Finite Group Databasebeta

← #29583252

Subgroups — #29583252

Order 15616.

594 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 77 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#21445764C212yesmaximal
2#21445736C211yesmaximal, characteristic
3#21443727C211yesmaximal, characteristic, complemented
4#21443672C212yesmaximal, complemented
5C61×#2608C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519705—22no—
8C2×#14519705—22no—
9C2×#14519691—22no—
10#16839726C2212yes—
11#16839625C2211yescharacteristic
12C61×#396—22no—
13C61×#389—22no—
14C61×#387—22no—
15C61×#381C2211yescharacteristic
16C4×C22⋊C4×C61C2212yes—
17C2×#11758222—22no—
18C2×#11758222—22no—
19C2×#11758222—44no—
20C22×Dic122Q811yescharacteristic
21C22×D244Q811yescharacteristic
22C22×C22⋊D61—21no—
23C22×C4×D61—21no—
24C2×#11758194—22no—
25C2×C22⋊Dic61—21no—
26#14519705—44no—
27#14519705—44no—
28#14519691—44no—
29C22⋊#92418—22no—
30C22⋊#92418—22no—
31#14519684—22no—
32C23×C4×C61C2311yescharacteristic
33C2×C4⋊C4×C61—22no—
34C2×C4⋊C4×C61—22no—
35C2×C4⋊C4×C61—22no—
36C2×C22⋊C4×C61—22no—
37C2×C22⋊C4×C61D412yes—
38C2×C42×C61D412yes—
39C2×C42×C61D412yes—
40#95×C61—22no—
41#95×C61—22no—
42#95×C61—22no—
43C24×C61C2×D411yescharacteristic
44C23×D61—21no—
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61—42no—
49C22×C4×C61C2×D412yes—
50C22×C4×C61C2×Q811yescharacteristic
51C22×C4×C61#4112yes—
52C22×C4×C61#4111yescharacteristic
53C2×C22⋊D61—24no—
54C2×C22⋊D61—42no—
55C2×C22⋊D61—41no—
56C2×C22⋊D61—41no—
57C22×Dic61—21no—
58C22×Dic61—41no—
59#11758222—24no—
60#11758222—24no—
61#11758222—44no—
62#11758222—44no—
63#11758222—42no—
64#11758222—42no—
65C2×D244—22no—
66C2×D244—42no—
67C2×C4×D61—42no—
68C2×C4×D61—42no—
69C2×C4×D61—42no—
70C2×Dic122—22no—
71C2×Dic122—42no—
72C4⋊C4×C61—44no—
73C4⋊C4×C61—44no—
74C4⋊C4×C61—82no—
75C22⋊C4×C61—44no—
76C22⋊C4×C61—42no—
77C42×C61—42no—
78C42×C61—42no—
79C22⋊Dic61—42no—
80#11758194—42no—
81#11758194—42no—
82C2×#92418—42no—
83C23×C61—22no—
84C23×C61—42no—
85C23×C61#12011yescharacteristic
86C23×C61C22⋊Q811yescharacteristic
87C23×C61#12411yescharacteristic
88C22×D61—22no—
89C22×D61—41no—
90C22×D61—41no—
91C22×D61—82no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—24no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61C22⋊Q812yes—
107C2×C4×C61#12412yes—
108C22⋊D61—44no—
109C22⋊D61—44no—
110C2×Dic61—22no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—81no—
114C2×Dic61—82no—
115D244—24no—
116D244—42no—
117C4×D61—44no—
118C4×D61—44no—
119Dic122—24no—
120Dic122—42no—
121#92418—82no—
122#10388—611nomaximal, Sylow
123C22×C61—22no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—22no—
127C22×C61—22no—
128C22×C61—82no—
129C22×C61#39311yescharacteristic
130C22×C61B(4,𝔽2)12yes—
131C2×D61—42no—
132C2×D61—82no—
133C2×D61—82no—
134C2×D61—82no—
135C4×C61—22no—
136C4×C61—22no—
137C4×C61—44no—
138C4×C61—82no—
139C4×C61—82no—
140Dic61—42no—
141Dic61—161no—
142#2608D6111yescharacteristic, complemented
143#1789—612no—
144#1782—611no—
145#1306—611no—
146#1284—612no—
