Finite Group Databasebeta

← #29590584

Subgroups — #29590584

Order 15616.

652 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 87 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#21450765C211yesmaximal, characteristic
2C4×#14519692C211yesmaximal, characteristic, complemented
3#21443817C214yesmaximal, complemented
4C61×#1465C211yesmaximal, characteristic, complemented
5C2×#14519774C411yescharacteristic, complemented
6C2×#14519773—21no—
7C2×#14519751—21no—
8C2×#14519751—21no—
9C2×#14519692—21no—
10#16839671C412yescomplemented
11#16839671C2212yes—
12#16839663—21no—
13C4×C4⋊Dic61C411yescharacteristic, complemented
14C4×C22⋊Dic61—21no—
15#16839529C2212yes—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#407—21no—
18C61×#324—21no—
19C61×#323C2212yes—
20C22×D244—21no—
21C22×D244—21no—
22C22×D244C2×C412yes—
23C22×C22⋊D61—22no—
24C22×C22⋊D61—22no—
25#14519774—41no—
26#14519774—41no—
27#14519773—44no—
28#14519751—44no—
29#14519751—44no—
30C2×C4⋊Dic61—21no—
31C2×C22⋊Dic61—22no—
32C2×#11758198—22no—
33C2×#11758198—22no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—41no—
36#14519692—24no—
37#14519684—41no—
38#14519684C2×C412yes—
39C22×D4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C2×M16×C61—41no—
43C4×D4×C61—41no—
44C4×D4×C61—41no—
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C22⋊C4×C61C2×C411yescharacteristic
47C2×C22⋊C4×C61C2311yescharacteristic
48C2×C42×C61C2×C411yescharacteristic
49C22⋊C8×C61—22no—
50C22⋊C8×C61—41no—
51C24×C61C22⋊C411yescharacteristic
52C24×C61C2×D411yescharacteristic
53C23×D61—22no—
54C2×D4×C61—22no—
55C2×D4×C61—22no—
56C2×D4×C61—41no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C2×C22⋊D61—44no—
69C2×C22⋊D61—44no—
70C2×C22⋊D61—44no—
71C2×C22⋊D61—44no—
72C2×C22⋊D61—82no—
73C2×C22⋊D61—82no—
74C22×Dic61—22no—
75C22×Dic61—42no—
76C2×D244—42no—
77C2×D244—42no—
78C2×D244—42no—
79C2×D244—42no—
80C2×D244—42no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—42no—
84C2×D244—82no—
85M16×C61—82no—
86C2×C8×C61—42no—
87C4⋊C4×C61—41no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C42×C61—21no—
91C42×C61—21no—
92C22⋊Dic61—24no—
93C22⋊Dic61—42no—
94#11758198—44no—
95#11758198—44no—
96C4⋊Dic61—24no—
97C4×Dic61—42no—
98C4×Dic61—82no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61C2×C22⋊C411yescharacteristic
109C23×C61AGL(1,ℤ/8)12yes—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61—42no—
113C22×D61—42no—
114C22×D61—42no—
115C22×D61—42no—
116C22×D61—42no—
117D4×C61—44no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—81no—
130C2×C4×C61—81no—
131C2×C4×C61#9911yescharacteristic
132C2×C4×C61#9911yescharacteristic
133C2×C4×C61#10412yes—
134C22⋊D61—88no—
135C22⋊D61—88no—
136C2×Dic61—24no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—44no—
140C2×Dic61—82no—
141C2×Dic61—82no—
142D244—82no—
143D244—84no—
144D244—84no—
145D244—82no—
146C8×C61—82no—
147#11776—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—22no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61C2×#9911yescharacteristic
161C22×C61#41811yescharacteristic
162C22×C61C2×#10411yescharacteristic
163C2×D61—82no—
164C2×D61—82no—
165C2×D61—82no—
166C2×D61—82no—
167C2×D61—82no—
168C2×D61—82no—
169C2×D61—82no—
170C4×C61—21no—
171C4×C61—22no—
172C4×C61—41no—
173C4×C61—41no—
174C4×C61—41no—
175C4×C61#35312yes—
176Dic61—44no—
177Dic61—84no—
178#2541—611no—
179C4×#99—611no—
180#1465D6111yescharacteristic, complemented
181#1343—614no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61—42no—
185C2×C61—81no—
186C2×C61#150711yescharacteristic
187C2×C61#212211yescharacteristic
