Finite Group Databasebeta

← #30601817

Subgroups — #30601817

Order 15616.

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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#21461676C211yesmaximal, characteristic
2C2×#16840615C214yesmaximal
3#21460995C211yesmaximal, characteristic
4#21454172C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×#14519834C2212yes—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840615—28no—
12C2×#14519777C411yescharacteristic, complemented
13C2×#14519757C2212yes—
14C2×C42×D61C2211yescharacteristic
15#16839667C412yescomplemented
16#16839661—21no—
17#16839661—21no—
18#16839619—21no—
19#16839619—21no—
20C61×#384C411yescharacteristic, complemented
21C23×#6923—22no—
22C22×#92420—22no—
23#14519844—42no—
24#14519834—24no—
25(C2×#6923)⋊C4—22no—
26(C2×#6923)⋊C4—42no—
27(C2×#6923)⋊C4—42no—
28(C2×#6923)⋊C4D414yes—
29C22×Dic122C2×C412yes—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61C2311yescharacteristic
33#14519777—42no—
34#14519757—24no—
35#14519757—24no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—22no—
39C2×#11758198—22no—
40C2×C4×Dic61C2×C411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C42×D61—42no—
44#14519684—22no—
45#14519684—22no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C412yes—
49C2×C42×C61C2×C411yescharacteristic
50C23×D61C2×D411yescharacteristic
51C22×#6923—24no—
52C22×#6923—42no—
53C22×#6923—42no—
54C22×#6923—42no—
55C22×#6923—42no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×C4×C61—22no—
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C22×Dic61—22no—
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C22⋊C411yescharacteristic
65C22×Dic61C2×D411yescharacteristic
66C2×C4×D61—22no—
67C2×C4×D61—22no—
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61SD1612yes—
74C2×C4×D61Q1612yes—
75C2×C4×D61C2×D412yes—
76C2×C4×D61#4112yes—
77C2×Dic122—24no—
78C2×Dic122—42no—
79C2×Dic122—42no—
80C2×#92420—42no—
81C2×#92420—42no—
82C2×#92420—42no—
83C4⋊C4×C61—24no—
84C4⋊C4×C61—42no—
85C4⋊C4×C61—42no—
86C42×C61—22no—
87#11758198—44no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90C4⋊Dic61—42no—
91C4⋊Dic61—42no—
92C4×Dic61—22no—
93C4×Dic61—41no—
94C4×Dic61—41no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—21no—
100C22×D61#12112yes—
101C22×D61C2×SD1611yescharacteristic
102C22×D61C2×Q1611yescharacteristic
103C22×D61#13712yes—
104C2×#6923—44no—
105C2×#6923—44no—
106C2×#6923—44no—
107C2×#6923—82no—
108C2×#6923—82no—
109C2×#6923—82no—
110C2×#6923—82no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61#9912yes—
121C2×C4×C61#10312yes—
122C2×C4×C61C4×D412yes—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61#10312yes—
134C2×Dic61#12411yescharacteristic
135C2×Dic61#12711yescharacteristic
136C4×D61—42no—
137C4×D61—42no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—44no—
141Dic122—84no—
142#92420—82no—
143#9714—611nomaximal, Sylow
144C22×C61#39011yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61#41011yescharacteristic
147C22×C61C2×#10311yescharacteristic
148C22×C61#41911yescharacteristic
149C22×C61#42111yescharacteristic
150C22×C61#42111yescharacteristic
151C2×D61—22no—
152C2×D61—22no—
153C2×D61—22no—
154C2×D61—22no—
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61#46112yes—
160C2×D61#46212yes—
161#6923—84no—
162C4×C61—22no—
163C4×C61—22no—
164C4×C61—22no—
165C4×C61—82no—
166Dic61—21no—
167Dic61—21no—
168Dic61—22no—
169Dic61—41no—
170Dic61—82no—
171#2857—611no—
172#1797—611no—
173C2×#328—614no—
174#1434—611no—
175C2×C61#151111yescharacteristic
176C2×C61#151511yescharacteristic
