Finite Group Databasebeta

← #30604341

Subgroups — #30604341

Order 15616.

416 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 70 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#21460761C212yesmaximal, complemented
4#21460760C212yesmaximal, complemented
5#21451021C211yesmaximal, characteristic
6#16840732—21no—
7C2×C22⋊#92420—21no—
8#16840607C2212yes—
9#16840607C2212yes—
10#16840605—21no—
11#16840605—21no—
12C2×#14519768C411yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14#16839685—21no—
15#16839685—21no—
16#16839665C411yescharacteristic
17#16839637—21no—
18#16839637—21no—
19#16839632C2211yescharacteristic
20#16839627C2211yescharacteristic
21C4×#11758196C411yescharacteristic
22C61×#390C411yescharacteristic
23C2×#11758217—41no—
24C22×#92420—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C22⋊#92420—41no—
28C22⋊#92420—41no—
29C23×Dic61D411yescharacteristic
30C23×Dic61D411yescharacteristic
31#14519768—41no—
32#14519768—41no—
33D4×Dic61—41no—
34D4×Dic61—41no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C4⋊Dic61C2311yescharacteristic
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61D411yescharacteristic
45C2×C22⋊Dic61D411yescharacteristic
46C2×C4×Dic61C2×C411yescharacteristic
47C2×C4×Dic61C2×C411yescharacteristic
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—81no—
71C22×Dic61—81no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76#11758217—82no—
77C2×#92420—42no—
78C2×#92420—42no—
79C2×#92420—82no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C42×C61—22no—
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—81no—
87C22⋊Dic61—81no—
88C4⋊Dic61—41no—
89C4⋊Dic61—41no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92C4⋊Dic61—81no—
93C4⋊Dic61—81no—
94#11758196—42no—
95#11758196—42no—
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—22no—
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—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—41no—
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
139#92420—82no—
140#92420—82no—
141#11572—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61#40811yescharacteristic
155C22×C61C2×#9911yescharacteristic
156C22×C61C2×#10111yescharacteristic
157C4×C61—22no—
158C4×C61—42no—
159C4×C61—81no—
160Dic61—21no—
161Dic61—21no—
162Dic61—41no—
163Dic61—41no—
164Dic61—41no—
165Dic61—81no—
166Dic61—81no—
167#2566—611no—
168#1465—611no—
169#1459—611no—
170#1325—612no—
171#1324—612no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—42no—
175C2×C61—81no—
176C2×C61#147211yescharacteristic
177C2×C61#212711yescharacteristic
178C2×C61#213211yescharacteristic
179C2×#125—611no—
180C2×C4×D4—611no—
181#407—1221no—
182C2×C22⋊C8—1221no—
183#396—1221no—
184#396—1221no—
185#390—611no—
186#390#692311yescharacteristic
187#385—1221no—
188#385—1221no—
189#384—611no—
190#382—611no—
191C4×C4⋊C4—611no—
192#324—612no—
193#324—612no—
194#323—1221no—
195#323—1221no—
196C61#1157211yescharacteristic, complemented, Sylow
197C22×D4C2×#692311yescharacteristic
198C23×C4—611no—
199C23×C4—611no—
200C2×M16—2441no—
201C22×C8—2441no—
202#125—2441no—
203#125—2441no—
204C4×D4—2441no—
205C4×D4—2441no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4C2×#692311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—611no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C42—611no—
221C2×C42—611no—
222C2×C42C2×#692311yescharacteristic
223C22⋊C8—1222no—
224C22⋊C8—1222no—
225C22⋊C8—2441no—
226C22⋊C8—2441no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230C24C22⋊#692311yescharacteristic
231C24C22⋊#692311yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—4881no—
248C22×C4—4881no—
249C22×C4C22×#692311yescharacteristic
250M16—4882no—
251C2×C8—4882no—
252C2×C8—2442no—
253C2×C8—2442no—
254C4⋊C4—41no—
255C4⋊C4—4881no—
256C4⋊C4—4881no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C4⋊C4—2442no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C4⋊C4—2442no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—4881no—
266C22⋊C4—4881no—
267C22⋊C4—2441no—
268C22⋊C4—2441no—
269C42—22no—
270C42—1221no—
271C42—1221no—
272C42—1222no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1451983812yes—
283C23C2×C22⋊#692311yescharacteristic
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—4881no—
299C2×C4—4881no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4#1451986412yes—
313C8—4882no—
314C8—4882no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22#1684073511yescharacteristic
328C22D61⋊#9911yescharacteristic
329C22C2×#1451983811yescharacteristic
330C4—22no—
331C4—42no—
332C4—81no—
333C4—1221no—
334C4—1221no—
335C4—4881no—
336C4—4881no—
337C4—2441no—
338C4—2441no—
339C4—2441no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2146109811yescharacteristic
345C2#2146209711yescharacteristic
346C2#2146211111yescharacteristic
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#11572
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. #21460761C2semidirect
4. #21460760C2semidirect
196. C61#11572semidirect

Maximal subgroups

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

subgroupindex
#214610752
#214610552
#214607612
#214607602
#214510212
#1157261