Finite Group Databasebeta

← #34060677

Subgroups — #34060677

Order 19072.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×#18011990C211yesmaximal, characteristic
2C2×#18011974C212yesmaximal, complemented
3D149⋊(C22⋊C8)C211yesmaximal, characteristic
4#23750498C212yesmaximal, complemented
5C22⋊Q8×D149C211yesmaximal, characteristic
6C2×D149⋊C8C2212yes—
7#18011990—24no—
8C2×C4⋊#10597010C2212yes—
9C2×C22⋊#10597010—21no—
10C2×C4×#10597010—21no—
11#18011974—24no—
12#18011968—21no—
13C22⋊#11924753—21no—
14C2×Q8×D149C2211yescharacteristic
15C22×C4×D149C2211yescharacteristic
16#18011891C411yescharacteristic, complemented
17#18011890C411yescharacteristic, complemented
18#18011889—21no—
19#18011888—21no—
20#18011887C411yescharacteristic, complemented
21C4⋊C4×D149—21no—
22C4⋊C4×D149C2211yescharacteristic
23C22⋊C4×D149—21no—
24C22⋊Q8×C149C411yescharacteristic, complemented
25C22×#10597010—22no—
26D149⋊C8—22no—
27D149⋊C8—22no—
28C2×#11924753—22no—
29C4⋊#10597010—22no—
30C4⋊#10597010D414yes—
31C22⋊#10597010—24no—
32C4×#10597010—24no—
33C23×D149D411yescharacteristic
34C2×Dic298—21no—
35C2×Dic298C2×C411yescharacteristic
36Q8×D149—22no—
37Q8×D149—22no—
38C22×Dic149D411yescharacteristic
39C2×C4×D149—22no—
40C2×C4×D149—22no—
41C2×C4×D149—21no—
42C2×C4×D149—21no—
43C2×C4×D149C2×C411yescharacteristic
44C2×C4×D149D411yescharacteristic
45C2×C4×D149D411yescharacteristic
46C2×C4×D149C2311yescharacteristic
47C4⋊Dic149—21no—
48C4⋊Dic149C2×C411yescharacteristic
49#14558715—21no—
50#14558715—21no—
51#14558715—21no—
52C22⋊Dic149—21no—
53#14558713—21no—
54#14558713—21no—
55C2×Q8×C149C2×C411yescharacteristic
56C22×C4×C149C2×C411yescharacteristic
57C4⋊C4×C149—21no—
58C4⋊C4×C149C2×C411yescharacteristic
59C22⋊C4×C149—21no—
60C2×#10597010—24no—
61C2×#10597010—42no—
62C2×#10597010—42no—
63#11924753—42no—
64C22×D149—21no—
65C22×D149—21no—
66C22×D149—22no—
67C22×D149—22no—
68C22×D149C22⋊C411yescharacteristic
69C22×D149C2×D411yescharacteristic
70Dic298—22no—
71Dic298—42no—
72C2×Dic149—22no—
73C2×Dic149—21no—
74C2×Dic149—21no—
75C2×Dic149C22⋊C411yescharacteristic
76C2×Dic149C2×D411yescharacteristic
77C4×D149—21no—
78C4×D149—21no—
79C4×D149—22no—
80C4×D149—22no—
81C4×D149—41no—
82C4×D149—41no—
83C4×D149—42no—
84C4×D149C2×D412yes—
85C4×D149#4112yes—
86C23×C149C22⋊C411yescharacteristic
87Q8×C149—22no—
88C2×C4×C149—22no—
89C2×C4×C149—21no—
90C2×C4×C149—21no—
91C2×C4×C149C22⋊C411yescharacteristic
92C2×C4×C149C22×C411yescharacteristic
93C22×C149—21no—
94C22×C149—21no—
95C22×C149—21no—
96C22×C149C2×C22⋊C411yescharacteristic
97C2×D149—21no—
98C2×D149—21no—
99C2×D149—21no—
100C2×D149—42no—
101C2×D149—42no—
102C2×D149#12011yescharacteristic
103C2×D149#12311yescharacteristic
104C2×D149#13512yes—
105C2×D149#13712yes—
106#10597010—44no—
107C4×C149—21no—
108C4×C149—41no—
109C4×C149—41no—
110C4×C149C4×D412yes—
111Dic149—21no—
112Dic149—41no—
113Dic149—41no—
114Dic149#12112yes—
115C2×C149—41no—
116C2×C149#38611yescharacteristic
117C2×C149#41611yescharacteristic
118C2×C149#41911yescharacteristic
