Finite Group Databasebeta

← #11898399

Subgroups — #11898399

Order 1152.

572 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 99 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#10595413C212yesmaximal, complemented
2Q8×#622C211yesmaximal, characteristic, complemented
3#10593450C211yesmaximal, characteristic, complemented
4#10593445C212yesmaximal, complemented
5#10593440C214yesmaximal, complemented
6#10593437C212yesmaximal, complemented
7#10593424C212yesmaximal
8C4×#3774C211yesmaximal, characteristic
9C2×#3776C2212yes—
10C2×#3776C2212yes—
11C2×#3775C2214yes—
12C2×#3774C2211yescharacteristic
13Q8×C3⋊Dic3C2211yescharacteristic
14C3×#946—22no—
15#63919C2212yes—
16#63915C2214yes—
17C4×#625C2212yes—
18#63861—24no—
19#63841—22no—
20#63838C2214yes—
21C4⋊#622C2211yescharacteristic
22C4⋊#622C2212yes—
23C4×#622C2211yescharacteristic
24C4×#622C2212yes—
25#63743C2211yescharacteristic
26#63740C2214yes—
27#63739C2212yes—
28#5641—62no—
29C2×C32×Q8D411yescharacteristic
30C2×C3×C4×D3—22no—
31C2×#627C2314yes—
32C2×#625C2312yes—
33C2×#622C2312yes—
34C2×#622C2311yescharacteristic
35C2×#622C2311yescharacteristic
36#3776—24no—
37#3776—42no—
38#3775—44no—
39#3774—22no—
40C4⋊(C3⋊Dic3)D411yescharacteristic
41C4⋊(C3⋊Dic3)D412yes—
42C4×C3⋊Dic3C2312yes—
43C4×C3⋊Dic3C2311yescharacteristic
44C3×(C2×D3)⋊C4—24no—
45C3×C4⋊Dic3—22no—
46C3×Dic3⋊C4—24no—
47#3724—22no—
48#3724C2312yes—
49(C2×D3)⋊Dic3—24no—
50(C2×D3)⋊Dic3—22no—
51Dic32—24no—
52#3393—91nomaximal, Sylow
53C3×C22⋊Q8—62no—
54#946—22no—
55Q8×Dic3—62no—
56C4×#67—62no—
57#926—62no—
58#898—64no—
59#892—62no—
60#888—62no—
61#887—64no—
62#885—64no—
63#879—64no—
64C22×C3×D3—22no—
65C32×Q8—22no—
66C2×C32×C4C2×D412yes—
67C2×C32×C4C2×D411yescharacteristic
68C2×C3⋊Dic3C2×D412yes—
69C2×C3⋊Dic3C2×D411yescharacteristic
70C2×C3⋊Dic3C2411yescharacteristic
71C2×C3×Dic3—22no—
72C2×C3×Dic3—24no—
73C3×C4×D3—44no—
74#627—24no—
75#627—24no—
76#625—22no—
77#625—22no—
78#622—22no—
79#622—21no—
80#622#4112yes—
81#556—92no—
82#495—92no—
83#494—91no—
84#464—94no—
85#461—92no—
86C8×Q8—91no—
87C4×Q16—92no—
88C4×D8—91no—
89C2×C3×Q8—22no—
90C22×C3×C4—62no—
91C2×#67—62no—
92C2×C4×D3—22no—
93C2×Dic6—64no—
94C3×C4⋊C4—64no—
95C3×C4⋊C4—62no—
96C3×C22⋊C4—64no—
97#217—62no—
98#217—64no—
99(C2×D3)⋊C4—24no—
100(C2×D3)⋊C4—64no—
101(C2×D3)⋊C4—62no—
102C4⋊Dic3—22no—
103C4⋊Dic3—64no—
104C4⋊Dic3—62no—
105Dic3⋊C4—24no—
106Dic3⋊C4—62no—
107Dic3⋊C4—64no—
108C4×Dic3—64no—
109C4×Dic3—62no—
110C4×Dic3—64no—
111C4×Dic3—64no—
112C22×C32C22×D411yescharacteristic
113C2×C3×D3—42no—
114C2×C3×D3—42no—
115C2×C3×D3—42no—
116C32×C4—22no—
117C32×C4#13512yes—
118C3⋊Dic3—22no—
119C3⋊Dic3—21no—
120C3⋊Dic3C2×#4111yescharacteristic
