Finite Group Databasebeta

← #29575025

Subgroups — #29575025

Order 15616.

928 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16839679C211yesmaximal, characteristic
2#21448422C211yesmaximal, characteristic, complemented
3#21445751C212yesmaximal
4#21443699C212yesmaximal, complemented
5C2×C61×#324C211yesmaximal, characteristic, complemented
6C23×D244C2211yescharacteristic
7C22×#11758198—21no—
8C22×#11758198—21no—
9C2×#14519699—24no—
10#16839679—22no—
11#16839679C414yescomplemented
12#16839648—24no—
13#16839625—21no—
14#16839622C2212yes—
15C4×C22⋊Dic61—21no—
16C2×#14519684C2211yescharacteristic
17C22×C4⋊C4×C61C2211yescharacteristic
18C2×C22⋊C8×C61C2212yes—
19C61×#324—22no—
20C24×D61—21no—
21C22×D244—22no—
22C22×D244—22no—
23C22×D244—21no—
24C22×D244—44no—
25C22×D244C2×C412yes—
26C23×Dic61—21no—
27C2×C4⋊Dic61D414yes—
28C2×C22⋊Dic61—22no—
29C2×#11758198—24no—
30C2×#11758198—24no—
31C2×#11758198—44no—
32C2×#11758198—44no—
33C2×C4×Dic61—22no—
34#14519699—48no—
35#14519684—22no—
36#14519684—24no—
37#14519684—21no—
38#14519684—22no—
39#14519684C2×C412yes—
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C22×C8×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61C2×C412yes—
47C2×C4⋊C4×C61D412yes—
48C22⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C24×C61C2×D411yescharacteristic
51C23×D61—24no—
52C23×D61—22no—
53C23×D61—42no—
54C23×D61—42no—
55C23×D61—42no—
56C23×D61—41no—
57C23×D61—41no—
58C23×D61—42no—
59C23×D61—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—22no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22⋊C412yes—
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D412yes—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61#4112yes—
74C22×Dic61—22no—
75C22×Dic61—24no—
76C22×Dic61—42no—
77C22×Dic61—42no—
78C22×Dic61—42no—
79C2×D244—28no—
80C2×D244—24no—
81C2×D244—44no—
82C2×D244—48no—
83C2×D244—44no—
84C2×D244—44no—
85C2×D244—42no—
86C2×C8×C61—44no—
87C2×C8×C61—42no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—24no—
90C22⋊Dic61—42no—
91C22⋊Dic61—42no—
92#11758198—48no—
93#11758198—48no—
94C4⋊Dic61—44no—
95C4×Dic61—82no—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61C2×C22⋊C411yescharacteristic
99C23×C61#12011yescharacteristic
100C23×C61#12311yescharacteristic
101C23×C61#13512yes—
102C23×C61AGL(1,ℤ/8)12yes—
103C22×D61—24no—
104C22×D61—42no—
105C22×D61—44no—
106C22×D61—48no—
107C22×D61—48no—
108C22×D61—42no—
109C22×D61—42no—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61—42no—
113C22×D61—44no—
114C22×D61—84no—
115C22×D61—84no—
116C22×D61—82no—
117C22×D61—82no—
118C22×D61—82no—
119C22×D61—82no—
120C2×C4×C61—24no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—81no—
133C2×C4×C61C4×D412yes—
134C2×C4×C61#12112yes—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—44no—
139C2×Dic61—44no—
140C2×Dic61—44no—
141C2×Dic61—84no—
142D244—28no—
143D244—48no—
144D244—44no—
145C8×C61—82no—
146#9018—611nomaximal, Sylow
147C22×C61—24no—
148C22×C61—24no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61#38611yescharacteristic
154C22×C61#41611yescharacteristic
155C22×C61#41811yescharacteristic
156C22×C61#44912yes—
157C22×C61#45412yes—
158C22×C61B(4,𝔽2)12yes—
159C22×C61#48212yes—
160C2×D61—44no—
161C2×D61—44no—
162C2×D61—44no—
163C2×D61—82no—
164C2×D61—82no—
165C2×D61—88no—
166C2×D61—88no—
167C2×D61—82no—
168C2×D61—82no—
169C4×C61—22no—
170C4×C61—22no—
171C4×C61—42no—
172C4×C61—42no—
173Dic61—82no—
174Dic61—84no—
175C2×#394—611no—
176#2313—611no—
177#1786—612no—
178C2×#324D6111yescharacteristic, complemented
179#1294—612no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—22no—
183C2×C61#159714yes—
