Finite Group Databasebeta

← #30601187

Subgroups — #30601187

Order 15616.

610 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462257C211yesmaximal, characteristic
2#21461726C212yesmaximal
3#21461068C211yesmaximal, characteristic
4#21460666C212yesmaximal
5#21450107C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840733C2212yes—
8C4⋊C4×#6923—24no—
9C4×C22⋊#6923—21no—
10C4×C22⋊#6923C2212yes—
11#16840632—24no—
12#16840608—21no—
13#16840607—21no—
14C2×#14519735C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16#16839700C411yescharacteristic, complemented
17#16839694C411yescharacteristic, complemented
18#16839692C411yescharacteristic, complemented
19#16839649—21no—
20#16839645—21no—
21#16839636—21no—
22#16839624—21no—
23#16839624C2211yescharacteristic
24C61×#400C411yescharacteristic, complemented
25C2×#11758213—22no—
26#14519856—24no—
27#14519856—24no—
28C2×C4⋊#6923—22no—
29C2×C22⋊#6923—22no—
30C2×C4×#6923—22no—
31C2×C4×#6923—22no—
32C2×C4×#6923D414yes—
33#14519834—22no—
34C22⋊#92420—22no—
35(C2×#6923)⋊C4—22no—
36(C2×#6923)⋊C4—22no—
37C22×C4×D61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61C2311yescharacteristic
40#14519735—22no—
41#14519735—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196C2×C411yescharacteristic
47C2×#11758196D411yescharacteristic
48C4⋊C4×D61—22no—
49C4⋊C4×D61—22no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×C4×Dic61—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61—21no—
63#95×C61C2×C411yescharacteristic
64C23×D61C2×D411yescharacteristic
65C22×#6923—24no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—42no—
83C2×C4×D61—42no—
84C2×C4×D61C2×D412yes—
85C2×C4×D61#4112yes—
86C22⋊#6923—24no—
87C22⋊#6923—24no—
88C2×#92420—42no—
89C4⋊#6923—28no—
90C4×#6923—28no—
91C4×#6923—44no—
92C4×#6923—82no—
93#11758213—44no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96#11758198—42no—
97#11758198—41no—
98#11758198—41no—
99C4⋊Dic61—22no—
100C4⋊Dic61—22no—
101C4⋊Dic61—42no—
102#11758196—22no—
103#11758196—22no—
104#11758196—42no—
105C4×Dic61—42no—
106C23×C61C2×C22⋊C411yescharacteristic
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—22no—
110C22×D61—41no—
111C22×D61#12011yescharacteristic
112C22×D61#12311yescharacteristic
113C2×#6923—28no—
114C2×#6923—44no—
115C2×#6923—44no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61#10412yes—
127C2×C4×C61C4×D412yes—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61#10412yes—
140C2×Dic61#12112yes—
141C4×D61—22no—
142C4×D61—22no—
143C4×D61—24no—
144C4×D61—24no—
145C4×D61—42no—
146C4×D61—42no—
147C4×D61—81no—
148#92420—82no—
149#9478—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#38611yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C22×C61C2×#10411yescharacteristic
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—42no—
161C2×D61—42no—
162C2×D61#45812yes—
163#6923—48no—
164C4×C61—22no—
165C4×C61—22no—
166C4×C61—41no—
167C4×C61—81no—
168C4×C61—81no—
169Dic61—22no—
170Dic61—22no—
171Dic61—41no—
172Dic61—81no—
173Dic61—81no—
174#2484—611no—
175#2274—611no—
176#1826—612no—
177#1463—611no—
178#1284—612no—
179C2×C61—21no—
180C2×C61—21no—
181C2×C61#189011yescharacteristic
182C2×C61#189611yescharacteristic
183C2×C61#191211yescharacteristic
184D61—22no—
185D61—41no—
186C2×#117—611no—
187C2×#117—1221no—
188C22×C4⋊C4—611no—
189#407—612no—
190#400—611no—
191#400#692311yescharacteristic, complemented
