Finite Group Databasebeta

← #13166283

Subgroups — #13166283

Order 1728.

760 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 102 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×C3⋊D3×Dic6C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic
3C2×C3×C4⋊#157C211yesmaximal, characteristic
4#11722722C214yesmaximal
5#10593837—31nomaximal
6C22×C3⋊#157C2212yes—
7C22×C3×#157C2212yes—
8C2×#6308C411yescharacteristic, complemented
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6C411yescharacteristic, complemented
11C2×#6293C412yescomplemented
12C2×Dic3×C3⋊D3C2212yes—
13C3⋊D3×Dic6—22no—
14C3⋊D3×Dic6—22no—
15C4×C3⋊#157—22no—
16C3×C4⋊#157—22no—
17C2×Q8×C3⋊D3—31no—
18C2×D3×Dic6—62no—
19C2×C4⋊#157D311yescharacteristic
20C2×C4×#157—31no—
21#63781—34no—
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×C3⋊#157—22no—
24C2×C3⋊#157—22no—
25C2×C3⋊#157Q814yescomplemented
26C2×C3×#157—22no—
27C2×C3×#157—24no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C3⋊(C3⋊Dic3)C2×C412yes—
30#6308—22no—
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C3×C4×C3⋊D3D412yes—
33C3×C4×C3⋊D3D412yes—
34C2×C32×Dic3C2×C412yes—
35C32×Dic6—22no—
36#6293—24no—
37Dic3×C3⋊D3—22no—
38Dic3×C3⋊D3—22no—
39#4973—91nomaximal
40C22×#157—32no—
41C22×#157C2×D312yes—
42C2×C32×Q8—31no—
43Q8×C3⋊D3—62no—
44Q8×C3⋊D3—62no—
45C2×C4×C3⋊D3—32no—
46C2×C4×C3⋊D3C2×D311yescharacteristic
47C2×#634—31no—
48C2×#634—32no—
49C2×#634—62no—
50C2×#634—122no—
51C2×C3×C4×D3—62no—
52C2×C3×Dic6—22no—
53C2×#627—64no—
54C2×D3×Dic3—64no—
55D3×Dic6—68no—
56D3×Dic6—68no—
57C4⋊#157—22no—
58C4×#157—62no—
59C22×C33C22×C411yescharacteristic
60C2×C3×C3⋊D3C2×D411yescharacteristic
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C2×C3×C3⋊D3C2×Q812yes—
63C2×C3×C3⋊D3#4112yes—
64C3⋊#157—24no—
65C3×#157—42no—
66C33×C4C22⋊C412yes—
67C3⋊(C3⋊Dic3)—22no—
68C3×C3⋊Dic3C22⋊C412yes—
69C32×Dic3—22no—
70C2×D3×Q8—182no—
71C22×Dic6—91no—
72C2×C3×C4⋊C4—91no—
73C2×C4×Dic3—91no—
74C3⋊#95—94no—
75C22×C3⋊D3C22×D311yescharacteristic
76C22×C3×D3—62no—
77C2×#157—24no—
78C2×#157—22no—
79C2×#157—34no—
80C2×#157—62no—
81C2×#157—62no—
82C32×Q8—62no—
83C2×C32×C4—22no—
84C2×C32×C4—32no—
85C2×C32×C4—42no—
86C2×C32×C4C4×D311yescharacteristic
87C2×C3⋊Dic3—32no—
88C2×C3⋊Dic3—64no—
89C2×C3⋊Dic3—124no—
90C2×C3⋊Dic3C4×D311yescharacteristic
91C4×C3⋊D3—62no—
92C4×C3⋊D3—62no—
93C4×C3⋊D3D1212yes—
94C4×C3⋊D3#6712yes—
95#634—62no—
96#634—64no—
97#634—68no—
98#634—128no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—62no—
101C3×C4×D3—64no—
102C3×C4×D3—64no—
103C3×Dic6—28no—
104#627—616no—
105D3×Dic3—68no—
106D3×Dic3—68no—
107#391—271noSylow
108C2×C33C2×C22⋊C411yescharacteristic
109C2×C33C4×Q812yes—
110C3×C3⋊D3C22⋊Q812yes—
111C3×C3⋊D3#12411yescharacteristic
112C3×C3⋊D3C4⋊Q811yescharacteristic
113C2×C3×Q8—62no—
114C22×C3×C4—91no—
115C22×C3×C4—92no—
116C22×Dic3—92no—
117C22×Dic3—92no—
118D3×Q8—188no—
119D3×Q8—188no—
120C2×C4×D3—62no—
121C2×C4×D3—184no—
122C2×Dic6—91no—
123C2×Dic6—92no—
124C2×Dic6—184no—
125C2×Dic6—182no—
126C2×Dic6—182no—
127C2×Dic6—182no—
128C2×Dic6—362no—
129C2×Dic6#15711yescharacteristic, complemented
130C3×C4⋊C4—182no—
131C4×Dic3—182no—
132C22×C32—22no—
133C22×C32—42no—
134C22×C32C2×C4×D311yescharacteristic
