Finite Group Databasebeta

← #29568332

Subgroups — #29568332

Order 15616.

624 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 119 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#21448383C211yesmaximal, characteristic
2#21445934C212yesmaximal, complemented
3#21443640C211yesmaximal, characteristic, complemented
4#21443639C212yesmaximal, complemented
5C61×#1270C211yesmaximal, characteristic, complemented
6C2×#14519773C411yescharacteristic
7C2×D4×Dic61C411yescharacteristic
8C2×#14519739C411yescharacteristic
9C2×#14519737—22no—
10C2×#14519706—22no—
11#16839664C2211yescharacteristic
12#16839662—22no—
13#16839631C411yescharacteristic
14C42×Dic61C2211yescharacteristic
15#16839517C2212yes—
16#16839516—22no—
17C61×#390C2211yescharacteristic
18C61×#336C2212yes—
19C22×D244C2×C411yescharacteristic
20C22×C22⋊D61—22no—
21C23×Dic61—21no—
22C22×C4×D61—21no—
23C22×#92418—22no—
24#14519773—22no—
25#14519773—22no—
26D4×Dic61—42no—
27#14519739—42no—
28#14519737—44no—
29#14519737—44no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—22no—
32C2×C22⋊Dic61—21no—
33C2×#11758198—22no—
34C2×#11758198—21no—
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D412yes—
38C2×C4×Dic61D412yes—
39#14519706—44no—
40C22×D4×C61C2×C411yescharacteristic
41C22×C8×C61—22no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61—22no—
44C2×C42×C61C2311yescharacteristic
45C24×C61—21no—
46C23×D61—21no—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C2×D4×C61M1612yes—
51C22×C4×C61—21no—
52C22×C4×C61C22×C411yescharacteristic
53C22×C4×C61C2×D412yes—
54C2×C22⋊D61—28no—
55C2×C22⋊D61—44no—
56C2×C22⋊D61—44no—
57C22×Dic61—21no—
58C22×Dic61—42no—
59C22×Dic61—44no—
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C22⋊C412yes—
62C22×Dic61C22⋊C411yescharacteristic
63C2×D244—21no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244M1612yes—
67C2×C4×D61—42no—
68C2×C4×D61—44no—
69C2×C8×C61—42no—
70C2×C8×C61—42no—
71C2×C8×C61—42no—
72C4⋊C4×C61—41no—
73C22⋊C4×C61—42no—
74C22⋊C4×C61—42no—
75C42×C61—41no—
76C22⋊Dic61—42no—
77#11758198—42no—
78#11758198—42no—
79#11758198—42no—
80C4⋊Dic61—41no—
81#11758196—44no—
82C4×Dic61—21no—
83C4×Dic61—21no—
84C4×Dic61—22no—
85C4×Dic61—42no—
86C4×Dic61D814yes—
87C4×Dic61SD1614yes—
88C2×#92418—42no—
89C2×#92418—42no—
90C2×#92418—42no—
91C23×C61—22no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61C2×C22⋊C411yescharacteristic
97C22×D61—22no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—42no—
101C22×D61—42no—
102D4×C61—42no—
103D4×C61—42no—
104C2×C4×C61—22no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—22no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C2×C4×C61C2×D812yes—
113C2×C4×C61C2×SD1612yes—
114C22⋊D61—416no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—21no—
118C2×Dic61—22no—
119C2×Dic61—41no—
120C2×Dic61—42no—
121C2×Dic61—84no—
122C2×Dic61#10012yes—
123C2×Dic61#10214yes—
124C2×Dic61#10214yes—
125D244—42no—
126D244—42no—
127C4×D61—84no—
128C8×C61—82no—
129#92418—82no—
130#7651—611nomaximal, Sylow
131C22×C61—42no—
132C22×C61—44no—
133C22×C61—81no—
134C22×C61—81no—
135C22×C61—82no—
136C22×C61#40711yescharacteristic
137C22×C61#41011yescharacteristic
138C22×C61C2×#10011yescharacteristic
139C22×C61C2×#10212yes—
140C22×C61#42112yes—
141C2×D61—42no—
142C2×D61—44no—
143C2×D61—81no—
144C2×D61—81no—
145C2×D61—82no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—81no—
151Dic61—24no—
152Dic61—41no—
153Dic61—81no—
154Dic61#33114yes—
155C4×#127—611no—
156#1831—612no—
157#1270—611no—
