Finite Group Databasebeta

← #24918489

Subgroups — #24918489

Order 10624.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C4⋊Q8×D83C211yesmaximal, characteristic, complemented
2#19128233C211yesmaximal, characteristic
3#19128092C212yesmaximal, complemented
4#19128068C212yesmaximal, complemented
5#19127912C211yesmaximal, characteristic, complemented
6#19127908C211yesmaximal, characteristic, complemented
7#19127906C211yesmaximal, characteristic, complemented
8#19127858C212yesmaximal, complemented
9#19127838C212yesmaximal
10#19127757C211yesmaximal, characteristic, complemented
11#461×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2212yes—
13C2×Dic332C2212yes—
14C2×#13050580C2212yes—
15C2×#13050578C2212yes—
16#14581792C2211yescharacteristic
17#14581791—21no—
18#14581791—21no—
19#14581790C2212yes—
20#14581785—21no—
21#14581783C2211yescharacteristic
22#14581779C2211yescharacteristic
23#14581774—21no—
24#14581772C2211yescharacteristic
25#14581763C2211yescharacteristic
26C4×Dic166C2211yescharacteristic
27C4×C22⋊D83—21no—
28#14581751—21no—
29C4⋊C4×D83—21no—
30C4⋊C4×D83—21no—
31C42×D83C2211yescharacteristic
32C4⋊(C83⋊C8)C2211yescharacteristic
33#14581717C2212yes—
34#14581716C2212yes—
35#14581715C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581702C2212yes—
38C2×SD16×C83C2212yes—
39C4⋊Q8×C83C2211yescharacteristic
40C4×D4×C83C2211yescharacteristic
41C4⋊C8×C83C2211yescharacteristic
42#103×C83C2212yes—
43C2×Dic166—22no—
44C2×Dic166C2311yescharacteristic
45C2×Dic166C2312yes—
46Q8×D83—42no—
47Q8×D83—42no—
48C2×C22⋊D83—21no—
49C2×C4×D83—22no—
50C2×C4×D83D411yescharacteristic
51C2×C4×D83D411yescharacteristic
52C2×C4×D83C2311yescharacteristic
53Dic332—42no—
54#13050580—42no—
55#13050578—42no—
56C2×C83⋊C8C2312yes—
57C4⋊Dic83—21no—
58C4⋊Dic83—21no—
59C4⋊Dic83—21no—
60C4⋊Dic83—21no—
61C4⋊Dic83C2311yescharacteristic
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66C22⋊Dic83—21no—
67C22⋊Dic83—21no—
68C22⋊Dic83—21no—
69(C2×D83)⋊C4—21no—
70C4×Dic83—21no—
71C4×Dic83D411yescharacteristic
72C4×Dic83D411yescharacteristic
73C4×Dic83C2311yescharacteristic
74C2×Q8×C83C2312yes—
75C2×D4×C83C2311yescharacteristic
76C22×C4×C83—21no—
77SD16×C83—42no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83—21no—
81C4⋊C4×C83C2311yescharacteristic
82C22⋊C4×C83—21no—
83C42×C83C2311yescharacteristic
84C23×C83—21no—
85C22×D83C2×D411yescharacteristic
86Q8×C83—22no—
87Q8×C83—22no—
88D4×C83—21no—
89D4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—42no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C4×D83—22no—
104C4×D83—22no—
105C4×D83—21no—
106C4×D83—21no—
107C4×D83—44no—
108Dic166—22no—
109Dic166—21no—
110Dic166—21no—
111Dic166—22no—
112Dic166—44no—
113C8×C83—22no—
114C83⋊C8—22no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—41no—
118C22×C83C22×D411yescharacteristic
119C2×D83—21no—
120C2×D83—21no—
121C2×D83#13712yes—
122C4×C83—21no—
123C4×C83—42no—
124C4×C83—41no—
125C4×C83#13712yes—
126C4×C83H(5,𝔽2)11yescharacteristic
127C4×C83H(5,𝔽2)11yescharacteristic
128Dic83—21no—
129Dic83—21no—
130Dic83—21no—
131Dic83—42no—
132Dic83—41no—
133Dic83—41no—
134C2×C83—81no—
135C2×C83#53511yescharacteristic
136C2×C83C2×#13711yescharacteristic
