Finite Group Databasebeta

← #24918486

Subgroups — #24918486

Order 10624.

386 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128230C211yesmaximal, characteristic
3#19128096C211yesmaximal, characteristic, complemented
4#19128070C211yesmaximal, characteristic, complemented
5#19128069C211yesmaximal, characteristic, complemented
6#19127909C212yesmaximal, complemented
7#19127904C212yesmaximal, complemented
8#19127855C212yesmaximal, complemented
9#19127838C212yesmaximal
10#19127756C211yesmaximal, characteristic, complemented
11#497×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2212yes—
13C2×#13050580C2212yes—
14C2×#13050578C2212yes—
15#14581792C2211yescharacteristic
16#14581791—21no—
17#14581791—21no—
18#14581790C2212yes—
19#14581785—21no—
20#14581784C2211yescharacteristic
21#14581774—21no—
22#14581768C2211yescharacteristic
23#14581763C2211yescharacteristic
24Q8×Dic83C2211yescharacteristic
25D4×Dic83C2211yescharacteristic
26(C22⋊D83)⋊C4—21no—
27#14581748—21no—
28C4⋊C4×D83—21no—
29C4⋊C4×D83—21no—
30C42×D83C2211yescharacteristic
31Dic83⋊C8C2212yes—
32#14581717C2212yes—
33#14581717C2212yes—
34#14581716C2212yes—
35#14581716C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic
38C2×Q16×C83C2212yes—
39C2×SD16×C83C2212yes—
40C4⋊Q8×C83C2211yescharacteristic
41#124×C83C2211yescharacteristic
42#97×C83C2211yescharacteristic
43C2×Dic166—22no—
44C2×Dic166C2312yes—
45Q8×D83—42no—
46Q8×D83—42no—
47C2×C22⋊D83—21no—
48C22×Dic83—21no—
49C2×C4×D83—22no—
50C2×C4×D83D411yescharacteristic
51C2×C4×D83D411yescharacteristic
52C2×C4×D83C2311yescharacteristic
53#13050580—42no—
54#13050578—42no—
55C2×C83⋊C8C2312yes—
56C4⋊Dic83—21no—
57C4⋊Dic83—21no—
58C4⋊Dic83—21no—
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66C22⋊Dic83—21no—
67C22⋊Dic83—21no—
68(C2×D83)⋊C4—21no—
69C4×Dic83—21no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83C2311yescharacteristic
74C2×Q8×C83C2312yes—
75C2×D4×C83C2311yescharacteristic
76Q16×C83—42no—
77SD16×C83—42no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83—21no—
81C22⋊C4×C83—21no—
82C22⋊C4×C83—21no—
83C42×C83C2311yescharacteristic
84C23×C83—21no—
85C22×D83C2×D411yescharacteristic
86Q8×C83—22no—
87Q8×C83—22no—
88Q8×C83—22no—
89D4×C83—22no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—42no—
96C2×Dic83—21no—
97C2×Dic83—21no—
98C2×Dic83—22no—
99C2×Dic83—42no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C4×D83—22no—
104C4×D83—22no—
105C4×D83—22no—
106C4×D83—44no—
107Dic166—22no—
108Dic166—22no—
109Dic166—44no—
110C8×C83—22no—
111C83⋊C8—22no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83—41no—
115C22×C83C22×D411yescharacteristic
116C2×D83—21no—
117C2×D83—21no—
118C2×D83#13712yes—
119C4×C83—21no—
120C4×C83—21no—
121C4×C83—42no—
122C4×C83—41no—
123C4×C83H(5,𝔽2)12yes—
124Dic83—21no—
125Dic83—21no—
126Dic83—42no—
127Dic83—41no—
128Dic83—41no—
129Dic83#13712yes—
130C2×C83—81no—
131C2×C83#53511yescharacteristic
132C2×C83C2×#13711yescharacteristic
133C2×C83C2×#13711yescharacteristic
134D83—22no—
135#3218—831nomaximal, Sylow
136C83#321811yescharacteristic, complemented, Sylow
137C2×C4⋊Q8—831no—
138#529—831no—
139#497—831no—
140#497D8311yescharacteristic, complemented
141#491—831no—
142#491—831no—
143#462—832no—
144#461—832no—
145#451—832no—
146#448—832no—
147#432—831no—
148C22×Q8—832no—
149C2×Q16—832no—
150C2×Q16C2×D8312yes—
151C2×SD16—832no—
152C2×SD16C2×D8312yes—
153C4⋊Q8—832no—
154C4⋊Q8—831no—
155C4⋊Q8—1661no—
156C4⋊Q8—1661no—
157C4⋊Q8C2×D8311yescharacteristic
158#126—1661no—
159#125—831no—
160#124C2×D8311yescharacteristic
161#123—1661no—
162C22⋊Q8—831no—
163#121—831no—
164C4×Q8—831no—
165C4×D4—831no—
166C4×D4—1661no—
167#117—1661no—
168C2×C4⋊C4—1661no—
169C2×C4⋊C4—1661no—
170C2×C42—831no—
171C4⋊C8—832no—
172#103—832no—
173#103—832no—
174#103—832no—
175#103—832no—
176C22⋊C8—832no—
177#97—831no—
178#97C2×D8311yescharacteristic
179C2×Q8—3322no—
180C2×Q8—3322no—
181C2×Q8—832no—
182C2×Q8—1662no—
183C2×Q8C22×D8312yes—
184C2×Q8C22×D8311yescharacteristic
185C2×D4—1661no—
186C2×D4C22×D8311yescharacteristic
187C22×C4—831no—
188C22×C4—831no—
189C22×C4—831no—
190C22×C4—1662no—
191C22×C4—1661no—
192Q16—42no—
193Q16—3322no—
194SD16—42no—
195SD16—3322no—
196C2×C8—832no—
197C2×C8C22×D8312yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—831no—
201C4⋊C4—831no—
202C4⋊C4—1662no—
203C4⋊C4—1662no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—1661no—
213C22⋊C4—1661no—
214C22⋊C4—1661no—
215C42—831no—
216C42—831no—
217C42—831no—
218C42—1661no—
219C42C22×D8311yescharacteristic
220C23—21no—
221C23—831no—
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—3324no—
226Q8—1662no—
227Q8—1662no—
228D4—22no—
229D4—3322no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—3324no—
233C2×C4—3322no—
234C2×C4—831no—
235C2×C4—831no—
236C2×C4—831no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4D4×D8311yescharacteristic
244C2×C4D4×D8311yescharacteristic
245C2×C4C23×D8311yescharacteristic
246C8—22no—
247C8—1662no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—832no—
252C22—1661no—
253C22—1661no—
254C22C2×D4×D8311yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—3321no—
260C4—3321no—
261C4—3322no—
262C4—832no—
263C4—1661no—
264C4—1661no—
265C4#1458183712yes—
266C2—81no—
267C2—1662no—
268C2#1912826411yescharacteristic
269C2#137×D8311yescharacteristic
270C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#497×C83C2
socleC22×C83C22×D4
prime radical#14581763C22
hypercenter#497D83
hypocenterC83#3218
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D83C2semidirect
3. #19128096C2semidirect
4. #19128070C2semidirect
5. #19128069C2semidirect
6. #19127909C2semidirect
7. #19127904C2semidirect
8. #19127855C2semidirect
10. #19127756C2semidirect
11. #497×C83C2semidirect
136. C83#3218semidirect
140. #497D83semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D832
#191282302
#191280962
#191280702
#191280692
#191279092
#191279042
#191278552
#191278382
#191277562
#497×C832
#321883