Finite Group Databasebeta

← #35313702

Subgroups — #35313702

Order 21248.

392 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 119 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#24905787C212yesmaximal
2#24905407C212yesmaximal
3#24905397C212yesmaximal
4C83×#2327C211yesmaximal, characteristic
5#19127754—22no—
6#19127427—22no—
7#19127427C2214yes—
8C2×C4⋊Q8×C83C411yescharacteristic
9C2×C4×Q8×C83C411yescharacteristic
10C2×C4×Q8×C83C2211yescharacteristic
11#391×C83C411yescharacteristic
12#391×C83C2211yescharacteristic
13#387×C83—22no—
14#387×C83—22no—
15#382×C83C2211yescharacteristic
16#376×C83C411yescharacteristic
17C4⋊(C83⋊C8)—42no—
18C22⋊(C83⋊C8)—24no—
19C4×C83⋊C8—42no—
20C22×Q8×C83C2×C411yescharacteristic
21C22×Q8×C83C2×C411yescharacteristic
22C4⋊Q8×C83—22no—
23C4⋊Q8×C83C814yescomplemented
24C4×Q8×C83—42no—
25C4×Q8×C83—42no—
26C2×C4⋊C4×C83—22no—
27C2×C4⋊C4×C83—21no—
28C2×C4⋊C4×C83C2×C411yescharacteristic
29C2×C4⋊C4×C83D412yes—
30C2×C4⋊C4×C83C2311yescharacteristic
31C2×C42×C83—21no—
32C2×C42×C83C2×C411yescharacteristic
33C2×C42×C83C2×C411yescharacteristic
34C2×C42×C83C2×C411yescharacteristic
35C2×C42×C83D412yes—
36#95×C83—21no—
37#95×C83—21no—
38#95×C83—22no—
39#95×C83—22no—
40#95×C83—22no—
41C2×C83⋊C8—44no—
42C2×Q8×C83—21no—
43C2×Q8×C83—41no—
44C2×Q8×C83—41no—
45C2×Q8×C83—41no—
46C2×Q8×C83—41no—
47C2×Q8×C83C2×C812yes—
48C2×Q8×C83C2×C812yes—
49C2×Q8×C83M1612yes—
50C22×C4×C83—21no—
51C22×C4×C83—21no—
52C22×C4×C83—21no—
53C22×C4×C83—21no—
54C22×C4×C83C22⋊C412yes—
55C22×C4×C83C22⋊C411yescharacteristic
56C22×C4×C83C22⋊C411yescharacteristic
57C22×C4×C83C22×C411yescharacteristic
58C22×C4×C83C2×D412yes—
59C4⋊C4×C83—28no—
60C4⋊C4×C83—42no—
61C4⋊C4×C83—41no—
62C4⋊C4×C83—41no—
63C4⋊C4×C83—42no—
64C42×C83—22no—
65C42×C83—22no—
66C42×C83—22no—
67C42×C83—21no—
68C42×C83—41no—
69C42×C83—42no—
70C42×C83C2×C812yes—
71C23×C83C2×C22⋊C411yescharacteristic
72Q8×C83—42no—
73Q8×C83—42no—
74Q8×C83—42no—
75Q8×C83—42no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—41no—
85C2×C4×C83—41no—
86C2×C4×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—42no—
91C2×C4×C83—41no—
92C2×C4×C83—41no—
93C2×C4×C83—42no—
94C2×C4×C83—41no—
95C2×C4×C83—41no—
96C2×C4×C83C22⋊C814yes—
97C2×C4×C83#9912yes—
98C2×C4×C83C22×C811yescharacteristic
99C2×C4×C83C2×M1611yescharacteristic
100C83⋊C8—84no—
101C22×C83—21no—
102C22×C83—21no—
103C22×C83C2×C22⋊C811yescharacteristic
104C22×C83C2×#9911yescharacteristic
105C22×C83#41311yescharacteristic
106C4×C83—22no—
107C4×C83—24no—
108C4×C83—21no—
109C4×C83—21no—
110C4×C83—42no—
111C4×C83—42no—
112C4×C83—41no—
113C4×C83—42no—
114C4×C83—81no—
115C4×C83—81no—
116#11482—831nomaximal, Sylow
117C2×C83—21no—
118C2×C83—21no—
119C2×C83#145911yescharacteristic
120C2×C83#212911yescharacteristic
121C2×C83#213311yescharacteristic
122#2327D8311yescharacteristic
123#1461—832no—
124#1324—832no—
125#1319—832no—
126C83#1148211yescharacteristic, complemented, Sylow
127C2×C4⋊Q8Dic8311yescharacteristic
128C2×C4×Q8Dic8311yescharacteristic
129C2×C4×Q8C2×D8311yescharacteristic
130#431—1662no—
131#391Dic8311yescharacteristic
132#391C2×D8311yescharacteristic
133#387—22no—
134#387—22no—
135#382C2×D8311yescharacteristic
136#376Dic8311yescharacteristic
137#324—834no—
138#324—1662no—
139C22×Q8C2×Dic8311yescharacteristic
140C22×Q8C2×Dic8311yescharacteristic
141C4⋊Q8—22no—
142C4⋊Q8C83⋊C814yescomplemented
143C4×Q8—42no—
144C4×Q8—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic8311yescharacteristic
148C2×C4⋊C4C22⋊D8312yes—
149C2×C4⋊C4C22×D8311yescharacteristic
150C2×C42—21no—
151C2×C42C2×Dic8311yescharacteristic
152C2×C42C2×Dic8311yescharacteristic
153C2×C42C2×Dic8311yescharacteristic
154C2×C42C22⋊D8312yes—
155C4⋊C8—3322no—
156C22⋊C8—1664no—
157C4×C8—3322no—
158#95—21no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8C2×C83⋊C812yes—
169C2×Q8C2×C83⋊C812yes—
170C2×Q8#1305057412yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C22⋊Dic8312yes—
176C22×C4C22⋊Dic8311yescharacteristic
177C22×C4C22⋊Dic8311yescharacteristic
178C22×C4C22×Dic8311yescharacteristic
179C22×C4C2×C22⋊D8312yes—
180C2×C8—3324no—
181C4⋊C4—28no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C42—21no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—41no—
191C42—42no—
192C42C2×C83⋊C812yes—
193C23C2×C22⋊Dic8311yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4C22⋊(C83⋊C8)14yes—
219C2×C4#1458170712yes—
220C2×C4C22×C83⋊C811yescharacteristic
221C2×C4C2×#1305057411yescharacteristic
222C8—6644no—
223C22—21no—
224C22—21no—
225C22C2×C22⋊(C83⋊C8)11yescharacteristic
226C22C2×#1458170711yescharacteristic
227C22#1912768511yescharacteristic
228C4—21no—
229C4—21no—
230C4—24no—
231C4—22no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2#2490578111yescharacteristic
241C2#2490913811yescharacteristic
242C2#2490915211yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2327C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#2327D83
hypocenterC83#11482
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
23. C4⋊Q8×C83C8semidirect
126. C83#11482semidirect
142. C4⋊Q8C83⋊C8semidirect

Maximal subgroups

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

subgroupindex
#249057872
#249054072
#249053972
C83×#23272
#1148283