Finite Group Databasebeta

← #29576923

Subgroups — #29576923

Order 15616.

650 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455911C211yesmaximal, characteristic
2#21448181C211yesmaximal, characteristic, complemented
3#21446738C211yesmaximal, characteristic, complemented
4#21444326C211yesmaximal, characteristic, complemented
5#21444263C211yesmaximal, characteristic, complemented
6#21443676C211yesmaximal, characteristic, complemented
7C61×#1300C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×C4×D244—21no—
10C2×C4×D244C2211yescharacteristic
11#16840081—21no—
12#16840053C412yescomplemented
13#16839958C412yescomplemented
14#16839898—21no—
15#16839875—22no—
16#16839819C2211yescharacteristic
17C2×#14519703—22no—
18C4×C4⋊Dic61—21no—
19C4×C4⋊Dic61C2211yescharacteristic
20#16839609—22no—
21#16839609—21no—
22#95×D61—21no—
23#16839495—21no—
24#16839495—21no—
25#16839491—22no—
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C2×#11758222—21no—
30C22×D244D411yescharacteristic
31C22×C4×D61—22no—
32C22×C4×D61—21no—
33C2×D488—22no—
34#14519777C2×C411yescharacteristic, complemented
35#14519774C2×C411yescharacteristic, complemented
36#14519765—21no—
37#14519751—21no—
38#14519751—21no—
39C4×D244—28no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244—21no—
44C4×D244C2×C412yes—
45C4×D244D412yes—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61D411yescharacteristic
48C2×#11758198—22no—
49C2×#11758198—21no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61D412yes—
52#14519709—22no—
53C4⋊#92418—21no—
54#14519703—44no—
55#14519699—22no—
56#14519699—22no—
57C22⋊#92418—21no—
58#14519688—21no—
59#14519684—22no—
60#14519684—21no—
61C2×M16×C61—21no—
62C2×C42×C61—21no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61C2×C412yescomplemented
65C22⋊C8×C61—21no—
66#97×C61—21no—
67#97×C61—21no—
68C4×C8×C61—21no—
69#95×C61—21no—
70#95×C61D411yescharacteristic
71#95×C61D411yescharacteristic
72C23×D61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—41no—
79C22×Dic61—21no—
80C22×Dic61—22no—
81#11758222—42no—
82#11758222—42no—
83C2×D244—21no—
84C2×D244—24no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—21no—
88C2×D244C22⋊C412yes—
89C2×C4×D61—28no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—41no—
93C2×C4×D61—82no—
94C2×C4×D61—82no—
95C2×C4×D61—82no—
96C2×Dic122—21no—
97M16×C61—22no—
98M16×C61—41no—
99C2×C8×C61—22no—
100C2×C8×C61—41no—
101C42×C61—24no—
102C42×C61C22×C411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61#4111yescharacteristic
105C42×C61#4111yescharacteristic
106#11758198—28no—
107#11758198—22no—
108#11758198—22no—
109#11758198—41no—
110#11758198—41no—
111C4⋊Dic61—24no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61C22⋊C412yes—
116C4×Dic61—42no—
117C4×Dic61—82no—
118C2×#92418—41no—
119D488—44no—
120C23×C61#12011yescharacteristic
121C22×D61—24no—
122C22×D61—22no—
123C22×D61—42no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C22⋊D61—81no—
144C22⋊D61—81no—
145C2×Dic61—24no—
146C2×Dic61—22no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—41no—
151D244—24no—
152D244—22no—
153D244—42no—
154D244—42no—
155D244—81no—
156D244#10414yescomplemented
157C4×D61—82no—
158C4×D61—82no—
159C4×D61—88no—
160C4×D61—82no—
161Dic122—42no—
162C8×C61—42no—
163C8×C61—42no—
164#92418—81no—
165#9325—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61#38611yescharacteristic
169C22×C61#45311yescharacteristic
170C22×C61#45411yescharacteristic
171C2×D61—24no—
172C2×D61—44no—
173C2×D61—44no—
174C2×D61—42no—
175C2×D61—42no—
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—42no—
185C4×C61—42no—
186C4×C61C2×#10412yes—
187Dic61—44no—
188Dic61—42no—
189Dic61—81no—
190#3109—611no—
191#2275—611no—
192#1980—611no—
193#1505—611no—
194#1490—611no—
195#1300D6111yescharacteristic, complemented
196#1285—611no—
197C2×C61—21no—
198C2×C61—21no—
199C2×C61#188311yescharacteristic
200C2×C61#189611yescharacteristic
201C2×C61#190411yescharacteristic
202D61—44no—
203D61—161no—
204#532—611no—
205C2×C4×D4—611no—
206C2×C4×D4—1221no—
207#481—1221no—
208#476—612no—
209#459—612no—
