Finite Group Databasebeta

← #30598246

Subgroups — #30598246

Order 15616.

1718 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 152 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#21461676C212yesmaximal, complemented
2C2×#16840620C214yesmaximal, complemented
3C2×#127×D61C211yesmaximal, characteristic
4C2×#14519844C2214yescomplemented
5C2×#14519834—22no—
6#16840628—22no—
7#16840620—28no—
8C22×D4×D61—22no—
9C22×D4×D61C412yes—
10C2×#14519774C411yescharacteristic
11C2×#14519773—22no—
12C2×#14519773C2212yes—
13#127×D61—28no—
14C2×C42×D61C2211yescharacteristic
15C2×#127×C61C411yescharacteristic
16C22×#92420—24no—
17#14519844—28no—
18#14519834—44no—
19C24×D61—22no—
20C22×D244—22no—
21C22×D244C2×C412yes—
22C22×C22⋊D61—22no—
23C22×C22⋊D61—24no—
24C2×D4×D61—216no—
25C2×D4×D61—24no—
26C2×D4×D61—24no—
27C2×D4×D61—44no—
28C2×D4×D61—48no—
29C2×D4×D61—44no—
30C2×D4×D61—44no—
31C2×D4×D61—44no—
32C22×C4×D61C2×C411yescharacteristic
33C22×C4×D61D412yes—
34#14519774—24no—
35#14519773—216no—
36#14519773—24no—
37#14519773D418yes—
38C2×C4×Dic61D412yes—
39C2×C4×Dic61C2311yescharacteristic
40C42×D61—24no—
41C22×D4×C61—22no—
42C22×D4×C61C2×C412yes—
43#127×C61—24no—
44C2×C42×C61C2×C411yescharacteristic
45C24×C61—22no—
46C23×D61—28no—
47C23×D61—22no—
48C23×D61—44no—
49C23×D61—48no—
50C23×D61—44no—
51C23×D61—48no—
52C23×D61C22⋊C411yescharacteristic
53C2×D4×C61—28no—
54C2×D4×C61—22no—
55C2×D4×C61—44no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C2×D4×C61—44no—
59C2×D4×C61C22⋊C414yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C2×C22⋊D61—28no—
63C2×C22⋊D61—216no—
64C2×C22⋊D61—44no—
65C2×C22⋊D61—44no—
66C2×C22⋊D61—44no—
67C2×C22⋊D61—44no—
68C2×C22⋊D61—48no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D412yes—
71D4×D61—416no—
72D4×D61—416no—
73D4×D61—432no—
74D4×D61—432no—
75C2×D244—28no—
76C2×D244—22no—
77C2×D244—44no—
78C2×D244—42no—
79C2×D244—42no—
80C2×D244—44no—
81C2×D244C22⋊C414yes—
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—24no—
86C2×C4×D61—24no—
87C2×C4×D61D814yes—
88C2×C4×D61SD1614yes—
89C2×#92420—44no—
90C2×#92420—44no—
91C2×#92420—44no—
92C42×C61—22no—
93C4×Dic61—24no—
94C4×Dic61C2×D414yes—
95C23×C61—24no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—44no—
99C23×C61—44no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—22no—
104C22×D61—24no—
105C22×D61—416no—
106C22×D61—432no—
107C22×D61—42no—
108C22×D61—42no—
109C22×D61—44no—
110C22×D61—44no—
111C22×D61—816no—
112C22×D61—88no—
113C22×D61—88no—
114C22×D61#10012yes—
115C22×D61#10214yes—
116D4×C61—44no—
117D4×C61—44no—
118D4×C61—416no—
119C2×C4×C61—22no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—22no—
123C2×C4×C61#10214yes—
124C2×C4×C61C2×C22⋊C412yes—
125C22⋊D61—416no—
126C22⋊D61—416no—
127C22⋊D61—416no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61#10012yes—
131C2×Dic61#12014yes—
132C2×Dic61C2×D812yes—
133C2×Dic61C2×SD1612yes—
134C2×Dic61AGL(1,ℤ/8)14yes—
135D244—44no—
136D244—44no—
137D244—416no—
138C4×D61—28no—
139C4×D61—28no—
140C4×D61—28no—
141#92420—84no—
142#8339—611nomaximal, Sylow
143C22×C61—44no—
144C22×C61—48no—
145C22×C61—82no—
146C22×C61—84no—
147C22×C61—82no—
148C22×C61#37911yescharacteristic
149C22×C61C2×#10011yescharacteristic
150C22×C61C2×#10011yescharacteristic
151C22×C61C2×#10212yes—
152C22×C61#41812yes—
153C2×D61—24no—
154C2×D61—22no—
155C2×D61—22no—
156C2×D61—24no—
157C2×D61—44no—
158C2×D61—48no—
159C2×D61—832no—
160C2×D61—82no—
161C2×D61—84no—
162C2×D61—82no—
163C2×D61#33114yescomplemented
164C4×C61—22no—
165C4×C61—24no—
166Dic61—22no—
167Dic61#44714yes—
168Dic61#45014yes—
169C22×#127—611no—
170#1797—612no—
