Finite Group Databasebeta

← #13167307

Subgroups — #13167307

Order 1728.

448 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 79 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×C3⋊#622C211yesmaximal, characteristic
2#11723715C211yesmaximal, characteristic
3C3×#63957C211yesmaximal, characteristic
4#11722779C212yesmaximal, complemented
5#11722778C212yesmaximal, complemented
6#10593511—31nomaximal
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3⋊#622C414yescomplemented
9C2×C3⋊#622C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C3×#3757—21no—
12C32×#217—21no—
13#88815C2212yes—
14#88813—21no—
15C3×#3724C2212yes—
16C3×Dic32—21no—
17C22×#622—31no—
18C3×#877—61no—
19#63957—31no—
20#63957D311yescharacteristic
21#63955—61no—
22#63743—32no—
23#63742—32no—
24C23×C33D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2×C412yes—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3Q812yes—
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C2×C32×Dic3—22no—
31C3⋊#622C2×C412yes—
32C3⋊#622C2×C412yes—
33#4998—91nomaximal
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—61no—
36C22×C3×Dic3—61no—
37C2×#622—34no—
38C2×#622—31no—
39C2×#622—31no—
40C32×C22⋊C4—61no—
41#3757—21no—
42C3×#217—21no—
43C3×#217—21no—
44C3×#217—41no—
45C3×#217—61no—
46C3×#217—61no—
47C3×#217—121no—
48C3×Dic3⋊C4—62no—
49C3×C4×Dic3—62no—
50#3724—32no—
51#3724C2×D312yes—
52Dic3⋊Dic3—62no—
53Dic32—21no—
54Dic32—62no—
55Dic32—61no—
56C22×C33—22no—
57C22×C33C4⋊C412yes—
58C22×C33C2×D411yescharacteristic
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3C4⋊C412yes—
61C3×C3⋊Dic3C22×C411yescharacteristic
62C3×C3⋊Dic3C2×Q811yescharacteristic
63C32×Dic3—42no—
64C3×#117—91no—
65#926—91no—
66C22×C3⋊C8—91no—
67#877—61no—
68#877—181no—
69#808—92no—
70#807—92no—
71C23×C32—21no—
72C23×C32—21no—
73C23×C32—41no—
74C23×C32—41no—
75C23×C32#6711yescharacteristic
76C2×C32×C4—62no—
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3Dic612yes—
79C2×C3⋊Dic3C4×D312yes—
80C2×C3⋊Dic3#6711yescharacteristic
81C2×C3⋊Dic3C22×D311yescharacteristic
82C2×C3×Dic3—22no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—122no—
96#622—32no—
97#622—32no—
98#427—271noSylow
99C2×C33—22no—
100C2×C33C2×C4⋊C411yescharacteristic
101C2×C33#13512yes—
102C22×C3×C4—91no—
103C22×Dic3—61no—
104C22×Dic3—61no—
105C3×C4⋊C4—92no—
106C3×C22⋊C4—61no—
107C3×C22⋊C4—61no—
108C3×C22⋊C4—121no—
109C3×C22⋊C4—181no—
110C3×C42—181no—
111#217—21no—
112#217—61no—
113#217—61no—
114#217—121no—
115C4⋊Dic3—92no—
116Dic3⋊C4—62no—
117Dic3⋊C4—182no—
118C4×Dic3—62no—
119C4×Dic3—181no—
120C4×Dic3—182no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—91no—
123C2×C3⋊C8—94no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—22no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—41no—
130C22×C32—41no—
131C22×C32—42no—
132C22×C32—42no—
133C22×C32—42no—
134C22×C32—42no—
135C22×C32—82no—
136C22×C32—82no—
137C22×C32Dic3⋊C412yes—
138C22×C32C2×#6711yescharacteristic
139C32×C4—122no—
140C3⋊Dic3—42no—
141C3⋊Dic3Dic3⋊C412yes—
142C3⋊Dic3C2×Dic611yescharacteristic
143C3⋊Dic3C2×C4×D311yescharacteristic
144C3×Dic3—42no—
145C3×Dic3—42no—
146C3×Dic3—62no—
147C3×Dic3—62no—
148C3×Dic3—61no—
149C3×Dic3—61no—
150C3×Dic3—61no—
151C3×Dic3—61no—
152C3×Dic3—82no—
153C3×Dic3—122no—
154C3×Dic3—122no—
155C3×Dic3—242no—
156C22×C8—271no—
157#117—91no—
158#117—271no—
159#107—272no—
160#106—272no—
161C33#42711yescharacteristic, complemented, Sylow
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3GO+(4,𝔽2)11yescharacteristic
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—92no—
172C2×C3×C4—92no—
173C2×C3×C4—122no—
174C2×C3×C4—182no—
175C2×Dic3—22no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—122no—
187C3⋊C8—92no—
188C3⋊C8—92no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—22no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—42no—
197C2×C32—42no—
198C2×C32—42no—
199C2×C32—42no—
200C2×C32—82no—
201C2×C32—82no—
202C2×C32C2×Dic3⋊C411yescharacteristic
203C2×C32#95012yes—
204C22×C4—91no—
205C2×C8—271no—
206C2×C8—271no—
207C2×C8—274no—
208C4⋊C4—92no—
209C4⋊C4—272no—
210C22⋊C4—61no—
211C22⋊C4—181no—
212C42—181no—
213C42—541no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—82no—
226C22×C3—82no—
227C22×C3#377312yes—
228C22×C3C2×GO+(4,𝔽2)11yescharacteristic
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—92no—
232C3×C4—122no—
233C3×C4—122no—
234C3×C4—242no—
235C3×C4—362no—
236Dic3—42no—
237Dic3—62no—
238Dic3—62no—
239Dic3—61no—
240Dic3—61no—
241Dic3—61no—
242Dic3—61no—
243Dic3—122no—
244Dic3—122no—
245Dic3—242no—
246C32—21no—
247C32—21no—
248C32—41no—
249C32—41no—
250C32#499811yescharacteristic, complemented
251C23D32⋊D311yescharacteristic
252C2×C4—62no—
253C2×C4—91no—
254C2×C4—91no—
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—182no—
258C8—272no—
259C8—272no—
260C2×C3—22no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—22no—
264C2×C3—22no—
265C2×C3—42no—
266C2×C3—42no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3—82no—
272C2×C3—82no—
273C2×C3C2×#377311yescharacteristic
274C2×C3#6423712yes—
275C22—22no—
276C22#8893412yes—
277C22C2×D32⋊D311yescharacteristic
278C4—91no—
279C4—92no—
280C4—91no—
281C4—122no—
282C4—362no—
283C3—21no—
284C3—21no—
285C3—41no—
286C3—41no—
287C3#1059351111yescharacteristic, complemented
288C2—22no—
289C2C2×#8893411yescharacteristic
290C2#1172432412yes—
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D32⋊D3
hypocenterC33#427
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722779C2semidirect
5. #11722778C2semidirect
8. C2×C3⋊#622C4semidirect
161. C33#427semidirect
250. C32#4998semidirect
287. C3#10593511semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
#117237152
C3×#639572
#117227792
#117227782
#105935113
#49989