Finite Group Databasebeta

← #24915792

Subgroups — #24915792

Order 10624.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C2×#14581775C211yesmaximal, characteristic, complemented
2C2×#14581764C211yesmaximal, characteristic
3C22×C4×Dic83C211yesmaximal, characteristic, complemented
4#19127601C212yesmaximal, complemented
5#19127596C212yesmaximal, complemented
6#19127595C212yesmaximal, complemented
7#19127591C212yesmaximal, complemented
8#19127581C211yesmaximal, characteristic, complemented
9#19127578C211yesmaximal, characteristic, complemented
10#19127559C211yesmaximal, characteristic, complemented
11C2×#123×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15C23×Dic83C2211yescharacteristic
16#14581775—24no—
17#14581764—24no—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83—24no—
28C2×C4×Dic83C2212yes—
29C2×C4×Dic83C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C83⋊#95C2212yes—
33C22×D4×C83C2211yescharacteristic
34C23×C4×C83C2211yescharacteristic
35#123×C83—24no—
36C2×C4⋊C4×C83C2212yes—
37C2×C22⋊C4×C83C2211yescharacteristic
38C2×C22⋊C4×C83C2212yes—
39C23×D83C2311yescharacteristic
40C2×C22⋊D83—22no—
41C2×C22⋊D83—22no—
42C2×C22⋊D83—22no—
43C2×C22⋊D83—22no—
44C2×C22⋊D83—24no—
45C22×Dic83—22no—
46C22×Dic83—24no—
47C22×Dic83—24no—
48C22×Dic83D414yes—
49C22×Dic83C2311yescharacteristic
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2312yes—
53C22×Dic83C2311yescharacteristic
54Dic83⋊C4—24no—
55Dic83⋊C4—22no—
56Dic83⋊C4—24no—
57C22⋊Dic83—22no—
58C22⋊Dic83—22no—
59C22⋊Dic83—24no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—24no—
63C4×Dic83—24no—
64C4×Dic83—24no—
65C24×C83C2311yescharacteristic
66C24×C83C2311yescharacteristic
67C2×D4×C83—22no—
68C2×D4×C83—22no—
69C2×D4×C83—22no—
70C22×C4×C83—22no—
71C22×C4×C83—22no—
72C22×C4×C83D414yescomplemented
73C22×C4×C83C2312yes—
74C22×C4×C83C2311yescharacteristic
75C22×C4×C83C2312yes—
76C4⋊C4×C83—24no—
77C22⋊C4×C83—22no—
78C22⋊C4×C83—24no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—21no—
82C23×C83—21no—
83C23×C83—21no—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83C2×D412yes—
87C23×C83C2×D412yes—
88C23×C83#4112yes—
89C23×C83C2411yescharacteristic
90C22×D83—21no—
91C22×D83—21no—
92C22×D83—22no—
93C22×D83—22no—
94C22×D83—21no—
95D4×C83—44no—
96C2×C4×C83—28no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83C2×D414yes—
106C22⋊D83—44no—
107C22⋊D83—44no—
108C2×Dic83—28no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—28no—
113C2×Dic83—22no—
114C2×Dic83—21no—
115C2×Dic83—21no—
116C2×Dic83—22no—
117C2×Dic83—22no—
118C2×Dic83—21no—
119C2×Dic83C2×D412yes—
120C2×Dic83C2×D412yes—
121C2×Dic83#4114yes—
122C2×Dic83#4114yes—
123C22×C83—22no—
124C22×C83—22no—
125C22×C83—24no—
126C22×C83—24no—
127C22×C83—41no—
128C22×C83—41no—
129C22×C83—41no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83#12114yes—
133C22×C83C22×D411yescharacteristic
134C22×C83C22×D411yescharacteristic
135C22×C83C2×#4112yes—
136C22×C83C2×#4112yes—
137C22×C83C2×#4111yescharacteristic
138C2×D83—41no—
139C2×D83—41no—
140C2×D83—41no—
141C2×D83—42no—
142C2×D83—42no—
143C4×C83—24no—
144C4×C83—41no—
145C4×C83—42no—
146Dic83—22no—
147Dic83—24no—
148Dic83—41no—
149Dic83—42no—
150Dic83#12314yescomplemented
151C2×C83—24no—
152C2×C83—81no—
153C2×C83C2×#12111yescharacteristic
154C2×C83C2×#12311yescharacteristic
155C2×C83#52511yescharacteristic
156C2×C83#52912yes—
157C2×C83#53212yes—
158D83—81no—
159#2854—831nomaximal, Sylow