135C2×C3⋊D3C2×D1211yescharacteristic
136C2×C3⋊D3#23712yes—
137C2×C3⋊D3D3×Q812yes—
138C2×C3⋊D3C2×#6711yescharacteristic
139C2×C3×D3—62no—
140C2×C3×D3—62no—
141C2×C3×D3—64no—
142C2×C3×D3—64no—
143#157—42no—
144#157—64no—
145C32×C4—24no—
146C32×C4—44no—
147C32×C4—62no—
148C32×C4(C2×D3)⋊C412yes—
149C3⋊Dic3—62no—
150C3⋊Dic3—68no—
151C3⋊Dic3—128no—
152C3⋊Dic3(C2×D3)⋊C412yes—
153C3×Dic3—28no—
154C3×Dic3—64no—
155C22×Q8—271no—
156C2×C4⋊C4—91no—
157C2×C42—271no—
158#95—274no—
159C33#39111yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—62no—
162C3×Q8—68no—
163C2×C3×C4—22no—
164C2×C3×C4—42no—
165C2×C3×C4—64no—
166C2×C3×C4—91no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4—182no—
170C2×C3×C4—184no—
171C2×C3×C4C2×#15711yescharacteristic
172C2×Dic3—62no—
173C2×Dic3—94no—
174C2×Dic3—92no—
175C2×Dic3—184no—
176C2×Dic3—182no—
177C2×Dic3—182no—
178C2×Dic3—182no—
179C2×Dic3—182no—
180C2×Dic3—364no—
181C2×Dic3C2×#15712yes—
182C4×D3—64no—
183C4×D3—64no—
184C4×D3—188no—
185C4×D3—188no—
186Dic6—22no—
187Dic6—184no—
188Dic6—1816no—
189Dic6—188no—
190Dic6—182no—
191Dic6—368no—
192C2×C32—24no—
193C2×C32—22no—
194C2×C32—42no—
195C2×C32—44no—
196C2×C32#88712yes—
197C2×C32C2×(C2×D3)⋊C411yescharacteristic
198C3⋊D3#89712yes—
199C3⋊D3#93611yescharacteristic
200C3⋊D3#94411yescharacteristic
201C3×D3—64no—
202C3×D3—62no—
203C3×D3—62no—
204C2×Q8—31no—
205C2×Q8—272no—
206C2×Q8—271no—
207C2×Q8—542no—
208C2×Q8—542no—
209C22×C4—91no—
210C22×C4—92no—
211C22×C4—272no—
212C22×C4—272no—
213C4⋊C4—182no—
214C42—542no—
215C22×C3—22no—
216C22×C3—42no—
217C22×C3—92no—
218C22×C3—92no—
219C22×C3—91no—
220C22×C3—91no—
221C22×C3C22×#15711yescharacteristic
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—64no—
225C2×D3—64no—
226C3×C4—24no—
227C3×C4—44no—
228C3×C4—68no—
229C3×C4—92no—
230C3×C4—362no—
231C3×C4C22⋊#15712yes—
232Dic3—22no—
233Dic3—64no—
234Dic3—188no—
235Dic3—184no—
236Dic3—182no—
237Dic3—368no—
238C32—22no—
239C32—42no—
240C32#497311yescharacteristic, complemented
241C23—91no—
242Q8—62no—
243Q8—544no—
244Q8—542no—
245C2×C4—32no—
246C2×C4—91no—
247C2×C4—92no—
248C2×C4—92no—
249C2×C4—182no—
250C2×C4—184no—
251C2×C4—274no—
252C2×C4—272no—
253C2×C4—542no—
254C2×C4—542no—
255C2×C4—542no—
256C2×C4—542no—
257C2×C4D3×#15711yescharacteristic
258C2×C3—24no—
259C2×C3—22no—
260C2×C3—44no—
261C2×C3—42no—
262C2×C3—92no—
263C2×C3—91no—
264C2×C3—91no—
265C2×C3Q8×#15712yes—
266C2×C3C2×C22⋊#15711yescharacteristic
267D3—64no—
268D3—62no—
269D3—62no—
270C22—92no—
271C22—92no—
272C22—91no—
273C22—91no—
274C22C2×D3×#15711yescharacteristic
275C4—62no—
276C4—92no—
277C4—362no—
278C4—544no—
279C4—542no—
280C4(C2×D3)⋊#15712yes—
281C3—22no—
282C3—42no—
283C3#1059383711yescharacteristic, complemented
284C2—91no—
285C2—92no—
286C2—91no—
287C2#1172424512yes—
288C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#391
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
8. C2×#6308C4semidirect
10. C2×C32×Dic6C4semidirect
11. C2×#6293C4semidirect
25. C2×C3⋊#157Q8semidirect
129. C2×Dic6#157semidirect
159. C33#391semidirect
240. C32#4973semidirect
283. C3#10593837semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C4×C3⋊#1572
C2×C3×C4⋊#1572
#117227222
#105938373
#49739