Finite Group Databasebeta

← #29572554

Subgroups — #29572554

Order 15616.

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

normal subgroups only

This group has 656 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
1C2×#16840373C211yesmaximal, characteristic
2#21444545C211yesmaximal, characteristic, complemented
3#21444445C212yesmaximal, complemented
4#21444151C212yesmaximal
5C61×#1558C211yesmaximal, characteristic, complemented
6#16840373—24no—
7#16840373—22no—
8#16840373—22no—
9C2×#14519772—22no—
10C2×#14519769C2211yescharacteristic
11C2×#14519767—22no—
12C2×#14519753C412yes—
13C2×Q8×Dic61C412yes—
14C2×#14519734C2211yescharacteristic
15#16839819C2212yes—
16#16839511—22no—
17#16839503—24no—
18#16839487—22no—
19#16839485—22no—
20C2×C4⋊Q8×C61C2211yescharacteristic
21C61×#432C2212yes—
22C61×#333—22no—
23C22×C4×D61C2×C411yescharacteristic
24#14519772—28no—
25#14519772—28no—
26#14519769—21no—
27#14519769—21no—
28#14519769D414yes—
29#14519767—28no—
30#14519767—28no—
31#14519753—24no—
32#14519753—24no—
33Q8×Dic61—24no—
34Q8×Dic61—24no—
35#14519734—21no—
36#14519734—22no—
37#14519734—21no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61C2×C412yes—
40C2×#11758196—22no—
41C2×#11758196—22no—
42C2×#11758196D412yes—
43C2×#11758198—22no—
44C2×#11758198D412yes—
45C2×C4×Dic61—22no—
46C2×C4×Dic61C2×C411yescharacteristic
47C4⋊#92418—22no—
48C22⋊#92418—22no—
49#14519688—22no—
50C22×Q8×C61C2×C412yes—
51C4⋊Q8×C61—22no—
52C4⋊Q8×C61D412yes—
53C4⋊Q8×C61D412yes—
54C2×C4⋊C4×C61—22no—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58#97×C61—22no—
59C23×D61C22⋊C411yescharacteristic
60C2×Q8×C61—22no—
61C2×Q8×C61—44no—
62C2×Q8×C61C22⋊C414yes—
63C22×C4×C61—22no—
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D412yes—
66C22×Dic61—22no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C412yes—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—42no—
72C2×C8×C61—42no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—24no—
75C42×C61C2×D411yescharacteristic
76C42×C61C2×D412yes—
77C42×C61C2×D411yescharacteristic
78#11758198—28no—
79#11758198—22no—
80#11758198—22no—
81C4⋊Dic61—28no—
82C4⋊Dic61—22no—
83C4⋊Dic61C22⋊C414yes—
84#11758196—28no—
85#11758196—24no—
86#11758196—24no—
87#11758196—22no—
88#11758196—22no—
89C4×Dic61—28no—
90C4×Dic61—21no—
91C4×Dic61—21no—
92C2×#92418—42no—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—21no—
95C22×D61—22no—
96C22×D61—42no—
97Q8×C61—44no—
98Q8×C61—44no—
99C2×C4×C61—22no—
100C2×C4×C61—24no—
101C2×C4×C61—42no—
102C2×C4×C61—44no—
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61#12012yes—
106C2×C4×C61#12012yes—
107C2×Dic61—22no—
108C2×Dic61—24no—
109C2×Dic61—21no—
110C2×Dic61—22no—
111C2×Dic61—44no—
112C2×Dic61—42no—
113C4×D61—42no—
114C4×D61—42no—
115C8×C61—82no—
116#92418—82no—
117#8527—611nomaximal, Sylow
118C22×C61—22no—
119C22×C61#37911yescharacteristic
120C22×C61#41312yes—
121C2×D61—42no—
122C2×D61—41no—
123C2×D61—82no—
124C4×C61—22no—
125C4×C61—22no—
126C4×C61—21no—
127C4×C61—21no—
128C4×C61—44no—
129Dic61—42no—
130Dic61—44no—
131Dic61—41no—
132C2×#541—611no—
133#1558—611no—
