Finite Group Databasebeta

← #29837610

Subgroups — #29837610

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458251C211yesmaximal, characteristic, complemented
2#21458166C211yesmaximal, characteristic
3#21448092C211yesmaximal, characteristic, complemented
4#21448020C212yesmaximal, complemented
5#21448012C212yesmaximal, complemented
6#21447957C211yesmaximal, characteristic, complemented
7#21447935C212yesmaximal, complemented
8C4×#14519720C211yesmaximal, characteristic, complemented
9#21447650C211yesmaximal, characteristic, complemented
10#21447606C212yesmaximal, complemented
11C61×#2225C211yesmaximal, characteristic, complemented
12#16840485—21no—
13#16840415—21no—
14C2×#14519720C2211yescharacteristic
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840102—22no—
18#16840077—22no—
19#16840047—21no—
20#16840039—21no—
21#16840000C2212yes—
22#16839999C2212yes—
23C4×D488C2211yescharacteristic
24C8×D244C2211yescharacteristic
25#16839799C2212yes—
26#107×D61C2212yes—
27#16839589C2211yescharacteristic
28#16839588C2212yes—
29#16839587C2212yes—
30#16839587C2212yes—
31#16839566C2211yescharacteristic
32#16839564C2212yes—
33#16839562C2211yescharacteristic
34#16839541C2212yes—
35C4×#11758185C2211yescharacteristic
36C61×#500C2211yescharacteristic
37C4×Q16×C61C2211yescharacteristic
38C61×#367C2212yes—
39C61×C4⋊C16C2211yescharacteristic
40C61×#358C2212yes—
41C2×#11758226—41no—
42C2×D488C2311yescharacteristic
43C2×C8×D61C2312yes—
44#14519777—21no—
45#14519772—42no—
46#14519765—41no—
47#14519762—42no—
48#14519753—22no—
49#14519753—41no—
50#14519752—22no—
51C4×Dic122—41no—
52C4×D244—21no—
53C4×D244D411yescharacteristic
54#14519735—22no—
55C4⋊C4×D61—22no—
56#14519720—22no—
57#14519716—24no—
58C2×#11758185C2311yescharacteristic
59C2×#11758185C2311yescharacteristic
60#14519711—21no—
61#14519711C2312yes—
62#14519711C2311yescharacteristic
63#14519710—22no—
64#14519708—22no—
65C4⋊#92418D411yescharacteristic
66#14519701—22no—
67#14519699—21no—
68#14519697—22no—
69#14519689D412yes—
70C2×Q16×C61C2311yescharacteristic
71C4⋊Q8×C61—21no—
72C4×Q8×C61—21no—
73C2×C16×C61C2312yes—
74#107×C61—21no—
75#107×C61C2312yes—
76#107×C61C2311yescharacteristic
77#103×C61—21no—
78C4×C8×C61C2311yescharacteristic
79C2×Q8×C61—21no—
80C2×Q8×C61—41no—
81#11758226—84no—
82C2×D244—21no—
83C2×D244—42no—
84C2×D244D811yescharacteristic
85C2×C4×D61—42no—
86C2×C4×D61—41no—
87C2×C4×D61C2×D412yes—
88C2×Dic122—41no—
89Q16×C61—21no—
90Q16×C61—21no—
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2411yescharacteristic
93C4⋊C4×C61—21no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C42×C61—41no—
98C42×C61C2×D411yescharacteristic
99#11758198—42no—
100#11758198—42no—
101#11758198—41no—
102#11758198—42no—
103#11758198D812yes—
104C4⋊Dic61—21no—
105C4⋊Dic61—22no—
106C4⋊Dic61—41no—
107C4⋊Dic61D811yescharacteristic
108#11758196—42no—
109#11758196—42no—
110C4×Dic61—42no—
111C2×#92418C2×D412yes—
112D488—21no—
113D488—21no—
114C8×D61—22no—
115C8×D61—22no—
116C16×C61—22no—
117#11758185—21no—
118#11758185#4112yescomplemented
119C22×D61—41no—
120C22×D61C2×D812yes—
121Q8×C61—41no—
122Q8×C61—41no—
123Q8×C61—82no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61C2×D811yescharacteristic
128C2×C4×C61C2×D811yescharacteristic
129C2×C4×C61C22×D411yescharacteristic
130C2×Dic61—42no—
131C2×Dic61—41no—
132C2×Dic61C2×D812yes—
133D244—22no—
134D244—41no—
135D244—41no—
136D244—84no—
137C4×D61—22no—
138C4×D61—22no—
139C4×D61—84no—
140C4×D61—82no—
141Dic122—82no—
142C8×C61—21no—
143C8×C61C2×#4111yescharacteristic
144C8×C612−1+411yescharacteristic
145#92418—22no—
146#33920—611nomaximal, Sylow
147C22×C61C22×D811yescharacteristic
148C2×D61—22no—
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—21no—
155C4×C61—81no—
156C4×C61—82no—
157C4×C61—81no—
158C4×C61#50912yes—
159C4×C61#54711yescharacteristic
160C4×C61#57811yescharacteristic
161Dic61—22no—
162Dic61—82no—
163Dic61—81no—
164#3342—611no—