158#1270—612no—
159#1270D6111yescharacteristic, complemented
160C2×C61—82no—
161C2×C61#146611yescharacteristic
162C2×C61#148612yes—
163C2×C61#151712yes—
164C2×C61#153611yescharacteristic
165C2×C61C2×#33111yescharacteristic
166D61—82no—
167C2×#127—611no—
168C2×C4×D4—611no—
169C2×C4×D4—611no—
170C2×C4×D4—1222no—
171C2×C4⋊C8—1222no—
172#390—611no—
173#390—1222no—
174#390C2×D6111yescharacteristic
175#384—611no—
176C43—611no—
177#336—612no—
178#336—1222no—
179#336C2×D6112yes—
180C61#765111yescharacteristic, complemented, Sylow
181C22×D4—611no—
182C22×D4—1222no—
183C22×D4C4×D6111yescharacteristic
184C23×C4—1221no—
185C23×C4—1221no—
186C22×C8—22no—
187C22×C8—1222no—
188#127—1222no—
189#127—1222no—
190C4×D4—2444no—
191C4×D4—2444no—
192C4×D4—2442no—
193C4×D4—2442no—
194C2×C4⋊C4—611no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4C4×D6111yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1221no—
199C2×C22⋊C4—1221no—
200C2×C22⋊C4—1222no—
201C2×C42—612no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—612no—
205C2×C42C22×D6111yescharacteristic
206C4⋊C8—2444no—
207C24—21no—
208C24—1221no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—612no—
213C2×D4—1228no—
214C2×D4—1221no—
215C2×D4—2441no—
216C2×D4—2441no—
217C2×D4—2444no—
218C2×D4—2444no—
219C2×D4#1175818912yes—
220C22×C4—21no—
221C22×C4—611no—
222C22×C4—612no—
223C22×C4—611no—
224C22×C4—1221no—
225C22×C4—2444no—
226C22×C4—2442no—
227C22×C4—2444no—
228C22×C4—2442no—
229C22×C4C2×C4×D6111yescharacteristic
230C22×C4D4×D6112yes—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—2442no—
235C2×C8—2442no—
236C2×C8—2442no—
237C4⋊C4—41no—
238C4⋊C4—2441no—
239C4⋊C4—2444no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—2442no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C22⋊C4—2442no—
246C42—41no—
247C42—614no—
248C42—614no—
249C42—1222no—
250C42—1221no—
251C42—1221no—
252C42—2442no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—1222no—
259C23—2441no—
260C23—2441no—
261C23—2442no—
262C23—2442no—
263C23C22⋊C4×D6111yescharacteristic
264D4—42no—
265D4—42no—
266D4—2442no—
267D4—2442no—
268D4—24416no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—612no—
276C2×C4—614no—
277C2×C4—614no—
278C2×C4—1222no—
279C2×C4—1222no—
280C2×C4—1221no—
281C2×C4—1222no—
282C2×C4—4884no—
283C2×C4—4884no—
284C2×C4—2441no—
285C2×C4—2442no—
286C2×C4M16×D6111yescharacteristic
287C2×C4C2×#1175818911yescharacteristic
288C2×C4D8×D6112yes—
289C2×C4SD16×D6112yes—
290C8—82no—
291C8—4882no—
292C22—42no—
293C22—44no—
294C22—82no—
295C22—81no—
296C22—81no—
297C22—4882no—
298C22—4881no—
299C22—4881no—
300C22—2442no—
301C22—2444no—
302C22#1683972411yescharacteristic
303C22#1683973411yescharacteristic
304C22#100×D6111yescharacteristic
305C22#102×D6112yes—
306C22#1683977712yes—
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—614no—
313C4—1224no—
314C4—4881no—
315C4—2441no—
316C2—82no—
317C2—4882no—
318C2#2144415811yescharacteristic
319C2#2144424212yes—
320C2#2144437712yes—
321C2#2144445211yescharacteristic
322C2#331×D6111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C4
cosocleC2×C42×C61C23
hypercenter#1270D61
hypocenterC61#7651
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21445934C2semidirect
3. #21443640C2semidirect
4. #21443639C2semidirect
5. C61×#1270C2semidirect
159. #1270D61semidirect
180. C61#7651semidirect

Maximal subgroups

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

subgroupindex
#214483832
#214459342
#214436402
#214436392
C61×#12702
#765161