134#1558D6111yescharacteristic, complemented
135#1534—612no—
136#1464—612no—
137C2×C61—22no—
138C2×C61#178511yescharacteristic
139C2×C61#179612yes—
140D61—81no—
141#541—1224no—
142#541—1222no—
143#541—1222no—
144C2×C4⋊Q8C2×D6111yescharacteristic
145C2×#126—1222no—
146C2×#125—611no—
147C2×#125—1222no—
148C2×C22⋊Q8—612no—
149C2×C4×Q8—612no—
150C2×#117—611no—
151#432—612no—
152#432C2×D6112yes—
153#333—22no—
154#333—1222no—
155#330—1224no—
156#324—1222no—
157#323—1222no—
158C61#852711yescharacteristic, complemented, Sylow
159C22×Q8C4×D6112yes—
160C23×C4—611no—
161C4⋊Q8—22no—
162C4⋊Q8D24412yes—
163C4⋊Q8C22⋊D6112yes—
164#126—1228no—
165#126—1228no—
166#125—614no—
167#125—1228no—
168#125—1228no—
169#125—1221no—
170#125—1221no—
171C22⋊Q8—1224no—
172C22⋊Q8—1224no—
173C4×Q8—1224no—
174C4×Q8—1224no—
175#117—1221no—
176#117—1222no—
177#117—1221no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—612no—
180C2×C4⋊C4—612no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—1222no—
183C2×C4⋊C4—1222no—
184C2×C22⋊C4—612no—
185C2×C22⋊C4—1222no—
186C2×C42—611no—
187C2×C42—1222no—
188C2×C42C22×D6111yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—1222no—
191C22⋊C8—22no—
192C22⋊C8—1222no—
193#97—22no—
194#97—1222no—
195C24—611no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8#1175819814yes—
199C22×C4—22no—
200C22×C4—611no—
201C22×C4—612no—
202C22×C4—1222no—
203C22×C4—1221no—
204C22×C4—1221no—
205C22×C4—2442no—
206C22×C4C2×C4×D6111yescharacteristic
207C22×C4D4×D6112yes—
208C2×C8—42no—
209C2×C8—2442no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—614no—
213C4⋊C4—1228no—
214C4⋊C4—1228no—
215C4⋊C4—1222no—
216C4⋊C4—1222no—
217C4⋊C4—1224no—
218C4⋊C4—1222no—
219C4⋊C4—1224no—
220C22⋊C4—1228no—
221C22⋊C4—1222no—
222C22⋊C4—1222no—
223C42—1228no—
224C42—1221no—
225C42—1221no—
226C42C2×D24411yescharacteristic
227C42D4×D6112yes—
228C42C2×C22⋊D6111yescharacteristic
229C23—1222no—
230C23—1221no—
231C23—2442no—
232C23C22⋊C4×D6111yescharacteristic
233Q8—44no—
234Q8—44no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—1222no—
240C2×C4—1222no—
241C2×C4—1224no—
242C2×C4—1221no—
243C2×C4—2442no—
244C2×C4—2442no—
245C2×C4—2444no—
246C2×C4—2442no—
247C2×C4C22⋊C4×D6111yescharacteristic
248C2×C4C2×#1175819811yescharacteristic
249C2×C4#1451974512yes—
250C2×C4#1451974712yes—
251C8—82no—
252C8—4882no—
253C22—22no—
254C22—4882no—
255C22—2441no—
256C22—2442no—
257C22#1683961711yescharacteristic
258C22#1683974312yes—
259C4—21no—
260C4—21no—
261C4—22no—
262C4—22no—
263C4—44no—
264C4—2441no—
265C4—2442no—
266C4—2444no—
267C2—22no—
268C2—4881no—
269C2#2144574411yescharacteristic
270C2#2144579112yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1558C2
socleC22×C61#379
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1558D61
hypocenterC61#8527
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #21444545C2semidirect
3. #21444445C2semidirect
5. C61×#1558C2semidirect
134. #1558D61semidirect
158. C61#8527semidirect

Maximal subgroups

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

subgroupindex
C2×#168403732
#214445452
#214444452
#214441512
C61×#15582
#852761