Finite Group Databasebeta

← #29657957

Subgroups — #29657957

Order 15616.

834 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 179 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#21457202C212yesmaximal
2#21456144C211yesmaximal, characteristic
3#21444977C214yesmaximal, complemented
4#21444915C214yesmaximal, complemented
5#21444557C211yesmaximal, characteristic, complemented
6#21444274C212yesmaximal, complemented
7C61×#3556C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840119—22no—
10#16840114C2212yes—
11#16840110C2212yes—
12#16840057C2214yes—
13#16840039—22no—
14#16840010—24no—
15#16839964—22no—
16#16839962C2214yes—
17#16839786C2212yes—
18#16839756—22no—
19C2×C4×#92418C2211yescharacteristic
20#16839508C2214yes—
21Q8⋊#92418C2218yes—
22C61×#564—22no—
23C61×#557—22no—
24C61×#557—22no—
25C61×#557C2214yes—
26C61×#541—24no—
27C61×#541—22no—
28C61×#533C2212yes—
29C61×#533C2211yescharacteristic
30C2×#11758222—21no—
31C2×#11758201—24no—
32C2×#11758194—22no—
33C22×#92418—21no—
34#14519777C2311yescharacteristic
35#14519774C2311yescharacteristic
36#14519765—21no—
37#14519751—22no—
38C4×D244—22no—
39#14519710—24no—
40C4⋊#92418—22no—
41C4⋊#92418C2314yes—
42#14519701—24no—
43#14519700—24no—
44#14519697—24no—
45#14519697—24no—
46C4×#92418—21no—
47C4×#92418C2312yes—
48C4⋊Q8×C61—22no—
49C4⋊Q8×C61—24no—
50C4⋊Q8×C61D412yes—
51C4⋊Q8×C61D412yes—
52C4⋊Q8×C61C2312yes—
53#126×C61—28no—
54#125×C61—22no—
55#125×C61—28no—
56#125×C61—28no—
57#125×C61—24no—
58#125×C61—28no—
59#125×C61—22no—
60#125×C61D412yes—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61D414yes—
64C4×Q8×C61—24no—
65C4×Q8×C61—24no—
66C4×Q8×C61—24no—
67C4×Q8×C61—22no—
68C4×Q8×C61—22no—
69C4×Q8×C61C2314yes—
70#117×C61—22no—
71#117×C61D412yes—
72C2×C42×C61C2311yescharacteristic
73C2×Q8×C61—22no—
74C2×Q8×C61—22no—
75C2×Q8×C61C2×D414yes—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D412yes—
78C2×C22⋊D61—41no—
79#11758222—44no—
80C2×D244—21no—
81C2×D244—22no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84C4⋊C4×C61—28no—
85C4⋊C4×C61—28no—
86C4⋊C4×C61—28no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—24no—
91C4⋊C4×C61—24no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61C2×D414yes—
95C4⋊C4×C61C2×D414yes—
96C22⋊C4×C61—24no—
97C22⋊C4×C61—22no—
98C42×C61—28no—
99C42×C61—22no—
100C42×C61—24no—
101C42×C61C2×D412yes—
102C42×C61C2×D411yescharacteristic
103C42×C61C2411yescharacteristic
104#11758201—84no—
105#11758198—42no—
106C4⋊Dic61—22no—
107#11758194—82no—
108C2×#92418—22no—
109C2×#92418—24no—
110C2×#92418—42no—
111C23×C61#12011yescharacteristic
112C22×D61—41no—
113Q8×C61—24no—
114Q8×C61—24no—
115Q8×C61—42no—
116Q8×C61—42no—
117Q8×C61—44no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—24no—
122C2×C4×C61—24no—
123C2×C4×C61—24no—
124C2×C4×C61#12012yes—
125C2×C4×C61#12011yescharacteristic
126C2×C4×C61C22×D411yescharacteristic
127C2×C4×C61C22×D412yes—
128C22⋊D61—82no—
129C2×Dic61—41no—
130D244—42no—
131D244—42no—
132D244—42no—
133C4×D61—82no—
134Dic122—42no—
135#92418—42no—
136#92418—44no—
137#18698—611nomaximal, Sylow
138C22×C61—22no—
139C22×C61—21no—
140C22×C61C2×#12011yescharacteristic
141C2×D61—82no—
142C2×D61—81no—
143C4×C61—22no—
144C4×C61—21no—
145C4×C61—44no—
146C4×C61—44no—
147C4×C61—44no—
148C4×C61#45412yes—
