Finite Group Databasebeta

← #13167522

Subgroups — #13167522

Order 1728.

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

normal subgroups only

This group has 410 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
1C2×#88943C212yesmaximal
2C4⋊#6296C211yesmaximal, characteristic
3C4×#6296C211yesmaximal, characteristic
4C3×C4⋊#622C211yesmaximal, characteristic
5#11722794C212yesmaximal
6#10593447—31nomaximal
7C2×#6296C2212yes—
8C2×#6296C2211yescharacteristic
9C2×C3×#622C2212yes—
10#88943—24no—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4×C3⋊Dic3—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14#88815—21no—
15#88815—21no—
16#88815C2211yescharacteristic
17C2×#3776—32no—
18#63919—31no—
19#63918—61no—
20C4×#627—31no—
21C4×#627—61no—
22C4⋊#622D311yescharacteristic
23#63740—32no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30#6296—21no—
31#6296—22no—
32#6296—22no—
33#6296D412yes—
34C3×#622—22no—
35#4849—91nomaximal
36C2×#627—31no—
37C2×#627—32no—
38C2×#627—61no—
39C2×#627—62no—
40C2×#622C2×D312yes—
41#3776—64no—
42C4⋊(C3⋊Dic3)—21no—
43C4×C3⋊Dic3—21no—
44C4×C3⋊Dic3C2×D311yescharacteristic
45C3×C4⋊Dic3—61no—
46C3×C4⋊Dic3—61no—
47C3×C4⋊Dic3—121no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—61no—
52C3×C4×Dic3—121no—
53#3724—31no—
54#3724—61no—
55#3724—61no—
56Dic3⋊Dic3—61no—
57Dic3⋊Dic3—61no—
58Dic3⋊Dic3—62no—
59Dic3⋊Dic3—61no—
60C22×C33C2×D411yescharacteristic
61C33×C4Q1612yescomplemented
62C3×C3⋊Dic3—22no—
63C3×C3⋊Dic3—21no—
64C3×C3⋊Dic3—42no—
65C3×C3⋊Dic3—41no—
66C3×C3⋊Dic3C2×D411yescharacteristic
67C3×C3⋊Dic3#4111yescharacteristic
68C2×Dic12—92no—
69#888—181no—
70#869—91no—
71C4×Dic6—91no—
72C4×Dic6—181no—
73C3×C4⋊C8—91no—
74#816—92no—
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4—41no—
78C2×C32×C4—41no—
79C2×C32×C4D1211yescharacteristic
80C2×C3⋊Dic3—21no—
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3D1211yescharacteristic
84C2×C3⋊Dic3C22×D311yescharacteristic
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—121no—
96C2×C3×Dic3—121no—
97C2×C3×Dic3—122no—
98#627—32no—
99#627—62no—
100#627—62no—
101#627—61no—
102#627—62no—
103#627—62no—
104#627—122no—
105#627—122no—
106#622—22no—
107#462—271noSylow
108C2×C33#12111yescharacteristic
109C2×C33C2×Q1611yescharacteristic
110C2×C33#13711yescharacteristic
111C2×Dic6—91no—
112C2×Dic6—92no—
113C2×Dic6—181no—
114C2×Dic6—182no—
115C2×C3×C8—92no—
116C3×C4⋊C4—181no—
117C3×C42—91no—
118C3×C42—181no—
119C4⋊Dic3—61no—
120C4⋊Dic3—61no—
121C4⋊Dic3—91no—
122C4⋊Dic3—121no—
123C4⋊Dic3—181no—
124C4⋊Dic3—181no—
125Dic3⋊C4—181no—
126Dic3⋊C4—181no—
127Dic3⋊C4—181no—
128Dic3⋊C4—182no—
129C4×Dic3—61no—
130C4×Dic3—61no—
131C4×Dic3—61no—
132C4×Dic3—61no—
133C4×Dic3—121no—
134Dic12—184no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32C2×D1211yescharacteristic
140C32×C4—22no—
141C32×C4—22no—
142C32×C4—42no—
143C32×C4—42no—
144C32×C4Dic1212yescomplemented
