Finite Group Databasebeta

← #29592127

Subgroups — #29592127

Order 15616.

476 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 71 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#21447735C212yesmaximal, complemented
2#21446829C211yesmaximal, characteristic, complemented
3#21443828C212yesmaximal, complemented
4#21443743C211yesmaximal, characteristic, complemented
5C61×#3186C211yesmaximal, characteristic
6C22×Dic244C2211yescharacteristic, complemented
7C2×#14519756—21no—
8C2×#14519721—22no—
9C2×#14519702—21no—
10#16839565—22no—
11C22⋊#11758185C2212yescomplemented
12#16839525C2211yescharacteristic, complemented
13C2×#121×C61—21no—
14C61×#480—22no—
15C61×#466—21no—
16C61×#466C2212yes—
17C61×#447—22no—
18C2×C22⋊C8×C61C2211yescharacteristic
19C22×Dic122—21no—
20C2×Dic244—22no—
21C2×Dic244—42no—
22#14519756—42no—
23#14519756—42no—
24C2×C4⋊Dic61—21no—
25C2×#11758196—41no—
26C2×C22⋊Dic61—41no—
27#14519721—44no—
28C2×#11758185—22no—
29#14519702—42no—
30C22×D4×C61—21no—
31C22×D4×C61—41no—
32C23×C4×C61D411yescharacteristic
33C2×D8×C61—21no—
34C2×D8×C61D412yes—
35C22×C8×C61D411yescharacteristic
36C22×C8×C61C2311yescharacteristic
37#121×C61—22no—
38#121×C61—22no—
39#121×C61—41no—
40#121×C61—41no—
41C2×C4⋊C4×C61—21no—
42C2×C22⋊C4×C61—41no—
43#107×C61—21no—
44#107×C61D412yes—
45#102×C61—22no—
46#102×C61—22no—
47C22⋊C8×C61—22no—
48C24×C61—41no—
49C24×C61D811yescharacteristic
50C2×D4×C61—22no—
51C2×D4×C61—41no—
52C2×D4×C61—42no—
53C2×D4×C61—42no—
54C2×D4×C61—81no—
55C2×D4×C61—81no—
56C2×D4×C61—82no—
57C2×D4×C61—82no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61—41no—
63C22×C4×C61D811yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×Dic61—41no—
66C22×Dic61—41no—
67C2×Dic122—22no—
68C2×Dic122—41no—
69C2×Dic122—41no—
70C2×Dic122—41no—
71C2×Dic122—41no—
72C2×Dic122—41no—
73D8×C61—42no—
74D8×C61—81no—
75C2×C8×C61—21no—
76C2×C8×C61—41no—
77C2×C8×C61—41no—
78C2×C8×C61C2×D412yes—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C22⋊Dic61—44no—
84C4⋊Dic61—22no—
85C4⋊Dic61—41no—
86#11758196—44no—
87Dic244—24no—
88Dic244—42no—
89#11758185—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—42no—
95C23×C61—82no—
96C23×C61—82no—
97C23×C61—82no—
98C23×C61SD3212yescomplemented
99C23×C61C2×D811yescharacteristic
100D4×C61—42no—
101D4×C61—42no—
102D4×C61—82no—
103D4×C61—84no—
104D4×C61—82no—
105D4×C61—82no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61—81no—
114C2×C4×C61—82no—
115C2×C4×C61#12011yescharacteristic
116C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—81no—
120C2×Dic61—81no—
121C2×Dic61—81no—
122C2×Dic61—81no—
123Dic122—42no—
124Dic122—42no—
125Dic122—81no—
126Dic122—81no—
127C8×C61—22no—
128C8×C61—41no—
129#12105—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—42no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61—82no—
138C22×C61—82no—
139C22×C61—82no—
140C22×C61—164no—
141C22×C61#44711yescharacteristic
142C22×C61C2×SD3211yescharacteristic
143C22×C61#50911yescharacteristic
144C4×C61—21no—
145C4×C61—41no—
146C4×C61—82no—
147C4×C61#45311yescharacteristic
148C4×C61B(4,𝔽2)11yescharacteristic
149Dic61—82no—
150Dic61—82no—
151#3186D6111yescharacteristic
152#2185—612no—
153#2001—611no—
