Finite Group Databasebeta

← #30606572

Subgroups — #30606572

Order 15616.

460 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462543C211yesmaximal, characteristic
2#21462091C211yesmaximal, characteristic
3#21461578C212yesmaximal, complemented
4#21460921C212yesmaximal, complemented
5#21447501C211yesmaximal, characteristic
6C2×#14519865C2211yescharacteristic
7C2×#14519864C411yescharacteristic, complemented
8#16840795—22no—
9#16840789—22no—
10#16840767C411yescharacteristic, complemented
11#16840743C2212yes—
12#16840737C2211yescharacteristic
13#16840736C411yescharacteristic, complemented
14#16840728—22no—
15C4×D61⋊C8C411yescharacteristic, complemented
16#16840669C412yes—
17#16840666C412yes—
18#16840645—22no—
19#16840311C2211yescharacteristic
20#16839745C2212yes—
21#16839560C412yescomplemented
22#16839559C411yescharacteristic, complemented
23C61×#355C411yescharacteristic, complemented
24#14519865—41no—
25#14519865—41no—
26#14519864—41no—
27#14519864—41no—
28C2×#11758213C2×C411yescharacteristic
29C2×#11758213C2×C411yescharacteristic
30C2×#11758213D412yes—
31C2×#11758217—21no—
32C2×D61⋊C8C2×C411yescharacteristic
33C2×D61⋊C8C2×C411yescharacteristic
34C2×D61⋊C8D412yes—
35C22×#92420—21no—
36#14519856—22no—
37#14519844—22no—
38#14519836—22no—
39#14519836C2×C414yes—
40#14519834—22no—
41C22⋊#92420—22no—
42#14519831—42no—
43C4×#92420—21no—
44C4×#92420—21no—
45C8×#6923—42no—
46C2×#11758226C2311yescharacteristic
47C2×#11758224C2×C411yescharacteristic
48M16×D61—22no—
49#14519775C2×C411yescharacteristic
50#14519773C2×C411yescharacteristic
51#14519765C2×C411yescharacteristic
52#14519750—22no—
53#14519737—22no—
54C42×D61C2×C411yescharacteristic
55#14519696C2×C412yes—
56#14519693—22no—
57#124×C61C2×C411yescharacteristic
58#101×C61C2×C412yes—
59C2×Q8×C61C22×C411yescharacteristic
60C2×D4×C61C4211yescharacteristic
61C2×C22⋊D61—21no—
62C2×C22⋊D61—22no—
63C22×Dic61—21no—
64#11758226—41no—
65#11758226—41no—
66#11758224—41no—
67#11758224—41no—
68C2×D244C22×C411yescharacteristic
69C2×D244#4112yes—
70C2×C4×D61—21no—
71C2×C4×D61C22×C411yescharacteristic
72C2×C4×D61C2×D412yes—
73C2×Dic122C4211yescharacteristic
74C22⋊#6923—24no—
75#11758217—42no—
76C2×#92420—21no—
77C2×#92420—21no—
78C2×#92420—41no—
79C2×#92420—41no—
80C2×#92420C22⋊C411yescharacteristic
81C2×#92420C22⋊C412yes—
82C2×#92420C22⋊C411yescharacteristic
83C4⋊#6923—42no—
84C4×#6923—42no—
85#11758213—41no—
86#11758213—42no—
87#11758213—41no—
88D61⋊C8—22no—
89D61⋊C8—22no—
90D61⋊C8—21no—
91D61⋊C8—21no—
92D61⋊C8—41no—
93M16×C61—22no—
94C22⋊C4×C61—22no—
95C42×C61C4211yescharacteristic
96#11758198—22no—
97#11758196—22no—
98C4×Dic61—21no—
99C4×Dic61C4211yescharacteristic
100#11758194—22no—
101#11758189—42no—
102C8×D61—42no—
103C23×C61—21no—
104C22×D61#11711yescharacteristic
105C22×D61C4×D412yes—
106C2×#6923—44no—
107Q8×C61—41no—
108D4×C61—41no—
109C2×C4×C61—21no—
110C2×C4×C61C2×C4211yescharacteristic
111C2×C4×C61C4×D412yes—
112C22⋊D61—44no—
113C22⋊D61—42no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61C2×C22⋊C411yescharacteristic
119D244—41no—
120D244—42no—
121C4×D61—22no—
122C4×D61—21no—
123C4×D61—22no—
124C4×D61—21no—
125C4×D61—42no—
126Dic122—41no—
127#92420—22no—
128#92420—21no—
129#92420—41no—
130#92420—41no—
131#92420#9911yescharacteristic
132#92420#9911yescharacteristic
133C8×C61—42no—
134#92418—42no—
135#12633—611nomaximal, Sylow
136C22×C61—42no—
137C22×C61—41no—
138C22×C61C4×C22⋊C411yescharacteristic
139C2×D61—21no—
140C2×D61—22no—
141C2×D61—41no—
142C2×D61—42no—
143#6923—84no—
