Finite Group Databasebeta

← #29590687

Subgroups — #29590687

Order 15616.

502 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 87 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#21445896C211yesmaximal, characteristic, complemented
2#21444153C211yesmaximal, characteristic, complemented
3#21443816C214yesmaximal, complemented
4C61×#2619C211yesmaximal, characteristic
5#16839819—21no—
6#16839625—21no—
7#16839529—21no—
8#16839529C2212yes—
9#16839488—21no—
10#16839486C2212yes—
11C2×#127×C61C411yescharacteristic, complemented
12C2×#121×C61—21no—
13C2×#117×C61C2211yescharacteristic
14C61×#394—21no—
15C61×#394C412yescomplemented
16C61×#392—21no—
17C61×#392C2212yes—
18C61×#390—21no—
19C61×#382C411yescharacteristic, complemented
20C2×C22⋊Dic61—22no—
21C4⋊#92418—41no—
22C22⋊#92418—22no—
23#14519688—41no—
24#14519684—41no—
25#14519684—41no—
26C22×D4×C61—22no—
27C22×D4×C61—21no—
28C22×D4×C61—21no—
29C22×D4×C61C2×C412yes—
30C23×C4×C61D411yescharacteristic
31#127×C61—41no—
32#127×C61—41no—
33#121×C61—44no—
34#117×C61—41no—
35#117×C61—41no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61C2×C411yescharacteristic
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61C2311yescharacteristic
44C2×C42×C61C2×C411yescharacteristic
45C2×C42×C61D411yescharacteristic
46#95×C61—21no—
47#95×C61C2×C412yes—
48C24×C61—22no—
49C24×C61C2×D411yescharacteristic
50C2×D4×C61—42no—
51C2×D4×C61—42no—
52C2×D4×C61—42no—
53C2×D4×C61—42no—
54C2×D4×C61—44no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61—82no—
58C2×D4×C61—82no—
59C2×D4×C61—82no—
60C2×D4×C61—82no—
61C22×C4×C61—22no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—41no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—42no—
73C4⋊C4×C61—41no—
74C4⋊C4×C61—41no—
75C4⋊C4×C61—42no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C22⋊C4×C61—44no—
79C22⋊C4×C61—82no—
80C22⋊C4×C61—82no—
81C42×C61—21no—
82C42×C61—21no—
83C42×C61—21no—
84C42×C61—21no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C2×#92418—42no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61—42no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C23×C61AGL(1,ℤ/8)12yes—
101D4×C61—82no—
102D4×C61—82no—
103D4×C61—84no—
104D4×C61—84no—
105D4×C61—88no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#9911yescharacteristic
126C2×Dic61—82no—
127C2×Dic61—82no—
128C2×Dic61—82no—
129#92418—82no—
130#11800—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—22no—
134C22×C61—42no—
135C22×C61—41no—
136C22×C61—41no—
137C22×C61—81no—
138C22×C61—82no—
139C22×C61—82no—
140C22×C61—82no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—82no—
144C22×C61—82no—
145C22×C61C2×#9911yescharacteristic
146C22×C61#41811yescharacteristic
147C22×C61#42111yescharacteristic
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—41no—
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—41no—
154C4×C61—41no—
155C4×C61—82no—
156Dic61—162no—
157#2619D6111yescharacteristic
158#1820—611no—
159#1464—611no—
160#1343—614no—
161C2×C61—21no—
162C2×C61—21no—
163C2×C61—42no—
164C2×C61—162no—
165C2×C61#151011yescharacteristic
166C2×C61#212211yescharacteristic
167C2×C61#212811yescharacteristic
168C2×#127Dic6111yescharacteristic, complemented
169C2×#121—21no—
170C2×#117C2×D6111yescharacteristic
171#432—1221no—
172#394—21no—
173#394Dic6112yescomplemented
174#392—21no—
175#392C2×D6112yes—
176#390—21no—
177#382Dic6111yescharacteristic, complemented
178#381—1221no—
179#342—612no—
180#342—1221no—
181#324—1221no—
182#323—612no—
183C61#1180011yescharacteristic, complemented, Sylow
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C2×Dic6112yes—
188C23×C4C22⋊D6111yescharacteristic
189#127—41no—
190#127—41no—
191#121—44no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic6111yescharacteristic
196C2×C4⋊C4C22⋊D6111yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4C22⋊D6111yescharacteristic
202C2×C22⋊C4C22×D6111yescharacteristic
203C2×C42C2×Dic6111yescharacteristic
204C2×C42C22⋊D6111yescharacteristic
205C4⋊C8—2441no—
206C22⋊C8—1222no—
207#97—2441no—
208#95—21no—
209#95—2441no—
210#95—2441no—
211#95C2×Dic6112yes—
212C24—22no—
213C24C2×C22⋊D6111yescharacteristic
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—2442no—
231C22×C4C22⋊Dic6111yescharacteristic
232C22×C4C22⋊Dic6111yescharacteristic
233C22×C4C22⋊Dic6111yescharacteristic
234C22×C4C22⋊Dic6111yescharacteristic
235C22×C4C22×Dic6111yescharacteristic
236C22×C4C2×C22⋊D6111yescharacteristic
237C2×C8—2442no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—2442no—
247C22⋊C4—2442no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23C2×C22⋊Dic6111yescharacteristic
264C23#1451980312yes—
265D4—82no—
266D4—82no—
267D4—84no—
268D4—84no—
269D4—88no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—4882no—
289C2×C4—4882no—
290C2×C4—4882no—
291C2×C4#1451969211yescharacteristic
292C2×C4#1451969211yescharacteristic
293C8—4882no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C2×#1451969211yescharacteristic
309C22#1683976711yescharacteristic
310C22#1683977911yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—9762no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—162no—
324C2#2144434811yescharacteristic
325C2#2144746911yescharacteristic
326C2#2144749111yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
Fitting, prime radicalC61×#2619C2
socleC22×C61C2×#99
cosocleC2×C22⋊C4×C61C23
hypercenter#2619D61
hypocenterC61#11800
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #21445896C2semidirect
2. #21444153C2semidirect
3. #21443816C2semidirect
11. C2×#127×C61C4semidirect
15. C61×#394C4semidirect
19. C61×#382C4semidirect
168. C2×#127Dic61semidirect
173. #394Dic61semidirect
177. #382Dic61semidirect
183. C61#11800semidirect

Maximal subgroups

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

subgroupindex
#214458962
#214441532
#214438162
C61×#26192
#1180061