Finite Group Databasebeta

← #29575320

Subgroups — #29575320

Order 15616.

570 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 155 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#21448909C211yesmaximal, characteristic
2#21446768C212yesmaximal
3C2×#16839511C211yesmaximal, characteristic
4#21443659C212yesmaximal
5C2×C61×#494C211yesmaximal, characteristic
6C2×C4⋊#92418C2212yes—
7#16839669—22no—
8#16839661—22no—
9C4⋊C4×Dic61—22no—
10C4×C4⋊Dic61C2212yes—
11#16839524—22no—
12#16839524—22no—
13#16839511—22no—
14C22×Q16×C61—22no—
15C2×C4⋊Q8×C61C2211yescharacteristic
16C2×C4⋊Q8×C61C2211yescharacteristic
17C61×#494—22no—
18C61×#494C414yescomplemented
19C2×C4×C8×C61C2211yescharacteristic
20C22×#92418—22no—
21C2×C4⋊Dic61—22no—
22C2×C4⋊Dic61D412yes—
23C2×C4⋊Dic61D412yes—
24C2×C4×Dic61—22no—
25C2×C4×Dic61—22no—
26C4⋊#92418—22no—
27C4⋊#92418—22no—
28#14519684—24no—
29#14519684—22no—
30C22×Q8×C61—21no—
31C22×Q8×C61D412yes—
32C2×Q16×C61—28no—
33C2×Q16×C61—44no—
34C2×Q16×C61—44no—
35C22×C8×C61—21no—
36C4⋊Q8×C61—21no—
37C4⋊Q8×C61—22no—
38C4⋊Q8×C61—22no—
39C4⋊Q8×C61C2×C412yes—
40C4⋊Q8×C61C2×C412yes—
41C4⋊Q8×C61D412yes—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C42×C61C2311yescharacteristic
45C4×C8×C61—21no—
46C4×C8×C61C2×C412yes—
47C2×Q8×C61—22no—
48C2×Q8×C61—22no—
49C2×Q8×C61—42no—
50C2×Q8×C61—42no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C2×Q8×C61—41no—
54C2×Q8×C61C22⋊C414yes—
55C22×C4×C61—22no—
56C22×C4×C61—41no—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D412yes—
59C22×Dic61—22no—
60C22×Dic61—22no—
61Q16×C61—416no—
62C2×C8×C61—22no—
63C2×C8×C61—22no—
64C2×C8×C61—22no—
65C4⋊C4×C61—24no—
66C4⋊C4×C61—24no—
67C4⋊C4×C61—24no—
68C4⋊C4×C61—42no—
69C42×C61C22×C411yescharacteristic
70C42×C61C2×D411yescharacteristic
71C42×C61#4112yes—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74C4⋊Dic61—22no—
75C4⋊Dic61—82no—
76C4⋊Dic61Q1614yescomplemented
77C4×Dic61—42no—
78C4×Dic61—42no—
79C4×Dic61—82no—
80C2×#92418—42no—
81C2×#92418—42no—
82C2×#92418—42no—
83C23×C61#12011yescharacteristic
84Q8×C61—44no—
85Q8×C61—44no—
86Q8×C61—82no—
87Q8×C61—82no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—24no—
91C2×C4×C61—42no—
92C2×C4×C61—42no—
93C2×C4×C61—42no—
94C2×C4×C61—81no—
95C2×C4×C61—81no—
96C2×C4×C61C2×C22⋊C411yescharacteristic
97C2×C4×C61C4×D412yes—
98C2×C4×C61#12112yes—
99C2×C4×C61#12311yescharacteristic
100C2×C4×C61C2×Q1612yes—
101C2×C4×C61#13512yes—
102C2×C4×C61#13712yes—
103C2×C4×C61#13712yes—
104C2×Dic61—24no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61—42no—
108C2×Dic61—42no—
109C2×Dic61—42no—
110C8×C61—24no—
111#92418—82no—
112#9066—611nomaximal, Sylow
113C22×C61#38611yescharacteristic
114C22×C61#45112yes—
115C22×C61#45212yes—
116C22×C61#45311yescharacteristic
117C22×C61#45311yescharacteristic
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—44no—
121C4×C61—82no—
122C4×C61C4×Q1612yes—
123C4×C61#44112yes—
124C4×C61#46212yes—
125C4×C61#46412yes—
126Dic61—44no—
127Dic61—82no—
128C2×#494D6111yescharacteristic
