Finite Group Databasebeta

← #30604778

Subgroups — #30604778

Order 15616.

504 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462048C211yesmaximal, characteristic, complemented
2C2×#16840636C211yesmaximal, characteristic, complemented
3#21460808C212yesmaximal, complemented
4#21460806C212yesmaximal, complemented
5#21455401C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C2×#14519851—21no—
8#16840657—21no—
9#16840657C2212yes—
10#16840636—21no—
11#16840636C2212yes—
12C2×#14519799C411yescharacteristic, complemented
13C22×C8×D61C2211yescharacteristic
14#16840274—21no—
15#16840004C411yescharacteristic, complemented
16#16840004C2211yescharacteristic
17#16839999C411yescharacteristic, complemented
18#16839999C2211yescharacteristic
19#16839992—21no—
20#16839992—21no—
21#16839756—21no—
22C61×#427C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×C4⋊#6923—24no—
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923—42no—
27C2×C4⋊#6923—41no—
28#14519851—42no—
29C2×#11758187—22no—
30C2×#11758222—21no—
31C22×C4×D61D411yescharacteristic
32#14519799—42no—
33C2×Dic244C2×C411yescharacteristic, complemented
34C2×#11758190—21no—
35C2×D488C2×C411yescharacteristic, complemented
36C22×#92418D411yescharacteristic
37C2×C8×D61—42no—
38C2×C8×D61C2×C411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41C2×C8×D61C2311yescharacteristic
42#14519761—41no—
43#14519757—21no—
44#14519757—21no—
45#14519752—21no—
46#14519752—21no—
47#14519747—41no—
48C4×D244—41no—
49C4×D244—41no—
50#14519700—21no—
51#14519700—21no—
52#14519697—21no—
53#14519697—21no—
54C22×C8×C61C2×C411yescharacteristic
55#117×C61—21no—
56#107×C61C2×C411yescharacteristic, complemented
57#107×C61C2×C411yescharacteristic, complemented
58#106×C61—21no—
59C23×D61D811yescharacteristic
60C22×#6923—44no—
61C22×#6923—42no—
62C22×#6923—82no—
63C22×#6923—82no—
64C22×C4×C61C22⋊C411yescharacteristic
65C2×C22⋊D61—41no—
66C22×Dic61SD1611yescharacteristic
67#11758222—42no—
68#11758222—42no—
69C2×D244—21no—
70C2×D244—41no—
71C2×D244—41no—
72C2×C4×D61—22no—
73C2×C4×D61—22no—
74C2×C4×D61—41no—
75C2×C4×D61C22⋊C411yescharacteristic
76C2×C4×D61D811yescharacteristic
77C2×C4×D61SD1611yescharacteristic
78C2×C4×D61C2×D411yescharacteristic
79C2×Dic122—21no—
80C4⋊#6923—24no—
81C4⋊#6923—44no—
82C4⋊#6923—42no—
83C2×C8×C61—22no—
84C2×C8×C61C22⋊C411yescharacteristic
85C2×C8×C61C22×C411yescharacteristic
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C22⋊C4×C61—41no—
89C42×C61—41no—
90#11758198—41no—
91#11758198—41no—
92#11758198—41no—
93#11758198—41no—
94C4⋊Dic61—41no—
95C4⋊Dic61—41no—
96C2×#92418—22no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C2×D411yescharacteristic
99Dic244—41no—
100D488—41no—
101#11758190—42no—
102C8×D61—41no—
103C8×D61—41no—
104C8×D61—41no—
105C8×D61—41no—
106#11758187—42no—
107C23×C61#10211yescharacteristic
108C22×D61—22no—
109C22×D61—22no—
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—41no—
113C22×D61#10211yescharacteristic
114C22×D61C2×D811yescharacteristic
115C2×#6923—44no—
116C2×#6923—88no—
117C2×#6923—82no—
118C2×#6923—82no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61#9912yes—
122C2×C4×C61#10211yescharacteristic
123C2×C4×C61#10412yes—
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C22⋊D61—82no—
126C2×Dic61—22no—
127C2×Dic61—41no—
128C2×Dic61#10211yescharacteristic
129C2×Dic61C2×SD1611yescharacteristic
130D244—41no—
131D244—41no—
132D244—82no—
133D244—81no—
134D244—81no—
135C4×D61—22no—
136C4×D61—22no—
137C4×D61—21no—
138C4×D61—21no—
139C4×D61—21no—
140C4×D61—21no—
141C4×D61—82no—
142Dic122—41no—
143Dic122—41no—
144C8×C61—21no—
145C8×C61—21no—
146#92418—21no—
147#92418—21no—
148#11704—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61#32712yes—
152C22×C61C2×#10211yescharacteristic
153C2×D61—24no—
154C2×D61—24no—
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—81no—
160C2×D61—81no—
161C2×D61—81no—
162C2×D61#36012yes—
163C2×D61#50911yescharacteristic
164C2×D61#51011yescharacteristic
165#6923—84no—
166C4×C61—81no—
167C4×C61—81no—
168C4×C61#32712yes—
169C4×C61C2×#9911yescharacteristic
170C4×C61C2×#10411yescharacteristic
171Dic61—21no—
172Dic61—21no—
173Dic61—81no—
174#3050—611no—
175#2112—611no—
176C2×#340—611no—
