Finite Group Databasebeta

← #29577952

Subgroups — #29577952

Order 15616.

880 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449175C211yesmaximal, characteristic
2#21445739C212yesmaximal, complemented
3#21444157C211yesmaximal, characteristic, complemented
4#21443724C212yesmaximal, complemented
5C61×#2973C211yesmaximal, characteristic, complemented
6C23×D244C2211yescharacteristic
7C2×#14519751—21no—
8C2×#14519745C412yes—
9C2×#14519693—24no—
10#16839726C2212yes—
11#16839679—21no—
12#16839671—21no—
13#16839664—21no—
14#16839622C2211yescharacteristic
15#16839617C412yes—
16#16839488—21no—
17#16839486—21no—
18C2×C4×D4×C61C2211yescharacteristic
19C61×#435—24no—
20C61×#432—21no—
21C61×#407C2212yes—
22C2×C22⋊C8×C61—21no—
23C24×D61C2×C412yes—
24C22×D244—22no—
25C22×D244—21no—
26C22×D244—21no—
27C22×D244—22no—
28C22×D244—44no—
29C22×C22⋊D61—22no—
30C2×#11758194—22no—
31#14519751—44no—
32#14519745—44no—
33C2×C4⋊Dic61—21no—
34C2×C22⋊Dic61C2×C412yes—
35C2×#11758198—22no—
36C2×#11758198—22no—
37C2×#11758198—22no—
38#14519693—48no—
39C22⋊#92418—22no—
40C22⋊#92418—22no—
41#14519684—21no—
42C22×D4×C61C2×C411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C23×C4×C61C2311yescharacteristic
45C2×M16×C61D414yescomplemented
46C22×C8×C61—22no—
47C4×D4×C61—22no—
48C4×D4×C61—22no—
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C4⋊C8×C61—22no—
54C22⋊C8×C61—22no—
55C22⋊C8×C61—22no—
56C22⋊C8×C61—22no—
57#97×C61—22no—
58C24×C61C22⋊C411yescharacteristic
59C24×C61C22×C411yescharacteristic
60C23×D61—22no—
61C23×D61—24no—
62C23×D61—44no—
63C23×D61—42no—
64C23×D61—42no—
65C23×D61—82no—
66C2×D4×C61—22no—
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C2×D412yes—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4112yes—
80C2×C22⋊D61—44no—
81C2×C22⋊D61—44no—
82C2×C22⋊D61—82no—
83C22×Dic61—22no—
84C2×D244—28no—
85C2×D244—24no—
86C2×D244—44no—
87C2×D244—42no—
88C2×D244—44no—
89C2×D244—48no—
90C2×D244—41no—
91C2×D244—42no—
92C2×D244—41no—
93M16×C61—44no—
94C2×C8×C61—24no—
95C2×C8×C61—82no—
96C4⋊C4×C61—22no—
97C22⋊C4×C61—22no—
98C22⋊C4×C61—22no—
99C42×C61—21no—
100C42×C61—21no—
101C22⋊Dic61—42no—
102#11758198—44no—
103#11758198—44no—
104#11758198—82no—
105C4⋊Dic61—42no—
106#11758194—44no—
107C2×#92418—42no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61#10012yescomplemented
117C23×C61C2×C22⋊C411yescharacteristic
118C23×C61C4×D412yes—
119C22×D61—24no—
120C22×D61—28no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—44no—
124C22×D61—42no—
125C22×D61—42no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—88no—
129C22×D61—88no—
130C22×D61—84no—
131C22×D61—84no—
132D4×C61—44no—
133C2×C4×C61—24no—
134C2×C4×C61—22no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—41no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—81no—
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12112yes—
148C2×C4×C61#12311yescharacteristic
149C22⋊D61—88no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153D244—48no—
154D244—44no—
155D244—44no—
156C8×C61—44no—
157#92418—82no—
158#9503—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—81no—
171C22×C61#38611yescharacteristic
172C22×C61C2×#10011yescharacteristic
173C22×C61#41311yescharacteristic
174C2×D61—44no—
175C2×D61—44no—
176C2×D61—44no—
177C2×D61—82no—
178C2×D61—82no—
179C2×D61—88no—
180C2×D61—82no—
181C2×D61—82no—
182C2×D61—88no—
183C4×C61—22no—
184C4×C61—21no—
185C4×C61—41no—
186C4×C61—41no—
187C4×C61—41no—
188C4×C61B(4,𝔽2)12yes—
189Dic61—82no—
190#2973D6111yescharacteristic, complemented
191#2408—611no—
192#1783—612no—
193#1465—611no—
194#1305—612no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61—42no—
198C2×C61—42no—
199C2×C61#188911yescharacteristic
200C2×C61#190011yescharacteristic
201C2×C61#191411yescharacteristic
202D61—84no—
