Finite Group Databasebeta

← #30747191

Subgroups — #30747191

Order 17152.

1174 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 179 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#22560119C212yesmaximal
2C2×#16863122C214yesmaximal, complemented
3C22×C22⋊Q8×C67C211yesmaximal, characteristic
4C2×C22⋊(C67⋊C8)—24no—
5C2×C22⋊(C67⋊C8)C2214yes—
6#16863122—28no—
7C23×Q8×C67C411yescharacteristic
8C24×C4×C67C2211yescharacteristic
9C2×C22⋊Q8×C67—24no—
10C2×C22⋊Q8×C67—28no—
11C22×C4⋊C4×C67C411yescharacteristic
12C22×C4⋊C4×C67C2212yes—
13C22×C22⋊C4×C67C412yes—
14C22×C67⋊C8—24no—
15C22⋊(C67⋊C8)—28no—
16C22⋊(C67⋊C8)—48no—
17C25×C67C2×C411yescharacteristic
18C22×Q8×C67—24no—
19C22×Q8×C67—22no—
20C22×Q8×C67—44no—
21C23×C4×C67—24no—
22C23×C4×C67—44no—
23C23×C4×C67C2×C412yes—
24C23×C4×C67C2×C411yescharacteristic
25C23×C4×C67C2×C412yes—
26C23×C4×C67D414yes—
27C23×C4×C67C2311yescharacteristic
28C22⋊Q8×C67—232no—
29C2×C4⋊C4×C67—24no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—24no—
32C2×C4⋊C4×C67—24no—
33C2×C4⋊C4×C67D418yes—
34C2×C22⋊C4×C67—24no—
35C2×C22⋊C4×C67—24no—
36C2×C22⋊C4×C67—24no—
37C2×C67⋊C8—48no—
38C2×C67⋊C8—44no—
39C24×C67—24no—
40C24×C67—22no—
41C24×C67—24no—
42C24×C67—42no—
43C24×C67C22⋊C412yes—
44C24×C67C22×C411yescharacteristic
45C2×Q8×C67—28no—
46C2×Q8×C67—416no—
47C2×Q8×C67—44no—
48C2×Q8×C67—44no—
49C22×C4×C67—22no—
50C22×C4×C67—24no—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67—22no—
55C22×C4×C67—28no—
56C22×C4×C67—28no—
57C22×C4×C67—28no—
58C22×C4×C67—44no—
59C22×C4×C67—42no—
60C22×C4×C67—416no—
61C22×C4×C67—44no—
62C22×C4×C67C22⋊C414yes—
63C22×C4×C67C22⋊C412yes—
64C22×C4×C67C22⋊C414yes—
65C22×C4×C67C2×D412yes—
66C22×C4×C67C2×D414yes—
67C4⋊C4×C67—28no—
68C4⋊C4×C67—216no—
69C22⋊C4×C67—216no—
70C23×C67—22no—
71C23×C67—22no—
72C23×C67—28no—
73C23×C67—22no—
74C23×C67—24no—
75C23×C67—24no—
76C23×C67—22no—
77C23×C67—416no—
78C23×C67—44no—
79C23×C67—44no—
80C23×C67#10114yes—
81C23×C67C2×C22⋊C412yes—
82C23×C67C2×C22⋊C411yescharacteristic
83C23×C67#13114yes—
84Q8×C67—416no—
85C2×C4×C67—28no—
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—24no—
89C2×C4×C67—24no—
90C2×C4×C67—24no—
91C2×C4×C67—28no—
92C2×C4×C67—44no—
93C2×C4×C67—416no—
94C2×C4×C67—44no—
95C2×C4×C67—44no—
96C2×C4×C67—44no—
97C2×C4×C67—44no—
98C2×C4×C67—44no—
99C2×C4×C67#9914yes—
100C2×C4×C67#12014yes—
101C67⋊C8—84no—
102C22×C67—28no—
103C22×C67—22no—
104C22×C67—22no—
105C22×C67—28no—
106C22×C67—24no—
107C22×C67—416no—
108C22×C67—44no—
109C22×C67—42no—
110C22×C67—42no—
111C22×C67#37911yescharacteristic
112C22×C67#40814yes—
113C22×C67#40812yes—
114C22×C67C2×#9912yes—
115C22×C67C2×#10112yes—
116C4×C67—24no—
117C4×C67—24no—
118C4×C67—44no—
119C4×C67—44no—
120#8308—671nomaximal, Sylow
121C2×C67—24no—