165#3334—611no—
166#2252—611no—
167#2236—612no—
168#2234—612no—
169#2225D6111yescharacteristic, complemented
170#2224—611no—
171#2220—612no—
172C4×SD32—611no—
173#2169—611no—
174#2159—612no—
175C2×C61#333411yescharacteristic
176C2×C61C2×#50911yescharacteristic
177C2×C61#341711yescharacteristic
178D61—42no—
179D61—161no—
180#556—1221no—
181#547—1221no—
182C2×SD32—611no—
183C2×M32—612no—
184#500—611no—
185#500C2×D6111yescharacteristic
186#485—1222no—
187#480—1222no—
188#475—1221no—
189#474—1221no—
190#467—612no—
191#466—612no—
192C4×Q16C2×D6111yescharacteristic
193C4×D8—611no—
194C8×D4—611no—
195#427—612no—
196C2×#107—612no—
197#367—611no—
198#367C2×D6112yes—
199#366—612no—
200#366—612no—
201#366—612no—
202C4⋊C16C2×D6111yescharacteristic
203#358—611no—
204#358C2×D6112yes—
205#357—612no—
206#357—611no—
207C22⋊C16—612no—
208C4×C16—611no—
209C61#3392011yescharacteristic, complemented, Sylow
210C2×#41—2441no—
211C2×Q16C22×D6111yescharacteristic
212C2×D8—611no—
213C22×C8—612no—
214C4⋊Q8—21no—
215C4⋊Q8—1221no—
216#126—2442no—
217#124—2441no—
218#123—2442no—
219C22⋊Q8—1222no—
220C22⋊Q8—2441no—
221#121—1222no—
222C4×Q8—21no—
223C4×Q8—2441no—
224C4×D4—611no—
225C4×D4—1221no—
226#117—1222no—
227C2×C4⋊C4—1222no—
228SD32—1222no—
229M32—1224no—
230C2×C16—611no—
231C2×C16—611no—
232C2×C16C22×D6112yes—
233#107—21no—
234#107—612no—
235#107—611no—
236#107—1222no—
237#107—1221no—
238#107C22×D6111yescharacteristic
239#107C22×D6112yes—
240#106—1222no—
241C4⋊C8—611no—
242#103—21no—
243#103—1222no—
244#102—1222no—
245#102—1221no—
246C22⋊C8—612no—
247C4×C8C22×D6111yescharacteristic
248#41—4884no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—2441no—
252C2×D4—611no—
253C2×D4—1221no—
254C2×D4—2442no—
255C22×C4—612no—
256C22×C4—2442no—
257C22×C4—2441no—
258Q16—21no—
259Q16—21no—
260D8—1221no—
261D8—1221no—
262C2×C8—612no—
263C2×C8—1222no—
264C2×C8—1222no—
265C2×C8D4×D6111yescharacteristic
266C2×C8C23×D6111yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—611no—
272C4⋊C4—1221no—
273C4⋊C4—1222no—
274C4⋊C4—2442no—
275C4⋊C4—2442no—
276C4⋊C4—2441no—
277C22⋊C4—612no—
278C22⋊C4—2442no—
279C22⋊C4—2442no—
280C22⋊C4—2442no—
281C22⋊C4—2441no—
282C42—41no—
283C42—2442no—
284C42D4×D6111yescharacteristic
285C16—22no—
286C16—612no—
287C16—1221no—
288C23—612no—
289C23—2441no—
290Q8—41no—
291Q8—41no—
292Q8—82no—
293Q8—4882no—
294D4—1222no—
295D4—4884no—
296D4—2441no—
297D4—2441no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—612no—
302C2×C4—1222no—
303C2×C4—1222no—
304C2×C4—4884no—
305C2×C4—4882no—
306C2×C4—2442no—
307C2×C4—2441no—
308C2×C4D8×D6111yescharacteristic
309C2×C4D8×D6111yescharacteristic
310C2×C4C2×D4×D6111yescharacteristic
311C8—21no—
312C8—1222no—
313C8#41×D6111yescharacteristic
314C8#1451982511yescharacteristic
315C22—1222no—
316C22—1222no—
317C22—1222no—
318C22—4881no—
319C22—4881no—
320C22—4881no—
321C22C2×D8×D6111yescharacteristic
322C4—21no—
323C4—81no—
324C4—81no—
325C4—82no—
326C4—1222no—
327C4—4881no—
328C4—4882no—
329C4#1684019912yes—
330C4#1684041411yescharacteristic
331C4#1684057811yescharacteristic
332C2—9761no—
333C2—2442no—
334C2#2145816511yescharacteristic
335C2C2×#1684019911yescharacteristic
336C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2225C2
socleC22×C61C22×D8
prime radical#21458166C2
hypercenter#2225D61
hypocenterC61#33920
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
1. #21458251C2semidirect
3. #21448092C2semidirect
4. #21448020C2semidirect
5. #21448012C2semidirect
6. #21447957C2semidirect
7. #21447935C2semidirect
8. C4×#14519720C2semidirect
9. #21447650C2semidirect
10. #21447606C2semidirect
11. C61×#2225C2semidirect
118. #11758185#41semidirect
169. #2225D61semidirect
209. C61#33920semidirect

Maximal subgroups

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

subgroupindex
#214582512
#214581662
#214480922
#214480202
#214480122
#214479572
#214479352
C4×#145197202
#214476502
#214476062
C61×#22252
#3392061