Finite Group Databasebeta

← #30604995

Subgroups — #30604995

Order 15616.

446 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×#16840642C211yesmaximal, characteristic
2#21461069C211yesmaximal, characteristic
3#21460818C212yesmaximal, complemented
4#21460803C212yesmaximal, complemented
5#21450102C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840732—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C22×#11758196C2211yescharacteristic
13#16839700C411yescharacteristic
14#16839700C2211yescharacteristic
15#16839693C411yescharacteristic
16#16839693C2211yescharacteristic
17#16839646—21no—
18#16839646—21no—
19#16839634—21no—
20#16839625C411yescharacteristic
21#16839623—21no—
22C2×#117×C61C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C2×#11758217—41no—
28C22×#92420—41no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×#11758196—41no—
36C2×#11758196—41no—
37C2×#11758196C2×C411yescharacteristic
38C2×#11758196D411yescharacteristic
39C2×#11758196D411yescharacteristic
40C2×#11758196C2311yescharacteristic
41C2×C22⋊Dic61—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51C23×C4×C61C2×C411yescharacteristic
52#117×C61—41no—
53#117×C61—41no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C2×C42×C61—21no—
58C24×C61C22⋊C411yescharacteristic
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—22no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—42no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80#11758217—24no—
81#11758217—44no—
82#11758217—42no—
83#11758217—82no—
84C2×#92420—44no—
85C2×#92420—42no—
86C2×#92420—42no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C42×C61—42no—
92C22⋊Dic61—42no—
93#11758196—41no—
94#11758196—41no—
95#11758196—41no—
96#11758196—41no—
97#11758196—42no—
98#11758196—41no—
99#11758196—41no—
100C23×C61—22no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—41no—
104C23×C61#9911yescharacteristic
105C23×C61#9911yescharacteristic
106C23×C61C2×C22⋊C411yescharacteristic
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—42no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—81no—
138C2×Dic61#10412yes—
139C2×Dic61AGL(1,ℤ/8)11yescharacteristic
140C2×Dic61#13711yescharacteristic
141#92420—44no—
142#92420—82no—
143#11764—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61#35212yes—
154C22×C61C2×#9911yescharacteristic
155C22×C61#41711yescharacteristic
156C22×C61C2×#10411yescharacteristic
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61—81no—
161Dic61—21no—
162Dic61—22no—
163Dic61—21no—
164Dic61—41no—
165Dic61—41no—
166Dic61—81no—
167Dic61—81no—
168#2484—611no—
169C2×#343—611no—
170#1463—611no—
171#1353—612no—
172#1346—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—22no—
176C2×C61—41no—
177C2×C61#150411yescharacteristic
178C2×C61C2×#35211yescharacteristic
179C2×C61#213311yescharacteristic
180C22×M16—1221no—
181C2×#117#692311yescharacteristic
182C22×C4⋊C4—611no—
183#407—1221no—
184#400—611no—
185#400—611no—
186#398—611no—
187#398—611no—
188#387—1221no—
189#387—1221no—
190#385—1221no—
191#381—611no—
192#381—1221no—
193#343—612no—
194#343—1221no—
195#324—612no—
196#324—1221no—
197C61#1176411yescharacteristic, complemented, Sylow
198C23×C4—611no—
199C23×C4—611no—
200C23×C4C2×#692311yescharacteristic
201C2×M16—1222no—
202C2×M16—1224no—
203C2×M16—2441no—
204C2×M16—2442no—
205C2×M16—2441no—
206C22×C8—2441no—
207#117—41no—
208#117—41no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—2441no—
216C2×C4⋊C4—2441no—
217C2×C4⋊C4C2×#692311yescharacteristic
218C2×C4⋊C4C2×#692311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—1221no—
221C2×C42—21no—
222C22⋊C8—1222no—
223C22⋊C8—2441no—
224#95—611no—
225#95—611no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233C24C22⋊#692311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—1222no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—2442no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255M16—1224no—
256M16—4882no—
257M16—2442no—
258M16—2444no—
259C2×C8—2442no—
260C2×C8—2444no—
261C2×C8—2442no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C4⋊C4—2442no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—2442no—
274C42—42no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1451983511yescharacteristic
280C23#1451983511yescharacteristic
281C23C2×C22⋊#692311yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—612no—
296C2×C4—1222no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—1222no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—4881no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2442no—
314C2×C4—2441no—
315C2×C4—2441no—
316C8—4882no—
317C8—2444no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1684066412yes—
328C22C2×#1451983511yescharacteristic
329C22#1684075311yescharacteristic
330C22D61⋊#10411yescharacteristic
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—1221no—
336C4—1222no—
337C4—1221no—
338C4—4881no—
339C4—4881no—
340C4—2441no—
341C4—2441no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2146120511yescharacteristic
347C2C2×#1684066411yescharacteristic
348C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×#11758196C23
hypercenterC2×#117#6923
hypocenterC61#11764
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21460818C2semidirect
4. #21460803C2semidirect
197. C61#11764semidirect

Maximal subgroups

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

subgroupindex
C2×#168406422
#214610692
#214608182
#214608032
#214501022
#1176461