119D149—41no—
120D149#45212yes—
121D149#48512yes—
122C149#187311yescharacteristic, complemented, Sylow
123#1873—1491nomaximal, Sylow
124C2×C22⋊Q8—1491no—
125C2×#117—1491no—
126C2×#103—1492no—
127C2×C22⋊C8—1491no—
128#340—1492no—
129C22×Q8—1491no—
130C23×C4—1491no—
131C22×C8—1492no—
132C22⋊Q8—2981no—
133C22⋊Q8—2981no—
134C22⋊Q8—1491no—
135C22⋊Q8—1491no—
136C22⋊Q8—1491no—
137C22⋊Q8#1059701011yescharacteristic, complemented
138#117—2984no—
139C2×C4⋊C4—2981no—
140C2×C4⋊C4—1492no—
141C2×C4⋊C4—1491no—
142C2×C22⋊C4—2981no—
143C2×C22⋊C4—2981no—
144C2×C42—2981no—
145#103—2984no—
146C22⋊C8—2981no—
147C22⋊C8—2981no—
148C24—1491no—
149C2×Q8—2981no—
150C2×Q8—2982no—
151C2×Q8—2982no—
152C2×Q8—1491no—
153C2×Q8C2×#1059701011yescharacteristic
154C22×C4—2982no—
155C22×C4—2982no—
156C22×C4—2981no—
157C22×C4—2981no—
158C22×C4—2982no—
159C22×C4—1491no—
160C22×C4—1491no—
161C22×C4—1491no—
162C22×C4—1491no—
163C22×C4—1491no—
164C22×C4C2×#1059701011yescharacteristic
165C2×C8—2982no—
166C2×C8—2982no—
167C2×C8—2982no—
168C4⋊C4—21no—
169C4⋊C4—2981no—
170C4⋊C4—2981no—
171C4⋊C4—2981no—
172C4⋊C4—2982no—
173C4⋊C4—2981no—
174C4⋊C4—1494no—
175C4⋊C4—1491no—
176C4⋊C4C2×#1059701011yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—2981no—
179C22⋊C4—2981no—
180C22⋊C4—2984no—
181C22⋊C4—2981no—
182C42—2984no—
183C23—2981no—
184C23—2981no—
185C23—2982no—
186C23—2982no—
187C23—1491no—
188C23—1491no—
189C23C22⋊#1059701011yescharacteristic
190Q8—22no—
191Q8—2982no—
192Q8—5962no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—2981no—
197C2×C4—2981no—
198C2×C4—2982no—
199C2×C4—2982no—
200C2×C4—2982no—
201C2×C4—2984no—
202C2×C4—2981no—
203C2×C4—2981no—
204C2×C4—5962no—
205C2×C4—5961no—
206C2×C4—5961no—
207C2×C4—5962no—
208C2×C4—5962no—
209C2×C4—1491no—
210C2×C4—1491no—
211C2×C4—1492no—
212C2×C4—1492no—
213C2×C4C22⋊#1059701011yescharacteristic
214C2×C4C22×#1059701011yescharacteristic
215C8—5962no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—2981no—
220C22—2981no—
221C22—2981no—
222C22—5962no—
223C22—5962no—
224C22—1491no—
225C22—1491no—
226C22—1492no—
227C22—1492no—
228C22C2×C22⋊#1059701011yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—2981no—
233C4—5964no—
234C4—5961no—
235C4—5961no—
236C4—1492no—
237C4D4×#1059701012yes—
238C2—41no—
239C2—5961no—
240C2—1492no—
241C2—1492no—
242C2#2375057311yescharacteristic
243C2#2375061511yescharacteristic
244C2#2375061811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
FittingC22⋊Q8×C149C4
socleC22×C149C2×C22⋊C4
prime radicalC23×D149D4
cosocleC2×C4×D149C23
hypercenterC22⋊Q8#10597010
hypocenterC149#1873
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#18011974C2semidirect
4. #23750498C2semidirect
16. #18011891C4semidirect
17. #18011890C4semidirect
20. #18011887C4semidirect
24. C22⋊Q8×C149C4semidirect
122. C149#1873semidirect
137. C22⋊Q8#10597010semidirect

Maximal subgroups

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

subgroupindex
C2×#180119902
C2×#180119742
D149⋊(C22⋊C8)2
#237504982
C22⋊Q8×D1492
#1873149