Finite Group Databasebeta

← #30599438

Subgroups — #30599438

Order 15616.

466 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#14519835C211yesmaximal, characteristic
2#21461074C211yesmaximal, characteristic
3#21460700C212yesmaximal, complemented
4#21460666C212yesmaximal, complemented
5#21448793C211yesmaximal, characteristic
6C2×#14519835—21no—
7#16840733C2212yes—
8C4×C22⋊#6923C2212yes—
9#16840640—21no—
10#16840608—21no—
11#16840605—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839662—21no—
14#16839656—21no—
15#16839653—21no—
16#16839626C411yescharacteristic, complemented
17#16839625C411yescharacteristic, complemented
18#16839624C2211yescharacteristic
19C4×#11758196—21no—
20#16839613C411yescharacteristic, complemented
21#16839609C2211yescharacteristic
22C4×C22⋊C4×C61C411yescharacteristic, complemented
23C2×#11758213—22no—
24C2×C22⋊#6923—22no—
25C2×C4×#6923—22no—
26#14519835—24no—
27#14519834—22no—
28C22⋊#92420—22no—
29(C2×#6923)⋊C4—22no—
30C22×C22⋊D61D411yescharacteristic
31C22×C4×D61C2311yescharacteristic
32C4×C22⋊D61—22no—
33C4×C22⋊D61—22no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196D411yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61D411yescharacteristic
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198D411yescharacteristic
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C23×C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61—21no—
52C2×C42×C61—21no—
53#95×C61C2×C411yescharacteristic
54#95×C61C2×C411yescharacteristic
55C24×C61C22⋊C411yescharacteristic
56C23×D61C2×D411yescharacteristic
57C22×#6923—42no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—81no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C2×C22⋊D61—22no—
66C2×C22⋊D61—22no—
67C2×C22⋊D61—41no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C2×C4×D61—81no—
74C2×C4×D61#4112yes—
75C2×C4×D61#4112yes—
76C22⋊#6923—24no—
77C22⋊#6923—42no—
78C2×#92420—42no—
79C4×#6923—82no—
80#11758213—82no—
81C22⋊C4×C61—42no—
82C42×C61—81no—
83C22⋊Dic61—22no—
84C22⋊Dic61—81no—
85#11758198—22no—
86#11758198—42no—
87#11758198—81no—
88#11758196—22no—
89#11758196—42no—
90#11758196—42no—
91C4×Dic61—21no—
92C4×Dic61—21no—
93C4×Dic61—42no—
94C4×Dic61—81no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61#12312yes—
106C2×#6923—44no—
107C2×#6923—82no—
108C2×#6923—82no—
109C2×C4×C61—22no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—81no—
119C2×C4×C61—81no—
120C2×C4×C61#11711yescharacteristic
121C2×C4×C61#11711yescharacteristic
122C22⋊D61—44no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
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#10412yes—
135C2×Dic61#12312yes—
136C4×D61—42no—
137C4×D61—42no—
138#92420—82no—
139#8779—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—81no—
146C22×C61—81no—
147C22×C61#38111yescharacteristic
148C22×C61C2×#10411yescharacteristic
149C22×C61#42111yescharacteristic
150C2×D61—22no—
151C2×D61—22no—
152C2×D61—22no—
153C2×D61—81no—
154C2×D61—81no—
155#6923—84no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—42no—
159C4×C61—41no—
160C4×C61—81no—
161Dic61—22no—
162Dic61—22no—
163Dic61—41no—
164Dic61—81no—
165Dic61—81no—
166#2357—611no—
167C4×#99—611no—
168#1465—611no—
169#1297—612no—
170#1284—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—81no—
174C2×C61#182611yescharacteristic
175C2×C61#182711yescharacteristic
176C2×C61#182811yescharacteristic
177D61—42no—
178C2×C4×D4—611no—
179C2×#99—1221no—
180#407—612no—
181#390—1221no—
182#388—1221no—
183#388—1221no—
184#382—611no—
185#381—611no—
186#381—611no—
187C4×C4⋊C4—611no—
188C4×C4⋊C4—1221no—
189C4×C22⋊C4—611no—
190C4×C22⋊C4—612no—
191C4×C22⋊C4#692311yescharacteristic, complemented
192#342—1221no—
193#324—1221no—
194#323—1221no—
195C61#877911yescharacteristic, complemented, Sylow
196C22×D4—611no—
197C23×C4—611no—
198C23×C4C2×#692311yescharacteristic
199C2×M16—1222no—
200C4×D4—1222no—
201C4×D4—1222no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—611no—
207C2×C22⋊C4—611no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C42—21no—
213C2×C42—611no—
214C2×C42—1221no—
215C2×C42—1222no—
216C2×C42—1221no—
217#99—1224no—
218C22⋊C8—1222no—
219C22⋊C8—1222no—
220#95—611no—
221#95—611no—
222#95—1221no—
223#95—1221no—
224#95—1222no—
225#95—1221no—
226#95C2×#692311yescharacteristic
227#95C2×#692311yescharacteristic
228C24—611no—
229C24C22⋊#692311yescharacteristic
230C2×D4—1222no—
231C2×D4—1222no—
232C2×D4—2441no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—81no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—611no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—4881no—
246C22×C4—2442no—
247C22×C4C22⋊#692311yescharacteristic
248C22×C4C22×#692311yescharacteristic
249M16—4882no—
250C2×C8—2442no—
251C4⋊C4—1222no—
252C4⋊C4—2442no—
253C4⋊C4—2442no—
254C22⋊C4—42no—
255C22⋊C4—1222no—
256C22⋊C4—1222no—
257C22⋊C4—1224no—
258C22⋊C4—4881no—
259C22⋊C4—4881no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C42—81no—
263C42—1221no—
264C42—1221no—
265C42—4881no—
266C42—4882no—
267C42—2442no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—612no—
274C23—1221no—
275C23—1221no—
276C23—2441no—
277C23—2441no—
278C23C2×C22⋊#692311yescharacteristic
279D4—2444no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—612no—
292C2×C4—612no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—4882no—
297C2×C4—4882no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2444no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4#1451985611yescharacteristic
310C2×C4#1451985611yescharacteristic
311C8—4882no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22—81no—
319C22—1222no—
320C22—1222no—
321C22—1222no—
322C22—4881no—
323C22—4881no—
324C22#1684070511yescharacteristic
325C22D61⋊#10411yescharacteristic
326C22#1684075911yescharacteristic
327C4—21no—
328C4—21no—
329C4—42no—
330C4—41no—
331C4—81no—
332C4—1222no—
333C4—1222no—
334C4—4881no—
335C4—4881no—
336C4—4884no—
337C4—2441no—
338C2—21no—
339C2—21no—
340C2—81no—
341C2—2442no—
342C2#2146172711yescharacteristic
343C2#2146172811yescharacteristic
344C2#2146173011yescharacteristic
subgroupG/H
centerC22#16840705
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC4×C22⋊C4×C61C4
socleC22×C61#381
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenterC4×C22⋊C4#6923
hypocenterC61#8779
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460700C2semidirect
4. #21460666C2semidirect
16. #16839626C4semidirect
17. #16839625C4semidirect
20. #16839613C4semidirect
22. C4×C22⋊C4×C61C4semidirect
191. C4×C22⋊C4#6923semidirect
195. C61#8779semidirect

Maximal subgroups

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

subgroupindex
C4×#145198352
#214610742
#214607002
#214606662
#214487932
#877961