Finite Group Databasebeta

← #29669345

Subgroups — #29669345

Order 15616.

728 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456557C211yesmaximal, characteristic
2#21456538C212yesmaximal, complemented
3#21456215C212yesmaximal, complemented
4#21445473C212yesmaximal, complemented
5#21445467C212yesmaximal, complemented
6#21445461C212yesmaximal, complemented
7#21445432C212yesmaximal, complemented
8#21444122C211yesmaximal, characteristic, complemented
9C61×#3139C211yesmaximal, characteristic, complemented
10#16840312C2212yes—
11#16840165C2212yes—
12#16840160C2211yescharacteristic
13#16840139C2211yescharacteristic
14#16840136C2212yes—
15#16840124—24no—
16#16840114C2212yes—
17#16840114C2212yes—
18#16840112C2212yes—
19#16840110C2212yes—
20#16840110C2212yes—
21#16840110C2212yes—
22#16840109C2212yes—
23#16840015—22no—
24#16840010—24no—
25#16839875—22no—
26#16839860—24no—
27#16839761—24no—
28C4×#11758194C2212yes—
29#16839484C2212yes—
30#16839483C2211yescharacteristic
31C8×#92418C2211yescharacteristic
32C61×#533C2211yescharacteristic
33C2×C4⋊Q8×C61C2211yescharacteristic
34C61×#491—22no—
35C61×#487C2212yes—
36C61×#487C2212yes—
37C61×#419—22no—
38C4×M16×C61C2211yescharacteristic
39C2×#11758222—22no—
40C2×#11758201—24no—
41C2×D488—22no—
42C2×#11758194D414yes—
43#14519777C2312yes—
44#14519774C2312yes—
45#14519765—22no—
46#14519751—22no—
47#14519751—22no—
48C4×D244—22no—
49C4×D244—22no—
50#14519710—24no—
51#14519709—22no—
52#14519701—24no—
53#14519700—24no—
54#14519700—24no—
55#14519699—24no—
56#14519697—24no—
57#14519697—24no—
58#14519697—24no—
59#14519688D414yes—
60C4×#92418C2312yes—
61C4×#92418C2312yes—
62C22×Q8×C61—21no—
63C2×M16×C61D412yes—
64C4⋊Q8×C61—22no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61C2312yes—
67C4⋊Q8×C61C2312yes—
68#125×C61—21no—
69C22⋊Q8×C61—22no—
70C4×Q8×C61—22no—
71#117×C61—21no—
72C2×C4⋊C4×C61—22no—
73C2×C42×C61C2311yescharacteristic
74#103×C61—24no—
75#103×C61—24no—
76#97×C61D412yes—
77C4×C8×C61C2311yescharacteristic
78C4×C8×C61C2311yescharacteristic
79C2×Q8×C61—22no—
80C2×Q8×C61—22no—
81C2×Q8×C61—44no—
82C2×Q8×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—42no—
87#11758222—48no—
88C2×D244—22no—
89C2×D244—22no—
90C2×D244—22no—
91C2×C4×D61—42no—
92C2×Dic122—22no—
93M16×C61—42no—
94C2×C8×C61C2×D412yes—
95C2×C8×C61C2×D412yes—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—42no—
101C4⋊C4×C61—42no—
102C22⋊C4×C61—41no—
103C42×C61—41no—
104C42×C61C2×D412yes—
105C42×C61C2×D411yescharacteristic
106C42×C61C2411yescharacteristic
107#11758201—48no—
108#11758198—42no—
109#11758198—42no—
110C4⋊Dic61—22no—
111C4⋊Dic61—22no—
112#11758194—44no—
113C2×#92418C2×D414yes—
114C2×#92418C2×D414yes—
115D488—44no—
116C23×C61#12711yescharacteristic
117C22×D61—42no—
118Q8×C61—44no—
119Q8×C61—44no—
120Q8×C61—82no—
121C2×C4×C61—24no—
122C2×C4×C61—22no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—82no—
126C2×C4×C61#12711yescharacteristic
127C2×C4×C61#12712yes—
128C2×C4×C61C22×D412yes—
129C2×C4×C61C22×D411yescharacteristic
130C22⋊D61—84no—
131C2×Dic61—42no—
132D244—44no—
133D244—44no—
134D244—44no—
135C4×D61—84no—
136Dic122—44no—
137C8×C61—22no—
138C8×C61—22no—
139#92418—24no—
140#92418—24no—
141#19754—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61C2×#12711yescharacteristic
145C2×D61—82no—
146C2×D61—82no—
147C2×D61—82no—
148C4×C61—22no—
149C4×C61—21no—
150C4×C61—82no—
151C4×C61—82no—
152C4×C61#57512yes—
153C4×C61#57514yes—
154Dic61—82no—
155#3184—611no—
156#3182—612no—
