Finite Group Databasebeta

← #30601762

Subgroups — #30601762

Order 15616.

770 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461546C211yesmaximal, characteristic, complemented
2C2×#16840614C214yesmaximal, complemented
3#21460995C211yesmaximal, characteristic, complemented
4#21454165C211yesmaximal, characteristic
5C2×#14519834C2212yes—
6C2×#14519832—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840614—28no—
12C2×#14519774C411yescharacteristic, complemented
13C2×#14519752—21no—
14C2×#14519752—21no—
15C2×#14519752C2212yes—
16C2×C42×D61C2211yescharacteristic
17#16839663—21no—
18#16839663—21no—
19#16839663C412yescomplemented
20C61×#384C411yescharacteristic, complemented
21C23×#6923—22no—
22C22×#92420—22no—
23#14519834—22no—
24#14519834D414yescomplemented
25#14519832—42no—
26(C2×#6923)⋊C4—24no—
27(C2×#6923)⋊C4—24no—
28(C2×#6923)⋊C4—42no—
29C22×D244—22no—
30C22×D244—21no—
31C22×D244—21no—
32C22×D244C2×C412yes—
33C22×C4×D61D411yescharacteristic
34C22×C4×D61D411yescharacteristic
35C22×C4×D61C2311yescharacteristic
36#14519774—42no—
37#14519752—24no—
38#14519752—24no—
39#14519752—44no—
40#14519752—44no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C42×D61—42no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61C2×C412yes—
51C2×C42×C61C2×C411yescharacteristic
52C23×D61—22no—
53C23×D61C2×D411yescharacteristic
54C22×#6923—24no—
55C22×#6923—42no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×#6923—42no—
59C22×#6923—42no—
60C22×#6923—42no—
61C22×C4×C61—22no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×D244—24no—
69C2×D244—42no—
70C2×D244—42no—
71C2×D244—42no—
72C2×D244—42no—
73C2×D244—42no—
74C2×D244—42no—
75C2×D244—42no—
76C2×D244—42no—
77C2×D244—44no—
78C2×D244—44no—
79C2×D244—42no—
80C2×D244—42no—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—41no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61C2×D412yes—
90C2×C4×D61#4112yes—
91C2×#92420—24no—
92C2×#92420—42no—
93C2×#92420—42no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61—42no—
96C4⋊C4×C61—42no—
97C42×C61—22no—
98#11758198—44no—
99#11758198—44no—
100#11758198—44no—
101C4×Dic61—22no—
102C4×Dic61—41no—
103C4×Dic61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—42no—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61—42no—
113C22×D61—42no—
114C22×D61—42no—
115C22×D61—42no—
116C22×D61#12411yescharacteristic
117C22×D61#12711yescharacteristic
118C22×D61AGL(1,ℤ/8)12yes—
119C22×D61AGL(1,ℤ/8)12yes—
120C2×#6923—44no—
121C2×#6923—44no—
122C2×#6923—44no—
123C2×#6923—82no—
124C2×#6923—82no—
125C2×#6923—82no—
126C2×#6923—82no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61#9912yes—
137C2×C4×C61#10412yes—
138C2×C4×C61C4×D412yes—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61#12112yes—
148D244—84no—
149D244—84no—
150D244—84no—
151D244—84no—
152D244—84no—
153C4×D61—42no—
154C4×D61—42no—
155C4×D61—44no—
156C4×D61—44no—
157#92420—44no—
158#9684—611nomaximal, Sylow
159C22×C61#39011yescharacteristic
160C22×C61C2×#9911yescharacteristic
161C22×C61#41011yescharacteristic
162C22×C61#41811yescharacteristic
163C22×C61#41811yescharacteristic
164C22×C61C2×#10411yescharacteristic
165C22×C61#42111yescharacteristic
166C2×D61—21no—
167C2×D61—21no—
168C2×D61—21no—
169C2×D61—21no—
170C2×D61—22no—
171C2×D61—22no—
172C2×D61—22no—
173C2×D61—22no—
174C2×D61—82no—
175C2×D61—82no—
176C2×D61—82no—
177C2×D61—82no—
178C2×D61—82no—
179C2×D61—82no—
180C2×D61—82no—
181C2×D61#49012yes—
182C2×D61#49612yes—
183#6923—84no—
184C4×C61—22no—
185C4×C61—22no—
186C4×C61—22no—
187C4×C61—82no—
188Dic61—22no—
189Dic61—41no—
190Dic61—41no—
191#2856—611no—
192#1749—611no—
193C2×#327—614no—
194#1434—611no—
195C2×C61#150511yescharacteristic
196C2×C61#150711yescharacteristic
197C2×C61#151011yescharacteristic
198C2×C61#151411yescharacteristic
199C2×C61#196111yescharacteristic
200C2×C61#196411yescharacteristic
201C2×C61#198011yescharacteristic
202D61—22no—
