Finite Group Databasebeta

← #29571777

Subgroups — #29571777

Order 15616.

782 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#16840469C211yesmaximal, characteristic
2#21444505C212yesmaximal, complemented
3#21444247C212yesmaximal, complemented
4#21444140C211yesmaximal, characteristic
5C61×#1800C211yesmaximal, characteristic
6#16840469—24no—
7#16840469—24no—
8C2×#14519777C2211yescharacteristic
9C2×#14519768—21no—
10C2×#14519756C412yes—
11C2×#14519754—22no—
12C2×C4×Dic122C2211yescharacteristic
13C2×D4×Dic61—21no—
14C22×C4⋊Dic61C412yes—
15#16839815C2212yes—
16#16839507—24no—
17#16839492—24no—
18#16839486—22no—
19C2×C4×D4×C61C2211yescharacteristic
20C61×#423C2212yes—
21C2×#100×C61—21no—
22C61×#343—21no—
23C22×Dic122—21no—
24C22×Dic122C2×C411yescharacteristic
25C23×Dic61—22no—
26#14519777—22no—
27#14519777D414yes—
28#14519768—28no—
29#14519756—24no—
30#14519756—24no—
31#14519754—216no—
32C4×Dic122—22no—
33C4×Dic122D414yes—
34D4×Dic61—28no—
35C2×C4⋊Dic61—22no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—24no—
38C2×C4⋊Dic61—24no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61C2×C412yes—
41C2×#11758196—22no—
42C2×#11758196—21no—
43C2×C22⋊Dic61—22no—
44C2×C4×Dic61—21no—
45C4⋊#92418—22no—
46C22⋊#92418—22no—
47C4×#92418—22no—
48C22×D4×C61D411yescharacteristic
49C23×C4×C61C2×C412yes—
50C2×M16×C61—22no—
51C4×D4×C61—22no—
52C4×D4×C61—22no—
53C2×C4⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61D412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C4⋊C8×C61—22no—
58#100×C61—42no—
59C24×C61C22⋊C412yes—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61SD1612yes—
63C2×D4×C61Q1612yes—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—44no—
67C22×C4×C61C22⋊C412yes—
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—22no—
73C22×Dic61—21no—
74C22×Dic61—28no—
75C22×Dic61—84no—
76C2×Dic122—21no—
77C2×Dic122—24no—
78C2×Dic122—42no—
79C2×Dic122—82no—
80C2×Dic122C22⋊C412yes—
81M16×C61—42no—
82M16×C61—42no—
83C2×C8×C61—42no—
84C4⋊C4×C61—22no—
85C22⋊C4×C61—24no—
86C42×C61C2×D412yes—
87C42×C61C2×D412yes—
88C22⋊Dic61—28no—
89C4⋊Dic61—28no—
90C4⋊Dic61—21no—
91C4⋊Dic61—22no—
92C4⋊Dic61—24no—
93C4⋊Dic61C22⋊C412yes—
94C4⋊Dic61C22⋊C414yes—
95#11758196—28no—
96#11758196—24no—
97C4×Dic61—24no—
98C2×#92418—42no—
99C23×C61—22no—
100C23×C61—22no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61#10314yes—
104C23×C61C2×C22⋊C411yescharacteristic
105D4×C61—24no—
106D4×C61—24no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—44no—
111C2×C4×C61—44no—
112C2×C4×C61—42no—
113C2×C4×C61C2×C22⋊C411yescharacteristic
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C2×C4×C61#12012yes—
116C2×C4×C61#12012yes—
117C2×C4×C61C2×SD1611yescharacteristic
118C2×C4×C61C2×Q1611yescharacteristic
119C2×C4×C61#13712yes—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—24no—
123C2×Dic61—42no—
124C2×Dic61—44no—
125C2×Dic61—42no—
126C2×Dic61—88no—
127Dic122—24no—
128Dic122—42no—
129Dic122—84no—
130C8×C61—82no—
131#92418—82no—
132#8395—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—24no—
136C22×C61—44no—
137C22×C61—44no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61#37911yescharacteristic
141C22×C61C2×#10311yescharacteristic
142C22×C61#41911yescharacteristic
143C4×C61—22no—
144C4×C61—21no—
145C4×C61—42no—
146C4×C61#41312yes—
147C4×C61#44812yes—
148C4×C61#45112yes—
149Dic61—44no—
150Dic61—42no—
151Dic61—42no—
152C2×#554—611no—