171C2×#331—614no—
172C2×C61—84no—
173C2×C61C2×#33111yescharacteristic
174C2×C61#154512yes—
175C2×C61#155511yescharacteristic
176C2×C61#178311yescharacteristic
177C2×C61#178612yes—
178D61—24no—
179D61—84no—
180C23×D4—612no—
181C23×D4—1222no—
182C2×#127—612no—
183C2×#127—611no—
184C2×#127—1222no—
185C2×#127—1228no—
186C2×#127#692311yescharacteristic
187C22×C42—611no—
188C2×C4⋊C8—614no—
189C2×C22⋊C8—1222no—
190#336—1222no—
191#331—1228no—
192C61#833911yescharacteristic, complemented, Sylow
193C25—1222no—
194C22×D4—22no—
195C22×D4—612no—
196C22×D4—1224no—
197C22×D4—1222no—
198C22×D4—1224no—
199C22×D4—12216no—
200C22×D4—1224no—
201C22×D4—1222no—
202C22×D4—2444no—
203C22×D4—2444no—
204C22×D4—2444no—
205C22×D4—2448no—
206C22×D4—2444no—
207C22×D4C2×#692312yes—
208C23×C4—612no—
209C23×C4—611no—
210C22×C8—1224no—
211#127—24no—
212#127—618no—
213#127—1224no—
214#127—1224no—
215#127—12216no—
216C2×C42—611no—
217C2×C42—612no—
218C2×C42—1224no—
219C2×C42C2×#692311yescharacteristic
220C4⋊C8—1228no—
221C22⋊C8—2444no—
222C24—22no—
223C24—611no—
224C24—1228no—
225C24—1222no—
226C24—2448no—
227C24—2444no—
228C24—2444no—
229C24—2448no—
230C2×D4—28no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—614no—
237C2×D4—1228no—
238C2×D4—12216no—
239C2×D4—1228no—
240C2×D4—1222no—
241C2×D4—2442no—
242C2×D4—2442no—
243C2×D4—2444no—
244C2×D4—2444no—
245C2×D4—2444no—
246C2×D4—2448no—
247C2×D4—2444no—
248C2×D4—24432no—
249C2×D4—24432no—
250C2×D4—24416no—
251C2×D4—24416no—
252C2×D4—2444no—
253C2×D4—2444no—
254C2×D4C22⋊#692314yes—
255C22×C4—612no—
256C22×C4—611no—
257C22×C4—614no—
258C22×C4—614no—
259C22×C4—1224no—
260C22×C4—1222no—
261C22×C4—1224no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4C22⋊#692312yes—
265C22×C4C22×#692311yescharacteristic
266C2×C8—2444no—
267C2×C8—2444no—
268C2×C8—2444no—
269C42—22no—
270C42—614no—
271C42—1224no—
272C23—24no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—612no—
278C23—614no—
279C23—1221no—
280C23—1222no—
281C23—1221no—
282C23—1224no—
283C23—4888no—
284C23—48816no—
285C23—4888no—
286C23—24416no—
287C23—24432no—
288C23—2442no—
289C23—2442no—
290C23—2444no—
291C23—2444no—
292C23C2×C22⋊#692311yescharacteristic
293D4—44no—
294D4—44no—
295D4—416no—
296D4—2444no—
297D4—2444no—
298D4—24416no—
299D4—24416no—
300D4—24416no—
301D4—24416no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—612no—
307C2×C4—612no—
308C2×C4—614no—
309C2×C4—614no—
310C2×C4—612no—
311C2×C4—1228no—
312C2×C4—1228no—
313C2×C4—1228no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4#1451983914yes—
317C2×C4C2×C22⋊#692312yes—
318C8—4884no—
319C22—44no—
320C22—48no—
321C22—82no—
322C22—84no—
323C22—82no—
324C22—614no—
325C22—1224no—
326C22—1222no—
327C22—1222no—
328C22—1224no—
329C22—48832no—
330C22—4882no—
331C22—4884no—
332C22—4882no—
333C22—2444no—
334C22—2448no—
335C22#1684070311yescharacteristic
336C22D61⋊#10011yescharacteristic
337C22D61⋊#10011yescharacteristic
338C22C2×#1451983912yes—
339C22#1684075412yes—
340C4—22no—
341C4—24no—
342C4—614no—
343C4—614no—
344C4—1222no—
345C2—84no—
346C2—1224no—
347C2—4884no—
348C2D61⋊#33111yescharacteristic
349C2#2146132712yes—
350C2#2146134711yescharacteristic
351C2#2146165211yescharacteristic
352C2#2146165712yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#127×D61C2
cosocleC2×C4×Dic61C23
hypercenterC2×#127#6923
hypocenterC61#8339
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461676C2semidirect
2. C2×#16840620C2semidirect
4. C2×#14519844C22semidirect
163. C2×D61#331semidirect
192. C61#8339semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406202
C2×#127×D612
#833961