154#1347—612no—
155#1314—611no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—42no—
159C2×C61—162no—
160C2×C61#159511yescharacteristic
161C2×C61#218711yescharacteristic
162C2×C61#219111yescharacteristic
163C22×Q16—611no—
164C2×C22⋊Q8—1221no—
165C2×#121—21no—
166C2×SD32—1222no—
167#480—22no—
168#466—21no—
169#466C2×D6112yes—
170#447—22no—
171C2×#103—1221no—
172C2×C22⋊C8C2×D6111yescharacteristic
173#358—1222no—
174C22⋊C16—612no—
175#340—611no—
176C61#1210511yescharacteristic, complemented, Sylow
177C22×Q8—1221no—
178C22×D4—21no—
179C22×D4—41no—
180C23×C4C22⋊D6111yescharacteristic
181C2×Q16—1222no—
182C2×Q16—2442no—
183C2×D8—21no—
184C2×D8C22⋊D6112yes—
185C22×C8C22⋊D6111yescharacteristic
186C22×C8C22×D6111yescharacteristic
187C22⋊Q8—2442no—
188C22⋊Q8—2442no—
189#121—22no—
190#121—22no—
191#121—41no—
192#121—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—2441no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—2441no—
198SD32—2444no—
199C2×C16—1222no—
200#107—21no—
201#107C22⋊D6112yes—
202#103—2442no—
203#102—22no—
204#102—22no—
205C22⋊C8—22no—
206C24—41no—
207C24#1175819911yescharacteristic
208C2×Q8—1222no—
209C2×Q8—2441no—
210C2×Q8—2441no—
211C2×Q8—2441no—
212C2×Q8—2441no—
213C2×Q8—2441no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—2441no—
228C22×C4—2441no—
229C22×C4#1175819911yescharacteristic
230C22×C4C2×C22⋊D6111yescharacteristic
231Q16—1224no—
232Q16—2442no—
233D8—42no—
234D8—81no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8C2×C22⋊D6112yes—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—1222no—
242C4⋊C4—2441no—
243C4⋊C4—2444no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—2444no—
247C16—2442no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23#1451972112yescomplemented
257C23C2×#1175819911yescharacteristic
258Q8—4881no—
259Q8—4881no—
260Q8—2442no—
261Q8—2442no—
262D4—42no—
263D4—42no—
264D4—82no—
265D4—84no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—4881no—
278C2×C4—4881no—
279C2×C4—4881no—
280C2×C4—4881no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4#1451974611yescharacteristic
284C2×C4#1451980311yescharacteristic
285C8—22no—
286C8—41no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—164no—
298C22#1683989611yescharacteristic
299C22C2×#1451972111yescharacteristic
300C22#1684019811yescharacteristic
301C4—21no—
302C4—41no—
303C4—82no—
304C4—4882no—
305C4—4882no—
306C4#1683993011yescharacteristic
307C4#1683994911yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—162no—
312C2#2144473711yescharacteristic
313C2#2144774811yescharacteristic
314C2#2144777011yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
Fitting, prime radicalC61×#3186C2
socleC22×C61#447
hypercenter#3186D61
hypocenterC61#12105
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #21447735C2semidirect
2. #21446829C2semidirect
3. #21443828C2semidirect
4. #21443743C2semidirect
6. C22×Dic244C22semidirect
11. C22⋊#11758185C22semidirect
12. #16839525C22semidirect
98. C23×C61SD32semidirect
176. C61#12105semidirect
256. C23#14519721semidirect

Maximal subgroups

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

subgroupindex
#214477352
#214468292
#214438282
#214437432
C61×#31862
#1210561