Finite Group Databasebeta

← #29570137

Subgroups — #29570137

Order 15616.

954 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 159 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#407×D61C211yesmaximal, characteristic
2#21448200C212yesmaximal
3#21443705C212yesmaximal, complemented
4#21443704C211yesmaximal, characteristic, complemented
5C61×#1297C211yesmaximal, characteristic, complemented
6C23×C4×D61C2211yescharacteristic
7C2×M16×D61C412yescomplemented
8C2×#14519734—22no—
9#16839726C2211yescharacteristic
10#16839725C412yescomplemented
11#16839724—22no—
12C22⋊C8×D61—22no—
13#16839624C414yescomplemented
14#16839621C414yescomplemented
15#16839620—22no—
16#16839617—22no—
17C4×#11758198—22no—
18C4×C22⋊Dic61C2212yes—
19C61×#407C2211yescharacteristic
20C4×C22⋊C4×C61C2212yes—
21C24×D61D411yescharacteristic
22C23×Dic61D411yescharacteristic
23C22×C4×D61—44no—
24C22×C4×D61C2×C411yescharacteristic
25C22×C4×D61C2×C411yescharacteristic
26C22×C4×D61C2×C412yes—
27C22×C4×D61C2×C412yes—
28C22×C4×D61D412yes—
29C22×C4×D61D412yes—
30C22×C4×D61Q811yescharacteristic
31C22×C4×D61Q811yescharacteristic
32C2×#11758194C2×C412yes—
33C2×#11758189—22no—
34M16×D61—44no—
35C2×C8×D61—22no—
36#14519734—48no—
37C2×#11758196—24no—
38C2×C22⋊Dic61—22no—
39C2×#11758198—24no—
40C2×#11758198—24no—
41C2×C4×Dic61—22no—
42C22⋊#92418—22no—
43#14519689—24no—
44#14519684—24no—
45#14519684C2×C414yes—
46C23×C4×C61C2311yescharacteristic
47C2×M16×C61C2×C412yes—
48C2×C22⋊C4×C61—22no—
49C2×C42×C61—22no—
50C22⋊C8×C61—22no—
51#95×C61C2×C414yes—
52C24×C61C2×D411yescharacteristic
53C23×D61—42no—
54C23×D61—44no—
55C23×D61C22⋊C412yes—
56C23×D61C22⋊C412yes—
57C23×D61C4⋊C411yescharacteristic
58C23×D61C4⋊C411yescharacteristic
59C22×C4×C61—24no—
60C22×C4×C61—42no—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C22×C412yes—
63C22×C4×C61C2×D412yes—
64C22×C4×C61C2×Q811yescharacteristic
65C22×Dic61—24no—
66C22×Dic61—42no—
67C22×Dic61C22⋊C412yes—
68C22×Dic61C22⋊C412yes—
69C22×Dic61C4⋊C411yescharacteristic
70C22×Dic61C4⋊C411yescharacteristic
71C2×C4×D61—24no—
72C2×C4×D61—28no—
73C2×C4×D61—44no—
74C2×C4×D61—42no—
75C2×C4×D61—42no—
76C2×C4×D61—44no—
77C2×C4×D61—44no—
78C2×C4×D61—42no—
79C2×C4×D61C4211yescharacteristic
80C2×C4×D61C4211yescharacteristic
81C2×C4×D61C4212yes—
82C2×C4×D61C22⋊C412yes—
83C2×C4×D61C22⋊C411yescharacteristic
84C2×C4×D61C22⋊C411yescharacteristic
85C2×C4×D61C4⋊C412yes—
86C2×C4×D61C4⋊C412yes—
87C2×C4×D61C4⋊C412yes—
88C2×C4×D61C4⋊C412yes—
89M16×C61—42no—
90C2×C8×C61—22no—
91C22⋊C4×C61—44no—
92C42×C61—44no—
93C22⋊Dic61—44no—
94#11758198—48no—
95#11758198—48no—
96#11758196—48no—
97C4×Dic61—44no—
98#11758194—42no—
99C2×#92418—22no—
100#11758189—44no—
101C8×D61—44no—
102C23×C61—24no—
103C23×C61—41no—
104C23×C61C2×C22⋊C412yes—
105C23×C61C2×C4⋊C411yescharacteristic
106C22×D61—24no—
107C22×D61—24no—
108C22×D61—24no—
109C22×D61—22no—
110C22×D61—42no—
111C22×D61—41no—
112C22×D61—42no—
113C22×D61—42no—
114C22×D61—42no—
115C22×D61—42no—
116C22×D61—416no—
117C22×D61#9511yescharacteristic
118C22×D61#9512yes—
119C22×D61#9511yescharacteristic
120C2×C4×C61—22no—
121C2×C4×C61—24no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—44no—
125C2×C4×C61—44no—
126C2×C4×C61—44no—
127C2×C4×C61C2×C4211yescharacteristic
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C2×C4⋊C412yes—
130C2×Dic61—24no—
131C2×Dic61—22no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—44no—
135C2×Dic61—44no—
136C2×Dic61—44no—
137C2×Dic61#9511yescharacteristic
138C2×Dic61#9512yes—
139C2×Dic61#9511yescharacteristic
140C4×D61—22no—
141C4×D61—24no—
142C4×D61—22no—
143C4×D61—24no—
144C4×D61—22no—
145C4×D61—24no—
146C4×D61—42no—
147C4×D61—42no—
148C4×D61—42no—
149C4×D61#9914yes—
150C8×C61—42no—
151#92418—42no—
152#8049—611nomaximal, Sylow
153C22×C61—22no—
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—21no—
157C22×C61—44no—
158C22×C61C2×#9511yescharacteristic
159C22×C61#42114yes—
160C2×D61—22no—
161C2×D61—22no—
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—21no—
165C2×D61—22no—
166C2×D61—48no—
167C2×D61—44no—
168C2×D61—48no—
169C2×D61—44no—