177C2×C61#151811yescharacteristic
178C2×C61#152011yescharacteristic
179C2×C61#196311yescharacteristic
180C2×C61#196411yescharacteristic
181C2×C61#198111yescharacteristic
182D61—22no—
183D61—22no—
184C2×C4⋊Q8—611no—
185C2×C22⋊Q8—612no—
186C22×C42—611no—
187C2×C4⋊C8—1221no—
188C2×C22⋊C8—612no—
189#391—612no—
190#389—1221no—
191#389—1221no—
192#384#692311yescharacteristic, complemented
193#380—1221no—
194#380—1221no—
195C2×#95—612no—
196C2×#95—1221no—
197C2×#95—1221no—
198#336—1221no—
199#328—1228no—
200C61#971411yescharacteristic, complemented, Sylow
201C22×Q8—612no—
202C23×C4—611no—
203C23×C4—611no—
204C23×C4—611no—
205C23×C4—1222no—
206C22×C8—1222no—
207C4⋊Q8—2442no—
208C22⋊Q8—1224no—
209C22⋊Q8—1224no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1222no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4C2×#692312yes—
216C2×C22⋊C4—1222no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C2×C42—2442no—
221C2×C42C2×#692311yescharacteristic
222C4⋊C8—2442no—
223C22⋊C8—1224no—
224#95—614no—
225#95—1222no—
226#95—1222no—
227#95—1222no—
228#95—2442no—
229#95—2442no—
230C24—611no—
231C2×Q8—1224no—
232C2×Q8—2442no—
233C2×Q8—2442no—
234C22×C4—22no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—612no—
240C22×C4—612no—
241C22×C4—611no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1224no—
245C22×C4—1222no—
246C22×C4—2441no—
247C22×C4—2441no—
248C22×C4—2441no—
249C22×C4—2441no—
250C22×C4—2441no—
251C22×C4—2442no—
252C22×C4—2442no—
253C22×C4—2442no—
254C22×C4—2442no—
255C22×C4—2442no—
256C22×C4—2442no—
257C22×C4C22⋊#692311yescharacteristic
258C22×C4C22⋊#692311yescharacteristic
259C22×C4C22×#692311yescharacteristic
260C2×C8—2442no—
261C2×C8—2442no—
262C2×C8—2442no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—2442no—
267C4⋊C4—2442no—
268C4⋊C4—2442no—
269C4⋊C4—2442no—
270C22⋊C4—2444no—
271C42—22no—
272C42—1222no—
273C42—2441no—
274C42—2441no—
275C23—612no—
276C23—611no—
277C23—611no—
278C23—612no—
279C23—1221no—
280C23—1221no—
281C23—1221no—
282C23—1221no—
283C23C2×C22⋊#692311yescharacteristic
284Q8—4884no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—612no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—4882no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—4882no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4—2442no—
312C2×C4—2444no—
313C2×C4—2444no—
314C2×C4—2444no—
315C2×C4—2444no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4#1451983612yes—
320C2×C4#1451984012yes—
321C2×C4D4×#692312yes—
322C8—4882no—
323C22—612no—
324C22—612no—
325C22—1221no—
326C22—1221no—
327C22—1221no—
328C22—1221no—
329C22—1222no—
330C22—1222no—
331C22—1222no—
332C22—1222no—
333C22#1684071411yescharacteristic
334C22C2×#1451983611yescharacteristic
335C22#1684074311yescharacteristic
336C22C2×#1451984011yescharacteristic
337C22#1684075511yescharacteristic
338C22#1684076011yescharacteristic
339C22#1684076011yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—82no—
344C4—1222no—
345C4—1221no—
346C4—1221no—
347C4—4884no—
348C4—4882no—
349C4—2441no—
350C2—1222no—
351C2—1222no—
352C2#2146122711yescharacteristic
353C2#2146124211yescharacteristic
354C2#2146125111yescharacteristic
355C2#2146125811yescharacteristic
356C2#2146195711yescharacteristic
357C2#2146195911yescharacteristic
358C2#2146198511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#9714
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
12. C2×#14519777C4semidirect
15. #16839667C4semidirect
20. C61×#384C4semidirect
192. #384#6923semidirect
200. C61#9714semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406152
#214609952
#214541722
#971461