Finite Group Databasebeta

← #30601195

Subgroups — #30601195

Order 15616.

518 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 78 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#21462296C211yesmaximal, characteristic
2#21461726C212yesmaximal
3#21461068C211yesmaximal, characteristic
4#21460666C212yesmaximal
5#21450107C211yesmaximal, characteristic
6#16840733C2212yes—
7#16840711—24no—
8#16840705—21no—
9#16840704—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—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28C2×C4×#6923D414yes—
29#14519834—22no—
30C22⋊#92420—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—24no—
33(C2×#6923)⋊C4—22no—
34C22×C4×D61D411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61C2311yescharacteristic
37#14519735—22no—
38#14519735—22no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196C2×C411yescharacteristic
44C2×#11758196D411yescharacteristic
45C4⋊C4×D61—22no—
46C4⋊C4×D61—22no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×C4×Dic61—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684C2×C411yescharacteristic
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58#95×C61—21no—
59#95×C61—21no—
60#95×C61C2×C411yescharacteristic
61C23×D61C2×D411yescharacteristic
62C22×#6923—24no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—42no—
78C2×C4×D61—41no—
79C2×C4×D61—41no—
80C2×C4×D61—42no—
81C2×C4×D61C2×D412yes—
82C2×C4×D61#4112yes—
83C22⋊#6923—42no—
84C22⋊#6923—42no—
85C2×#92420—42no—
86C4⋊#6923—82no—
87C4×#6923—44no—
88#11758213—44no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—22no—
91#11758198—42no—
92#11758198—41no—
93#11758198—41no—
94C4⋊Dic61—22no—
95C4⋊Dic61—22no—
96C4⋊Dic61—42no—
97#11758196—22no—
98#11758196—22no—
99#11758196—42no—
100C4×Dic61—42no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—22no—
105C22×D61—41no—
106C22×D61#12011yescharacteristic
107C22×D61#12311yescharacteristic
108C2×#6923—44no—
109C2×#6923—44no—
110C2×#6923—44no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—42no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61C4×D412yes—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
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#12112yes—
136C4×D61—24no—
137C4×D61—42no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—42no—
141C4×D61—81no—
142#92420—82no—
143#9482—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61#38611yescharacteristic
147C22×C61#42111yescharacteristic
148C22×C61#42111yescharacteristic
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—21no—
153C2×D61—21no—
154C2×D61—42no—
155C2×D61—42no—
156C2×D61#45812yes—
157#6923—84no—
158C4×C61—22no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—81no—
162C4×C61—81no—
163Dic61—22no—
164Dic61—41no—
165Dic61—41no—
166Dic61—81no—
167Dic61—81no—
168#2484—611no—
169#2313—611no—
170#1826—612no—
171#1463—611no—
172#1284—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#189011yescharacteristic
176C2×C61#189711yescharacteristic
177C2×C61#191211yescharacteristic
178D61—22no—
179D61—41no—
180C2×#117—611no—
181C22×C4⋊C4—611no—
182#407—612no—
183#400—611no—
184#400#692311yescharacteristic, complemented
185#398—611no—
186#398—611no—
187#387—1221no—
188#387—1221no—
189#387—1224no—
190#385—1221no—
191#381—611no—
192#381—1221no—
193#381—1221no—
194#380—1221no—
195C4×C22⋊C4—612no—
196#339—1224no—
197#324—1221no—
198#324—1221no—
199C61#948211yescharacteristic, complemented, Sylow
200C23×C4—611no—
201C23×C4—611no—
202C23×C4—611no—
203C2×M16—1222no—
204#117—1222no—
205#117—1222no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4—1222no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4C2×#692311yescharacteristic
216C2×C4⋊C4C2×#692311yescharacteristic
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1222no—
220C2×C42—614no—
221C2×C42—1221no—
222C22⋊C8—1222no—
223C22⋊C8—1222no—
224#95—21no—
225#95—21no—
226#95—611no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1222no—
232#95—1222no—
233#95—1224no—
234#95—1221no—
235#95C2×#692311yescharacteristic
236C24—611no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1224no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—2441no—
252C22×C4—2441no—
253C22×C4—2442no—
254C22×C4—2442no—
255C22×C4C22⋊#692311yescharacteristic
256C22×C4C22⋊#692311yescharacteristic
257C22×C4C22×#692311yescharacteristic
258M16—2444no—
259C2×C8—2442no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—1222no—
263C4⋊C4—1222no—
264C4⋊C4—1222no—
265C4⋊C4—1222no—
266C4⋊C4—4882no—
267C4⋊C4—2442no—
268C4⋊C4—2442no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C22⋊C4—2442no—
272C22⋊C4—2442no—
273C22⋊C4—2442no—
274C42—2442no—
275C42—2444no—
276C23—611no—
277C23—611no—
278C23—1222no—
279C23—1221no—
280C23—1221no—
281C23—2441no—
282C23C2×C22⋊#692311yescharacteristic
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—612no—
295C2×C4—1224no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—4881no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2444no—
309C2×C4—2444no—
310C2×C4—2444no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4D4×#692312yes—
317C8—4882no—
318C22—21no—
319C22—21no—
320C22—612no—
321C22—1221no—
322C22—1221no—
323C22—1221no—
324C22—1221no—
325C22—1221no—
326C22—2442no—
327C22—2442no—
328C22#1684071011yescharacteristic
329C22#1684075911yescharacteristic
330C22#1684076111yescharacteristic
331C4—22no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—1222no—
337C4—4881no—
338C4—4884no—
339C4—4881no—
340C4—2441no—
341C4—2441no—
342C2—21no—
343C2—21no—
344C2—1222no—
345C2—2441no—
346C2#2146182911yescharacteristic
347C2#2146183811yescharacteristic
348C2#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#9482
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
184. #400#6923semidirect
199. C61#9482semidirect

Maximal subgroups

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

subgroupindex
#214622962
#214617262
#214610682
#214606662
#214501072
#948261