129#2369—611no—
130#1987—612no—
131C2×#333—611no—
132#1279—612no—
133C2×C61#163212yes—
134C2×C61#163311yescharacteristic
135C2×C61#163411yescharacteristic
136C2×C61#187812yes—
137C2×C61#188711yescharacteristic
138C22×Q16—22no—
139C2×C4⋊Q8C2×D6111yescharacteristic
140C2×C4⋊Q8C2×D6111yescharacteristic
141#494—22no—
142#494Dic6114yescomplemented
143C2×C4⋊C8—612no—
144C2×C4×C8C2×D6111yescharacteristic
145#391—1222no—
146#389—1222no—
147C4×C4⋊C4—612no—
148C4×C4⋊C4—1222no—
149#340—1222no—
150#340—1222no—
151#333—1222no—
152C61#906611yescharacteristic, complemented, Sylow
153C22×Q8—21no—
154C22×Q8C22⋊D6112yes—
155C2×Q16—28no—
156C2×Q16—44no—
157C2×Q16—44no—
158C22×C8—21no—
159C22×C8—1222no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8C2×Dic6112yes—
164C4⋊Q8C2×Dic6112yes—
165C4⋊Q8C22⋊D6112yes—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—612no—
169C2×C4⋊C4—612no—
170C2×C4⋊C4—1222no—
171C2×C42—1222no—
172C2×C42—1222no—
173C2×C42C22×D6111yescharacteristic
174C4⋊C8—1222no—
175C4⋊C8—1222no—
176C4×C8—21no—
177C4×C8C2×Dic6112yes—
178#95—1224no—
179#95—1222no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8C22⋊Dic6114yes—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—1222no—
191C22×C4—1222no—
192C22×C4D4×D6112yes—
193C22×C4C2×C22⋊D6111yescharacteristic
194Q16—416no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—2442no—
199C2×C8—2442no—
200C2×C8—2442no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—42no—
205C4⋊C4—614no—
206C4⋊C4—1222no—
207C4⋊C4—1222no—
208C4⋊C4—1222no—
209C4⋊C4—4882no—
210C42—4882no—
211C42—2442no—
212C42—2442no—
213C42#1175822412yes—
214C42C22×Dic6111yescharacteristic
215C42C2×C22⋊D6111yescharacteristic
216C23#1451974511yescharacteristic
217Q8—44no—
218Q8—44no—
219Q8—82no—
220Q8—82no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—1224no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4C2×C22⋊Dic6111yescharacteristic
236C2×C4D4×Dic6112yes—
237C2×C4#1451974912yes—
238C2×C4#1451975811yescharacteristic
239C2×C4Q16×D6112yes—
240C2×C4#1451980012yes—
241C2×C4#1451980912yes—
242C2×C4#1451980912yes—
243C8—24no—
244C8—4882no—
245C22#1683964211yescharacteristic
246C22#1683991612yes—
247C22#1683992312yes—
248C22#1683992911yescharacteristic
249C22#1683992911yescharacteristic
250C4—21no—
251C4—21no—
252C4—44no—
253C4—82no—
254C4—4882no—
255C4—2444no—
256C4Q16×Dic6112yes—
257C4#1683986912yes—
258C4#1683997012yes—
259C4#1683998612yes—
260C2#2144495012yes—
261C2#2144495711yescharacteristic
262C2#2144496211yescharacteristic
263C2#2144616212yes—
264C2#2144621811yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#494C2
socle, prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenterC2×#494D61
hypocenterC61#9066
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
18. C61×#494C4semidirect
76. C4⋊Dic61Q16semidirect
142. #494Dic61semidirect
152. C61#9066semidirect

Maximal subgroups

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

subgroupindex
#214489092
#214467682
C2×#168395112
#214436592
C2×C61×#4942
#906661