177#1348—612no—
178#1347—612no—
179C2×C61—22no—
180C2×C61C2×#32711yescharacteristic
181C2×C61#213911yescharacteristic
182C2×C61C2×#36011yescharacteristic
183D61—21no—
184D61—22no—
185D61—21no—
186D61—161no—
187C2×#135—611no—
188C23×C8—611no—
189#525—1221no—
190C22×C4⋊C4—1221no—
191C2×M32—1221no—
192#467—611no—
193#467—611no—
194#466—611no—
195#466—611no—
196#465—1221no—
197#465—1221no—
198#427#692311yescharacteristic, complemented
199#416—1221no—
200C22⋊C16—612no—
201C22⋊C16—1221no—
202#340—612no—
203#340—1221no—
204C61#1170411yescharacteristic, complemented, Sylow
205C2×#41—1221no—
206C23×C4—611no—
207C23×C4—2441no—
208#135—2442no—
209C2×Q16—611no—
210C2×SD16—1221no—
211C2×D8—611no—
212C22×C8—611no—
213C22×C8—611no—
214C22×C8—611no—
215C22×C8—611no—
216C22×C8—611no—
217C22×C8—2442no—
218C22×C8C2×#692311yescharacteristic
219#123—2441no—
220C22⋊Q8—1221no—
221C22⋊Q8—1221no—
222#121—1221no—
223#121—1221no—
224#120—2441no—
225C4×D4—2441no—
226C4×D4—2441no—
227#117—21no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1224no—
230C2×C4⋊C4—2441no—
231C2×C4⋊C4—2442no—
232M32—2442no—
233C2×C16—1222no—
234#107C2×#692311yescharacteristic, complemented
235#107C2×#692311yescharacteristic, complemented
236#106—21no—
237#103—1221no—
238#103—1221no—
239#102—1221no—
240#102—1221no—
241C24—611no—
242#41—2442no—
243#41—2442no—
244C2×Q8—1221no—
245C2×D4—1221no—
246C2×D4—2441no—
247C2×D4—2441no—
248C2×D4—2441no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—611no—
254C22×C4—1222no—
255C22×C4—1222no—
256C22×C4—4882no—
257C22×C4—4882no—
258C22×C4—2442no—
259C22×C4—2444no—
260C22×C4—2441no—
261C22×C4C22⋊#692311yescharacteristic
262Q16—2441no—
263SD16—2442no—
264D8—2441no—
265C2×C8—22no—
266C2×C8—611no—
267C2×C8—611no—
268C2×C8—1222no—
269C2×C8—2441no—
270C2×C8—2441no—
271C2×C8—2441no—
272C2×C8—2441no—
273C2×C8C22⋊#692311yescharacteristic
274C2×C8C22×#692311yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—1224no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C4⋊C4—2442no—
281C4⋊C4—2444no—
282C22⋊C4—41no—
283C22⋊C4—2441no—
284C22⋊C4—2441no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—41no—
288C16—2442no—
289C23—611no—
290C23—611no—
291C23—1221no—
292C23—1222no—
293C23—1221no—
294C23—1222no—
295C23—2441no—
296C23#1451983911yescharacteristic
297Q8—2441no—
298Q8—2441no—
299D4—4881no—
300D4—4881no—
301D4—4882no—
302D4—4882no—
303D4—2441no—
304D4—2441no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—1222no—
310C2×C4—1221no—
311C2×C4—1221no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1222no—
315C2×C4—1222no—
316C2×C4—4882no—
317C2×C4—4888no—
318C2×C4—4882no—
319C2×C4—4882no—
320C2×C4—2444no—
321C2×C4—2441no—
322C2×C4#1451983612yes—
323C2×C4#1451983911yescharacteristic
324C2×C4#1451984212yes—
325C2×C4C2×C22⋊#692311yescharacteristic
326C8—21no—
327C8—21no—
328C8—1221no—
329C8—1221no—
330C22—21no—
331C22—21no—
332C22—611no—
333C22—611no—
334C22—612no—
335C22—1224no—
336C22—1224no—
337C22—1221no—
338C22—1221no—
339C22—1221no—
340C22—1221no—
341C22—4881no—
342C22—4881no—
343C22—4881no—
344C22#1684061412yes—
345C22C2×#1451983911yescharacteristic
346C4—81no—
347C4—81no—
348C4—1221no—
349C4—1221no—
350C4—4884no—
351C4—4881no—
352C4#1684061412yes—
353C4C2×#1451983611yescharacteristic
354C4C2×#1451984211yescharacteristic
355C2—22no—
356C2—1221no—
357C2—1221no—
358C2—1222no—
359C2—9761no—
360C2C2×#1684061411yescharacteristic
361C2#2146212711yescharacteristic
362C2D61⋊#36011yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#427C4
socleC22×C61C2×#102
prime radical#21455401C2
cosocleC2×C8×D61C23
hypercenter#427#6923
hypocenterC61#11704
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #21462048C2semidirect
2. C2×#16840636C2semidirect
3. #21460808C2semidirect
4. #21460806C2semidirect
12. C2×#14519799C4semidirect
15. #16840004C4semidirect
17. #16839999C4semidirect
22. C61×#427C4semidirect
33. C2×Dic244C2×C4semidirect
35. C2×D488C2×C4semidirect
56. #107×C61C2×C4semidirect
57. #107×C61C2×C4semidirect
198. #427#6923semidirect
204. C61#11704semidirect
234. #107C2×#6923semidirect
235. #107C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214620482
C2×#168406362
#214608082
#214608062
#214554012
#1170461