Finite Group Databasebeta

← #29837653

Subgroups — #29837653

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458792C211yesmaximal, characteristic, complemented
2D8×Dic122C211yesmaximal, characteristic, complemented
3#21448042C211yesmaximal, characteristic, complemented
4#21448041C212yesmaximal, complemented
5#21447957C211yesmaximal, characteristic, complemented
6#21447954C212yesmaximal, complemented
7#21447852C212yesmaximal, complemented
8#21447721C211yesmaximal, characteristic, complemented
9D16×Dic61C212yesmaximal, complemented
10C4×#14519718C211yesmaximal, characteristic, complemented
11C61×#2206C211yesmaximal, characteristic, complemented
12#16840432—21no—
13D4×Dic122—21no—
14C2×#14519718C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840166C2212yes—
17#16840137C2212yes—
18#16840042—21no—
19#16840040—22no—
20#16839955—22no—
21#16839954—21no—
22C8×Dic122C2211yescharacteristic
23C4×D488C2211yescharacteristic
24D8×Dic61C2212yes—
25D8×Dic61C2212yes—
26#16839588C2212yes—
27#16839587C2211yescharacteristic
28#16839580C2212yes—
29#16839564C2212yes—
30#16839563C2212yes—
31#16839563C2212yes—
32#16839563C2211yescharacteristic
33#16839562C2211yescharacteristic
34C4×#11758185C2211yescharacteristic
35C16×Dic61C2212yes—
36C2×D16×C61C2212yes—
37C61×#493C2211yescharacteristic
38C4×D8×C61C2211yescharacteristic
39C61×C4⋊C16C2211yescharacteristic
40C61×#357C2212yes—
41C2×#11758224—41no—
42C22×Dic122—41no—
43C2×D488C2311yescharacteristic
44C2×#11758199—22no—
45#14519777—21no—
46#14519776—22no—
47#14519773—22no—
48#14519756—41no—
49#14519754—42no—
50#14519749—42no—
51#14519748—41no—
52C4×Dic122D411yescharacteristic
53C4×D244—21no—
54D4×Dic61—22no—
55D4×Dic61—22no—
56#14519718—22no—
57C2×#11758185C2311yescharacteristic
58C2×#11758185C2311yescharacteristic
59#14519711—21no—
60#14519711C2312yes—
61#14519711C2311yescharacteristic
62#14519710—22no—
63C4⋊#92418D411yescharacteristic
64#14519706D412yes—
65#14519699—21no—
66#14519698—22no—
67#14519698—22no—
68C8×Dic61C2312yes—
69C2×D8×C61—21no—
70C2×D8×C61C2312yes—
71C2×D8×C61C2311yescharacteristic
72#127×C61—21no—
73C4×D4×C61—21no—
74D16×C61—24no—
75C2×C16×C61C2312yes—
76#107×C61C2311yescharacteristic
77#102×C61—21no—
78C4×C8×C61C2311yescharacteristic
79C2×D4×C61—21no—
80C2×D4×C61—22no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C22×C4×C61—41no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88#11758224—84no—
89C2×D244—21no—
90C2×C4×D61—41no—
91C2×Dic122—41no—
92C2×Dic122—42no—
93C2×Dic122—84no—
94C2×Dic122D811yescharacteristic
95D8×C61—22no—
96D8×C61—21no—
97D8×C61—22no—
98D8×C61—81no—
99D8×C61#4112yescomplemented
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61C2411yescharacteristic
102C4⋊C4×C61—21no—
103C22⋊C4×C61—41no—
104C42×C61C2×D411yescharacteristic
105C22⋊Dic61—42no—
106C22⋊Dic61—42no—
107#11758199—82no—