153#1800D6111yescharacteristic
154#1549—612no—
155#1487—612no—
156#1460—611no—
157C2×C61—21no—
158C2×C61—21no—
159C2×C61—44no—
160C2×C61#178511yescharacteristic
161C2×C61#178711yescharacteristic
162C2×C61#179511yescharacteristic
163#554—1224no—
164#554—1224no—
165C2×C4⋊Q8—611no—
166C2×#125—1221no—
167C2×C22⋊Q8—612no—
168C2×C22⋊Q8—1222no—
169C2×C4×Q8—611no—
170C2×C4×D4—1221no—
171C2×C4×D4C2×D6111yescharacteristic
172C22×C4⋊C4—612no—
173#431—612no—
174#423C2×D6112yes—
175C2×#100—21no—
176#343—21no—
177#332—1224no—
178Q8⋊C8—1224no—
179#323—1222no—
180C61#839511yescharacteristic, complemented, Sylow
181C22×Q8—611no—
182C22×Q8—1221no—
183C22×D4C22⋊D6111yescharacteristic
184C23×C4—1222no—
185C23×C4C4×D6112yes—
186C2×M16—22no—
187C4⋊Q8—614no—
188C4⋊Q8—1222no—
189#125—1228no—
190C22⋊Q8—12216no—
191C22⋊Q8—1224no—
192C22⋊Q8—1224no—
193C4×Q8—614no—
194C4×Q8—1222no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—1228no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—612no—
200C2×C4⋊C4—1224no—
201C2×C4⋊C4—1224no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4C22⋊D6111yescharacteristic
207C2×C22⋊C4—1222no—
208C2×C22⋊C4D24412yes—
209C2×C42—1221no—
210C2×C42C22×D6111yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—1222no—
214#100—42no—
215C22⋊C8—1222no—
216C4×C8—1222no—
217C24#1175819812yes—
218C2×Q8—612no—
219C2×Q8—1221no—
220C2×Q8—1224no—
221C2×Q8—4882no—
222C2×Q8—2442no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4#1175820012yes—
226C2×D4#1175820212yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—1228no—
231C22×C4—1221no—
232C22×C4—1221no—
233C22×C4—1222no—
234C22×C4—4884no—
235C22×C4#1175819812yes—
236C22×C4C2×C4×D6111yescharacteristic
237C22×C4C2×D24411yescharacteristic
238C22×C4C2×C22⋊D6111yescharacteristic
239M16—42no—
240M16—42no—
241C2×C8—42no—
242C2×C8—2442no—
243C4⋊C4—22no—
244C4⋊C4—612no—
245C4⋊C4—614no—
246C4⋊C4—1228no—
247C4⋊C4—1228no—
248C4⋊C4—1221no—
249C4⋊C4—1222no—
250C4⋊C4—1224no—
251C4⋊C4—1224no—
252C22⋊C4—24no—
253C22⋊C4—1228no—
254C42—1224no—
255C42D4×D6112yes—
256C42D4×D6112yes—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23#1451970014yes—
262C23C2×#1175819811yescharacteristic
263Q8—1224no—
264Q8—4884no—
265Q8—2442no—
266D4—24no—
267D4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1224no—
277C2×C4—4888no—
278C2×C4—2442no—
279C2×C4—2444no—
280C2×C4—2442no—
281C2×C4C22⋊C4×D6111yescharacteristic
282C2×C4C22⋊C4×D6111yescharacteristic
283C2×C4#1451974512yes—
284C2×C4#1451974712yes—
285C2×C4C2×#1175820011yescharacteristic
286C2×C4C2×#1175820211yescharacteristic
287C2×C4#1451980712yes—
288C8—82no—
289C8—4882no—
290C22—21no—
291C22—21no—
292C22—24no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22#1683961711yescharacteristic
298C22C2×#1451970011yescharacteristic
299C22#1683977011yescharacteristic
300C4—21no—
301C4—22no—
302C4—42no—
303C4—2442no—
304C4—2442no—
305C4—2444no—
306C4#1683974312yes—
307C4#1683990012yes—
308C4#1683991712yes—
309C2—21no—
310C2—21no—
311C2—44no—
312C2#2144574411yescharacteristic
313C2#2144575311yescharacteristic
314C2#2144579011yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1800C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1800D61
hypocenterC61#8395
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21444505C2semidirect
3. #21444247C2semidirect
180. C61#8395semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
#214445052
#214442472
#214441402
C61×#18002
#839561