210#447—1221no—
211#442—1222no—
212#432—611no—
213#432C2×D6111yescharacteristic
214C2×#104—1222no—
215C4×M16C2×D6111yescharacteristic
216C4×C4⋊C4—611no—
217C4×C4⋊C4—1221no—
218C4×C22⋊C4—1222no—
219C4×C22⋊C4—1221no—
220#376C2×D6111yescharacteristic
221C2×#95—1221no—
222#327—1221no—
223#327—1221no—
224D4⋊C8—1222no—
225C61#932511yescharacteristic, complemented, Sylow
226C2×#41—1221no—
227C22×D4—611no—
228C23×C4—1222no—
229C23×C4—1221no—
230C2×D8—1222no—
231C2×M16—21no—
232C4⋊Q8—611no—
233#127—611no—
234#124—1221no—
235#121—1221no—
236#121—1221no—
237C4×D4—612no—
238C4×D4—612no—
239C4×D4—1228no—
240C4×D4—1221no—
241C4×D4—1221no—
242C4×D4—1221no—
243C4×D4—1221no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—1221no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1221no—
248C2×C42—21no—
249C2×C42—612no—
250C2×C42—1222no—
251C2×C42C22×D6111yescharacteristic
252#106—1222no—
253C4⋊C8—1221no—
254C4⋊C8C4×D6112yescomplemented
255#104—2444no—
256#102—1222no—
257#102—1222no—
258C22⋊C8—21no—
259C22⋊C8—1221no—
260#97—21no—
261#97—21no—
262#97—1221no—
263C4×C8—21no—
264#95—21no—
265#95—1222no—
266#95—1221no—
267#95D24411yescharacteristic
268#95D24411yescharacteristic
269C24—1221no—
270#41—2442no—
271#41—2442no—
272C2×Q8—1221no—
273C2×D4—612no—
274C2×D4—1224no—
275C2×D4—1221no—
276C2×D4—1221no—
277C2×D4—1221no—
278C2×D4—1221no—
279C2×D4—2441no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—1228no—
283C22×C4—1222no—
284C22×C4—1221no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—4882no—
288C22×C4—4882no—
289C22×C4—4882no—
290C22×C4—2441no—
291C22×C4C2×D24411yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C22×C4D4×D6111yescharacteristic
294D8—2444no—
295M16—22no—
296M16—41no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—2441no—
300C4⋊C4—612no—
301C4⋊C4—1224no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C22⋊C4—1228no—
306C22⋊C4—1222no—
307C22⋊C4—1222no—
308C22⋊C4—2441no—
309C22⋊C4—2441no—
310C42—24no—
311C42—4882no—
312C42—2442no—
313C42C2×C4×D6111yescharacteristic
314C42#1175822211yescharacteristic
315C42D4×D6111yescharacteristic
316C42#1175822611yescharacteristic
317C23—1224no—
318C23—1222no—
319C23—2442no—
320C23—2441no—
321C23—2441no—
322C23—2441no—
323C23#1451974711yescharacteristic
324Q8—2442no—
325D4—614no—
326D4—1224no—
327D4—1222no—
328D4—4881no—
329D4—4881no—
330D4—4881no—
331D4—2442no—
332D4—2442no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—1224no—
344C2×C4—1222no—
345C2×C4—4882no—
346C2×C4—4882no—
347C2×C4—4882no—
348C2×C4—4888no—
349C2×C4—2441no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4C22⋊C4×D6111yescharacteristic
354C2×C4#1451973911yescharacteristic
355C2×C4C4×D24411yescharacteristic
356C2×C4#1451975011yescharacteristic
357C2×C4#1451975211yescharacteristic
358C2×C4#1451976211yescharacteristic
359C8—42no—
360C8—42no—
361C8—4881no—
362C22—21no—
363C22—21no—
364C22—1224no—
365C22—4881no—
366C22—4881no—
367C22—4881no—
368C22—2444no—
369C22—2444no—
370C22—2442no—
371C22—2442no—
372C22#1683964011yescharacteristic
373C22#1683993111yescharacteristic
374C22#1683993611yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—42no—
382C4—4881no—
383C4—2444no—
384C4—2442no—
385C4#104×D6112yes—
386C2—21no—
387C2—21no—
388C2—9761no—
389C2—2444no—
390C2#2144619611yescharacteristic
391C2#2144626111yescharacteristic
392C2#2144630511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1300C2
socleC22×C61#386
prime radical#21455911C2
cosocleC2×C42×C61C23
hypercenter#1300D61
hypocenterC61#9325
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
2. #21448181C2semidirect
3. #21446738C2semidirect
4. #21444326C2semidirect
5. #21444263C2semidirect
6. #21443676C2semidirect
7. C61×#1300C2semidirect
12. #16840053C4semidirect
13. #16839958C4semidirect
34. #14519777C2×C4semidirect
35. #14519774C2×C4semidirect
64. C4⋊C8×C61C2×C4semidirect
156. D244#104semidirect
195. #1300D61semidirect
225. C61#9325semidirect
254. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214559112
#214481812
#214467382
#214443262
#214442632
#214436762
C61×#13002
#932561