203C23×D4—611no—
204C2×#121—1221no—
205C2×#120—612no—
206C2×C4×D4C2×D6111yescharacteristic
207#435—24no—
208#432—21no—
209C2×#100—1224no—
210#407—612no—
211#407C2×D6112yes—
212C2×C22⋊C8—21no—
213#394—1221no—
214#392—1221no—
215#390—1221no—
216#380—611no—
217#379—612no—
218#324—1221no—
219#323—1221no—
220C61#950311yescharacteristic, complemented, Sylow
221C25—612no—
222C22×D4—1222no—
223C22×D4—1221no—
224C22×D4—1222no—
225C22×D4—1221no—
226C22×D4—1222no—
227C22×D4—2444no—
228C22×D4C4×D6111yescharacteristic
229C23×C4C22⋊D6111yescharacteristic
230C23×C4C22×D6111yescharacteristic
231C2×M16—1222no—
232C2×M16C22⋊D6114yescomplemented
233C22×C8—22no—
234#121—2444no—
235#120—2444no—
236C4×D4—22no—
237C4×D4—22no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4D24411yescharacteristic
240C2×C22⋊C4—612no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4C4×D6111yescharacteristic
245C2×C22⋊C4D24411yescharacteristic
246C2×C42C22⋊D6111yescharacteristic
247C4⋊C8—22no—
248#100—2448no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—1222no—
253C22⋊C8—1222no—
254#97—22no—
255#95—1221no—
256C24—1224no—
257C24—1222no—
258C24—4882no—
259C24—2442no—
260C24—2442no—
261C24—2444no—
262C24#1175819811yescharacteristic
263C24C2×C4×D6111yescharacteristic
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—41no—
267C2×D4—1228no—
268C2×D4—1224no—
269C2×D4—4882no—
270C2×D4—2442no—
271C2×D4—2444no—
272C2×D4—2444no—
273C2×D4—2448no—
274C2×D4—2444no—
275C2×D4—2444no—
276C2×D4—2441no—
277C2×D4—2442no—
278C2×D4—2441no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—1222no—
284C22×C4#1175819811yescharacteristic
285C22×C4#1175819811yescharacteristic
286C22×C4#1175819811yescharacteristic
287C22×C4C2×D24411yescharacteristic
288C22×C4#1175822212yes—
289C22×C4C2×C22⋊D6112yes—
290C22×C4C2×C22⋊D6111yescharacteristic
291M16—44no—
292M16—2444no—
293C2×C8—24no—
294C2×C8—82no—
295C2×C8—2442no—
296C4⋊C4—22no—
297C4⋊C4—2442no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—4882no—
301C22⋊C4—2444no—
302C22⋊C4—2444no—
303C22⋊C4—2442no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—1228no—
315C23—1224no—
316C23—4884no—
317C23—4884no—
318C23—4888no—
319C23—4888no—
320C23—2442no—
321C23—2442no—
322C23—2442no—
323C23—2444no—
324C23—2442no—
325C23—2442no—
326C23—2442no—
327C23#1451969312yescomplemented
328C23C2×#1175819811yescharacteristic
329C23C4×C22⋊D6112yes—
330D4—44no—
331D4—4888no—
332D4—2448no—
333D4—2444no—
334D4—2444no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2442no—
351C2×C4#1451974611yescharacteristic
352C2×C4#1451975112yes—
353C2×C4#1451976011yescharacteristic
354C8—44no—
355C8—4882no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22—4882no—
369C22—4882no—
370C22—4888no—
371C22—4888no—
372C22—4882no—
373C22—4882no—
374C22—2444no—
375C22—2444no—
376C22—2444no—
377C22#1683964111yescharacteristic
378C22C2×#1451969311yescharacteristic
379C22#1683974411yescharacteristic
380C4—22no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—4882no—
386C4#1683994912yes—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—4884no—
392C2#2144622711yescharacteristic
393C2#2144628211yescharacteristic
394C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2973C2
socleC22×C61#386
prime radical#21449175C2
cosocleC23×C4×C61C23
hypercenter#2973D61
hypocenterC61#9503
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21445739C2semidirect
3. #21444157C2semidirect
4. #21443724C2semidirect
5. C61×#2973C2semidirect
45. C2×M16×C61D4semidirect
116. C23×C61#100semidirect
190. #2973D61semidirect
220. C61#9503semidirect
232. C2×M16C22⋊D61semidirect
327. C23#14519693semidirect

Maximal subgroups

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

subgroupindex
#214491752
#214457392
#214441572
#214437242
C61×#29732
#950361