Finite Group Databasebeta

← #29577821

Subgroups — #29577821

Order 15616.

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

normal subgroups only

This group has 734 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#21450095C211yesmaximal, characteristic
2#21448171C211yesmaximal, characteristic
3#21445919C211yesmaximal, characteristic, complemented
4#21445916C211yesmaximal, characteristic, complemented
5#21444145C211yesmaximal, characteristic
6#21443671C211yesmaximal, characteristic, complemented
7C61×#1284C211yesmaximal, characteristic, complemented
8C2×#14519736C2211yescharacteristic
9C22×#11758198C2211yescharacteristic
10C2×C22⋊C4×D61—21no—
11#16839726C2211yescharacteristic
12#16839699C412yescomplemented
13#16839689C412yescomplemented
14#16839654—22no—
15#16839637—21no—
16#16839617—21no—
17C22⋊C4×Dic61—21no—
18C4×#11758198—22no—
19C4×C22⋊Dic61C2211yescharacteristic
20#16839609—22no—
21#16839523—22no—
22#16839522—22no—
23#16839485—22no—
24C61×#407C2211yescharacteristic
25C61×#381C2211yescharacteristic
26C4×C22⋊C4×C61C2211yescharacteristic
27C24×D61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C22×C4×D61—22no—
30C2×#11758194—21no—
31#14519736—21no—
32#14519736—21no—
33#14519736—21no—
34#14519736—21no—
35C2×C4⋊Dic61—22no—
36C2×C4⋊Dic61C2×C411yescharacteristic, complemented
37C2×C4⋊Dic61C2×C411yescharacteristic, complemented
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C2×#11758198C2×C412yes—
48C2×#11758198D412yes—
49C22⋊C4×D61—28no—
50C2×C4×Dic61—21no—
51C2×C4×Dic61—22no—
52C2×C4×Dic61D412yes—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519684—22no—
56#14519684—21no—
57#14519684—22no—
58C23×C4×C61C2311yescharacteristic
59C2×M16×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—21no—
62C2×C42×C61D412yes—
63C22⋊C8×C61—21no—
64C22⋊C8×C61—21no—
65#95×C61—21no—
66#95×C61—21no—
67#95×C61C2×C412yescomplemented
68C24×C61C2×D411yescharacteristic
69C23×D61—21no—
70C23×D61—21no—
71C23×D61—24no—
72C23×D61—42no—
73C23×D61—81no—
74C23×D61C22⋊C412yes—
75C22×C4×C61—22no—
76C22×C4×C61—22no—
77C22×C4×C61—42no—
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C22×Dic61—22no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—81no—
90C22×Dic61C22⋊C412yes—
91C2×C4×D61—28no—
92C2×C4×D61—82no—
93C2×C4×D61—82no—
94M16×C61—42no—
95C2×C8×C61—41no—
96C22⋊C4×C61—24no—
97C22⋊C4×C61—42no—
98C42×C61—42no—
99C22⋊Dic61—24no—
100C22⋊Dic61—41no—
101C22⋊Dic61—41no—
102#11758198—28no—
103#11758198—22no—
104#11758198—22no—
105#11758198—22no—
106#11758198—22no—
107#11758198—44no—
108#11758198—82no—
109C4⋊Dic61—21no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—82no—
114C4×Dic61—41no—
115C4×Dic61—41no—
116C4×Dic61—42no—
117C4×Dic61—82no—
118#11758194—42no—
119C2×#92418—41no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—41no—
123C23×C61C2×C22⋊C411yescharacteristic
124C23×C61#12011yescharacteristic
125C23×C61#12311yescharacteristic
126C22×D61—22no—
127C22×D61—22no—
128C22×D61—22no—
129C22×D61—24no—
130C22×D61—22no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—88no—
136C22×D61—81no—
137C22×D61—81no—
138C22×D61#10414yes—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—24no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×Dic61—24no—
156C2×Dic61—22no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C2×Dic61—81no—
166C4×D61—88no—
167C8×C61—81no—
168#92418—81no—
169#9477—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—44no—
178C22×C61#38611yescharacteristic
179C22×C61C2×#10412yes—
180C22×C61#45812yes—
181C2×D61—24no—
182C2×D61—44no—
183C2×D61—44no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—44no—
187C2×D61—82no—
188C2×D61—84no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—44no—
194C4×C61—82no—
195Dic61—44no—
196Dic61—42no—
197Dic61—81no—
198Dic61—81no—
199#2482—611no—
200#2273—611no—
201#1826—611no—
202#1826—611no—
203#1462—611no—
204#1284—611no—
205#1284D6111yescharacteristic, complemented
206C2×C61—22no—
207C2×C61—22no—
208C2×C61—41no—
209C2×C61#189011yescharacteristic
