Finite Group Databasebeta

← #29572161

Subgroups — #29572161

Order 15616.

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

normal subgroups only

This group has 746 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#21459622C211yesmaximal, characteristic
2#21446618C211yesmaximal, characteristic
3#21444520C212yesmaximal, complemented
4#21444435C212yesmaximal, complemented
5C61×#1466C211yesmaximal, characteristic
6#16840471—24no—
7#16840469—22no—
8D4×Dic122—22no—
9#16840393C412yes—
10#16840388—22no—
11#16840319C2212yes—
12C2×#14519756—22no—
13#16840237C412yes—
14C2×C4⋊#92418C2211yescharacteristic
15C2×C22⋊#92418—22no—
16C2×#14519688C2211yescharacteristic
17#16839509—22no—
18#16839507—22no—
19#16839503—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#432C2212yes—
22C61×#323—22no—
23C22×Dic122—22no—
24C22×#92418—21no—
25C22×#92418—21no—
26#14519777D412yes—
27#14519777D412yes—
28#14519776—24no—
29#14519769D414yes—
30#14519768—24no—
31#14519756—24no—
32#14519756—24no—
33#14519756—24no—
34#14519756—22no—
35#14519756—22no—
36#14519754—28no—
37#14519754—28no—
38C4×Dic122—22no—
39C4×Dic122—22no—
40D4×Dic61—24no—
41D4×Dic61—24no—
42#14519736C2×C414yes—
43C2×C4⋊Dic61—22no—
44C2×#11758196—24no—
45C2×C22⋊Dic61—24no—
46C4⋊#92418—22no—
47C22⋊#92418—44no—
48#14519688—22no—
49C22×D4×C61C2×C411yescharacteristic
50C23×C4×C61D412yes—
51C4×D4×C61—22no—
52C4×D4×C61—22no—
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61D412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58#97×C61—22no—
59C24×C61C22⋊C412yes—
60C2×D4×C61—24no—
61C2×D4×C61—22no—
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61—42no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—24no—
71C22×Dic61—24no—
72C2×Dic122—22no—
73C2×Dic122—24no—
74C2×Dic122—22no—
75C2×Dic122—82no—
76C2×Dic122—82no—
77C2×C8×C61—42no—
78C4⋊C4×C61—22no—
79C22⋊C4×C61—24no—
80C42×C61C2×D412yes—
81C42×C61C2×D412yes—
82C22⋊Dic61—28no—
83C22⋊Dic61—24no—
84C4⋊Dic61—24no—
85C4⋊Dic61—22no—
86C4⋊Dic61—22no—
87C4⋊Dic61C22⋊C414yes—
88C4⋊Dic61C22⋊C414yes—
89#11758196—28no—
90#11758196—24no—
91#11758196—24no—
92C4×Dic61—24no—
93C2×#92418—22no—
94C2×#92418—41no—
95C2×#92418—41no—
96C2×#92418—42no—
97C2×#92418—41no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61#10114yes—
103C23×C61C2×C22⋊C411yescharacteristic
104D4×C61—28no—
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—82no—
114C2×C4×C61C2×C22⋊C412yes—
115C2×C4×C61#12012yes—
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61#12011yescharacteristic
118C2×Dic61—24no—
119C2×Dic61—24no—
120C2×Dic61—84no—
121C2×Dic61—84no—
122Dic122—42no—
123Dic122—42no—
124Dic122—82no—
125Dic122—84no—
126C8×C61—82no—
127#92418—42no—
128#92418—81no—
129#8457—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—24no—
133C22×C61—44no—
134C22×C61—42no—
135C22×C61—42no—
136C22×C61—82no—
137C22×C61#37911yescharacteristic
138C22×C61C2×#10112yes—
139C4×C61—21no—
140C4×C61—22no—
141C4×C61—22no—
142C4×C61—42no—
143C4×C61#45311yescharacteristic
144C4×C61#45511yescharacteristic
145Dic61—44no—
146Dic61—44no—
147#3501—611no—
148#1955—611no—
149#1552—612no—
150#1531—612no—
151#1466D6111yescharacteristic