137C2×C83C2×#13711yescharacteristic
138D83—22no—
139#3218—831nomaximal, Sylow
140C83#321811yescharacteristic, complemented, Sylow
141C2×C4⋊Q8—831no—
142#529—831no—
143#497—832no—
144#491—832no—
145#462—831no—
146#462—831no—
147#461—831no—
148#461D8311yescharacteristic, complemented
149#451—832no—
150#448—832no—
151#432—831no—
152C22×Q8—832no—
153C2×Q16—832no—
154C2×Q16—832no—
155C2×SD16—832no—
156C2×SD16C2×D8312yes—
157C4⋊Q8—832no—
158C4⋊Q8—831no—
159C4⋊Q8—1661no—
160C4⋊Q8—1661no—
161C4⋊Q8C2×D8311yescharacteristic
162#126—1661no—
163#125—831no—
164#124—831no—
165#123—1661no—
166C22⋊Q8—831no—
167#121—831no—
168C4×Q8—831no—
169C4×D4—1661no—
170C4×D4C2×D8311yescharacteristic
171#117—1661no—
172C2×C4⋊C4—1661no—
173C2×C4⋊C4—1661no—
174C2×C42—831no—
175C4⋊C8—831no—
176C4⋊C8C2×D8311yescharacteristic
177#103—832no—
178#103—832no—
179#103—832no—
180#103C2×D8312yes—
181C22⋊C8—832no—
182#97—832no—
183C2×Q8—3322no—
184C2×Q8—3322no—
185C2×Q8—832no—
186C2×Q8—831no—
187C2×Q8—1662no—
188C2×Q8C22×D8312yes—
189C2×D4—1661no—
190C2×D4C22×D8311yescharacteristic
191C22×C4—21no—
192C22×C4—831no—
193C22×C4—831no—
194C22×C4—831no—
195C22×C4—1662no—
196Q16—3322no—
197Q16—3322no—
198SD16—42no—
199SD16—3322no—
200C2×C8—832no—
201C2×C8C22×D8312yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—831no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4—1661no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4C22×D8311yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1661no—
218C22⋊C4—1661no—
219C42—831no—
220C42—831no—
221C42—831no—
222C42—1661no—
223C42C22×D8311yescharacteristic
224C23—21no—
225C23—831no—
226Q8—22no—
227Q8—22no—
228Q8—3324no—
229Q8—1661no—
230Q8—1662no—
231Q8—1662no—
232Q8—1661no—
233D4—21no—
234D4—21no—
235D4—3322no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3324no—
240C2×C4—831no—
241C2×C4—831no—
242C2×C4—831no—
243C2×C4—1662no—
244C2×C4—1661no—
245C2×C4—1661no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1661no—
249C2×C4—1661no—
250C2×C4D4×D8311yescharacteristic
251C2×C4D4×D8311yescharacteristic
252C2×C4C23×D8311yescharacteristic
253C8—22no—
254C8—1662no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—832no—
259C22—1661no—
260C22—1661no—
261C22C2×D4×D8311yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—3321no—
266C4—3322no—
267C4—3321no—
268C4—1661no—
269C4—1661no—
270C4—1661no—
271C4#1458182112yes—
272C4#1458183711yescharacteristic
273C4#1458183811yescharacteristic
274C2—81no—
275C2—1662no—
276C2#1912826711yescharacteristic
277C2#137×D8311yescharacteristic
278C2C2×#1458182111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radical#14581763C22
hypercenter#461D83
hypocenterC83#3218
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D83C2semidirect
3. #19128092C2semidirect
4. #19128068C2semidirect
5. #19127912C2semidirect
6. #19127908C2semidirect
7. #19127906C2semidirect
8. #19127858C2semidirect
10. #19127757C2semidirect
11. #461×C83C2semidirect
140. C83#3218semidirect
148. #461D83semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D832
#191282332
#191280922
#191280682
#191279122
#191279082
#191279062
#191278582
#191278382
#191277572
#461×C832
#321883