157#3141—612no—
158#3139D6111yescharacteristic, complemented
159#1719—612no—
160#1718—612no—
161#1717—612no—
162#1711—612no—
163#1455—611no—
164C2×C61—41no—
165C2×C61#315311yescharacteristic
166C2×C61#315312yes—
167D61—162no—
168#533C2×D6111yescharacteristic
169#532—612no—
170C2×C4⋊Q8C2×D6111yescharacteristic
171#500—612no—
172#498—611no—
173#493—611no—
174#492—612no—
175#491—22no—
176#489—1224no—
177#487—612no—
178#487—612no—
179#487—612no—
180#487C2×D6112yes—
181#487C2×D6112yes—
182#486—612no—
183#486—612no—
184#486—612no—
185#486—612no—
186#469—1222no—
187#468—1224no—
188#442—1222no—
189#440—1224no—
190#419—22no—
191#417—1224no—
192C4×M16—612no—
193C4×M16C2×D6111yescharacteristic
194#322—611no—
195#322—612no—
196C82—611no—
197C61#1975411yescharacteristic, complemented, Sylow
198C2×#41—1222no—
199C22×Q8—21no—
200C2×SD16—1224no—
201C2×D8—1222no—
202C2×M16—614no—
203C2×M16C22⋊D6112yes—
204C4⋊Q8—22no—
205C4⋊Q8—21no—
206C4⋊Q8—612no—
207C4⋊Q8C22×D6112yes—
208C4⋊Q8C22×D6112yes—
209#127—612no—
210#125—21no—
211#124—1222no—
212C22⋊Q8—22no—
213#121—1222no—
214#121—1222no—
215C4×Q8—22no—
216C4×D4—1222no—
217C4×D4—1222no—
218#117—21no—
219C2×C4⋊C4—22no—
220C2×C42C22×D6111yescharacteristic
221#107—1224no—
222#106—1222no—
223#103—24no—
224#103—24no—
225#103—1224no—
226#103—1224no—
227#103—1224no—
228#102—1224no—
229#102—1224no—
230#102—1224no—
231#102—1224no—
232#97—614no—
233#97C22⋊D6112yes—
234C4×C8—612no—
235C4×C8—612no—
236C4×C8C22×D6111yescharacteristic
237C4×C8C22×D6111yescharacteristic
238#41—2448no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—41no—
243C2×Q8—1222no—
244C2×D4—1222no—
245C2×D4—1222no—
246C2×D4—1222no—
247C2×D4—2442no—
248C22×C4—41no—
249C22×C4—2442no—
250C22×C4D4×D6112yes—
251C22×C4C2×C22⋊D6111yescharacteristic
252SD16—2448no—
253D8—2444no—
254M16—42no—
255M16—2444no—
256C2×C8—614no—
257C2×C8—614no—
258C2×C8C2×C22⋊D6112yes—
259C2×C8C2×C22⋊D6112yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—1222no—
267C4⋊C4—1222no—
268C22⋊C4—41no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C42—41no—
272C42D4×D6112yes—
273C42C2×C22⋊D6111yescharacteristic
274C42C23×D6111yescharacteristic
275C23—2442no—
276C23#1451977311yescharacteristic
277Q8—44no—
278Q8—44no—
279Q8—82no—
280Q8—2444no—
281D4—4884no—
282D4—2444no—
283D4—2444no—
284D4—2444no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—82no—
290C2×C4—4884no—
291C2×C4—2442no—
292C2×C4#1451977311yescharacteristic
293C2×C4#1451977312yes—
294C2×C4C2×D4×D6112yes—
295C2×C4C22×C22⋊D6111yescharacteristic
296C8—22no—
297C8—22no—
298C8—1224no—
299C8—1224no—
300C22—21no—
301C22—22no—
302C22—4882no—
303C22—4882no—
304C22—4882no—
305C22C2×#1451977311yescharacteristic
306C4—21no—
307C4—22no—
308C4—82no—
309C4—82no—
310C4—4882no—
311C4#1684056012yes—
312C4#1684056114yes—
313C2—41no—
314C2—9762no—
315C2#2145628311yescharacteristic
316C2#2145628612yes—
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#127
prime radical#21456557C2
hypercenter#3139D61
hypocenterC61#19754
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #21456538C2semidirect
3. #21456215C2semidirect
4. #21445473C2semidirect
5. #21445467C2semidirect
6. #21445461C2semidirect
7. #21445432C2semidirect
8. #21444122C2semidirect
9. C61×#3139C2semidirect
158. #3139D61semidirect
197. C61#19754semidirect

Maximal subgroups

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

subgroupindex
#214565572
#214565382
#214562152
#214454732
#214454672
#214454612
#214454322
#214441222
C61×#31392
#1975461