Finite Group Databasebeta

← #30605386

Subgroups — #30605386

Order 15616.

472 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461679C211yesmaximal, characteristic
2#21461058C211yesmaximal, characteristic
3#21460799C214yesmaximal, complemented
4#21450121C211yesmaximal, characteristic
5#16840779—21no—
6C2×#14519835—21no—
7#16840704—21no—
8#16840640C2212yes—
9#16840606C2212yes—
10#16840605—21no—
11C2×#14519760—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839693C412yescomplemented
14#16839688C2212yes—
15#16839658—21no—
16#16839625—21no—
17#16839622C411yescharacteristic, complemented
18#16839619—21no—
19C22×C4⋊C4×C61C411yescharacteristic, complemented
20C2×C4⋊#6923—41no—
21C2×C22⋊#6923—22no—
22#14519844—41no—
23#14519835—42no—
24C22⋊#92420—22no—
25C4×#92420—41no—
26(C2×#6923)⋊C4—41no—
27C22×C22⋊D61D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29#14519760—42no—
30#14519760—42no—
31C4×C22⋊D61—41no—
32C4×C22⋊D61—41no—
33C2×C4⋊Dic61—21no—
34C2×#11758196—22no—
35C2×#11758196C2×C411yescharacteristic
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61D411yescharacteristic
38C2×#11758198—22no—
39C2×#11758198C2311yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41#14519684—22no—
42#14519684—22no—
43#14519684C2×C412yes—
44C23×C4×C61—21no—
45C23×C4×C61C2×C411yescharacteristic
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—41no—
48C2×C4⋊C4×C61—41no—
49C2×C4⋊C4×C61C2×C412yes—
50C24×C61C22⋊C411yescharacteristic
51C23×D61C2×D411yescharacteristic
52C22×#6923—42no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—22no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61—42no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C2×C22⋊D61—41no—
63C2×C22⋊D61—41no—
64C2×C22⋊D61D812yes—
65C2×C22⋊D61SD1612yes—
66C22×Dic61—22no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—42no—
73C22⋊#6923—42no—
74C22⋊#6923—42no—
75C2×#92420—42no—
76C4⋊#6923—82no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—41no—
80C22⋊Dic61—41no—
81C22⋊Dic61—42no—
82#11758198—41no—
83#11758198—42no—
84#11758198—42no—
85C4⋊Dic61—42no—
86#11758196—41no—
87#11758196—44no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61#10212yes—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61C2×D811yescharacteristic
101C22×D61C2×SD1611yescharacteristic
102C2×#6923—82no—
103C2×#6923—82no—
104C2×#6923—82no—
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—84no—
114C2×C4×C61#9911yescharacteristic
115C2×C4×C61#9911yescharacteristic
116C22⋊D61—42no—
117C22⋊D61—42no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61#10212yes—
128C4×D61—42no—
129C4×D61—81no—
130#92420—82no—
131#11874—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—22no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61C2×#9911yescharacteristic
140C22×C61C2×#10211yescharacteristic
141C22×C61#42111yescharacteristic
142C2×D61—21no—
143C2×D61—21no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—41no—
147C2×D61—41no—
148C2×D61—81no—
149#6923—162no—
150C4×C61—21no—
151C4×C61—41no—
152C4×C61—82no—
153C4×C61#35212yes—
154Dic61—22no—
155Dic61—41no—
156Dic61—41no—
157Dic61—82no—
158#2486—611no—
159#1799—611no—
160#1460—611no—
161#1344—614no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—42no—
165C2×C61#151911yescharacteristic
166C2×C61C2×#35211yescharacteristic
167C2×C61#212911yescharacteristic
168D61—41no—
169D61—41no—
170C2×#123—1221no—
171C2×C4×D4—611no—
172C22×C4⋊C4#692311yescharacteristic, complemented
173#431—1221no—
174C2×#99—1221no—
175#398—612no—
176#397—612no—
177#389—1221no—
178#381—1221no—
179#380—611no—
180#380—1221no—
181#380—1221no—
182#342—612no—
183#323—612no—
184#323—1221no—
185C61#1187411yescharacteristic, complemented, Sylow
186C22×D4—611no—
187C23×C4—21no—
188C23×C4—611no—
189C23×C4C2×#692311yescharacteristic
190#123—2442no—
191#123—2442no—
192C4×D4—2441no—
193C4×D4—2441no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—611no—
198C2×C4⋊C4—1222no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—2441no—
201C2×C4⋊C4C2×#692312yes—
202C2×C22⋊C4—611no—
203C2×C22⋊C4—611no—
204C2×C22⋊C4—1222no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4—1222no—
207C2×C42—611no—
208C4⋊C8—2441no—
209#99—2442no—
210C22⋊C8—1222no—
211C4×C8—2441no—
212#95—612no—
213#95—1222no—
214#95—1222no—
215#95—2441no—
216C24—611no—
217C24C22⋊#692311yescharacteristic
218C2×D4—612no—
219C2×D4—612no—
220C2×D4—2441no—
221C2×D4—2441no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1222no—
235C22×C4—2442no—
236C22×C4—2442no—
237C22×C4C22⋊#692311yescharacteristic
238C22×C4C22×#692311yescharacteristic
239C2×C8—2442no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—4882no—
244C4⋊C4—2441no—
245C4⋊C4—2442no—
246C4⋊C4—2444no—
247C22⋊C4—2441no—
248C22⋊C4—2442no—
249C22⋊C4—2442no—
250C22⋊C4—2441no—
251C22⋊C4—2442no—
252C22⋊C4—2442no—
253C22⋊C4—2442no—
254C42—1221no—
255C42—1221no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—611no—
261C23—611no—
262C23—1221no—
263C23—1221no—
264C23—2441no—
265C23—2441no—
266C23#1451983912yes—
267C23C2×C22⋊#692311yescharacteristic
268D4—2442no—
269D4—2442no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—84no—
279C2×C4—612no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—4881no—
285C2×C4—4882no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4#1451983511yescharacteristic
295C2×C4#1451983511yescharacteristic
296C8—4882no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—1221no—
305C22—1221no—
306C22—4881no—
307C22—2441no—
308C22—2441no—
309C22—2441no—
310C22—2441no—
311C22C2×#1451983511yescharacteristic
312C22C2×#1451983911yescharacteristic
313C22#1684076011yescharacteristic
314C4—21no—
315C4—41no—
316C4—82no—
317C4—1222no—
318C4—9762no—
319C4—4882no—
320C4—2441no—
321C4—2441no—
322C4#1684066412yes—
323C2—21no—
324C2—21no—
325C2—42no—
326C2—2441no—
327C2—2441no—
328C2#2146125511yescharacteristic
329C2C2×#1684066411yescharacteristic
330C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#11874
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460799C2semidirect
13. #16839693C4semidirect
17. #16839622C4semidirect
19. C22×C4⋊C4×C61C4semidirect
172. C22×C4⋊C4#6923semidirect
185. C61#11874semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610582
#214607992
#214501212
#1187461