Finite Group Databasebeta

← #13167066

Subgroups — #13167066

Order 1728.

496 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 95 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×#88988C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic
3C3×#63900C211yesmaximal, characteristic
4#11722778C214yesmaximal, complemented
5#10593387—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×#6294—21no—
8C2×C3×#624—21no—
9C2×C3⋊#622C2212yes—
10#88988C414yescomplemented
11C3×C4⋊(C3⋊Dic3)—21no—
12C32×C4⋊Dic3—21no—
13#88815C2212yes—
14C3×#3724C2212yes—
15C2×#3787—31no—
16C3×D3×C4⋊C4—61no—
17#63900—31no—
18#63900D311yescharacteristic
19D3×C4⋊Dic3—61no—
20#63742—34no—
21C22×C3×C3⋊D3D411yescharacteristic
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×C4×C3⋊D3C2×C412yes—
26C3×C4×C3⋊D3Q812yes—
27C2×C32×Dic3—22no—
28#6294—42no—
29C3×#624—42no—
30C3⋊#622C2×C414yescomplemented
31C2×#808—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—61no—
34C2×C3×C4×D3—61no—
35C2×C3×C4×D3—62no—
36C2×#624—21no—
37C2×#624—61no—
38C2×D3×Dic3—62no—
39C2×#622—32no—
40#3787—34no—
41C32×C4⋊C4—61no—
42C4⋊(C3⋊Dic3)—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—41no—
46C3×C4⋊Dic3—61no—
47C3×C4⋊Dic3—61no—
48C3×C4⋊Dic3—121no—
49C3×Dic3⋊C4—62no—
50#3724—32no—
51#3724C2×D312yes—
52Dic3⋊Dic3—62no—
53C22×C33C2×D411yescharacteristic
54C2×C3×C3⋊D3C4⋊C412yes—
55C2×C3×C3⋊D3SD1612yes—
56C2×C3×C3⋊D3SD1612yes—
57C33×C4C4⋊C412yes—
58C3×C3⋊Dic3—42no—
59C3×C3⋊Dic3C22×C411yescharacteristic
60C3×C3⋊Dic3C2×Q811yescharacteristic
61C32×Dic3—42no—
62C2×C3×C4⋊C4—91no—
63C2×C4⋊Dic3—91no—
64C22×C3⋊C8—91no—
65D3×C4⋊C4—61no—
66D3×C4⋊C4—181no—
67#808—94no—
68C22×C3⋊D3#6711yescharacteristic
69C22×C3×D3—61no—
70C22×C3×D3—61no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C32×C4—41no—
74C2×C32×C4—41no—
75C2×C32×C4—62no—
76C2×C32×C4#6711yescharacteristic
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3C22×D311yescharacteristic
79C4×C3⋊D3Dic612yes—
80C4×C3⋊D3C4×D312yes—
81C2×C3×Dic3—22no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—122no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92C3×C4×D3—62no—
93C3×C4×D3—124no—
94#624—42no—
95#624—122no—
96D3×Dic3—124no—
97#622—34no—
98C2×#106—271noSylow
99C2×C33C2×C4⋊C411yescharacteristic
100C2×C33C2×SD1611yescharacteristic
101C2×C33C2×SD1611yescharacteristic
102C3×C3⋊D3#10614yescomplemented
103C22×C3×C4—91no—
104C22×C3×C4—181no—
105C22×Dic3—181no—
106C2×C4×D3—61no—
107C2×C4×D3—61no—
108C2×C4×D3—62no—
109C2×C4×D3—182no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—121no—
114C3×C4⋊C4—181no—
115C4⋊Dic3—21no—
116C4⋊Dic3—61no—
117C4⋊Dic3—61no—
118C4⋊Dic3—92no—
119C4⋊Dic3—121no—
120Dic3⋊C4—62no—
121Dic3⋊C4—182no—
122C2×C3⋊C8—92no—
123C2×C3⋊C8—94no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32C2×#6711yescharacteristic
129C2×C3⋊D3Dic3⋊C412yes—
130C2×C3⋊D3#21412yes—
131C2×C3⋊D3#21512yes—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C2×C3×D3—62no—
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—42no—
141C32×C4—42no—
142C32×C4—122no—
