—1222no—
249C2×SD16—2444no—
250C2×D8—1222no—
251C2×M16—1221no—
252C22×C8—24no—
253C22×C8—1221no—
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—21no—
264C4×D4—21no—
265C4×D4—1221no—
266C4×D4—1221no—
267C4×D4D24412yes—
268C4×D4C22×D6112yes—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4D24411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D6111yescharacteristic
274#107—1224no—
275C4⋊C8—612no—
276C4⋊C8—612no—
277C4⋊C8—1221no—
278C4⋊C8—1221no—
279C4⋊C8C22⋊D6112yes—
280C4⋊C8C22×D6112yes—
281#103—1224no—
282#103—1222no—
283#102—1224no—
284#102—1222no—
285C22⋊C8—21no—
286C22⋊C8—24no—
287C22⋊C8C22⋊D6112yes—
288C4×C8—21no—
289C4×C8C22×D6112yes—
290C24—21no—
291#41—2442no—
292#41—2442no—
293C2×Q8—1221no—
294C2×Q8—1222no—
295C2×Q8—2442no—
296C2×Q8—2442no—
297C2×Q8—2441no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—2441no—
304C2×D4D48814yes—
305C2×D4C2×D24412yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4C2×D24411yescharacteristic
313C22×C4#1175822211yescharacteristic
314C22×C4C2×C22⋊D6111yescharacteristic
315SD16—2448no—
316SD16—2444no—
317D8—2444no—
318M16—2442no—
319C2×C8—22no—
320C2×C8—84no—
321C2×C8—1222no—
322C2×C8—1222no—
323C2×C8—2442no—
324C2×C8C2×C22⋊D6114yes—
325C4⋊C4—21no—
326C4⋊C4—1222no—
327C4⋊C4—1221no—
328C4⋊C4—1222no—
329C4⋊C4—1221no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4C2×D24412yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C42C2×D24411yescharacteristic
338C42#1175822211yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—22no—
342C23—24no—
343C23—81no—
344C23—81no—
345C23—2441no—
346C23#1451975111yescharacteristic
347Q8—4882no—
348Q8—2442no—
349Q8—2441no—
350Q8—2442no—
351Q8—2441no—
352D4—24no—
353D4—22no—
354D4—4881no—
355D4—4882no—
356D4—2441no—
357D4—2441no—
358D4—2442no—
359D4C2×D48814yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—4882no—
367C2×C4—4882no—
368C2×C4—2441no—
369C2×C4—2442no—
370C2×C4#1451975111yescharacteristic
371C2×C4#1451975111yescharacteristic
372C2×C4#1451975111yescharacteristic
373C2×C4C2×D48812yes—
374C2×C4#1451979912yes—
375C2×C4C22×C22⋊D6111yescharacteristic
376C2×C4C22×D24411yescharacteristic
377C2×C4C2×#1175822211yescharacteristic
378C8—24no—
379C8—42no—
380C8—2442no—
381C8—2442no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—4881no—
389C22—4881no—
390C22—4881no—
391C22#1683999812yes—
392C22C2×#1451975111yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—4881no—
398C4—4882no—
399C4#1683999812yes—
400C4C22×D48811yescharacteristic
401C4C2×#1451979911yescharacteristic
402C4#1684057312yes—
403C2—22no—
404C2—44no—
405C2—9761no—
406C2#2145533111yescharacteristic
407C2C2×#1683999811yescharacteristic
408C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenter#2972D61
hypocenterC61#19392
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21456942C2semidirect
3. C2×#16839967C2semidirect
4. #21445256C2semidirect
5. #21445208C2semidirect
6. #21444594C2semidirect
7. #21444490C2semidirect
8. #21444466C2semidirect
9. C61×#2972C2semidirect
14. #16840167C22semidirect
15. #16840109C22semidirect
27. #16839515C22semidirect
28. #16839507C22semidirect
29. #16839507C22semidirect
30. #16839505C22semidirect
202. #2972D61semidirect
241. C61#19392semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214552832
C2×#168399672
#214452562
#214452082
#214445942
#214444902
#214444662
C61×#29722
#1939261