160C83#285411yescharacteristic, complemented, Sylow
161C2×#123—831no—
162C2×#123D8311yescharacteristic, complemented
163C2×#121—831no—
164C22×C42—831no—
165#390—832no—
166#389—832no—
167#388—832no—
168#388—832no—
169#386—831no—
170#386—831no—
171#380—831no—
172C22×D4—831no—
173C22×D4—831no—
174C22×D4C2×D8311yescharacteristic
175C23×C4—831no—
176C23×C4—831no—
177C23×C4C2×D8311yescharacteristic
178#123—24no—
179#123—1664no—
180#121—1664no—
181C2×C4⋊C4—831no—
182C2×C4⋊C4—832no—
183C2×C4⋊C4—832no—
184C2×C4⋊C4C2×D8312yes—
185C2×C22⋊C4—831no—
186C2×C22⋊C4—831no—
187C2×C22⋊C4—832no—
188C2×C22⋊C4—831no—
189C2×C22⋊C4—832no—
190C2×C22⋊C4—832no—
191C2×C22⋊C4C2×D8311yescharacteristic
192C2×C22⋊C4C2×D8312yes—
193C2×C42—832no—
194C2×C42—832no—
195C2×C42—1664no—
196#95—832no—
197#95—832no—
198#95—832no—
199C24—831no—
200C24C22×D8311yescharacteristic
201C24C22×D8311yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—1662no—
206C2×D4—1664no—
207C2×D4—1662no—
208C2×D4—1662no—
209C2×D4—1662no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—831no—
213C22×C4—834no—
214C22×C4—831no—
215C22×C4—832no—
216C22×C4—832no—
217C22×C4—831no—
218C22×C4—1664no—
219C22×C4—1664no—
220C22×C4—1662no—
221C22×C4C22⋊D8314yescomplemented
222C22×C4C22×D8312yes—
223C22×C4C22×D8311yescharacteristic
224C22×C4C22×D8312yes—
225C4⋊C4—24no—
226C4⋊C4—1662no—
227C4⋊C4—1664no—
228C4⋊C4—1664no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—1662no—
232C22⋊C4—1664no—
233C22⋊C4—1662no—
234C22⋊C4—1664no—
235C22⋊C4—1662no—
236C22⋊C4—1664no—
237C42—1664no—
238C42—1664no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—1661no—
247C23—1661no—
248C23—1662no—
249C23—1662no—
250C23—1661no—
251C23D4×D8312yes—
252C23C2×C22⋊D8312yes—
253C23#1305058912yes—
254C23C23×D8311yescharacteristic
255D4—44no—
256D4—3324no—
257D4—3324no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—832no—
268C2×C4—834no—
269C2×C4—832no—
270C2×C4—834no—
271C2×C4—1661no—
272C2×C4—1662no—
273C2×C4—1661no—
274C2×C4—1662no—
275C2×C4—1662no—
276C2×C4—1661no—
277C2×C4—1668no—
278C2×C4—1668no—
279C2×C4—1662no—
280C2×C4—1662no—
281C2×C4—1662no—
282C2×C4C2×C22⋊D8314yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—3321no—
293C22—3321no—
294C22—3321no—
295C22—3322no—
296C22—3322no—
297C22#1458176414yes—
298C22C2×D4×D8311yescharacteristic
299C22C22×C22⋊D8311yescharacteristic
300C22#41×D8312yes—
301C22#41×D8312yes—
302C22C2×#1305058911yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—3321no—
307C4—3322no—
308C4—834no—
309C4—1662no—
310C4—1664no—
311C2—24no—
312C2—81no—
313C2—6641no—
314C2C2×#1458176411yescharacteristic
315C2#123×D8311yescharacteristic
316C2#1912820911yescharacteristic
317C2#1912822812yes—
318C2#1912824712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
prime radicalC2×#14581764C2
hypercenterC2×#123D83
hypocenterC83#2854
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14581775C2semidirect
3. C22×C4×Dic83C2semidirect
4. #19127601C2semidirect
5. #19127596C2semidirect
6. #19127595C2semidirect
7. #19127591C2semidirect
8. #19127581C2semidirect
9. #19127578C2semidirect
10. #19127559C2semidirect
11. C2×#123×C83C2semidirect
72. C22×C4×C83D4semidirect
150. Dic83#123semidirect
160. C83#2854semidirect
162. C2×#123D83semidirect
221. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
C2×#145817642
C22×C4×Dic832
#191276012
#191275962
#191275952
#191275912
#191275812
#191275782
#191275592
C2×#123×C832
#285483