Finite Group Databasebeta

← #29584400

Subgroups — #29584400

Order 15616.

574 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 95 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#21456570C211yesmaximal, characteristic
2#21446654C212yesmaximal, complemented
3#21446245C212yesmaximal, complemented
4#21445996C211yesmaximal, characteristic, complemented
5C61×#1289C211yesmaximal, characteristic, complemented
6C23×#92418C2211yescharacteristic
7C2×#14519751C2212yes—
8#16840076—22no—
9#16839998—21no—
10#16839998—21no—
11#16839898—22no—
12C2×#14519706—21no—
13C2×#14519697—22no—
14C2×#14519691—22no—
15C2×#14519691—22no—
16C4×C22⋊Dic61C2211yescharacteristic
17#16839524—22no—
18#16839516—21no—
19C2×C22⋊C8×C61C2211yescharacteristic
20C61×#380C2212yes—
21C22×D244D412yes—
22C22×C22⋊D61—22no—
23C2×D488—21no—
24C2×D488—21no—
25C22×#92418—22no—
26C22×#92418—21no—
27C22×#92418—42no—
28#14519751—22no—
29#14519751—22no—
30#14519751—22no—
31#14519751—22no—
32C2×C4⋊Dic61D412yes—
33C2×C22⋊Dic61D411yescharacteristic
34C2×C22⋊Dic61D411yescharacteristic
35C2×#11758198—22no—
36C2×C4×Dic61—21no—
37#14519711—21no—
38#14519711—21no—
39#14519706—42no—
40#14519699—22no—
41#14519699—22no—
42#14519697—44no—
43#14519691—44no—
44#14519691—44no—
45#14519684—21no—
46C23×C4×C61C2311yescharacteristic
47C22×C8×C61D411yescharacteristic
48C22×C8×C61D411yescharacteristic
49C2×C4⋊C4×C61—22no—
50C2×C22⋊C4×C61—22no—
51C22⋊C8×C61—22no—
52#95×C61—22no—
53C24×C61C2×D411yescharacteristic
54C23×D61—22no—
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61—42no—
58C22×C4×C61C2×D412yes—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61#4111yescharacteristic
61C22×C4×C61#4112yes—
62C2×C22⋊D61—42no—
63C2×C22⋊D61—42no—
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C2×C22⋊D61—42no—
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—21no—
71C2×D244—22no—
72C2×D244—22no—
73C2×D244—42no—
74C2×D244—42no—
75C2×C8×C61—21no—
76C2×C8×C61—21no—
77C2×C8×C61—41no—
78C2×C8×C61—41no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—42no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C22⋊Dic61—21no—
84C22⋊Dic61—22no—
85C22⋊Dic61D812yescomplemented
86#11758198—42no—
87#11758198—42no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90C4×Dic61—41no—
91C4×Dic61—41no—
92C2×#92418—24no—
93C2×#92418—22no—
94C2×#92418—41no—
95C2×#92418—42no—
96C2×#92418—41no—
97D488—81no—
98D488—81no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61#12112yes—
102C23×C61#12111yescharacteristic
103C23×C61C2×D811yescharacteristic
104C23×C61#13511yescharacteristic
105C23×C61AGL(1,ℤ/8)12yes—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61—42no—
109C22×D61—82no—
110C22×D61—82no—
111C2×C4×C61—21no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—82no—
118C2×C4×C61—82no—
119C2×C4×C61—82no—
120C2×C4×C61#12011yescharacteristic
121C2×C4×C61#12312yes—
122C2×C4×C61#12411yescharacteristic
123C22⋊D61—82no—
124C22⋊D61—82no—
125C22⋊D61—82no—
126C22⋊D61—82no—
127C2×Dic61—22no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136D244—44no—
137D244—42no—
138D244—42no—
139C8×C61—41no—
140C8×C61—41no—
141#92418—42no—
142#10564—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—42no—
150C22×C61#39411yescharacteristic
151C22×C61#45912yes—
152C22×C61#46312yes—
153C22×C61#46611yescharacteristic
154C22×C61#46911yescharacteristic
155C2×D61—82no—
156C2×D61—82no—
157C2×D61—82no—
158C2×D61—162no—
159C2×D61—162no—
160C4×C61—22no—
161C4×C61—21no—
162C4×C61—162no—
163C4×C61B(4,𝔽2)11yescharacteristic
164C4×C61#45811yescharacteristic
165Dic61—42no—
166Dic61—81no—
167Dic61—82no—
168#3186—611no—
169#1964—612no—
170#1893—612no—
171#1850—611no—
172#1289D6111yescharacteristic, complemented
