Finite Group Databasebeta

← #29574050

Subgroups — #29574050

Order 15616.

418 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 107 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#21446618C211yesmaximal, characteristic
2#21444605C212yesmaximal, complemented
3#21444140C211yesmaximal, characteristic
4#21443717C212yesmaximal, complemented
5C61×#2973C211yesmaximal, characteristic
6#16839815C2212yes—
7C2×C4⋊#92418C2211yescharacteristic
8C2×C22⋊#92418—21no—
9C2×C22⋊#92418—21no—
10C2×#14519688C2211yescharacteristic
11#16839514—22no—
12#16839512—22no—
13#16839486C412yes—
14#16839486C412yes—
15C2×C4×D4×C61C2211yescharacteristic
16C61×#435—22no—
17C61×#435—22no—
18C61×#432C2212yes—
19C61×#407—21no—
20C2×C22⋊C8×C61—21no—
21C22×#92418—21no—
22C22×#92418—21no—
23C4⋊#92418—22no—
24C4⋊#92418—22no—
25C22⋊#92418—42no—
26C22⋊#92418—42no—
27C22⋊#92418C2×C414yes—
28#14519688—22no—
29C4×#92418—22no—
30C22×D4×C61Q811yescharacteristic
31C23×C4×C61D411yescharacteristic
32C23×C4×C61D411yescharacteristic
33C2×M16×C61—21no—
34C2×M16×C61—21no—
35C22×C8×C61—21no—
36C22×C8×C61—21no—
37C4×D4×C61—22no—
38C4×D4×C61—22no—
39C2×C4⋊C4×C61Q811yescharacteristic
40C2×C22⋊C4×C61C2×C411yescharacteristic
41C2×C22⋊C4×C61C2×C411yescharacteristic
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61D412yes—
44C4⋊C8×C61D412yes—
45C22⋊C8×C61—22no—
46C22⋊C8×C61—22no—
47C22⋊C8×C61—22no—
48#97×C61—22no—
49C24×C61C4⋊C411yescharacteristic
50C24×C61C4⋊C411yescharacteristic
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61—41no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61C4⋊C411yescharacteristic
63C22×C4×C61C4⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×Q811yescharacteristic
67M16×C61—81no—
68M16×C61—81no—
69C2×C8×C61—22no—
70C2×C8×C61—22no—
71C2×C8×C61—81no—
72C2×C8×C61—81no—
73C4⋊C4×C61—22no—
74C22⋊C4×C61—22no—
75C22⋊C4×C61—22no—
76C42×C61C2×D412yes—
77C42×C61#4112yes—
78C2×#92418—22no—
79C2×#92418—24no—
80C2×#92418—41no—
81C2×#92418—41no—
82C2×#92418—41no—
83C2×#92418—41no—
84C2×#92418—41no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61#10812yes—
95C23×C61C2×C4⋊C411yescharacteristic
96D4×C61—44no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—81no—
111C2×C4×C61—81no—
112C2×C4×C61C4×D412yes—
113C2×C4×C61#12011yescharacteristic
114C2×C4×C61C22⋊Q812yes—
115C2×C4×C61#12311yescharacteristic
116C8×C61—42no—
117C8×C61—42no—
118#92418—44no—
119#92418—42no—
120#92418—81no—
121#8864—611nomaximal, Sylow
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—22no—
125C22×C61—42no—
126C22×C61—42no—
127C22×C61—41no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—81no—
133C22×C61—81no—
134C22×C61#38511yescharacteristic
135C22×C61C2×#10811yescharacteristic
136C22×C61#43011yescharacteristic
137C4×C61—22no—
138C4×C61—21no—
139C4×C61—22no—
140C4×C61—41no—
141C4×C61—41no—
142C4×C61#45311yescharacteristic
143C4×C61#45511yescharacteristic
144#2973D6111yescharacteristic
145#1955—611no—
146#1573—612no—
147#1460—611no—
148#1302—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—42no—
152C2×C61—81no—
153C2×C61#185511yescharacteristic
154C2×C61#185611yescharacteristic
155C2×C61#186811yescharacteristic
156C2×C4×D4C2×D6111yescharacteristic
157#435—22no—
158#435—22no—
159#432C2×D6112yes—
160#431—612no—
161C2×C4⋊C8—611no—
162#407—21no—
163C2×C22⋊C8—21no—
164C2×C22⋊C8—1221no—
165C2×C22⋊C8—1221no—
166C2×#97—611no—
167#335—1222no—
168#334—1222no—
169#323—612no—
170#323—612no—
171C61#886411yescharacteristic, complemented, Sylow
172C22×D4Dic12211yescharacteristic
173C23×C4C22⋊D6111yescharacteristic
174C23×C4C22⋊D6111yescharacteristic
175C2×M16—21no—
176C2×M16—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—1221no—
180C22×C8—1221no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4Dic12211yescharacteristic
184C2×C22⋊C4C4×D6111yescharacteristic
185C2×C22⋊C4C4×D6111yescharacteristic
186C2×C42C22×D6111yescharacteristic
187C4⋊C8—1222no—
188C4⋊C8—1222no—
189C4⋊C8C22⋊D6112yes—
190C4⋊C8C22⋊D6112yes—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—614no—
195C22⋊C8—2442no—
196C22⋊C8—2442no—
197#97—22no—
198#97—1222no—
199C4×C8—1222no—
200C24#1175819611yescharacteristic
201C24#1175819611yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4#1175819611yescharacteristic
214C22×C4#1175819611yescharacteristic
215C22×C4C2×Dic12211yescharacteristic
216C22×C4C2×C4×D6111yescharacteristic
217C22×C4C2×C22⋊D6111yescharacteristic
218M16—81no—
219M16—81no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—1222no—
225C2×C8—1224no—
226C2×C8—2441no—
227C2×C8—2441no—
228C2×C8—2441no—
229C2×C8—2441no—
230C2×C8—2441no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42#1175822212yes—
235C42C2×C22⋊D6112yes—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23#1451971212yes—
246C23C2×#1175819611yescharacteristic
247D4—44no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C4×C22⋊D6112yes—
264C2×C4#1451974611yescharacteristic
265C2×C4#1451975612yes—
266C2×C4#1451976011yescharacteristic
267C8—42no—
268C8—42no—
269C8—4881no—
270C8—2444no—
271C8—2442no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22—81no—
284C22#1683963511yescharacteristic
285C22C2×#1451971211yescharacteristic
286C22#1683981011yescharacteristic
287C4—22no—
288C4—21no—
289C4—22no—
290C4—41no—
291C4—41no—
292C4#1683993011yescharacteristic
293C4#1683994011yescharacteristic
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2144602411yescharacteristic
299C2#2144602911yescharacteristic
300C2#2144610511yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2973C2
socleC22×C61#385
prime radicalC22×D4×C61Q8
cosocleC2×C42×C61C23
hypercenter#2973D61
hypocenterC61#8864
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21444605C2semidirect
4. #21443717C2semidirect
171. C61#8864semidirect

Maximal subgroups

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

subgroupindex
#214466182
#214446052
#214441402
#214437172
C61×#29732
#886461