108#11758198—41no—
109C4⋊Dic61—21no—
110C4⋊Dic61—22no—
111C4⋊Dic61—41no—
112C4⋊Dic61D811yescharacteristic
113#11758196—42no—
114#11758196D812yes—
115C4×Dic61C2×D412yes—
116C2×#92418C2×D412yes—
117D488—21no—
118D488—21no—
119C16×C61—22no—
120#11758185—21no—
121#11758185—21no—
122C23×C61—41no—
123C23×C61—42no—
124C22×D61—41no—
125D4×C61—22no—
126D4×C61—42no—
127D4×C61—41no—
128D4×C61—42no—
129D4×C61—82no—
130D4×C61—161no—
131C2×C4×C61—41no—
132C2×C4×C61—82no—
133C2×C4×C61C2×D811yescharacteristic
134C2×C4×C61C2×D811yescharacteristic
135C2×C4×C61C22×D411yescharacteristic
136C22⋊D61—84no—
137C22⋊D61—162no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—84no—
141C2×Dic61—84no—
142C2×Dic61C2×D812yes—
143C2×Dic61C2×D812yes—
144D244—41no—
145D244—41no—
146C4×D61—82no—
147Dic122—82no—
148Dic122—84no—
149Dic122D1614yescomplemented
150C8×C61—21no—
151C8×C61C2×#4111yescharacteristic
152C8×C612−1+411yescharacteristic
153#92418—22no—
154#33925—611nomaximal, Sylow
155C22×C61—42no—
156C22×C61—81no—
157C22×C61—81no—
158C22×C61—82no—
159C22×C61—82no—
160C22×C61—82no—
161C22×C61C22×D811yescharacteristic
162C2×D61—81no—
163C2×D61—81no—
164C2×D61—81no—
165C4×C61—21no—
166C4×C61—81no—
167C4×C61C2×D1612yes—
168C4×C61#54811yescharacteristic
169C4×C61#57711yescharacteristic
170Dic61—22no—
171Dic61—82no—
172Dic61—81no—
173Dic61C2×D1614yes—
174#3391—611no—
175Q8×D8—611no—
176#2240—611no—
177#2240—612no—
178#2224—611no—
179#2224—612no—
180#2206—612no—
181#2206D6111yescharacteristic, complemented
182Q8×C16—611no—
183C4×D16—612no—
184C4×D16—611no—
185C2×C61—82no—
186C2×C61—162no—
187C2×C61#335811yescharacteristic
188C2×C61C22×D1611yescharacteristic
189C2×C61#341611yescharacteristic
190D61—161no—
191#550—1221no—
192D4×Q8—1221no—
193C2×D16—611no—
194C2×D16C2×D6112yes—
195#500—612no—
196#500—611no—
197#493—612no—
198#493C2×D6111yescharacteristic
199#474—1222no—
200#474—1221no—
201#459—1221no—
202#459—1222no—
203C8×Q8—611no—
204C4×D8—612no—
205C4×D8—612no—
206C4×D8—611no—
207C4×D8C2×D6111yescharacteristic
208#366—612no—
209#366—611no—
210C4⋊C16—612no—
211C4⋊C16C2×D6111yescharacteristic
212#357—611no—
213#357—612no—
214#357—612no—
215#357—612no—
216#357—611no—
217#357C2×D6112yes—
218C4×C16—612no—
219C4×C16—611no—
220C61#3392511yescharacteristic, complemented, Sylow
221C2×#41—2441no—
222C22×Q8—2441no—
223C2×D8—21no—
224C2×D8—611no—
225C2×D8—1222no—
226C2×D8C22×D6112yes—
227C2×D8C22×D6111yescharacteristic
228C4⋊Q8—1222no—
229C4⋊Q8—1221no—
230#127—21no—
231#127—1222no—
232C22⋊Q8—2442no—
233C22⋊Q8—2441no—
234#121—2441no—
235#121—2442no—
236C4×Q8—611no—
237C4×D4—21no—
238C4×D4—1222no—
239C4×D4—1222no—
240C4×D4—1221no—
241D16—24no—
242D16—1222no—
243C2×C16—611no—
244C2×C16—611no—
245C2×C16C22×D6112yes—
246#107—612no—
247#107—611no—
248#107—1222no—
249#107—1221no—
250#107C22×D6111yescharacteristic
