Finite Group Databasebeta

← #29575010

Subgroups — #29575010

Order 15616.

800 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#16839682C211yesmaximal, characteristic
2#21448437C211yesmaximal, characteristic
3#21445755C212yesmaximal
4#21443699C212yesmaximal
5C2×C61×#324C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C2×#14519736—21no—
8C22×#11758196—21no—
9C2×#14519702—24no—
10#16839682—22no—
11#16839682C414yescomplemented
12#16839661—24no—
13#16839622—21no—
14#16839622C2212yes—
15C2×#14519684—21no—
16C2×#14519684C2211yescharacteristic
17C22×C4⋊C4×C61C2211yescharacteristic
18C2×C22⋊C8×C61C2212yes—
19C61×#324—22no—
20C22×Dic122—22no—
21C22×Dic122—22no—
22C22×Dic122—21no—
23C22×Dic122—44no—
24C22×Dic122C2×C412yes—
25C23×Dic61—21no—
26C23×Dic61—21no—
27#14519736—42no—
28#14519736—42no—
29C2×C4⋊Dic61—22no—
30C2×C4⋊Dic61—22no—
31C2×C4⋊Dic61D414yes—
32C2×#11758196—24no—
33C2×#11758196—44no—
34C2×C22⋊Dic61—22no—
35C2×C4×Dic61—22no—
36#14519702—48no—
37#14519684—22no—
38#14519684—24no—
39#14519684—21no—
40#14519684—22no—
41#14519684C2×C412yes—
42C23×C4×C61D411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C23×C4×C61C2311yescharacteristic
45C22×C8×C61—22no—
46C2×C4⋊C4×C61—22no—
47C2×C4⋊C4×C61—22no—
48C2×C4⋊C4×C61C2×C412yes—
49C2×C4⋊C4×C61D412yes—
50C22⋊C8×C61—22no—
51C22⋊C8×C61—22no—
52C24×C61C2×D411yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61C22⋊C412yes—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61SD1612yes—
64C22×C4×C61Q1612yes—
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D412yes—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61#4112yes—
69C22×Dic61—24no—
70C22×Dic61—24no—
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—24no—
74C22×Dic61—42no—
75C22×Dic61—42no—
76C22×Dic61—41no—
77C22×Dic61—41no—
78C22×Dic61—41no—
79C22×Dic61—41no—
80C2×Dic122—28no—
81C2×Dic122—24no—
82C2×Dic122—44no—
83C2×Dic122—48no—
84C2×Dic122—44no—
85C2×Dic122—44no—
86C2×Dic122—42no—
87C2×C8×C61—44no—
88C2×C8×C61—42no—
89C4⋊C4×C61—24no—
90C4⋊C4×C61—24no—
91C22⋊Dic61—42no—
92C22⋊Dic61—42no—
93C4⋊Dic61—44no—
94C4⋊Dic61—44no—
95C4⋊Dic61—82no—
96#11758196—48no—
97C4×Dic61—44no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61#12011yescharacteristic
102C23×C61#12311yescharacteristic
103C23×C61C2×SD1611yescharacteristic
104C23×C61C2×Q1611yescharacteristic
105C23×C61AGL(1,ℤ/8)11yescharacteristic
106C23×C61#13711yescharacteristic
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—81no—
121C2×C4×C61#10314yescomplemented
122C2×C4×C61C4×D412yes—
123C2×C4×C61#12112yes—
124C2×Dic61—24no—
125C2×Dic61—24no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—44no—
129C2×Dic61—44no—
130C2×Dic61—44no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—44no—
134C2×Dic61—44no—
135C2×Dic61—44no—
136C2×Dic61—44no—
137Dic122—28no—
138Dic122—48no—
139Dic122—44no—
140C8×C61—82no—
141#9015—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#38611yescharacteristic
151C22×C61C2×#10311yescharacteristic
152C22×C61#41711yescharacteristic
153C22×C61#45011yescharacteristic
154C22×C61#45111yescharacteristic
155C22×C61#45612yes—
156C22×C61B(4,𝔽2)12yes—
157C22×C61#48311yescharacteristic
158C22×C61#48411yescharacteristic
159C4×C61—24no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—42no—
163Dic61—44no—
164Dic61—44no—
165Dic61—84no—
166C2×#395—611no—
167#2316—611no—
168#1787—612no—
169C2×#324D6111yescharacteristic
170#1294—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#160412yes—
176C2×C61#160512yes—
177C2×C61#187211yescharacteristic
178C2×C61#188611yescharacteristic