192#398—611no—
193#398—611no—
194#387—1221no—
195#387—1221no—
196#385—1221no—
197#381—611no—
198#381—1221no—
199C4×C4⋊C4—1224no—
200C4×C22⋊C4—612no—
201C4×C22⋊C4—1221no—
202#339—1224no—
203#324—1221no—
204#324—1221no—
205C61#947811yescharacteristic, complemented, Sylow
206C23×C4—611no—
207C23×C4—611no—
208C23×C4—611no—
209C2×M16—1222no—
210#117—1224no—
211#117—1224no—
212#117—1222no—
213#117—1222no—
214C2×C4⋊C4—611no—
215C2×C4⋊C4—611no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—611no—
218C2×C4⋊C4—1222no—
219C2×C4⋊C4—1222no—
220C2×C4⋊C4—1222no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4C2×#692311yescharacteristic
224C2×C4⋊C4C2×#692311yescharacteristic
225C2×C22⋊C4—1222no—
226C2×C22⋊C4—1221no—
227C2×C22⋊C4—1221no—
228C2×C42—614no—
229C2×C42—1222no—
230C2×C42—1222no—
231C2×C42—1221no—
232C22⋊C8—1222no—
233C22⋊C8—1222no—
234#95—21no—
235#95—21no—
236#95—611no—
237#95—1221no—
238#95—1221no—
239#95—1221no—
240#95—1221no—
241#95—1222no—
242#95—1222no—
243#95—1221no—
244#95C2×#692311yescharacteristic
245C24—611no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—612no—
251C22×C4—612no—
252C22×C4—611no—
253C22×C4—1222no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1222no—
259C22×C4—1224no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—2442no—
263C22×C4—2442no—
264C22×C4C22⋊#692311yescharacteristic
265C22×C4C22⋊#692311yescharacteristic
266C22×C4C22×#692311yescharacteristic
267M16—2444no—
268C2×C8—2442no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—1222no—
272C4⋊C4—1222no—
273C4⋊C4—1228no—
274C4⋊C4—1222no—
275C4⋊C4—1222no—
276C4⋊C4—2442no—
277C4⋊C4—2442no—
278C22⋊C4—1224no—
279C22⋊C4—1224no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C22⋊C4—2442no—
283C42—1228no—
284C42—4882no—
285C42—2442no—
286C42—2444no—
287C23—611no—
288C23—611no—
289C23—1222no—
290C23—1221no—
291C23—1221no—
292C23—2441no—
293C23C2×C22⋊#692311yescharacteristic
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—612no—
305C2×C4—612no—
306C2×C4—1224no—
307C2×C4—1222no—
308C2×C4—1222no—
309C2×C4—1224no—
310C2×C4—1221no—
311C2×C4—1221no—
312C2×C4—1228no—
313C2×C4—1221no—
314C2×C4—4881no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2444no—
325C2×C4—2444no—
326C2×C4—2441no—
327C2×C4#1451984212yes—
328C2×C4D4×#692312yes—
329C8—4882no—
330C22—21no—
331C22—21no—
332C22—612no—
333C22—1221no—
334C22—1221no—
335C22—1221no—
336C22—1221no—
337C22—1221no—
338C22—2442no—
339C22—2442no—
340C22#1684071011yescharacteristic
341C22D61⋊#10411yescharacteristic
342C22C2×#1451984211yescharacteristic
343C4—22no—
344C4—22no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—1222no—
349C4—1222no—
350C4—4881no—
351C4—4881no—
352C4—2448no—
353C4—2441no—
354C2—21no—
355C2—21no—
356C2—1222no—
357C2—2441no—
358C2#2146182911yescharacteristic
359C2#2146183611yescharacteristic
360C2#2146186711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#400C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#400#6923
hypocenterC61#9478
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
16. #16839700C4semidirect
17. #16839694C4semidirect
18. #16839692C4semidirect
24. C61×#400C4semidirect
191. #400#6923semidirect
205. C61#9478semidirect

Maximal subgroups

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

subgroupindex
#214622572
#214617262
#214610682
#214606662
#214501072
#947861