Finite Group Databasebeta

← #29576032

Subgroups — #29576032

Order 15616.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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#21454457C211yesmaximal, characteristic
2#21444504C212yesmaximal, complemented
3#21444133C211yesmaximal, characteristic
4#21443677C212yesmaximal, complemented
5C61×#2628C211yesmaximal, characteristic
6#16839823—24no—
7#16839815C2212yes—
8#16839786C2212yes—
9#16839726—21no—
10C4×#11758194—21no—
11#16839508—24no—
12#16839488—21no—
13#16839486—21no—
14C2×C4⋊Q8×C61C2211yescharacteristic
15C2×#117×C61C2211yescharacteristic
16C61×#400C412yes—
17C61×#399C412yes—
18C61×#395—21no—
19C61×#393—21no—
20C61×#389—21no—
21C61×#382C2211yescharacteristic
22C61×#380—21no—
23C2×#11758194—22no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26C4⋊#92418D414yes—
27C22⋊#92418—24no—
28C22⋊#92418—22no—
29#14519688—22no—
30C4×#92418—22no—
31C4×#92418—22no—
32C22×Q8×C61—21no—
33C23×C4×C61D411yescharacteristic
34C4⋊Q8×C61—22no—
35C4⋊Q8×C61—22no—
36#117×C61—22no—
37#117×C61—22no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61C2×C412yes—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C42×C61D411yescharacteristic
48C2×C42×C61C2311yescharacteristic
49#95×C61—22no—
50#95×C61—22no—
51#95×C61—21no—
52#95×C61C2×C412yes—
53C24×C61C22⋊C411yescharacteristic
54C2×Q8×C61—42no—
55C2×Q8×C61—82no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—22no—
61C22×C4×C61—22no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C4⋊C4×C61—22no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—24no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—42no—
74C4⋊C4×C61—82no—
75C22⋊C4×C61—22no—
76C22⋊C4×C61—22no—
77C22⋊C4×C61—82no—
78C42×C61—21no—
79C42×C61—21no—
80C42×C61C2×D412yes—
81C42×C61#4112yes—
82#11758194—42no—
83#11758194—42no—
84C2×#92418—24no—
85C2×#92418—42no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61C2×C22⋊C411yescharacteristic
92Q8×C61—84no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61C4×D412yes—
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61#12112yes—
113C2×C4×C61#12311yescharacteristic
114C2×C4×C61#13512yes—
115C2×C4×C61#13712yes—
116#92418—44no—
117#92418—82no—
118#9194—611nomaximal, Sylow
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—42no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61—41no—
125C22×C61—81no—
126C22×C61#38611yescharacteristic
127C22×C61#41611yescharacteristic
128C22×C61#41911yescharacteristic
129C4×C61—22no—
130C4×C61—22no—
131C4×C61—41no—
132C4×C61—41no—
133C4×C61—41no—
134C4×C61—82no—
135C4×C61—82no—
136C4×C61#45212yes—
137C4×C61#48512yes—
138#2921—611no—
139#2628D6111yescharacteristic
140#1549—612no—
141#1458—611no—
142#1286—612no—
143C2×C61—21no—
144C2×C61—21no—
145C2×C61—81no—
146C2×C61#187311yescharacteristic
147C2×C61#191011yescharacteristic
148C2×C61#191511yescharacteristic
149C2×C4⋊Q8C2×D6111yescharacteristic
150C2×#117C2×D6111yescharacteristic
151#433—1224no—
152#431—612no—
153#423—612no—
154#407—1221no—
155C4×M16—1221no—
156#400Dic6112yes—
157#399Dic6112yes—
158#395—21no—
159#393—21no—
160#389—21no—
161#382C2×D6111yescharacteristic
162#380—21no—
163#332—1224no—
164#324—1221no—
165#323—1221no—
166C61#919411yescharacteristic, complemented, Sylow
167C22×Q8—21no—
168C23×C4C22⋊D6111yescharacteristic
169C2×M16—1222no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×Dic6112yes—
178C2×C4⋊C4C2×Dic6111yescharacteristic
179C2×C4⋊C4C22⋊D6111yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic6111yescharacteristic
182C2×C22⋊C4C22⋊D6111yescharacteristic
183C2×C42C22⋊D6111yescharacteristic
184C2×C42C22×D6111yescharacteristic
185C4⋊C8—614no—
186C4⋊C8—1222no—
187C4⋊C8—1222no—
188C22⋊C8—1224no—
189C22⋊C8—1222no—
190#97—1222no—
191C4×C8—1222no—
192C4×C8—1222no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95C2×Dic6112yes—
197C24C22⋊Dic6111yescharacteristic
198C2×Q8—42no—
199C2×Q8—82no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊Dic6111yescharacteristic
207C22×C4C22⋊Dic6111yescharacteristic
208C22×C4C22⋊Dic6111yescharacteristic
209C22×C4C22×Dic6111yescharacteristic
210C22×C4C2×C22⋊D6111yescharacteristic
211C22×C4C2×C22⋊D6111yescharacteristic
212M16—2442no—
213M16—2442no—
214C2×C8—1224no—
215C2×C8—2442no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42D4×D6112yes—
229C42#1175822412yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic6111yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4D4×Dic6112yes—
255C2×C4#1451974511yescharacteristic
256C2×C4#1451974912yes—
257C2×C4#1451975811yescharacteristic
258C2×C4#1451979712yes—
259C2×C4#1451980812yes—
260C8—4882no—
261C8—2444no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1683964211yescharacteristic
270C22#1683975811yescharacteristic
271C22#1683977111yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#1683992212yes—
280C4#1684010012yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2144613211yescharacteristic
285C2#2144634711yescharacteristic
286C2#2144637211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2628C2
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#2628D61
hypocenterC61#9194
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #21444504C2semidirect
4. #21443677C2semidirect
166. C61#9194semidirect

Maximal subgroups

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

subgroupindex
#214544572
#214445042
#214441332
#214436772
C61×#26282
#919461