179C2×C61#188911yescharacteristic
180C23×Q8—611no—
181C2×#117—1221no—
182C22×C4⋊C4—1221no—
183C22×C4⋊C4C2×D6111yescharacteristic
184C2×#103—1224no—
185C2×C22⋊C8C2×D6112yes—
186#395—614no—
187#395—1222no—
188#389—1224no—
189#380—612no—
190#380—1221no—
191C2×#95—611no—
192C2×#95—1221no—
193#324—22no—
194C61#901511yescharacteristic, complemented, Sylow
195C22×Q8—612no—
196C22×Q8—1221no—
197C22×Q8—1222no—
198C22×Q8—1222no—
199C22×Q8—2444no—
200C23×C4—1221no—
201C23×C4—1221no—
202C23×C4D24411yescharacteristic
203C23×C4D24411yescharacteristic
204C23×C4C22×D6111yescharacteristic
205C22×C8—22no—
206#117—2442no—
207#117—2442no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—614no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4—1222no—
213C2×C4⋊C4—1224no—
214C2×C4⋊C4—2444no—
215C2×C4⋊C4C4×D6112yes—
216C2×C4⋊C4C22⋊D6112yes—
217C2×C22⋊C4—1222no—
218C2×C42—1222no—
219#103—2448no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222#95—612no—
223#95—1222no—
224#95—1224no—
225#95—1222no—
226#95—1221no—
227C24C2×D24411yescharacteristic
228C2×Q8—1228no—
229C2×Q8—1224no—
230C2×Q8—2444no—
231C2×Q8—2444no—
232C2×Q8—2444no—
233C2×Q8—2448no—
234C2×Q8—2442no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1224no—
244C22×C4—1224no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—1224no—
248C22×C4—2442no—
249C22×C4—2442no—
250C22×C4—2441no—
251C22×C4—2441no—
252C22×C4—2441no—
253C22×C4—2441no—
254C22×C4#1175819012yes—
255C22×C4Dic24412yes—
256C22×C4#1175819812yes—
257C22×C4#1175819812yes—
258C22×C4C2×C4×D6111yescharacteristic
259C22×C4D4×D6112yes—
260C22×C4#1175822412yes—
261C22×C4C2×C22⋊D6111yescharacteristic
262C2×C8—44no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—4882no—
267C4⋊C4—2444no—
268C4⋊C4—2444no—
269C4⋊C4—2448no—
270C22⋊C4—2442no—
271C22⋊C4—2442no—
272C42—2444no—
273C23—22no—
274C23—22no—
275C23C2×#1175819811yescharacteristic
276C23#1451974711yescharacteristic
277C23#1451976111yescharacteristic
278C23C2×#1175819011yescharacteristic
279C23C2×Dic24411yescharacteristic
280C23#1451980511yescharacteristic
281C23#1451981011yescharacteristic
282Q8—1228no—
283Q8—2448no—
284Q8—2444no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—1224no—
300C2×C4—1224no—
301C2×C4—2444no—
302C2×C4—2444no—
303C2×C4—2444no—
304C2×C4—2442no—
305C2×C4—2444no—
306C2×C4—2444no—
307C2×C4—2442no—
308C2×C4—2444no—
309C2×C4—2444no—
310C2×C4—2442no—
311C2×C4—2442no—
312C2×C4#1451970214yescomplemented
313C2×C4#1451973712yes—
314C2×C4#1451974912yes—
315C8—82no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#1683963911yescharacteristic
325C22C2×#1451970211yescharacteristic
326C22#1683976411yescharacteristic
327C22#1683991511yescharacteristic
328C22#1683992011yescharacteristic
329C22#1683994712yes—
330C22#1683995012yes—
331C22#1684009311yescharacteristic
332C22#1684009811yescharacteristic
333C4—22no—
334C4—22no—
335C4—24no—
336C4—42no—
337C4—4884no—
338C4—2444no—
339C4—2444no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2144479012yes—
345C2#2144479512yes—
346C2#2144612411yescharacteristic
347C2#2144621011yescharacteristic
348C2#2144622511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC23×C61#120
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9015
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
11. #16839682C4semidirect
121. C2×C4×C61#103semidirect
194. C61#9015semidirect
312. C2×C4#14519702semidirect

Maximal subgroups

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

subgroupindex
C2×#168396822
#214484372
#214457552
#214436992
C2×C61×#3242
#901561