Finite Group Databasebeta

← #30602184

Subgroups — #30602184

Order 15616.

600 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 126 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#21461675C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3#21460657C214yesmaximal
4C2×#16839661C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×C22⋊#92420C2212yes—
7C2×C22⋊#92420C2212yes—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628—21no—
11C22×C4⋊Dic61C412yes—
12C22×C4⋊Dic61C411yescharacteristic
13C22×C4×Dic61C2211yescharacteristic
14#16839661—21no—
15#16839661—21no—
16#16839661—21no—
17#16839661—21no—
18C2×#14519684C2212yes—
19C22×C4⋊C4×C61C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519844—42no—
23C22⋊#92420—22no—
24C22⋊#92420—24no—
25C22⋊#92420Q814yes—
26#14519832—42no—
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61C2×C412yes—
29C23×Dic61C2311yescharacteristic
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61—22no—
34C2×C4⋊Dic61—44no—
35C2×C4⋊Dic61—42no—
36C2×C4×Dic61—42no—
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41#14519684—22no—
42#14519684—22no—
43C23×C4×C61C2×C412yes—
44C23×C4×C61C2×C411yescharacteristic
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—24no—
48C24×C61C22×C411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—42no—
55C22×C4×C61—42no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61M1614yes—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—24no—
63C22×Dic61—41no—
64C22×Dic61—41no—
65C22×Dic61—41no—
66C22×Dic61—42no—
67C22×Dic61—42no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C22×Dic61C2×Q812yes—
73C22×Dic61#4112yes—
74C2×#92420—24no—
75C2×#92420—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61—24no—
82C4⋊Dic61—44no—
83C4⋊Dic61—42no—
84C4⋊Dic61—42no—
85C4⋊Dic61—44no—
86C4×Dic61—22no—
87C4×Dic61—22no—
88C4×Dic61—41no—
89C4×Dic61—41no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61C2×C22⋊C411yescharacteristic
95C23×C61C4×Q812yes—
96C23×C61C2×M1612yes—
97C23×C61#13112yes—
98C2×C4×C61—28no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—24no—
104C2×C4×C61—44no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—44no—
108C2×C4×C61#9912yes—
109C2×C4×C61#10112yes—
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—44no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—84no—
126C2×Dic61#10012yes—
127C2×Dic61C22⋊Q812yes—
128C2×Dic61#12411yescharacteristic
129C2×Dic61C4⋊Q811yescharacteristic
130#92420—44no—
131#92420—82no—
132#9883—611nomaximal, Sylow
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61#39111yescharacteristic
140C22×C61#40711yescharacteristic
141C22×C61#40811yescharacteristic
142C22×C61C2×#9911yescharacteristic
143C22×C61#41011yescharacteristic
144C22×C61C2×#10011yescharacteristic
145C22×C61C2×#10111yescharacteristic
146C22×C61#44614yes—
147C4×C61—22no—
148C4×C61—22no—
149C4×C61—44no—
150Dic61—22no—
151Dic61—22no—
152Dic61—21no—
153Dic61—21no—
154Dic61—41no—
155Dic61—82no—
156C2×#389—611no—
157#1797—611no—
158#1749—611no—
159#1280—614no—
160C2×C61—24no—
161C2×C61#146311yescharacteristic
162C2×C61#146611yescharacteristic
163C2×C61#147011yescharacteristic
164C2×C61#147111yescharacteristic
165C2×C61#198511yescharacteristic
166C2×C61#199611yescharacteristic
167C2×C61#199711yescharacteristic
168C22×C4⋊C4—612no—
169C22×C4⋊C4—611no—
170C22×C4⋊C4#692311yescharacteristic
171C22×C42—611no—
172C2×C4⋊C8—1221no—
173C2×C22⋊C8—612no—
174C2×C22⋊C8—612no—
175C2×#97—1221no—
176#389—1221no—
177#389—1221no—
178#389—1221no—
179#389—1221no—
180C2×#95—612no—
181#336—1221no—
182#336—1221no—
183C61#988311yescharacteristic, complemented, Sylow
184C23×C4—611no—
185C23×C4—611no—
186C23×C4—612no—
187C23×C4C2×#692312yes—
188C23×C4C2×#692311yescharacteristic
189C22×C8—1222no—
190C22×C8—1222no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—2442no—
199C2×C4⋊C4—2444no—
200C2×C42—611no—
201C2×C42—611no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—2442no—
205C4⋊C8—2442no—
206C22⋊C8—614no—
207C22⋊C8—1222no—
208C22⋊C8—1224no—
209#97—2442no—
210#95—1222no—
211#95—1222no—
212C24C22×#692311yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—611no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—612no—
224C22×C4—612no—
225C22×C4—611no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—1224no—
229C22×C4—1222no—
230C22×C4—2441no—
231C22×C4—2441no—
232C22×C4—2441no—
233C22×C4—2442no—
234C22×C4—2442no—
235C22×C4#1175821714yes—
236C22×C4C22⋊#692311yescharacteristic
237C22×C4C22⋊#692311yescharacteristic
238C2×C8—1224no—
239C2×C8—2442no—
240C2×C8—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8—2442no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—2442no—
247C4⋊C4—2444no—
248C4⋊C4—2444no—
249C4⋊C4—2442no—
250C42—1222no—
251C42—1222no—
252C42—2441no—
253C42—2441no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23C2×C22⋊#692311yescharacteristic
259C23Q8×#692312yes—
260C23C2×#1175821712yes—
261C23#1451986512yes—
262C2×C4—28no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—612no—
273C2×C4—611no—
274C2×C4—611no—
275C2×C4—612no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—4884no—
286C2×C4—2442no—
287C2×C4—2442no—
288C2×C4—2444no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4#1451983612yes—
293C2×C4#1451983812yes—
294C8—4882no—
295C8—2444no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22(C2×#6923)⋊Q811yescharacteristic
303C22#1684073211yescharacteristic
304C22#1684073711yescharacteristic
305C22C2×#1451983611yescharacteristic
306C22#1684074211yescharacteristic
307C22D61⋊#10011yescharacteristic
308C22C2×#1451983811yescharacteristic
309C22#1684081314yes—
310C4—22no—
311C4—22no—
312C4—44no—
313C4—1222no—
314C4—1221no—
315C4—1222no—
316C4—1221no—
317C4—4882no—
318C4—2441no—
319C2—24no—
320C2#2146106711yescharacteristic
321C2#2146107911yescharacteristic
322C2#2146109311yescharacteristic
323C2#2146109611yescharacteristic
324C2#2146199311yescharacteristic
325C2#2146200911yescharacteristic
326C2#2146201211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutator, prime radicalC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#9883
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
183. C61#9883semidirect

Maximal subgroups

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

subgroupindex
#214616752
#214615452
#214606572
C2×#168396612
#988361