Finite Group Databasebeta

← #70633

Subgroups — #70633

Order 384.

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

normal subgroups only

This group has 316 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
1C2×#892C211yesmaximal, characteristic
2C2×C3×#107C211yesmaximal, characteristic, complemented
3C2×(C2×D3)⋊C8C211yesmaximal, characteristic
4#4987C211yesmaximal, characteristic
5C2×#808C211yesmaximal, characteristic, complemented
6#4551C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#1854—31nomaximal, Sylow
9C22×C4×D3C2211yescharacteristic
10C22×C3×C8C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×(C2×D3)⋊C4—21no—
14C2×(C2×D3)⋊C4—21no—
15C2×C4⋊Dic3C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4×Dic3—21no—
18C22×C3⋊C8C2211yescharacteristic
19#892—22no—
20#892—22no—
21C3×#107—22no—
22C3⋊#95—21no—
23(C2×D3)⋊C8C414yescomplemented
24#808—22no—
25C2×#117—31no—
26C2×#107D311yescharacteristic, complemented
27C2×#106—31no—
28C2×C22⋊C8—31no—
29#380—31no—
30#340—31no—
31#340—31no—
32C23×D3D411yescharacteristic
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4C2311yescharacteristic
36C22×Dic3—21no—
37C22×Dic3—21no—
38C22×Dic3D411yescharacteristic
39C2×C4×D3—21no—
40C2×C4×D3—21no—
41C2×C4×D3—21no—
42C2×C4×D3—21no—
43C2×C4×D3C2×C412yes—
44C2×C4×D3Q812yes—
45C2×C3×C8—21no—
46C2×C3×C8—21no—
47C2×C3×C8C2×C412yes—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4D412yes—
52(C2×D3)⋊C4—24no—
53(C2×D3)⋊C4—42no—
54C4⋊Dic3—21no—
55C4⋊Dic3—21no—
56C4⋊Dic3—21no—
57C4⋊Dic3D412yes—
58C4×Dic3—24no—
59C2×C3⋊C8—21no—
60C2×C3⋊C8—21no—
61C2×C3⋊C8C2×C412yes—
62C23×C4—31no—
63C22×C8—31no—
64C22×C8C2×D311yescharacteristic
65#117—62no—
66#117—62no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C22⋊C4—61no—
72C2×C22⋊C4—61no—
73C2×C42—61no—
74#107—22no—
75#106—62no—
76C22⋊C8—34no—
77#95—61no—
78C23×C3C2×D411yescharacteristic
79C22×D3—21no—
80C22×D3—21no—
81C22×D3—21no—
82C22×D3—21no—
83C22×D3—21no—
84C22×D3—21no—
85C22×D3C4⋊C412yes—
86C2×C3×C4—22no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95C2×C3×C4C2×Q811yescharacteristic
96C2×C3×C4#4111yescharacteristic
97C2×C3×C4#4111yescharacteristic
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—22no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—41no—
106C2×Dic3C4⋊C412yes—
107C4×D3—22no—
108C4×D3—22no—
109C4×D3—22no—
110C4×D3—22no—
111C3×C8—22no—
112C3⋊C8—22no—
113C24—31no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—31no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—61no—
120C22×C4—61no—
121C22×C4—61no—
122C22×C4—61no—
123C22×C4—61no—
124C22×C4—61no—
125C22×C4C22×D311yescharacteristic
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—32no—
129C2×C8—61no—
130C2×C8—61no—
131C2×C8C4×D312yes—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—32no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4#6712yes—
140C22⋊C4—64no—
141C22⋊C4—122no—
142C42—64no—
143C22×C3C2×C4⋊C411yescharacteristic
144C22×C3#12011yescharacteristic
145C22×C3#12311yescharacteristic
146C22×C3#13511yescharacteristic
147C22×C3#13511yescharacteristic
148C22×C3AGL(1,ℤ/8)11yescharacteristic
149C22×C3#13711yescharacteristic
150C2×D3—22no—
151C2×D3—22no—
152C2×D3—22no—
153C2×D3—41no—
154C2×D3—41no—
155C2×D3—41no—
156C2×D3—41no—
157C3×C4—42no—
158C3×C4—81no—
159C3×C4C4×D411yescharacteristic
160C3×C4C4×D411yescharacteristic
161C3×C4C22⋊Q811yescharacteristic
162C3×C4C22⋊Q811yescharacteristic
163Dic3—22no—
164Dic3—42no—
165Dic3—81no—
166C23—32no—
167C23—61no—
168C23—61no—
169C23—61no—
170C23—61no—
171C23—61no—
172C23—61no—
173C23D3×D411yescharacteristic
174C2×C4—22no—
175C2×C4—32no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—121no—
189C2×C4—121no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4C2×C4×D311yescharacteristic
194C2×C4#23511yescharacteristic
195C2×C4D3×D411yescharacteristic
196C2×C4D3×Q811yescharacteristic
197C2×C4#23911yescharacteristic
198C2×C4C2×#6711yescharacteristic
199C8—22no—
200C8—62no—
201C2×C3#38511yescharacteristic
202C2×C3#42711yescharacteristic
203C2×C3#42811yescharacteristic
204C2×C3#44911yescharacteristic
205C2×C3#45211yescharacteristic
206C2×C3#48211yescharacteristic
207C2×C3#48511yescharacteristic
208D3—42no—
209C22—62no—
210C22—62no—
211C22—62no—
212C22—121no—
213C22—121no—
214C22—121no—
215C22—121no—
216C22D3×C4⋊C411yescharacteristic
217C22#89411yescharacteristic
218C22#91111yescharacteristic
219C22#91211yescharacteristic
220C22#91811yescharacteristic
221C22#91911yescharacteristic
222C22#93711yescharacteristic
223C4—42no—
224C4—62no—
225C4—81no—
226C4—122no—
227C4—241no—
228C4#89311yescharacteristic
229C4#89611yescharacteristic
230C4C4×#6711yescharacteristic
231C4#94611yescharacteristic
232C3#185411yescharacteristic, complemented, Sylow
233C2—122no—
234C2#488011yescharacteristic
235C2#488111yescharacteristic
236C2#488511yescharacteristic
237C2#488811yescharacteristic
238C2#498911yescharacteristic
239C2#515811yescharacteristic
240C2#518711yescharacteristic
subgroupG/H
centerC23D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#107C2
socleC23×C3C2×D4
prime radicalC23×D3D4
cosocleC22×C3×C4C23
hypercenterC2×#107D3
hypocenterC3#1854
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×C3×#107C2semidirect
5. C2×#808C2semidirect
6. #4551C2semidirect
7. #4529C2semidirect
23. (C2×D3)⋊C8C4semidirect
26. C2×#107D3semidirect
232. C3#1854semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×C3×#1072
C2×(C2×D3)⋊C82
#49872
C2×#8082
#45512
#45292
#18543