147C2×C61—22no—
148C2×C61—41no—
149C2×C61—41no—
150C2×C61#202911yescharacteristic
151C2×C61#203112yes—
152D61—82no—
153C22×#41—611no—
154C2×#104—1222no—
155C2×#104—1222no—
156C2×#99—1222no—
157#407—612no—
158#396—22no—
159#389—22no—
160#387—22no—
161#381—611no—
162#381C2×D6111yescharacteristic
163C4×C22⋊C4C2×D6112yes—
164C61#1038811yescharacteristic, complemented, Sylow
165C2×#41—1222no—
166C2×#41—1222no—
167C2×#41—2444no—
168C22×Q8—611no—
169C22×D4—611no—
170C22×D4—1221no—
171C23×C4—1221no—
172C23×C4C22×D6111yescharacteristic
173C2×M16—1222no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1221no—
179C2×C22⋊C4C22⋊D6112yes—
180C2×C42C22⋊D6112yes—
181C2×C42C22⋊D6112yes—
182#104—2444no—
183#104—2444no—
184#99—2444no—
185C22⋊C8—1222no—
186C22⋊C8—1222no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—1222no—
191C24—1221no—
192C24C2×C22⋊D6111yescharacteristic
193#41—1224no—
194#41—1224no—
195#41—2442no—
196#41—2442no—
197#41—2444no—
198#41—2444no—
199C2×Q8—1222no—
200C2×Q8—2442no—
201C2×D4—1224no—
202C2×D4—1222no—
203C2×D4—2442no—
204C2×D4—2442no—
205C2×D4—2441no—
206C2×D4—2441no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—1221no—
212C22×C4—2442no—
213C22×C4—2442no—
214C22×C4—2441no—
215C22×C4—2442no—
216C22×C4Q8×D6111yescharacteristic
217C22×C4#1175822612yes—
218C22×C4#1175822611yescharacteristic
219C22×C4C2×C22⋊D6112yes—
220M16—2442no—
221M16—2442no—
222C2×C8—2442no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—82no—
226C22⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—2442no—
229C42—42no—
230C42—42no—
231C23—22no—
232C23—42no—
233C23—1222no—
234C23—4882no—
235C23—2441no—
236C23—2441no—
237C23#1451974611yescharacteristic
238C23#1451975311yescharacteristic
239C23#1451976611yescharacteristic
240Q8—1224no—
241Q8—2442no—
242D4—1224no—
243D4—2442no—
244D4—2444no—
245D4—2444no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—1222no—
261C2×C4—4881no—
262C2×C4—4882no—
263C2×C4—2444no—
264C2×C4—2444no—
265C2×C4—2441no—
266C2×C4—2441no—
267C2×C4#1451975312yes—
268C2×C4#1451976612yes—
269C8—4882no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—82no—
276C22—4882no—
277C22—4882no—
278C22—4882no—
279C22—2442no—
280C22#1683967311yescharacteristic
281C22#1683994912yes—
282C4—22no—
283C4—22no—
284C4—44no—
285C4—82no—
286C4—82no—
287C4—9761no—
288C4—2442no—
289C2—22no—
290C2—41no—
291C2—41no—
292C2—4882no—
293C2#2144698211yescharacteristic
294C2#2144698812yes—
subgroupG/H
centerC22#16839673
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2608C2
socleC22×C61#393
prime radicalC22×#11758222C22
hypercenter#2608D61
hypocenterC61#10388
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2608

Complements

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

NH ≅ G/Ntype
3. #21443727C2semidirect
4. #21443672C2semidirect
5. C61×#2608C2semidirect
142. #2608D61semidirect
164. C61#10388semidirect

Maximal subgroups

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

subgroupindex
#214457642
#214457362
#214437272
#214436722
C61×#26082
#1038861