203D61—22no—
204D61—162no—
205C2×#127—611no—
206C2×#121—612no—
207C2×#121—1221no—
208C2×#121—1221no—
209C22×C42—611no—
210C2×C22⋊C8—612no—
211C2×#97—1221no—
212#390—612no—
213#390—1221no—
214#390—1221no—
215#384#692311yescharacteristic, complemented
216C2×#95—612no—
217C2×#95—1221no—
218C2×#95—1221no—
219#336—1221no—
220#327—1228no—
221C61#968411yescharacteristic, complemented, Sylow
222C22×D4—612no—
223C22×D4—1222no—
224C22×D4—1221no—
225C22×D4—1221no—
226C23×C4—611no—
227C23×C4—611no—
228C23×C4—611no—
229C23×C4—1222no—
230C22×C8—1222no—
231#127—2442no—
232#121—1224no—
233#121—1224no—
234#121—2444no—
235#121—2444no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C2×#692312yes—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—2442no—
246C2×C42C2×#692311yescharacteristic
247C22⋊C8—614no—
248C22⋊C8—1222no—
249#97—2442no—
250#95—1224no—
251#95—1224no—
252#95—2442no—
253C24—611no—
254C24—1222no—
255C2×D4—1224no—
256C2×D4—2442no—
257C2×D4—2442no—
258C2×D4—2442no—
259C2×D4—2442no—
260C2×D4—2442no—
261C2×D4—2442no—
262C2×D4—2442no—
263C2×D4—2442no—
264C2×D4—2444no—
265C2×D4—2444no—
266C2×D4—2442no—
267C2×D4—2442no—
268C22×C4—22no—
269C22×C4—611no—
270C22×C4—612no—
271C22×C4—612no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—1224no—
279C22×C4—2441no—
280C22×C4—2441no—
281C22×C4—2441no—
282C22×C4—2441no—
283C22×C4—2442no—
284C22×C4—2442no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4C22⋊#692311yescharacteristic
290C22×C4C22⋊#692311yescharacteristic
291C22×C4C22×#692311yescharacteristic
292C2×C8—1224no—
293C2×C8—2442no—
294C2×C8—2442no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C22⋊C4—2444no—
299C22⋊C4—2444no—
300C22⋊C4—2444no—
301C42—22no—
302C42—1222no—
303C42—2441no—
304C42—2441no—
305C23—611no—
306C23—611no—
307C23—612no—
308C23—612no—
309C23—1221no—
310C23—1221no—
311C23—1221no—
312C23—1221no—
313C23—2442no—
314C23—2442no—
315C23—2442no—
316C23—2442no—
317C23—2442no—
318C23—2442no—
319C23—2442no—
320C23C2×C22⋊#692311yescharacteristic
321D4—4884no—
322D4—4884no—
323D4—4884no—
324D4—4884no—
325D4—4884no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—612no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—4882no—
345C2×C4—4882no—
346C2×C4—4882no—
347C2×C4—4882no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2444no—
351C2×C4—2444no—
352C2×C4—2444no—
353C2×C4—2444no—
354C2×C4—2444no—
355C2×C4#1451983612yes—
356C2×C4#1451984212yes—
357C2×C4D4×#692312yes—
358C8—2444no—
359C22—612no—
360C22—612no—
361C22—1221no—
362C22—1221no—
363C22—1222no—
364C22—1221no—
365C22—1221no—
366C22—1222no—
367C22—1222no—
368C22—1222no—
369C22—4882no—
370C22—4882no—
371C22—4882no—
372C22—4882no—
373C22—4882no—
374C22—4882no—
375C22—4882no—
376C22#1684071411yescharacteristic
377C22C2×#1451983611yescharacteristic
378C22#1684074311yescharacteristic
379C22#1684075411yescharacteristic
380C22#1684075411yescharacteristic
381C22C2×#1451984211yescharacteristic
382C22#1684076111yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—82no—
387C4—1222no—
388C4—4884no—
389C4—2441no—
390C4—2441no—
391C2—1222no—
392C2—1222no—
393C2—9762no—
394C2#2146121011yescharacteristic
395C2#2146121611yescharacteristic
396C2#2146122611yescharacteristic
397C2#2146124011yescharacteristic
398C2#2146195111yescharacteristic
399C2#2146195911yescharacteristic
400C2#2146198211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radical#21454165C2
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#9684
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
1. #21461546C2semidirect
2. C2×#16840614C2semidirect
3. #21460995C2semidirect
12. C2×#14519774C4semidirect
19. #16839663C4semidirect
20. C61×#384C4semidirect
24. #14519834D4semidirect
215. #384#6923semidirect
221. C61#9684semidirect

Maximal subgroups

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

subgroupindex
#214615462
C2×#168406142
#214609952
#214541652
#968461