170C2×D61#34212yes—
171C4×C61—21no—
172C4×C61—22no—
173C4×C61—84no—
174C4×C61C2×#9912yes—
175Dic61—21no—
176Dic61—22no—
177Dic61—84no—
178Dic61#34212yes—
179C2×#407—611no—
180#2280—612no—
181#1297—611no—
182#1297—612no—
183#1297D6111yescharacteristic, complemented
184C2×C61—24no—
185C2×C61—41no—
186C2×C61#173312yes—
187C2×C61C2×#34211yescharacteristic
188D61—22no—
189D61—42no—
190D61—41no—
191C24×C4—611no—
192C22×M16—612no—
193C2×#117—1222no—
194#407—611no—
195#407—612no—
196#407—1222no—
197#407C2×D6111yescharacteristic
198C2×C22⋊C8—1222no—
199#381—614no—
200#380—614no—
201#380—1222no—
202#379—1222no—
203C4×C22⋊C4—612no—
204C4×C22⋊C4—1222no—
205C4×C22⋊C4C2×D6112yes—
206C61#804911yescharacteristic, complemented, Sylow
207C25—611no—
208C23×C4—611no—
209C23×C4—611no—
210C23×C4—612no—
211C23×C4—612no—
212C23×C4—612no—
213C23×C4—611no—
214C23×C4—612no—
215C23×C4—611no—
216C23×C4—611no—
217C23×C4—2444no—
218C23×C4C22×D6111yescharacteristic
219C2×M16—612no—
220C2×M16—1222no—
221C2×M16—2444no—
222C2×M16C4×D6112yes—
223C22×C8—1222no—
224#117—2448no—
225C2×C4⋊C4—1224no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—1224no—
228C2×C22⋊C4—1224no—
229C2×C22⋊C4—1222no—
230C2×C42—22no—
231C2×C42—1222no—
232C22⋊C8—22no—
233C22⋊C8—1224no—
234C22⋊C8—1222no—
235#95—614no—
236#95—1224no—
237#95C4×D6114yes—
238C24—612no—
239C24—612no—
240C24—611no—
241C24—611no—
242C24—2444no—
243C24—2442no—
244C24D4×D6111yescharacteristic
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—611no—
248C22×C4—612no—
249C22×C4—611no—
250C22×C4—612no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—611no—
254C22×C4—612no—
255C22×C4—611no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—612no—
259C22×C4—612no—
260C22×C4—612no—
261C22×C4—1228no—
262C22×C4—1224no—
263C22×C4—1224no—
264C22×C4—2444no—
265C22×C4—2442no—
266C22×C4—2444no—
267C22×C4—2442no—
268C22×C4—2444no—
269C22×C4—2442no—
270C22×C4—2442no—
271C22×C4C2×C4×D6111yescharacteristic
272C22×C4C2×C4×D6112yes—
273C22×C4D4×D6112yes—
274C22×C4Q8×D6111yescharacteristic
275M16—42no—
276M16—2444no—
277M16—2442no—
278C2×C8—22no—
279C2×C8—1222no—
280C2×C8—2444no—
281C4⋊C4—2448no—
282C22⋊C4—44no—
283C22⋊C4—2448no—
284C22⋊C4—2448no—
285C22⋊C4—2444no—
286C42—44no—
287C42—2444no—
288C23—24no—
289C23—41no—
290C23—611no—
291C23—611no—
292C23—612no—
293C23—1224no—
294C23—1222no—
295C23—1224no—
296C23—1224no—
297C23—2442no—
298C23—2441no—
299C23—2442no—
300C23—2442no—
301C23—2442no—
302C23—2442no—
303C23—24416no—
304C23C22⋊C4×D6112yes—
305C23C4⋊C4×D6111yescharacteristic
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—611no—
314C2×C4—614no—
315C2×C4—612no—
316C2×C4—611no—
317C2×C4—1222no—
318C2×C4—1224no—
319C2×C4—1222no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—1224no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2441no—
330C2×C4—2444no—
331C2×C4—2444no—
332C2×C4—2444no—
333C2×C4C42×D6111yescharacteristic
334C2×C4C22⋊C4×D6111yescharacteristic
335C2×C4C4⋊C4×D6112yes—
336C8—42no—
337C8—2442no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—21no—
342C22—44no—
343C22—612no—
344C22—1222no—
345C22—1222no—
346C22—1222no—
347C22—1222no—
348C22—1221no—
349C22—1222no—
350C22—2448no—
351C22—2448no—
352C22—2444no—
353C22—2444no—
354C22#95×D6111yescharacteristic
355C22#1683977714yes—
356C4—21no—
357C4—22no—
358C4—84no—
359C4—612no—
360C4—1222no—
361C4—1221no—
362C4—4884no—
363C4#99×D6112yes—
364C2—24no—
365C2—41no—
366C2—1222no—
367C2—2442no—
368C2—2441no—
369C2#2144552712yes—
370C2#342×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1297C2
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1297D61
hypocenterC61#8049
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #21443705C2semidirect
4. #21443704C2semidirect
5. C61×#1297C2semidirect
7. C2×M16×D61C4semidirect
10. #16839725C4semidirect
13. #16839624C4semidirect
14. #16839621C4semidirect
183. #1297D61semidirect
206. C61#8049semidirect

Maximal subgroups

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

subgroupindex
#407×D612
#214482002
#214437052
#214437042
C61×#12972
#804961