Finite Group Databasebeta

← #69513

Subgroups — #69513

Order 384.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C22×#208C211yesmaximal, characteristic
2C3×#380C211yesmaximal, characteristic
3#5212C211yesmaximal, characteristic
4C2×#808C211yesmaximal, characteristic, complemented
5C2×#807C211yesmaximal, characteristic, complemented
6#4529C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#1809—31nomaximal, Sylow
9C23×C3×C4C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4—21no—
13C2×#217—21no—
14C2×C4⋊Dic3C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×#208—22no—
17C2×#208—22no—
18C2×#208C414yescomplemented
19C22×C3⋊C8C2211yescharacteristic
20C22×C3⋊C8C2211yescharacteristic
21C3×#95—21no—
22C3⋊#95—21no—
23#808—22no—
24#807—22no—
25C22×M16—31no—
26C2×#107—31no—
27C2×#106—31no—
28#380—31no—
29#380D311yescharacteristic
30#340—31no—
31#340—31no—
32C24×C3D411yescharacteristic
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4C2×C412yes—
40C22×C3×C4D411yescharacteristic
41C22×C3×C4Q812yes—
42C22×C3×C4C2311yescharacteristic
43C22×Dic3—21no—
44C22×Dic3—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C22⋊C4—42no—
50#217—42no—
51C4⋊Dic3—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55#208—28no—
56C2×C3⋊C8—21no—
57C2×C3⋊C8—21no—
58C2×C3⋊C8C2×C412yes—
59C2×C3⋊C8C2×C412yes—
60C2×C3⋊C8D412yes—
61C2×C3⋊C8D412yes—
62C23×C4C2×D311yescharacteristic
63C2×M16—34no—
64C2×M16—62no—
65C2×M16—62no—
66C22×C8—31no—
67C22×C8—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—61no—
74#107—62no—
75#106—62no—
76#95—21no—
77#95—61no—
78C23×C3—21no—
79C23×C3—21no—
80C23×C3—21no—
81C23×C3—21no—
82C23×C3—21no—
83C23×C3—21no—
84C23×C3C4⋊C412yes—
85C23×C3C2×D411yescharacteristic
86C2×C3×C4—21no—
87C2×C3×C4—21no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4C4⋊C412yes—
99C2×C3×C4C22×C411yescharacteristic
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×Q811yescharacteristic
103C2×C3×C4#4111yescharacteristic
104C2×C3×C4#4111yescharacteristic
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C3⋊C8—22no—
112C3⋊C8C22⋊C414yescomplemented
113C24#6711yescharacteristic
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—61no—
121C22×C4—61no—
122C22×C4Dic612yes—
123C22×C4C4×D312yes—
124C22×C4#6711yescharacteristic
125C22×C4C22×D311yescharacteristic
126M16—68no—
127C2×C8—32no—
128C2×C8—32no—
129C2×C8—32no—
130C2×C8—32no—
131C2×C8—61no—
132C2×C8—61no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—61no—
141C22⋊C4—42no—
142C22⋊C4—122no—
143C22×C3—22no—
144C22×C3—22no—
145C22×C3—22no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3C2×C4⋊C411yescharacteristic
151C22×C3#12111yescharacteristic
152C22×C3#12111yescharacteristic
153C22×C3AGL(1,ℤ/8)11yescharacteristic
154C22×C3AGL(1,ℤ/8)11yescharacteristic
155C22×C3#13711yescharacteristic
156C22×C3#13711yescharacteristic
157C3×C4—22no—
158C3×C4—81no—
159C3×C4—81no—
160C3×C4C2×C22⋊C411yescharacteristic
161C3×C4#11711yescharacteristic
162C3×C4C22⋊Q811yescharacteristic
163C3×C4C22⋊Q811yescharacteristic
164Dic3—81no—
165Dic3—81no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23Dic3⋊C412yes—
173C23C2×#6711yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—121no—
187C2×C4—121no—
188C2×C4—121no—
189C2×C4—121no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4Dic3⋊C412yes—
193C2×C4C2×Dic611yescharacteristic
194C2×C4C2×C4×D311yescharacteristic
195C2×C4D3×D411yescharacteristic
196C2×C4D3×D411yescharacteristic
197C2×C4#23711yescharacteristic
198C2×C4#23711yescharacteristic
199C8—34no—
200C8—62no—
201C2×C3—42no—
202C2×C3#38011yescharacteristic
203C2×C3#42811yescharacteristic
204C2×C3#42811yescharacteristic
205C2×C3#46811yescharacteristic
206C2×C3#46811yescharacteristic
207C2×C3#46911yescharacteristic
208C2×C3#47011yescharacteristic
209C22—22no—
210C22—22no—
211C22—22no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22C2×Dic3⋊C411yescharacteristic
217C22#93211yescharacteristic
218C22#93911yescharacteristic
219C22#93911yescharacteristic
220C22#94211yescharacteristic
221C22#94911yescharacteristic
222C22#95111yescharacteristic
223C4—22no—
224C4—81no—
225C4—81no—
226C4—241no—
227C4—241no—
228C4#87711yescharacteristic
229C4#87811yescharacteristic
230C4#87811yescharacteristic
231C4D3×C22⋊C411yescharacteristic
232C3#180911yescharacteristic, complemented, Sylow
233C2—42no—
234C2#494211yescharacteristic
235C2#496411yescharacteristic
236C2#499511yescharacteristic
237C2#503911yescharacteristic
238C2#504211yescharacteristic
239C2#504911yescharacteristic
240C2#505211yescharacteristic
subgroupG/H
centerC23C2×#67
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#380C2
socleC23×C3C2×D4
prime radicalC24×C3D4
cosocleC22×C3×C4C23
hypercenter#380D3
hypocenterC3#1809
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
4. C2×#808C2semidirect
5. C2×#807C2semidirect
6. #4529C2semidirect
7. #4529C2semidirect
18. C2×#208C4semidirect
112. C3⋊C8C22⋊C4semidirect
232. C3#1809semidirect

Maximal subgroups

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

subgroupindex
C22×#2082
C3×#3802
#52122
C2×#8082
C2×#8072
#45292
#18093