Finite Group Databasebeta

← #30604329

Subgroups — #30604329

Order 15616.

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

normal subgroups only

This group has 424 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
1#21461075C211yesmaximal, characteristic
2#21461055C211yesmaximal, characteristic
3#21460760C212yesmaximal, complemented
4#21460759C212yesmaximal, complemented
5#21450990C211yesmaximal, characteristic
6#16840732—21no—
7C2×C22⋊#92420—21no—
8#16840607C2212yes—
9#16840607C2212yes—
10#16840605—21no—
11#16840605—21no—
12C2×#14519776C411yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14#16839674—21no—
15#16839674—21no—
16#16839665C411yescharacteristic
17#16839628C2211yescharacteristic
18#16839627C2211yescharacteristic
19#16839620—21no—
20#16839620—21no—
21C4×#11758196C411yescharacteristic
22C61×#390C411yescharacteristic
23C2×#11758217—41no—
24C22×#92420—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C22⋊#92420—41no—
28C22⋊#92420—41no—
29C22×Dic122—21no—
30C23×Dic61D411yescharacteristic
31C23×Dic61D411yescharacteristic
32#14519776—41no—
33#14519776—41no—
34D4×Dic61—41no—
35D4×Dic61—41no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2311yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61D411yescharacteristic
43C2×C22⋊Dic61D411yescharacteristic
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61C2×C411yescharacteristic
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51C22×D4×C61C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C42×C61C2×C411yescharacteristic
56C24×C61C22⋊C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C2×D4×C61—22no—
59C2×D4×C61—22no—
60C2×D4×C61—41no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—42no—
77C2×Dic122—81no—
78C2×Dic122—81no—
79#11758217—82no—
80C2×#92420—42no—
81C2×#92420—42no—
82C2×#92420—82no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—81no—
85C22⋊C4×C61—81no—
86C42×C61—22no—
87C22⋊Dic61—41no—
88C22⋊Dic61—41no—
89C22⋊Dic61—81no—
90C22⋊Dic61—81no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93#11758196—42no—
94#11758196—41no—
95#11758196—41no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—22no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61#10112yes—
109C23×C61C2×C22⋊C411yescharacteristic
110D4×C61—44no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61#13112yes—
120C2×Dic61—22no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—81no—
136C2×Dic61—81no—
137C2×Dic61#9911yescharacteristic
138C2×Dic61#9911yescharacteristic
139Dic122—84no—
140#92420—82no—
141#92420—82no—
142#11569—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61#40811yescharacteristic
156C22×C61C2×#9911yescharacteristic
157C22×C61C2×#10111yescharacteristic
158C4×C61—22no—
159C4×C61—81no—
160C4×C61—81no—
161Dic61—21no—
162Dic61—41no—
163Dic61—41no—
164Dic61—41no—
165Dic61—81no—
166Dic61—81no—
167Dic61#35412yes—
168#2563—611no—
169#1465—611no—
170#1459—611no—
171#1324—612no—
172#1324—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—42no—
176C2×C61—81no—
177C2×C61#147211yescharacteristic
178C2×C61C2×#35411yescharacteristic
179C2×C61#213411yescharacteristic
180C2×C4⋊Q8—611no—
181C2×C4×D4—611no—
182#407—1221no—
183C2×C22⋊C8—1221no—
184#393—1221no—
185#393—1221no—
186#390—611no—
187#390#692311yescharacteristic
188#382—611no—
189#382—611no—
190#380—1221no—
191#380—1221no—
192C4×C4⋊C4—611no—
193#324—612no—
194#324—612no—
195#323—1221no—
196#323—1221no—
197C61#1156911yescharacteristic, complemented, Sylow
198C22×Q8—1221no—
199C22×D4C2×#692311yescharacteristic
200C23×C4—611no—
201C23×C4—611no—
202C2×M16—2441no—
203C22×C8—2441no—
204C4⋊Q8—2441no—
205C4⋊Q8—2441no—
206C4×D4—2441no—
207C4×D4—2441no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4C2×#692311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1221no—
219C2×C42—611no—
220C2×C42—611no—
221C2×C42C2×#692311yescharacteristic
222C22⋊C8—1222no—
223C22⋊C8—1222no—
224C22⋊C8—2441no—
225C22⋊C8—2441no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24C22⋊#692311yescharacteristic
232C24C22⋊#692311yescharacteristic
233C2×Q8—4881no—
234C2×Q8—4881no—
235C2×Q8—2442no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—2442no—
252C22×C4—2442no—
253C22×C4C22×#692311yescharacteristic
254M16—4882no—
255C2×C8—4882no—
256C2×C8—2442no—
257C2×C8—2442no—
258C4⋊C4—41no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C22⋊C4—4881no—
267C22⋊C4—4881no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C42—22no—
271C42—1221no—
272C42—1221no—
273C42—1222no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1451983812yes—
284C23C2×C22⋊#692311yescharacteristic
285Q8—4884no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—611no—
296C2×C4—611no—
297C2×C4—1222no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—4881no—
302C2×C4—4881no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4#1451986412yes—
315C8—4882no—
316C8—4882no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22#1684073511yescharacteristic
330C22D61⋊#9911yescharacteristic
331C22C2×#1451983811yescharacteristic
332C4—22no—
333C4—81no—
334C4—81no—
335C4—612no—
336C4—1221no—
337C4—4881no—
338C4—4881no—
339C4—2441no—
340C4—2441no—
341C4—2441no—
342C2—21no—
343C2—21no—
344C2—42no—
345C2—81no—
346C2#2146109811yescharacteristic
347C2D61⋊#35411yescharacteristic
348C2#2146211911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×C4⋊Dic61C23
hypercenter#390#6923
hypocenterC61#11569
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #21460760C2semidirect
4. #21460759C2semidirect
197. C61#11569semidirect

Maximal subgroups

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

subgroupindex
#214610752
#214610552
#214607602
#214607592
#214509902
#1156961