149C4×C61#57712yes—
150C4×C61#57812yes—
151Dic61—81no—
152#3556D6111yescharacteristic, complemented
153#3254—612no—
154#3131—611no—
155#1636—614no—
156#1626—614no—
157#1561—611no—
158#1493—612no—
159C2×C61—41no—
160C2×C61#301312yes—
161C2×C61C2×#45411yescharacteristic
162D61—161no—
163#564—22no—
164#557—22no—
165#557—22no—
166#557C2×D6114yes—
167#541—24no—
168#541—22no—
169#533C2×D6112yes—
170#533C2×D6111yescharacteristic
171#532—611no—
172#488—1222no—
173#487—612no—
174#486—612no—
175#477—614no—
176#474—1222no—
177#468—1224no—
178#461—1222no—
179#460—614no—
180#423—612no—
181#416—1222no—
182C2×C4×C8—611no—
183#332—614no—
184Q8⋊C8—618no—
185C61#1869811yescharacteristic, complemented, Sylow
186C2×#41—1221no—
187C2×SD16—1224no—
188C2×M16—1222no—
189C22×C8—1221no—
190C4⋊Q8—22no—
191C4⋊Q8—24no—
192C4⋊Q8—611no—
193C4⋊Q8C22⋊D6112yes—
194C4⋊Q8C22⋊D6112yes—
195C4⋊Q8C22×D6112yes—
196#127—611no—
197#126—28no—
198#125—22no—
199#125—28no—
200#125—28no—
201#125—24no—
202#125—28no—
203#125—22no—
204#125C22⋊D6112yes—
205#124—1221no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8C22⋊D6114yes—
209#121—1222no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8C22×D6114yes—
216C4×D4—1222no—
217#117—22no—
218#117C22⋊D6112yes—
219C2×C42C22×D6111yescharacteristic
220#107—1224no—
221C4⋊C8—614no—
222C4⋊C8—1222no—
223#103—1224no—
224#103—1224no—
225#102—1224no—
226#102—1224no—
227C4×C8—612no—
228C4×C8—1221no—
229#41—2444no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—1221no—
233C2×Q8C2×C22⋊D6114yes—
234C2×D4—1221no—
235C2×D4—1222no—
236C2×D4—2441no—
237C22×C4—2441no—
238C22×C4C2×C22⋊D6112yes—
239C22×C4C2×C22⋊D6111yescharacteristic
240SD16—4884no—
241M16—4882no—
242C2×C8—1222no—
243C2×C8—1224no—
244C2×C8—2442no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—1222no—
256C4⋊C4C2×C22⋊D6114yes—
257C4⋊C4C2×C22⋊D6114yes—
258C22⋊C4—24no—
259C22⋊C4—22no—
260C22⋊C4—2442no—
261C42—28no—
262C42—24no—
263C42—22no—
264C42C2×C22⋊D6111yescharacteristic
265C42C2×C22⋊D6112yes—
266C42C23×D6111yescharacteristic
267C23—2441no—
268C23#1451974611yescharacteristic
269Q8—24no—
270Q8—24no—
271Q8—42no—
272Q8—42no—
273Q8—44no—
274Q8—2442no—
275D4—4882no—
276D4—2442no—
277D4—2442no—
278D4—2442no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—4882no—
286C2×C4—2441no—
287C2×C4#1451974612yes—
288C2×C4#1451974611yescharacteristic
289C2×C4C22×C22⋊D6111yescharacteristic
290C2×C4C22×C22⋊D6112yes—
291C8—2442no—
292C8—2444no—
293C22—22no—
294C22—21no—
295C22—4882no—
296C22—4881no—
297C22C2×#1451974611yescharacteristic
298C4—22no—
299C4—21no—
300C4—44no—
301C4—44no—
302C4—44no—
303C4—4881no—
304C4#1683993512yes—
305C4#1684057012yes—
306C4#1684057712yes—
307C2—41no—
308C2—9761no—
309C2#2145513412yes—
310C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3556C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenter#3556D61
hypocenterC61#18698
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
3. #21444977C2semidirect
4. #21444915C2semidirect
5. #21444557C2semidirect
6. #21444274C2semidirect
7. C61×#3556C2semidirect
152. #3556D61semidirect
185. C61#18698semidirect

Maximal subgroups

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

subgroupindex
#214572022
#214561442
#214449772
#214449152
#214445572
#214442742
C61×#35562
#1869861