152C2×C61—22no—
153C2×C61—44no—
154C2×C61#178411yescharacteristic
155C2×C61#179211yescharacteristic
156C2×C61#179411yescharacteristic
157#554—1224no—
158#554—1222no—
159D4×Q8—1222no—
160#544—612no—
161#543—1222no—
162#533—612no—
163C2×C22⋊Q8—1222no—
164#518—612no—
165C2×C4×D4C2×D6111yescharacteristic
166#432C2×D6112yes—
167C2×C4⋊C8—611no—
168C2×C22⋊C8—1222no—
169C2×#97—611no—
170#332—1222no—
171#332—1222no—
172#330—1224no—
173#323—22no—
174C61#845711yescharacteristic, complemented, Sylow
175C22×Q8—1222no—
176C22×D4C4×D6111yescharacteristic
177C23×C4D24412yes—
178C22×C8—1221no—
179C22×C8—1221no—
180C4⋊Q8—612no—
181C4⋊Q8—612no—
182C4⋊Q8—1224no—
183#125—614no—
184#125—1224no—
185C22⋊Q8—1228no—
186C22⋊Q8—1228no—
187C22⋊Q8—1224no—
188C22⋊Q8—1224no—
189C22⋊Q8—1224no—
190C22⋊Q8—1222no—
191C22⋊Q8—1222no—
192C4×Q8—1222no—
193C4×Q8—1222no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—1224no—
197C4×D4—1224no—
198#117—614no—
199C2×C4⋊C4—1224no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4C4×D6111yescharacteristic
202C2×C22⋊C4—1224no—
203C2×C22⋊C4C22⋊D6112yes—
204C2×C42C22×D6111yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—1222no—
207C22⋊C8—22no—
208C22⋊C8—2444no—
209#97—22no—
210#97—1222no—
211C24#1175819812yes—
212C2×Q8—1224no—
213C2×Q8—1222no—
214C2×Q8—1222no—
215C2×Q8—4882no—
216C2×Q8—4882no—
217C2×D4—24no—
218C2×D4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—1224no—
224C22×C4—1224no—
225C22×C4#1175819812yes—
226C22×C4C2×C4×D6111yescharacteristic
227C22×C4C2×D24411yescharacteristic
228C22×C4C2×C22⋊D6111yescharacteristic
229C2×C8—42no—
230C2×C8—1222no—
231C2×C8—2442no—
232C2×C8—2441no—
233C2×C8—2441no—
234C2×C8—2441no—
235C4⋊C4—22no—
236C4⋊C4—614no—
237C4⋊C4—614no—
238C4⋊C4—1228no—
239C4⋊C4—1224no—
240C4⋊C4—1224no—
241C4⋊C4—1222no—
242C4⋊C4—1224no—
243C4⋊C4—1222no—
244C22⋊C4—24no—
245C22⋊C4—1228no—
246C22⋊C4—1224no—
247C42—1224no—
248C42D4×D6112yes—
249C42D4×D6112yes—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23#1451969514yes—
255C23C2×#1175819811yescharacteristic
256Q8—4882no—
257Q8—4884no—
258Q8—2442no—
259Q8—2442no—
260D4—28no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—1224no—
271C2×C4—1224no—
272C2×C4—4884no—
273C2×C4—4884no—
274C2×C4C22⋊C4×D6112yes—
275C2×C4#1451974512yes—
276C2×C4#1451974711yescharacteristic
277C2×C4#1451974711yescharacteristic
278C8—82no—
279C8—4881no—
280C8—2442no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22#1683961711yescharacteristic
289C22C2×#1451969512yes—
290C4—22no—
291C4—21no—
292C4—22no—
293C4—42no—
294C4—2444no—
295C4—2444no—
296C4#1683993211yescharacteristic
297C4#1683994211yescharacteristic
298C2—22no—
299C2—44no—
300C2#2144574211yescharacteristic
301C2#2144577511yescharacteristic
302C2#2144578511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1466C2
socleC22×C61#379
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#1466D61
hypocenterC61#8457
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #21444520C2semidirect
4. #21444435C2semidirect
174. C61#8457semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214466182
#214445202
#214444352
C61×#14662
#845761