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—22no—
176C2×C61—22no—
177C2×C61#204411yescharacteristic
178C2×C61#205311yescharacteristic
179C2×C61#205411yescharacteristic
180D61—162no—
181C23×C8—611no—
182C2×#121—612no—
183#480—1222no—
184#466—1221no—
185#466—1221no—
186#447—1222no—
187C2×C4⋊C8—1221no—
188C2×#102—1222no—
189C2×#99—1222no—
190C2×#99—1222no—
191C2×C22⋊C8C2×D6111yescharacteristic
192#380C2×D6112yes—
193C4×C22⋊C4—611no—
194#340—1222no—
195#336—1221no—
196C61#1056411yescharacteristic, complemented, Sylow
197C22×D4—612no—
198C22×D4—1222no—
199C23×C4C22×D6111yescharacteristic
200C2×D8—1221no—
201C2×D8—1221no—
202C22×C8—1221no—
203C22×C8—1222no—
204C22×C8—2442no—
205C22×C8D24411yescharacteristic
206C22×C8D24411yescharacteristic
207#121—1222no—
208#121—1222no—
209#121—1222no—
210#121—1222no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—612no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—611no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—1222no—
217C2×C42—1221no—
218#107—1221no—
219#107—1221no—
220C4⋊C8—2442no—
221#102—1222no—
222#102—1222no—
223#102—2444no—
224#99—2444no—
225#99—2444no—
226C22⋊C8—22no—
227#95—22no—
228#95—1221no—
229C24—1222no—
230C24D4×D6111yescharacteristic
231C2×D4—1222no—
232C2×D4—1222no—
233C2×D4—2442no—
234C2×D4—2442no—
235C2×D4—2442no—
236C2×D4—2442no—
237C2×D4—2442no—
238C2×D4—2442no—
239C2×D4—2442no—
240C2×D4—2442no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—1221no—
245C22×C4—1222no—
246C22×C4—1221no—
247C22×C4C2×D24411yescharacteristic
248C22×C4#1175822212yes—
249C22×C4D4×D6112yes—
250C22×C4#1175822611yescharacteristic
251D8—4881no—
252D8—4881no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—1224no—
258C2×C8—1222no—
259C2×C8—2441no—
260C2×C8—2442no—
261C2×C8—2441no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—1222no—
265C4⋊C4—1222no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—612no—
269C22⋊C4—1221no—
270C22⋊C4—1222no—
271C22⋊C4—2442no—
272C22⋊C4—2442no—
273C42—2441no—
274C42—2441no—
275C23—41no—
276C23—41no—
277C23—4882no—
278C23—4882no—
279C23—2442no—
280C23—2442no—
281C23—2442no—
282C23#1451975012yes—
283C23#1451975211yescharacteristic
284C23D8×D6111yescharacteristic
285C23#1451980011yescharacteristic
286C23#1451980412yes—
287D4—4882no—
288D4—4882no—
289D4—4882no—
290D4—4882no—
291D4—2444no—
292D4—2442no—
293D4—2442no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—1222no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4#1451974711yescharacteristic
313C2×C4#1451976212yes—
314C2×C4#1451976511yescharacteristic
315C8—41no—
316C8—41no—
317C8—2442no—
318C22—22no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—9762no—
326C22—9762no—
327C22—4882no—
328C22—4882no—
329C22—4882no—
330C22#1683968011yescharacteristic
331C22#1683995612yes—
332C22#1683998012yes—
333C22#1683999911yescharacteristic
334C22#1684001611yescharacteristic
335C4—22no—
336C4—21no—
337C4—162no—
338C4—4881no—
339C4—4882no—
340C4—2442no—
341C4#1683995011yescharacteristic
342C4#1683995311yescharacteristic
343C2—22no—
344C2—21no—
345C2—21no—
346C2—22no—
347C2—9762no—
348C2#2144707111yescharacteristic
349C2#2144712611yescharacteristic
350C2#2144713111yescharacteristic
subgroupG/H
centerC22#16839680
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1289C2
socleC22×C61#394
prime radical#21456570C2
hypercenter#1289D61
hypocenterC61#10564
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21446654C2semidirect
3. #21446245C2semidirect
4. #21445996C2semidirect
5. C61×#1289C2semidirect
85. C22⋊Dic61D8semidirect
172. #1289D61semidirect
196. C61#10564semidirect

Maximal subgroups

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

subgroupindex
#214565702
#214466542
#214462452
#214459962
C61×#12892
#1056461