184C2×C61#187511yescharacteristic
185C2×C61#188811yescharacteristic
186C2×C61#188911yescharacteristic
187D61—84no—
188C23×D4—611no—
189C22×C4⋊C4C2×D6111yescharacteristic
190C22×C22⋊C4—1221no—
191C22×C22⋊C4—1221no—
192C2×#102—1224no—
193C2×C22⋊C8C2×D6112yes—
194#394—614no—
195#394—1222no—
196#387—1224no—
197#381—1221no—
198#380—612no—
199C4×C22⋊C4—1221no—
200C2×#95—611no—
201#324—22no—
202C61#901811yescharacteristic, complemented, Sylow
203C25—1221no—
204C22×D4—612no—
205C22×D4—1222no—
206C22×D4—1222no—
207C22×D4—1221no—
208C22×D4—2444no—
209C23×C4—1221no—
210C23×C4D24411yescharacteristic
211C23×C4D24411yescharacteristic
212C23×C4C22×D6111yescharacteristic
213C22×C8—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—614no—
217C2×C4⋊C4C4×D6112yes—
218C2×C4⋊C4C22⋊D6112yes—
219C2×C22⋊C4—1222no—
220C2×C22⋊C4—1224no—
221C2×C22⋊C4—1224no—
222C2×C22⋊C4—2444no—
223C2×C22⋊C4—2444no—
224C2×C42—1222no—
225#102—2448no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228#95—612no—
229#95—1222no—
230#95—1224no—
231#95—1222no—
232#95—1221no—
233C24—1224no—
234C24—1222no—
235C24—2442no—
236C24—2442no—
237C24—2442no—
238C24—2441no—
239C24—2441no—
240C24—2442no—
241C24—2442no—
242C24C2×D24411yescharacteristic
243C2×D4—1228no—
244C2×D4—1224no—
245C2×D4—2444no—
246C2×D4—2444no—
247C2×D4—2448no—
248C2×D4—2444no—
249C2×D4—2442no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—1222no—
259C22×C4—1224no—
260C22×C4—2442no—
261C22×C4—2442no—
262C22×C4—2442no—
263C22×C4#1175819812yes—
264C22×C4#1175819812yes—
265C22×C4C2×C4×D6111yescharacteristic
266C22×C4D4×D6112yes—
267C22×C4#1175822412yes—
268C22×C4C2×C22⋊D6111yescharacteristic
269C2×C8—44no—
270C2×C8—42no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—2444no—
274C22⋊C4—2448no—
275C22⋊C4—2448no—
276C22⋊C4—2442no—
277C22⋊C4—2442no—
278C42—4882no—
279C23—22no—
280C23—22no—
281C23—1224no—
282C23—4882no—
283C23—4884no—
284C23—4884no—
285C23—4882no—
286C23—4882no—
287C23—4882no—
288C23—2442no—
289C23—2444no—
290C23—2442no—
291C23—2442no—
292C23—2448no—
293C23—2448no—
294C23—2444no—
295C23—2442no—
296C23—2442no—
297C23—2442no—
298C23C2×#1175819811yescharacteristic
299C23#1451974711yescharacteristic
300C23#1451976111yescharacteristic
301C23#1451979912yes—
302C23#1451980512yes—
303D4—1228no—
304D4—2448no—
305D4—2444no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—4884no—
320C2×C4—2442no—
321C2×C4—2442no—
322C2×C4—2444no—
323C2×C4—2444no—
324C2×C4—2442no—
325C2×C4—2444no—
326C2×C4#1451973712yes—
327C2×C4#1451974912yes—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—24no—
335C22—4882no—
336C22—4888no—
337C22—4882no—
338C22—4882no—
339C22—4888no—
340C22—4882no—
341C22—2444no—
342C22—2444no—
343C22—2444no—
344C22#1683963911yescharacteristic
345C22#1683975911yescharacteristic
346C22#1683976811yescharacteristic
347C22#1683990912yes—
348C22#1683993712yes—
349C22#1683995012yes—
350C22#1684008712yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—4882no—
356C4—4884no—
357C2—21no—
358C2—22no—
359C2—21no—
360C2—4884no—
361C2#2144475014yes—
362C2#2144613911yescharacteristic
363C2#2144622011yescharacteristic
364C2#2144622511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC23×C61#120
prime radicalC23×D244C22
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9018
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
2. #21448422C2semidirect
4. #21443699C2semidirect
5. C2×C61×#324C2semidirect
11. #16839679C4semidirect
178. C2×#324D61semidirect
202. C61#9018semidirect

Maximal subgroups

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

subgroupindex
C2×#168396792
#214484222
#214457512
#214436992
C2×C61×#3242
#901861