Finite Group Databasebeta

← #69597

Subgroups — #69597

Order 384.

374 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 89 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#5798C211yesmaximal, characteristic
2C3×#386C211yesmaximal, characteristic
3C2×#834C211yesmaximal, characteristic
4#5210C212yesmaximal, complemented
5#4527C212yesmaximal, complemented
6#1867—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C23×C3×C4C2211yescharacteristic
9C2×C3×C22⋊C4—21no—
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×#217—21no—
12C2×#217C2212yes—
13C2×#217C2212yes—
14D4×Dic3—22no—
15D4×Dic3—22no—
16#926—21no—
17C2×Dic3⋊C4—22no—
18C3×#95—21no—
19#834C414yescomplemented
20C3⋊#95—22no—
21#518—31no—
22C2×#99—31no—
23#386D311yescharacteristic
24#385—32no—
25#342—32no—
26C24×C3D411yescharacteristic
27C24×C3C2311yescharacteristic
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4C2×C412yes—
31C2×C3×D4Q812yes—
32C22×C3×C4—22no—
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4—42no—
36C22×C3×C4D411yescharacteristic
37C22×Dic3—22no—
38C22×Dic3—21no—
39C22×Dic3—21no—
40C22×Dic3D412yes—
41C22×Dic3D412yes—
42C3×C22⋊C4—22no—
43C3×C22⋊C4—42no—
44#217—22no—
45#217—22no—
46#217—22no—
47#217—22no—
48#217C2×C414yescomplemented
49C4⋊Dic3—21no—
50C4⋊Dic3—21no—
51Dic3⋊C4—44no—
52C4×Dic3—22no—
53C22×D4C2×D311yescharacteristic
54C23×C4C2×D311yescharacteristic
55C4×D4—62no—
56C4×D4—62no—
57#117—61no—
58C2×C4⋊C4—62no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4—61no—
63C2×C22⋊C4C2×D311yescharacteristic
64#99—34no—
65#95—21no—
66#95—62no—
67C23×C3—21no—
68C23×C3—21no—
69C23×C3—41no—
70C23×C3—41no—
71C23×C3—41no—
72C23×C3—41no—
73C23×C3C4⋊C412yescomplemented
74C23×C3C22×C411yescharacteristic
75C23×C3C2×D412yes—
76C23×C3C2×D411yescharacteristic
77C23×C3C2×Q811yescharacteristic
78C23×C3#4112yes—
79C3×D4—24no—
80C3×D4—24no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—24no—
84C2×C3×C4—22no—
85C2×C3×C4—42no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4C4⋊C412yescomplemented
92C2×Dic3—24no—
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—42no—
96C2×Dic3—42no—
97C2×Dic3—42no—
98C2×Dic3—42no—
99C2×Dic3—81no—
100C2×Dic3—81no—
101C24#6711yescharacteristic
102C24C22×D311yescharacteristic
103C2×D4—22no—
104C2×D4—22no—
105C2×D4Dic612yes—
106C2×D4C4×D312yes—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—32no—
111C22×C4—32no—
112C22×C4—42no—
113C22×C4—62no—
114C22×C4—61no—
115C22×C4—61no—
116C22×C4#6711yescharacteristic
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—124no—
120C22⋊C4—22no—
121C22⋊C4—34no—
122C22⋊C4—42no—
123C22⋊C4—62no—
124C22⋊C4—62no—
125C22⋊C4—62no—
126C22⋊C4—62no—
127C42—62no—
128C22×C3—22no—
129C22×C3—21no—
130C22×C3—22no—
131C22×C3—21no—
132C22×C3—22no—
133C22×C3—41no—
134C22×C3—41no—
135C22×C3—42no—
136C22×C3—42no—
137C22×C3—41no—
138C22×C3—41no—
139C22×C3—81no—
140C22×C3C2×C4⋊C411yescharacteristic
141C22×C3C4×D412yes—
142C22×C3#12011yescharacteristic
143C22×C3C22⋊Q812yes—
144C22×C3#12311yescharacteristic
145C3×C4—22no—
146C3×C4—42no—
147C3×C4—81no—
148Dic3—44no—
149Dic3—42no—
150Dic3—42no—
151C23—21no—
152C23—21no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23Dic3⋊C412yescomplemented
158C23C2×Dic611yescharacteristic
159C23C2×C4×D311yescharacteristic
160C23D3×D412yes—
161C23#23712yes—
162C23C2×#6711yescharacteristic
163D4—24no—
164D4—24no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—64no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—122no—
179C2×C4—122no—
180C2×C4—122no—
181C2×C4—122no—
182C2×C4—241no—
183C2×C4—241no—
184C2×C4Dic3⋊C412yescomplemented
185C2×C3—22no—
186C2×C3—21no—
187C2×C3—21no—
188C2×C3—22no—
189C2×C3—42no—
190C2×C3#38511yescharacteristic
191C2×C3B(4,𝔽2)11yescharacteristic
192C2×C3#45811yescharacteristic
193C22—21no—
194C22—22no—
195C22—21no—
196C22—22no—
197C22—22no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—81no—
205C22#87812yes—
206C22#67⋊C412yes—
207C22C2×Dic3⋊C411yescharacteristic
208C22#93511yescharacteristic
209C22#93711yescharacteristic
210C4—22no—
211C4—42no—
212C4—81no—
213C4—124no—
214C4—122no—
215C4—122no—
216C3#186711yescharacteristic, complemented, Sylow
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#494611yescharacteristic
223C2#503211yescharacteristic
224C2#505311yescharacteristic
subgroupG/H
centerC22#937
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#386C2
socleC22×C3#120
prime radicalC22×C3×D4C22
cosocleC24×C3C23
hypercenter#386D3
hypocenterC3#1867
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #5210C2semidirect
5. #4527C2semidirect
19. #834C4semidirect
48. #217C2×C4semidirect
73. C23×C3C4⋊C4semidirect
91. C2×C3×C4C4⋊C4semidirect
157. C23Dic3⋊C4semidirect
184. C2×C4Dic3⋊C4semidirect
216. C3#1867semidirect

Maximal subgroups

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

subgroupindex
#57982
C3×#3862
C2×#8342
#52102
#45272
#18673