Finite Group Databasebeta

← #30601746

Subgroups — #30601746

Order 15616.

686 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×#14519834C211yesmaximal, characteristic
2C2×#16840615C214yesmaximal
3#21460997C211yesmaximal, characteristic
4#21454172C211yesmaximal, characteristic
5C22×C4⋊#6923—21no—
6C2×#14519834C2212yes—
7C2×C4×#92420—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—
23C2×C4⋊#6923—24no—
24C2×C4⋊#6923—42no—
25C2×C4⋊#6923—42no—
26#14519834—22no—
27#14519834D414yes—
28C4×#92420—42no—
29(C2×#6923)⋊C4—24no—
30(C2×#6923)⋊C4—42no—
31C22×Dic122C2×C412yes—
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C22×C4×D61C2311yescharacteristic
35#14519777—42no—
36#14519757—24no—
37#14519757—24no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—22no—
41C2×#11758198—22no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C42×D61—42no—
46#14519684—22no—
47#14519684—22no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61C2×C412yes—
51C2×C42×C61C2×C411yescharacteristic
52C23×D61C2×D411yescharacteristic
53C22×#6923—24no—
54C22×#6923—42no—
55C22×#6923—42no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×#6923—42no—
59C22×#6923—42no—
60C22×C4×C61—22no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—22no—
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×C4×D61—22no—
69C2×C4×D61—22no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×C4×D61SD1612yes—
75C2×C4×D61SD1612yes—
76C2×C4×D61Q1612yes—
77C2×C4×D61Q1612yes—
78C2×C4×D61C2×D412yes—
79C2×C4×D61#4112yes—
80C2×Dic122—24no—
81C2×Dic122—42no—
82C2×Dic122—42no—
83C2×#92420—24no—
84C2×#92420—42no—
85C2×#92420—42no—
86C4⋊#6923—48no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—42no—
89C4⋊C4×C61—42no—
90C42×C61—22no—
91#11758198—42no—
92#11758198—42no—
93C4⋊Dic61—42no—
94C4⋊Dic61—42no—
95C4⋊Dic61—44no—
96C4×Dic61—22no—
97C4×Dic61—41no—
98C4×Dic61—41no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61#12411yescharacteristic
105C22×D61#12711yescharacteristic
106C22×D61C2×SD1611yescharacteristic
107C22×D61C2×SD1611yescharacteristic
108C22×D61C2×Q1611yescharacteristic
109C22×D61C2×Q1611yescharacteristic
110C2×#6923—44no—
111C2×#6923—44no—
112C2×#6923—44no—
113C2×#6923—82no—
114C2×#6923—82no—
115C2×#6923—82no—
116C2×#6923—82no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61#9912yes—
125C2×C4×C61#10312yes—
126C2×C4×C61#10312yes—
127C2×C4×C61#10412yes—
128C2×C4×C61C4×D412yes—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61#10312yes—
139C2×Dic61#10312yes—
140C2×Dic61#12112yes—
141C4×D61—24no—
142C4×D61—24no—
143C4×D61—42no—
144C4×D61—42no—
145C4×D61—42no—
146C4×D61—42no—
147Dic122—84no—
148#92420—44no—
149#9679—611nomaximal, Sylow
150C22×C61#39011yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61#41011yescharacteristic
153C22×C61C2×#10311yescharacteristic
154C22×C61C2×#10311yescharacteristic
155C22×C61C2×#10411yescharacteristic
156C22×C61#42111yescharacteristic
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—21no—
167C2×D61—21no—
168C2×D61—21no—
169C2×D61#48711yescharacteristic
170C2×D61#48711yescharacteristic
171C2×D61#49211yescharacteristic
172C2×D61#49411yescharacteristic
173#6923—84no—
174C4×C61—22no—
175C4×C61—22no—
176C4×C61—82no—
177C4×C61#32814yescomplemented
178Dic61—22no—
179Dic61—22no—
180Dic61—22no—
181Dic61—82no—
182#2857—611no—
183C4×C22⋊C8—611no—
184C2×#328—614no—
185C4⋊#95—611no—
186C2×C61C2×#32811yescharacteristic
187C2×C61#150311yescharacteristic
188C2×C61#151811yescharacteristic
189C2×C61#152011yescharacteristic
190C2×C61#195611yescharacteristic
191C2×C61#196411yescharacteristic
192C2×C61#198011yescharacteristic
193D61—21no—
194D61—21no—
195D61—21no—
196D61—21no—