121C3⋊Dic3H(5,𝔽2)11yescharacteristic
122C3×Dic3—44no—
123C3×Dic3—42no—
124C2×Q16—92no—
125C2×Q16—92no—
126C2×SD16—94no—
127C2×D8—91no—
128C4⋊Q8—92no—
129#126—184no—
130#124—94no—
131C22⋊Q8—62no—
132C4×Q8—91no—
133C4×Q8—94no—
134C4×D4—92no—
135#117—182no—
136#107—91no—
137C4⋊C8—91no—
138C4⋊C8—92no—
139#103—94no—
140#102—92no—
141C4×C8—91no—
142C4×C8—92no—
143C23×C3—62no—
144C22×D3—22no—
145C3×Q8—44no—
146C2×C3×C4—22no—
147C2×C3×C4—24no—
148C2×C3×C4—62no—
149C2×C3×C4—64no—
150C2×C3×C4—124no—
151#67—122no—
152#67—122no—
153C2×Dic3—22no—
154C2×Dic3—24no—
155C2×Dic3—62no—
156C2×Dic3—64no—
157C2×Dic3—62no—
158C4×D3—44no—
159Dic6—124no—
160Dic6—124no—
161C2×C32#55011yescharacteristic
162C2×C32C2×#13511yescharacteristic
163C2×C32#57811yescharacteristic
164C3×D3—82no—
165C2×Q8—94no—
166C2×Q8GO+(4,𝔽2)11yescharacteristic
167C2×D4—92no—
168C22×C4—62no—
169Q16—184no—
170Q16—362no—
171SD16—364no—
172D8—182no—
173C2×C8—91no—
174C2×C8—91no—
175C2×C8—92no—
176C4⋊C4—62no—
177C4⋊C4—64no—
178C4⋊C4—92no—
179C4⋊C4—91no—
180C4⋊C4—92no—
181C4⋊C4—182no—
182C22⋊C4—64no—
183C22⋊C4—184no—
184C22⋊C4—182no—
185C42—91no—
186C42—92no—
187C42—184no—
188C22×C3—22no—
189C22×C3—122no—
190C22×C3—122no—
191C22×C3—122no—
192C2×D3—42no—
193C2×D3—42no—
194C2×D3—42no—
195C3×C4—24no—
196C3×C4—44no—
197C3×C4—124no—
198C3×C4—122no—
199Dic3—44no—
200Dic3—42no—
201Dic3—62no—
202Dic3—62no—
203Dic3—124no—
204Dic3—122no—
205C32#339311yescharacteristic, complemented, Sylow
206C23—62no—
207Q8—22no—
208Q8—184no—
209Q8—184no—
210D4—182no—
211D4—182no—
212C2×C4—62no—
213C2×C4—64no—
214C2×C4—91no—
215C2×C4—92no—
216C2×C4—91no—
217C2×C4—124no—
218C2×C4C2×GO+(4,𝔽2)12yes—
219C2×C4C2×GO+(4,𝔽2)11yescharacteristic
220C8—92no—
221C8—182no—
222C8—181no—
223C2×C3—22no—
224C2×C3—22no—
225C2×C3—22no—
226C2×C3—242no—
227D3—82no—
228C22—122no—
229C22—122no—
230C22—122no—
231C22C22×GO+(4,𝔽2)11yescharacteristic
232C4—22no—
233C4—91no—
234C4—91no—
235C4—124no—
236C4—122no—
237C4—182no—
238C4—181no—
239C4#6423712yes—
240C3—22no—
241C2—242no—
242C2#1059652011yescharacteristic
243C2#1059652511yescharacteristic
244C2C2×#6423711yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C24
FittingC2×C32×Q8D4
socleC22×C32C22×D4
prime radicalC2×#3774C22
hypercenterC2×Q8GO+(4,𝔽2)
hypocenterC32#3393
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #10595413C2semidirect
2. Q8×#622C2semidirect
3. #10593450C2semidirect
4. #10593445C2semidirect
5. #10593440C2semidirect
6. #10593437C2semidirect
205. C32#3393semidirect

Maximal subgroups

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

subgroupindex
#105954132
Q8×#6222
#105934502
#105934452
#105934402
#105934372
#105934242
C4×#37742
#33939