122C2×C67—44no—
123C2×C67#147214yes—
124C2×C67#178011yescharacteristic
125C2×C67#178111yescharacteristic
126C2×C67#178411yescharacteristic
127C22×C22⋊Q8D6711yescharacteristic
128#1779—672no—
129C2×#324—674no—
130C67#830811yescharacteristic, complemented, Sylow
131C23×Q8Dic6711yescharacteristic
132C24×C4C2×D6711yescharacteristic
133C2×C22⋊Q8—24no—
134C2×C22⋊Q8—28no—
135C22×C4⋊C4Dic6711yescharacteristic
136C22×C4⋊C4C2×D6712yes—
137C22×C22⋊C4Dic6712yes—
138C2×C22⋊C8—674no—
139C2×C22⋊C8—1344no—
140#324—1348no—
141C25C2×Dic6711yescharacteristic
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—44no—
145C23×C4—24no—
146C23×C4—44no—
147C23×C4C2×Dic6712yes—
148C23×C4C2×Dic6711yescharacteristic
149C23×C4C2×Dic6712yes—
150C23×C4C22⋊D6714yes—
151C23×C4C22×D6711yescharacteristic
152C22×C8—1344no—
153C22⋊Q8—232no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4C22⋊D6718yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C22⋊C8—2688no—
163C22⋊C8—1348no—
164C24—24no—
165C24—22no—
166C24—24no—
167C24—42no—
168C24C22⋊Dic6712yes—
169C24C22×Dic6711yescharacteristic
170C2×Q8—28no—
171C2×Q8—416no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—28no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4C22⋊Dic6714yes—
188C22×C4C22⋊Dic6712yes—
189C22×C4C22⋊Dic6714yes—
190C22×C4C2×C22⋊D6712yes—
191C22×C4C2×C22⋊D6714yes—
192C2×C8—2688no—
193C2×C8—2684no—
194C4⋊C4—28no—
195C4⋊C4—216no—
196C22⋊C4—216no—
197C23—22no—
198C23—22no—
199C23—28no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—416no—
205C23—44no—
206C23—44no—
207C23#1452396214yes—
208C23C2×C22⋊Dic6712yes—
209C23C2×C22⋊Dic6711yescharacteristic
210C23#1452405014yes—
211Q8—416no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—28no—
219C2×C4—44no—
220C2×C4—416no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4#1452395814yes—
227C2×C4#1452401214yes—
228C8—5364no—
229C22—28no—
230C22—22no—
231C22—22no—
232C22—28no—
233C22—24no—
234C22—416no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22#1686325211yescharacteristic
239C22#1686336314yes—
240C22#1686336312yes—
241C22C2×#1452395812yes—
242C22C2×#1452396212yes—
243C4—24no—
244C4—24no—
245C4—44no—
246C4—44no—
247C2—24no—
248C2—44no—
249C2#2255857314yes—
250C2#2256012111yescharacteristic
251C2#2256012411yescharacteristic
252C2#2256013311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC22×C22⋊Q8×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC25×C67C2×C4
cosocleC23×C4×C67C23
hypercenterC22×C22⋊Q8D67
hypocenterC67#8308
LCSG ▷ C24×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16863122C2semidirect
130. C67#8308semidirect

Maximal subgroups

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

subgroupindex
#225601192
C2×#168631222
C22×C22⋊Q8×C672
#830867