143C32×C4Dic3⋊C412yes—
144C3⋊Dic3—42no—
145C3⋊Dic3C2×Dic611yescharacteristic
146C3⋊Dic3C2×C4×D311yescharacteristic
147C3×Dic3—42no—
148C3×Dic3—42no—
149C3×Dic3—61no—
150C3×Dic3—61no—
151C3×Dic3—61no—
152C3×Dic3—61no—
153C3×Dic3—82no—
154C3×Dic3—122no—
155C3×Dic3—122no—
156C3×Dic3—242no—
157C22×C8—271no—
158C2×C4⋊C4—91no—
159C2×C4⋊C4—271no—
160#106—274no—
161C33C2×#10611yescharacteristic, complemented, Sylow
162C23×C3—91no—
163C22×D3—61no—
164C22×D3—61no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—91no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—122no—
175C2×C3×C4—182no—
176C2×C3×C4—362no—
177C2×C3×C4GO+(4,𝔽2)11yescharacteristic
178C2×Dic3—22no—
179C2×Dic3—61no—
180C2×Dic3—61no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—122no—
184C2×Dic3—362no—
185C4×D3—62no—
186C4×D3—62no—
187C4×D3—62no—
188C4×D3—62no—
189C4×D3—124no—
190C4×D3—364no—
191C3⋊C8—94no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32C2×Dic3⋊C411yescharacteristic
205C2×C32C2×#21411yescharacteristic
206C2×C32C2×#21511yescharacteristic
207C3⋊D3#80814yescomplemented
208C3×D3—64no—
209C3×D3—64no—
210C22×C4—91no—
211C22×C4—181no—
212C22×C4—541no—
213C2×C8—272no—
214C2×C8—274no—
215C4⋊C4—61no—
216C4⋊C4—92no—
217C4⋊C4—181no—
218C4⋊C4—272no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—92no—
224C22×C3—92no—
225C22×C3—92no—
226C22×C3C2×GO+(4,𝔽2)11yescharacteristic
227C2×D3—62no—
228C2×D3—62no—
229C2×D3—62no—
230C2×D3—62no—
231C2×D3—62no—
232C2×D3—62no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—42no—
236C3×C4—42no—
237C3×C4—91no—
238C3×C4—91no—
239C3×C4—122no—
240C3×C4—122no—
241C3×C4—242no—
242C3×C4—362no—
243C3×C4#377312yes—
244Dic3—42no—
245Dic3—61no—
246Dic3—61no—
247Dic3—61no—
248Dic3—61no—
249Dic3—122no—
250Dic3—122no—
251Dic3—242no—
252C32—21no—
253C32—21no—
254C32—41no—
255C32—41no—
256C32C2×#80811yescharacteristic, complemented
257C23—91no—
258C2×C4—62no—
259C2×C4—91no—
260C2×C4—92no—
261C2×C4—92no—
262C2×C4—182no—
263C2×C4—362no—
264C2×C4—1082no—
265C2×C4D32⋊D311yescharacteristic
266C8—274no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—94no—
280C2×C3#6422711yescharacteristic
281C2×C3#6422711yescharacteristic
282C2×C3C2×#377311yescharacteristic
283D3—64no—
284D3—64no—
285C22—92no—
286C22—92no—
287C22—92no—
288C22C2×D32⋊D311yescharacteristic
289C4—91no—
290C4—91no—
291C4—122no—
292C4—362no—
293C4#8893412yes—
294C3—21no—
295C3—21no—
296C3—41no—
297C3—41no—
298C3#1059338711yescharacteristic, complemented
299C2—94no—
300C2#1172430811yescharacteristic
301C2#1172431011yescharacteristic
302C2C2×#8893411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#106
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722778C2semidirect
10. #88988C4semidirect
30. C3⋊#622C2×C4semidirect
102. C3×C3⋊D3#106semidirect
161. C33C2×#106semidirect
207. C3⋊D3#808semidirect
256. C32C2×#808semidirect
298. C3#10593387semidirect

Maximal subgroups

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

subgroupindex
C2×#889882
#117236892
C3×#639002
#117227782
#105933873
C2×#8089