Finite Group Databasebeta

← #29591033

Subgroups — #29591033

Order 15616.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450127C211yesmaximal, characteristic
2#21445807C211yesmaximal, characteristic
3#21443820C214yesmaximal, complemented
4C61×#1460C211yesmaximal, characteristic
5C2×#14519758—21no—
6C22×C4⋊Dic61C411yescharacteristic, complemented
7C2×#14519692—21no—
8#16839692C412yescomplemented
9#16839687C2212yes—
10#16839660—21no—
11#16839623—21no—
12#16839622—21no—
13#16839622—21no—
14#16839622C411yescharacteristic, complemented
15#16839529C2212yes—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#431—21no—
18C61×#323—21no—
19C61×#323C2212yes—
20C23×Dic61—21no—
21#14519758—42no—
22#14519758—42no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—41no—
26C2×C4⋊Dic61—41no—
27C2×C4⋊Dic61—41no—
28C2×C4⋊Dic61C2×C412yes—
29C2×#11758196—22no—
30C2×C22⋊Dic61—21no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33#14519692—42no—
34#14519684—22no—
35#14519684—22no—
36#14519684—41no—
37#14519684C2×C412yes—
38C22×D4×C61D411yescharacteristic
39C23×C4×C61C2×C411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C4×D4×C61—41no—
42C4×D4×C61—41no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C22⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61C2311yescharacteristic
46C2×C42×C61D411yescharacteristic
47C4⋊C8×C61—41no—
48C22⋊C8×C61—22no—
49C4×C8×C61—41no—
50C24×C61C22⋊C411yescharacteristic
51C24×C61C2×D411yescharacteristic
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61D812yes—
55C2×D4×C61SD1612yes—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C22×Dic61—42no—
73C2×C8×C61—42no—
74C4⋊C4×C61—41no—
75C22⋊C4×C61—41no—
76C22⋊C4×C61—41no—
77C42×C61—21no—
78C42×C61—21no—
79C22⋊Dic61—42no—
80C22⋊Dic61—42no—
81C22⋊Dic61—42no—
82C22⋊Dic61—42no—
83C22⋊Dic61—42no—
84C4⋊Dic61—42no—
85C4⋊Dic61—41no—
86C4⋊Dic61—41no—
87C4⋊Dic61—42no—
88C4⋊Dic61—82no—
89#11758196—44no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61#10212yes—
99C23×C61C2×C22⋊C411yescharacteristic
100C23×C61C2×D811yescharacteristic
101C23×C61C2×SD1611yescharacteristic
102D4×C61—42no—
103D4×C61—42no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—81no—
117C2×C4×C61#9911yescharacteristic
118C2×C4×C61#9911yescharacteristic
119C2×C4×C61#10212yes—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—84no—
127C2×Dic61—82no—
128C2×Dic61—82no—
129C2×Dic61—82no—
130C8×C61—82no—
131#11874—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—22no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—42no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—42no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—81no—
146C22×C61C2×#9911yescharacteristic
147C22×C61C2×#10211yescharacteristic
148C22×C61#42111yescharacteristic
149C4×C61—21no—
150C4×C61—22no—
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—41no—
154C4×C61#35212yes—
155Dic61—82no—
156Dic61—82no—
157Dic61—162no—
158#2486—611no—
159#1799—611no—
160#1460D6111yescharacteristic
161#1344—614no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—41no—
165C2×C61—42no—
166C2×C61—41no—
167C2×C61#151911yescharacteristic
168C2×C61C2×#35211yescharacteristic
169C2×C61#212911yescharacteristic
170C2×#123—1221no—
171C2×C4×D4C2×D6111yescharacteristic
172C22×C4⋊C4—611no—
173#431—21no—
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—21no—
184#323C2×D6112yes—
185C61#1187411yescharacteristic, complemented, Sylow
186C22×D4D24411yescharacteristic
187C23×C4—1221no—
188C23×C4C4×D6111yescharacteristic
189C23×C4D24411yescharacteristic
190#123—2442no—
191#123—2442no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—612no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—2441no—
199C2×C4⋊C4—2441no—
200C2×C4⋊C4—2441no—
201C2×C4⋊C4C4×D6111yescharacteristic
202C2×C22⋊C4—1221no—
203C2×C22⋊C4—1222no—
204C2×C22⋊C4—1222no—
205C2×C22⋊C4C22⋊D6111yescharacteristic
206C2×C22⋊C4C22×D6111yescharacteristic
207C2×C42C22⋊D6111yescharacteristic
208C4⋊C8—41no—
209#99—2442no—
210C22⋊C8—22no—
211C4×C8—41no—
212#95—612no—
213#95—1222no—
214#95—1222no—
215#95—2441no—
216C24#1175819811yescharacteristic
217C24C2×D24411yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1175819012yes—
221C2×D4D48812yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—1222no—
229C22×C4—1222no—
230C22×C4—2442no—
231C22×C4—2442no—
232C22×C4—2442no—
233C22×C4—2442no—
234C22×C4#1175819811yescharacteristic
235C22×C4#1175819811yescharacteristic
236C22×C4#1175819811yescharacteristic
237C22×C4C2×C4×D6111yescharacteristic
238C22×C4C2×C22⋊D6111yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—4882no—
242C4⋊C4—2442no—
243C4⋊C4—2442no—
244C4⋊C4—2441no—
245C4⋊C4—2441no—
246C4⋊C4—2444no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—2442no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C22⋊C4—2442no—
253C22⋊C4—2442no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1451969912yes—
265C23C2×#1175819811yescharacteristic
266C23C2×D48811yescharacteristic
267C23C2×#1175819011yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—4882no—
284C2×C4—4882no—
285C2×C4—4882no—
286C2×C4—4884no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4#1451969111yescharacteristic
294C2×C4#1451969111yescharacteristic
295C2×C4#1451969912yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1451969111yescharacteristic
312C22C2×#1451969911yescharacteristic
313C22#1683977811yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—9762no—
320C4—4882no—
321C4—4882no—
322C4#1683955312yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2144438811yescharacteristic
329C2C2×#1683955311yescharacteristic
330C2#2144749311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1460C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1460D61
hypocenterC61#11874
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #21443820C2semidirect
6. C22×C4⋊Dic61C4semidirect
8. #16839692C4semidirect
14. #16839622C4semidirect
185. C61#11874semidirect

Maximal subgroups

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

subgroupindex
#214501272
#214458072
#214438202
C61×#14602
#1187461