210C2×C61#189611yescharacteristic
211C2×C61#191111yescharacteristic
212D61—44no—
213C2×#117—611no—
214C22×C22⋊C4—611no—
215C22×C22⋊C4—1221no—
216#407—611no—
217#407C2×D6111yescharacteristic
218#399—612no—
219#397—612no—
220#388—1222no—
221#385—1221no—
222#381C2×D6111yescharacteristic
223#379—1221no—
224C4×C22⋊C4—611no—
225C4×C22⋊C4—1222no—
226C4×C22⋊C4—1221no—
227C4×C22⋊C4—1222no—
228C4×C22⋊C4C2×D6111yescharacteristic
229#339—1222no—
230#339—1222no—
231#323—1222no—
232C61#947711yescharacteristic, complemented, Sylow
233C25—611no—
234C23×C4—611no—
235C23×C4—1222no—
236C23×C4C22×D6111yescharacteristic
237C2×M16—21no—
238C2×M16—1221no—
239#117—1221no—
240#117—1221no—
241#117—1221no—
242#117—1221no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—1222no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—612no—
249C2×C22⋊C4—612no—
250C2×C22⋊C4—1228no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1222no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1221no—
260C2×C42—612no—
261C2×C42—1222no—
262C2×C42—1221no—
263C2×C42D24412yes—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268#95—21no—
269#95—21no—
270#95—1222no—
271#95—1222no—
272#95—1221no—
273#95C4×D6112yescomplemented
274C24—612no—
275C24—1224no—
276C24—1221no—
277C24—1221no—
278C24—4881no—
279C24—2442no—
280C24D4×D6111yescharacteristic
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—612no—
285C22×C4—1228no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—4882no—
292C22×C4—4882no—
293C22×C4—4881no—
294C22×C4C2×C4×D6111yescharacteristic
295C22×C4C2×D24411yescharacteristic
296C22×C4#1175822211yescharacteristic
297C22×C4D4×D6111yescharacteristic
298C22×C4D4×D6111yescharacteristic
299C22×C4#1175822611yescharacteristic
300M16—42no—
301M16—2442no—
302C2×C8—41no—
303C2×C8—2441no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—1221no—
308C4⋊C4—4882no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—1228no—
312C22⋊C4—1224no—
313C22⋊C4—1222no—
314C22⋊C4—1222no—
315C22⋊C4—1222no—
316C22⋊C4—1222no—
317C22⋊C4—4882no—
318C22⋊C4—2444no—
319C22⋊C4—2441no—
320C22⋊C4—2441no—
321C42—42no—
322C42—4882no—
323C42—2442no—
324C42—2441no—
325C42—2441no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—614no—
330C23—1224no—
331C23—1222no—
332C23—1222no—
333C23—1222no—
334C23—1222no—
335C23—4881no—
336C23—4881no—
337C23—4888no—
338C23—2442no—
339C23—2442no—
340C23—2442no—
341C23—2442no—
342C23C22⋊C4×D6111yescharacteristic
343C23#1451974711yescharacteristic
344C23#1451976211yescharacteristic
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—1224no—
358C2×C4—1222no—
359C2×C4—4888no—
360C2×C4—4881no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4#1451973911yescharacteristic
370C2×C4C4×D24411yescharacteristic
371C2×C4#1451975011yescharacteristic
372C2×C4#1451975211yescharacteristic
373C8—81no—
374C8—4881no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—44no—
383C22—1224no—
384C22—4882no—
385C22—4884no—
386C22—2444no—
387C22—2444no—
388C22—2444no—
389C22—2442no—
390C22—2442no—
391C22#1683964011yescharacteristic
392C22#104×D6112yes—
393C22#1683995312yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—44no—
399C4—82no—
400C4—4881no—
401C4—4881no—
402C4—2444no—
403C4—2442no—
404C2—22no—
405C2—22no—
406C2—41no—
407C2—2444no—
408C2#2144623111yescharacteristic
409C2#2144626111yescharacteristic
410C2#2144635011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1284C2
socleC22×C61#386
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1284D61
hypocenterC61#9477
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
3. #21445919C2semidirect
4. #21445916C2semidirect
6. #21443671C2semidirect
7. C61×#1284C2semidirect
12. #16839699C4semidirect
13. #16839689C4semidirect
36. C2×C4⋊Dic61C2×C4semidirect
37. C2×C4⋊Dic61C2×C4semidirect
67. #95×C61C2×C4semidirect
205. #1284D61semidirect
232. C61#9477semidirect
273. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214500952
#214481712
#214459192
#214459162
#214441452
#214436712
C61×#12842
#947761