Finite Group Databasebeta

← #68313

Subgroups — #68313

Order 384.

208 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 61 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#5622C211yesmaximal, characteristic
2#4719C211yesmaximal, characteristic
3#4651C211yesmaximal, characteristic
4C3×#328C211yesmaximal, characteristic
5#4538C211yesmaximal, characteristic
6#4467C211yesmaximal, characteristic
7#4452C211yesmaximal, characteristic
8#1515—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10Q8×Dic3C411yescharacteristic, complemented
11#944C411yescharacteristic, complemented
12#924C2211yescharacteristic
13C2×C4×Dic3C2211yescharacteristic
14#889—21no—
15#888C411yescharacteristic, complemented
16#887—21no—
17#887C411yescharacteristic, complemented
18#878—21no—
19#877—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×#95C2211yescharacteristic
22C3⋊#95—21no—
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26#815—21no—
27Dic3⋊C8—21no—
28#533—31no—
29#432—31no—
30#383—31no—
31#328—31no—
32#328—31no—
33#328—31no—
34#328D311yescharacteristic
35C2×C3×Q8C2×C411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—21no—
39C22×Dic3D411yescharacteristic
40C22×Dic3D411yescharacteristic
41C2×Dic6—21no—
42C2×Dic6C2×C411yescharacteristic
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4C2×C411yescharacteristic
46C3×C22⋊C4—21no—
47#217—21no—
48C4⋊Dic3—21no—
49C4⋊Dic3C2×C411yescharacteristic
50Dic3⋊C4—21no—
51Dic3⋊C4—21no—
52Dic3⋊C4—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C4×Dic3—21no—
57C4×Dic3C2×C411yescharacteristic
58C4×Dic3C2×C411yescharacteristic
59C4×Dic3D411yescharacteristic
60C4×Dic3D411yescharacteristic
61C2×C3⋊C8—21no—
62C4⋊Q8—31no—
63C4⋊Q8—31no—
64#125—61no—
65C22⋊Q8—31no—
66C22⋊Q8—61no—
67C22⋊Q8C2×D311yescharacteristic
68C4×Q8—31no—
69C4×Q8—31no—
70C4×Q8—61no—
71#117—61no—
72C2×C42—31no—
73C4⋊C8—61no—
74C22⋊C8—31no—
75C22⋊C8C2×D311yescharacteristic
76#97—61no—
77#95—31no—
78#95—61no—
79#95—61no—
80#95C2×D311yescharacteristic
81C23×C3C2×D411yescharacteristic
82C3×Q8—41no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4C22×C411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91C2×Dic3—21no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—41no—
96C2×Dic3—41no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3C22⋊C411yescharacteristic
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C411yescharacteristic
102C2×Dic3C22⋊C411yescharacteristic
103Dic6—42no—
104Dic6—41no—
105C3×C8—41no—
106C3⋊C8—41no—
107C2×Q8—31no—
108C2×Q8—61no—
109C2×Q8C4×D311yescharacteristic
110C22×C4—21no—
111C22×C4—31no—
112C22×C4—31no—
113C22×C4—61no—
114C22×C4C22×D311yescharacteristic
115C2×C8—21no—
116C2×C8—61no—
117C4⋊C4—21no—
118C4⋊C4—31no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4C4×D311yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—61no—
128C42—31no—
129C42—31no—
130C42—31no—
131C42—31no—
132C42—61no—
133C42—61no—
134C22×C3—21no—
135C22×C3—21no—
136C22×C3C2×C22⋊C411yescharacteristic
137C22×C3#13711yescharacteristic
138C22×C3#13711yescharacteristic
139C3×C4—21no—
140C3×C4—21no—
141C3×C4—41no—
142C3×C4—41no—
143C3×C4—81no—
144Dic3—21no—
145Dic3—21no—
146Dic3—21no—
147Dic3—21no—
148Dic3—41no—
149Dic3—41no—
150Dic3—81no—
151C23D3×D411yescharacteristic
152Q8—41no—
153Q8—121no—
154Q8—122no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—121no—
170C2×C4—121no—
171C2×C4—121no—
172C2×C4—121no—
173C2×C4C2×C4×D311yescharacteristic
174C2×C4D3×D411yescharacteristic
175C8—41no—
176C8—121no—
177C2×C3—21no—
178C2×C3—21no—
179C2×C3#41011yescharacteristic
180C2×C3#41911yescharacteristic
181C2×C3#42111yescharacteristic
182C22—21no—
183C22—21no—
184C22D3×C22⋊C411yescharacteristic
185C22#91511yescharacteristic
186C22#91811yescharacteristic
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—61no—
192C4—61no—
193C4—61no—
194C4—61no—
195C4—81no—
196C4—121no—
197C4—121no—
198C4—241no—
199C3#151511yescharacteristic, complemented, Sylow
200C2—21no—
201C2—21no—
202C2#474011yescharacteristic
203C2#480211yescharacteristic
204C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3C2×C22⋊C4
prime radicalC23×C3C2×D4
cosocleC22×C3×C4C23
hypercenter#328D3
hypocenterC3#1515
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
10. Q8×Dic3C4semidirect
11. #944C4semidirect
15. #888C4semidirect
17. #887C4semidirect
199. C3#1515semidirect

Maximal subgroups

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

subgroupindex
#56222
#47192
#46512
C3×#3282
#45382
#44672
#44522
#15153