197C2×C4⋊Q8—611no—
198C2×C22⋊Q8—612no—
199C22×C4⋊C4—1221no—
200C22×C42—611no—
201C2×C22⋊C8—612no—
202C2×C4×C8—1221no—
203#391—612no—
204#389—1221no—
205#389—1221no—
206#384#692311yescharacteristic, complemented
207#380—1221no—
208#380—1221no—
209C2×#95—612no—
210C2×#95—1221no—
211#336—1221no—
212#328—1228no—
213C61#967911yescharacteristic, complemented, Sylow
214C22×Q8—612no—
215C23×C4—611no—
216C23×C4—611no—
217C23×C4—611no—
218C23×C4—1222no—
219C22×C8—1222no—
220C4⋊Q8—2442no—
221C22⋊Q8—1224no—
222C22⋊Q8—1224no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1222no—
226C2×C4⋊C4—1224no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—2442no—
230C2×C4⋊C4—2442no—
231C2×C4⋊C4C2×#692312yes—
232C2×C22⋊C4—1222no—
233C2×C42—611no—
234C2×C42—611no—
235C2×C42—611no—
236C2×C42—2442no—
237C2×C42C2×#692311yescharacteristic
238C22⋊C8—614no—
239C22⋊C8—1222no—
240C4×C8—2442no—
241#95—1222no—
242#95—1222no—
243#95—1224no—
244#95—2442no—
245C24—611no—
246C2×Q8—1224no—
247C2×Q8—2442no—
248C2×Q8—2442no—
249C22×C4—22no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—612no—
259C22×C4—1222no—
260C22×C4—1222no—
261C22×C4—1224no—
262C22×C4—1222no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4—2441no—
267C22×C4—2442no—
268C22×C4—2442no—
269C22×C4—2442no—
270C22×C4—2442no—
271C22×C4—2442no—
272C22×C4—2442no—
273C22×C4C22⋊#692311yescharacteristic
274C22×C4C22⋊#692311yescharacteristic
275C22×C4C22×#692311yescharacteristic
276C2×C8—1224no—
277C2×C8—2442no—
278C2×C8—2442no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—2442no—
283C4⋊C4—2444no—
284C4⋊C4—2448no—
285C4⋊C4—2442no—
286C22⋊C4—2442no—
287C22⋊C4—2442no—
288C42—22no—
289C42—1222no—
290C42—2441no—
291C42—2441no—
292C23—611no—
293C23—611no—
294C23—611no—
295C23—611no—
296C23—611no—
297C23—611no—
298C23—1221no—
299C23—1221no—
300C23—1221no—
301C23—1221no—
302C23C2×C22⋊#692311yescharacteristic
303Q8—4884no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—612no—
312C2×C4—612no—
313C2×C4—612no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4—4882no—
323C2×C4—4882no—
324C2×C4—4882no—
325C2×C4—4882no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2444no—
331C2×C4—2444no—
332C2×C4—2444no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4#1451983612yes—
337C2×C4#1451984012yes—
338C2×C4#1451984012yes—
339C2×C4#1451984112yes—
340C2×C4D4×#692312yes—
341C8—2444no—
342C22—611no—
343C22—611no—
344C22—611no—
345C22—611no—
346C22—1221no—
347C22—1221no—
348C22—1221no—
349C22—1221no—
350C22—1221no—
351C22—1221no—
352C22—1221no—
353C22—1221no—
354C22—1221no—
355C22—1221no—
356C22—1221no—
357C22—1221no—
358C22#1684071411yescharacteristic
359C22C2×#1451983611yescharacteristic
360C22#1684074311yescharacteristic
361C22C2×#1451984011yescharacteristic
362C22C2×#1451984011yescharacteristic
363C22C2×#1451984111yescharacteristic
364C22#1684076011yescharacteristic
365C4—22no—
366C4—22no—
367C4—82no—
368C4—1222no—
369C4—1222no—
370C4—1222no—
371C4—4884no—
372C4—4882no—
373C4#1684061514yescomplemented
374C2—1221no—
375C2—1221no—
376C2—1221no—
377C2—1221no—
378C2C2×#1684061511yescharacteristic
379C2#2146120211yescharacteristic
380C2#2146125111yescharacteristic
381C2#2146125811yescharacteristic
382C2#2146194611yescharacteristic
383C2#2146195911yescharacteristic
384C2#2146198311yescharacteristic
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#9679
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
177. C4×C61#328semidirect
206. #384#6923semidirect
213. C61#9679semidirect
373. C4#16840615semidirect

Maximal subgroups

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

subgroupindex
C4×#145198342
C2×#168406152
#214609972
#214541722
#967961