144C4×C61—22no—
145C4×C61—21no—
146C4×C61—41no—
147Dic61—22no—
148Dic61—41no—
149Dic61C2×#9911yescharacteristic
150Dic61#41011yescharacteristic
151#2931—611no—
152#2132—611no—
153#2126—611no—
154#1757—612no—
155#1395—612no—
156C2×C61—21no—
157C2×C61—81no—
158C2×C61C4×#9911yescharacteristic
159D61—42no—
160D61—41no—
161C2×#131—611no—
162C2×#131—611no—
163#532—611no—
164#436—1222no—
165C8×D4—1222no—
166#423—611no—
167#413—612no—
168#410—612no—
169#408—611no—
170#408—611no—
171#405—1222no—
172C2×C4×C8—611no—
173#356—612no—
174#355—611no—
175#355#692311yescharacteristic, complemented
176#354—612no—
177#353—612no—
178#344—1222no—
179C61#1263311yescharacteristic, complemented, Sylow
180C2×#41—611no—
181C2×#41—611no—
182#131—2441no—
183#131—2441no—
184#131—2441no—
185#131—2441no—
186C2×M16—611no—
187C2×M16—612no—
188C2×M16—611no—
189C2×M16—1221no—
190C2×M16—1222no—
191C22×C8—611no—
192C22×C8—612no—
193C22×C8—611no—
194C22×C8—1221no—
195C4⋊Q8—611no—
196#127—611no—
197#124—611no—
198#124C2×#692311yescharacteristic
199#121—1222no—
200C4×D4—1222no—
201#117—1222no—
202C2×C42—611no—
203C4⋊C8—1222no—
204#101—612no—
205#101C2×#692312yes—
206#100—1222no—
207#99—614no—
208#99—1222no—
209C22⋊C8—1222no—
210C22⋊C8—1222no—
211#97—2442no—
212C4×C8—1221no—
213C4×C8—1221no—
214C4×C8—2442no—
215#41—2441no—
216#41—2441no—
217#41—2441no—
218#41—2441no—
219C2×Q8—611no—
220C2×Q8C22×#692311yescharacteristic
221C2×D4—611no—
222C2×D4—612no—
223C2×D4—1222no—
224C2×D4—1221no—
225C2×D4C4×#692311yescharacteristic
226C22×C4—611no—
227C22×C4—612no—
228C22×C4—1221no—
229C22×C4—1221no—
230M16—22no—
231M16—1222no—
232M16—2442no—
233M16—2441no—
234M16—2442no—
235M16—2442no—
236M16—2441no—
237C2×C8—611no—
238C2×C8—611no—
239C2×C8—612no—
240C2×C8—1222no—
241C2×C8—1222no—
242C2×C8—1221no—
243C2×C8—1221no—
244C2×C8—1221no—
245C2×C8—1221no—
246C2×C8—2442no—
247C2×C8—2441no—
248C2×C8—2441no—
249C2×C8—2441no—
250C4⋊C4—1222no—
251C4⋊C4—2442no—
252C22⋊C4—22no—
253C22⋊C4—1222no—
254C22⋊C4—1224no—
255C42—611no—
256C42—1221no—
257C42—2442no—
258C42C4×#692311yescharacteristic
259C23—21no—
260C23—611no—
261C23—612no—
262Q8—41no—
263Q8—2441no—
264D4—41no—
265D4—2442no—
266D4—2441no—
267D4—2444no—
268D4—2442no—
269C2×C4—21no—
270C2×C4—611no—
271C2×C4—1222no—
272C2×C4—1222no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—2442no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2444no—
281C2×C4C2×C4×#692311yescharacteristic
282C2×C4D4×#692312yes—
283C8—42no—
284C8—611no—
285C8—611no—
286C8—1222no—
287C8—1221no—
288C8—2442no—
289C8—2441no—
290C8—2441no—
291C22—42no—
292C22—41no—
293C22—1222no—
294C22—1221no—
295C22—2441no—
296C22—2442no—
297C22C22⋊C4×#692311yescharacteristic
298C4—22no—
299C4—21no—
300C4—41no—
301C4—611no—
302C4—611no—
303C4—1222no—
304C4—4884no—
305C4—2441no—
306C2—21no—
307C2—81no—
308C2—2442no—
309C2—2441no—
310C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#355C4
socleC2×C61C4×#99
prime radical#16840311C22
cosocleC2×#11758226C23
hypercenter#355#6923
hypocenterC61#12633
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
3. #21461578C2semidirect
4. #21460921C2semidirect
7. C2×#14519864C4semidirect
10. #16840767C4semidirect
13. #16840736C4semidirect
15. C4×D61⋊C8C4semidirect
21. #16839560C4semidirect
22. #16839559C4semidirect
23. C61×#355C4semidirect
175. #355#6923semidirect
179. C61#12633semidirect

Maximal subgroups

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

subgroupindex
#214625432
#214620912
#214615782
#214609212
#214475012
#1263361