188C2×C61C2×#35311yescharacteristic
189D61—162no—
190C2×#127—611no—
191C2×#127—1221no—
192C2×#121—1221no—
193C2×#121—1221no—
194C2×C4×D4C2×D6111yescharacteristic
195C2×#99—1221no—
196#407—21no—
197#392—612no—
198#392—612no—
199#390—1221no—
200C4×C4⋊C4—611no—
201C4×C22⋊C4—1221no—
202#342—612no—
203#324—21no—
204#323C2×D6112yes—
205C61#1177611yescharacteristic, complemented, Sylow
206C22×D4—612no—
207C22×D4—1222no—
208C22×D4—1222no—
209C22×D4—1221no—
210C22×D4—1221no—
211C22×D4D24411yescharacteristic
212C23×C4D24411yescharacteristic
213C23×C4C22⋊D6111yescharacteristic
214C2×M16—41no—
215#127—2441no—
216#127—2441no—
217#127—2444no—
218#121—2444no—
219#121—2444no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4C22⋊D6111yescharacteristic
224C2×C22⋊C4—1222no—
225C2×C22⋊C4—1222no—
226C2×C22⋊C4—1222no—
227C2×C22⋊C4C4×D6111yescharacteristic
228C2×C22⋊C4C22×D6111yescharacteristic
229C2×C42—1221no—
230C2×C42—2441no—
231C2×C42C4×D6111yescharacteristic
232#99—1224no—
233C22⋊C8—22no—
234C22⋊C8—41no—
235#95—612no—
236#95—2441no—
237C24—1222no—
238C24#1175819811yescharacteristic
239C24C2×D24411yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—4882no—
244C2×D4—4882no—
245C2×D4—4882no—
246C2×D4—2442no—
247C2×D4—2442no—
248C2×D4—2442no—
249C2×D4—2442no—
250C2×D4—2442no—
251C2×D4—2444no—
252C2×D4—2444no—
253C2×D4—2444no—
254C2×D4—2444no—
255C2×D4—2441no—
256C2×D4—2441no—
257C2×D4—2442no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—1222no—
265C22×C4—2442no—
266C22×C4#1175819811yescharacteristic
267C22×C4#1175819811yescharacteristic
268C22×C4#1175819811yescharacteristic
269C22×C4C2×C4×D6111yescharacteristic
270C22×C4C2×C22⋊D6111yescharacteristic
271M16—82no—
272C2×C8—42no—
273C4⋊C4—41no—
274C4⋊C4—1224no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—1224no—
278C22⋊C4—2444no—
279C22⋊C4—2444no—
280C22⋊C4—2442no—
281C42—21no—
282C42—21no—
283C42—4882no—
284C42—2442no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—2442no—
295C23—2442no—
296C23—2442no—
297C23—2442no—
298C23—2442no—
299C23—2442no—
300C23—2442no—
301C23C2×#1175819811yescharacteristic
302C23#1451980512yes—
303D4—44no—
304D4—4882no—
305D4—4884no—
306D4—4884no—
307D4—4888no—
308D4—4888no—
309D4—4882no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—1224no—
324C2×C4—4882no—
325C2×C4—4882no—
326C2×C4—2444no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4#1451969111yescharacteristic
330C2×C4#1451969111yescharacteristic
331C2×C4#1451970312yes—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—4882no—
346C22—4882no—
347C22—4882no—
348C22—4882no—
349C22—4882no—
350C22—4882no—
351C22—4882no—
352C22C2×#1451969111yescharacteristic
353C22#1683976811yescharacteristic
354C22C2×#1451970311yescharacteristic
355C4—21no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—4884no—
361C4—2444no—
362C4#1683955412yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—9762no—
368C2#2144433611yescharacteristic
369C2#2144747011yescharacteristic
370C2C2×#1683955411yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1465C2
socleC22×C61C2×#99
prime radical#21450765C2
cosocleC2×C22⋊C4×C61C23
hypercenter#1465D61
hypocenterC61#11776
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. C4×#14519692C2semidirect
3. #21443817C2semidirect
4. C61×#1465C2semidirect
5. C2×#14519774C4semidirect
10. #16839671C4semidirect
13. C4×C4⋊Dic61C4semidirect
180. #1465D61semidirect
205. C61#11776semidirect

Maximal subgroups

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

subgroupindex
#214507652
C4×#145196922
#214438172
C61×#14652
#1177661