251C4⋊C8—612no—
252C4⋊C8—611no—
253#102—21no—
254#102—1222no—
255#102—1222no—
256#102—1221no—
257C4×C8—612no—
258C4×C8C22×D6111yescharacteristic
259#41—4884no—
260C2×Q8—611no—
261C2×Q8—4884no—
262C2×Q8—2442no—
263C2×Q8—2441no—
264C2×D4—21no—
265C2×D4—22no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—1221no—
269C2×D4—2442no—
270C2×D4—2442no—
271C22×C4—41no—
272C22×C4—2442no—
273C22×C4—2442no—
274C22×C4—2441no—
275D8—22no—
276D8—21no—
277D8—22no—
278D8—81no—
279D8—1221no—
280D8—1221no—
281D8—4882no—
282D8#1175822412yescomplemented
283C2×C8—612no—
284C2×C8D4×D6111yescharacteristic
285C2×C8C23×D6111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—612no—
288C4⋊C4—611no—
289C4⋊C4—1221no—
290C4⋊C4—1222no—
291C4⋊C4—2442no—
292C4⋊C4—2441no—
293C22⋊C4—41no—
294C22⋊C4—2442no—
295C22⋊C4—2442no—
296C22⋊C4—2441no—
297C42—612no—
298C42D4×D6111yescharacteristic
299C16—22no—
300C16—1221no—
301C16—1221no—
302C23—41no—
303C23—42no—
304C23—2441no—
305Q8—614no—
306Q8—4884no—
307Q8—4882no—
308D4—22no—
309D4—41no—
310D4—42no—
311D4—42no—
312D4—82no—
313D4—161no—
314D4—9762no—
315D4—4884no—
316D4—2441no—
317D4—2441no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—612no—
321C2×C4—612no—
322C2×C4—4884no—
323C2×C4—4884no—
324C2×C4—4882no—
325C2×C4—2441no—
326C2×C4—2442no—
327C2×C4D8×D6111yescharacteristic
328C2×C4D8×D6111yescharacteristic
329C2×C4C2×D4×D6111yescharacteristic
330C8—21no—
331C8—1222no—
332C8C2×#1175822411yescharacteristic
333C8#1451982411yescharacteristic
334C22—42no—
335C22—81no—
336C22—81no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—4881no—
341C22—4881no—
342C22—4881no—
343C22C2×D8×D6111yescharacteristic
344C4—21no—
345C4—81no—
346C4—614no—
347C4—1222no—
348C4—4881no—
349C4—4882no—
350C4D16×D6112yes—
351C4#1684041811yescharacteristic
352C4#1684057211yescharacteristic
353C2—82no—
354C2—162no—
355C2—9761no—
356C2#2145844911yescharacteristic
357C2C2×D16×D6111yescharacteristic
358C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2206C2
socleC22×C61C22×D8
hypercenter#2206D61
hypocenterC61#33925
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2206

Complements

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

NH ≅ G/Ntype
1. #21458792C2semidirect
2. D8×Dic122C2semidirect
3. #21448042C2semidirect
4. #21448041C2semidirect
5. #21447957C2semidirect
6. #21447954C2semidirect
7. #21447852C2semidirect
8. #21447721C2semidirect
9. D16×Dic61C2semidirect
10. C4×#14519718C2semidirect
11. C61×#2206C2semidirect
99. D8×C61#41semidirect
149. Dic122D16semidirect
181. #2206D61semidirect
220. C61#33925semidirect
282. D8#11758224semidirect

Maximal subgroups

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

subgroupindex
#214587922
D8×Dic1222
#214480422
#214480412
#214479572
#214479542
#214478522
#214477212
D16×Dic612
C4×#145197182
C61×#22062
#3392561