145C3⋊Dic3—22no—
146C3⋊Dic3—21no—
147C3⋊Dic3—41no—
148C3⋊Dic3—42no—
149C3⋊Dic3D3×D411yescharacteristic
150C3⋊Dic3#23911yescharacteristic
151C3×Dic3—61no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—62no—
156C3×Dic3—62no—
157C3×Dic3—62no—
158C3×Dic3—121no—
159C3×Dic3—122no—
160C3×Dic3—122no—
161C3×Dic3—121no—
162C3×Dic3—122no—
163C3×Dic3—122no—
164C3×Dic3—121no—
165C3×Dic3—242no—
166C2×Q16—272no—
167C4⋊Q8—271no—
168C4×Q8—271no—
169C4⋊C8—91no—
170#103—272no—
171C33#46211yescharacteristic, complemented, Sylow
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×C3×C4—182no—
181C2×C3×C4GO+(4,𝔽2)11yescharacteristic
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—62no—
190C2×Dic3—61no—
191C2×Dic3—62no—
192C2×Dic3—121no—
193C2×Dic3—121no—
194C2×Dic3—122no—
195Dic6—92no—
196Dic6—182no—
197Dic6—181no—
198Dic6—182no—
199Dic6—182no—
200Dic6—182no—
201Dic6—362no—
202Dic6—362no—
203C3×C8—182no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32#89511yescharacteristic
217C2×C32C2×Dic1211yescharacteristic
218C2×C32#90911yescharacteristic
219C2×Q8—271no—
220C2×Q8—272no—
221Q16—544no—
222C2×C8—92no—
223C4⋊C4—181no—
224C4⋊C4—271no—
225C4⋊C4—541no—
226C4⋊C4—541no—
227C42—91no—
228C42—181no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3C2×GO+(4,𝔽2)11yescharacteristic
234C3×C4—22no—
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—42no—
238C3×C4—91no—
239C3×C4—91no—
240C3×C4—182no—
241C3×C4—181no—
242C3×C4—362no—
243C3×C4—361no—
244C3×C4#377612yescomplemented
245Dic3—61no—
246Dic3—61no—
247Dic3—61no—
248Dic3—61no—
249Dic3—62no—
250Dic3—62no—
251Dic3—62no—
252Dic3—121no—
253Dic3—122no—
254Dic3—122no—
255Dic3—121no—
256Dic3—122no—
257Dic3—121no—
258Dic3—122no—
259Dic3—242no—
260C32—21no—
261C32—21no—
262C32—41no—
263C32—41no—
264C32#484911yescharacteristic, complemented
265Q8—272no—
266Q8—541no—
267Q8—542no—
268Q8—542no—
269C2×C4—91no—
270C2×C4—91no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—182no—
274C2×C4#157⋊D311yescharacteristic
275C8—182no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3#6422611yescharacteristic
289C2×C3#6423011yescharacteristic
290C2×C3C2×#377611yescharacteristic
291C22C2×#157⋊D311yescharacteristic
292C4—91no—
293C4—91no—
294C4—181no—
295C4—182no—
296C4—361no—
297C4—362no—
298C4#8894312yescomplemented
299C3—21no—
300C3—21no—
301C3—41no—
302C3—41no—
303C3#1059344711yescharacteristic, complemented
304C2#1172433811yescharacteristic
305C2#1172434311yescharacteristic
306C2C2×#8894311yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#462
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
61. C33×C4Q16semidirect
144. C32×C4Dic12semidirect
171. C33#462semidirect
244. C3×C4#3776semidirect
264. C32#4849semidirect
298. C4#88943semidirect
303. C3#10593447semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
C4⋊#62962
C4×#62962
C3×C4⋊#6222
#117227942
#105934473
#48499