Finite Group Databasebeta

← #13167377

Subgroups — #13167377

Order 1728.

398 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#11724340C212yesmaximal, complemented
2#11723651C211yesmaximal, characteristic, complemented
3#11723647C211yesmaximal, characteristic, complemented
4C3×#63771C211yesmaximal, characteristic, complemented
5#11722781C212yesmaximal, complemented
6#10593301—31nomaximal
7C4⋊#1113C2212yes—
8#89042C2211yescharacteristic
9C3×#3788C2212yes—
10#88942—22no—
11#88941—22no—
12C3×#3745—21no—
13C3×C8×C3⋊D3—21no—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88810—21no—
16#88809—21no—
17#88809C2211yescharacteristic
18#64224—32no—
19#63803—31no—
20#63803—61no—
21#63797—31no—
22D3×D24—61no—
23#63771D311yescharacteristic, complemented
24#63727—32no—
25C2×#1113—22no—
26C3×C4⋊(C3⋊D3)—22no—
27C3×C4×C3⋊D3—21no—
28C3×C4×C3⋊D3C2311yescharacteristic
29#6296D411yescharacteristic
30#6295—21no—
31#6295—22no—
32#6295D411yescharacteristic
33C3×#622—22no—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)—21no—
36C3×C3⋊(C3⋊C8)D411yescharacteristic
37#4897—91nomaximal
38#3802—32no—
39D3×D12—62no—
40#3798—31no—
41#3798—61no—
42#3788C2×D312yes—
43#3775—62no—
44#3774—62no—
45#3745—21no—
46C8×C3⋊D3—21no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×D24—61no—
49C3×D24—61no—
50C3×D24—121no—
51C3×D3×C8—61no—
52C3×D3×C8—61no—
53C3×D3×C8—61no—
54C3×D3×C8—61no—
55C3×D3×C8—121no—
56#3714—62no—
57#3713—61no—
58#3712—31no—
59#3712—61no—
60#3711—61no—
61#3711—61no—
62#3711—61no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3#4111yescharacteristic
66#1113—82no—
67C33×C4C2×D411yescharacteristic
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3C2×D411yescharacteristic
70D3×D8—181no—
71#908—92no—
72C2×D24—91no—
73#901—91no—
74#901—181no—
75C3×#108—91no—
76#823—92no—
77C2×D32—62no—
78C2×D32—122no—
79C4⋊(C3⋊D3)—22no—
80C4×C3⋊D3—21no—
81C4×C3⋊D3C22×D311yescharacteristic
82C3×D12—62no—
83C3×D12—62no—
84C3×D12—122no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—121no—
90#627—31no—
91#627—61no—
92#626—61no—
93#626—61no—
94#626—61no—
95#626—122no—
96#625—31no—
97#625—61no—
98#625—62no—
99#622—22no—
100C32×C8—21no—
101C32×C8—21no—
102C32×C8—41no—
103C32×C8—41no—
104C32×C8D1211yescharacteristic
105C3⋊(C3⋊C8)—21no—
106C3⋊(C3⋊C8)D1211yescharacteristic
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109C3×C3⋊C8—61no—
110C3×C3⋊C8—61no—
111C3×C3⋊C8—121no—
112#472—271noSylow
113C2×C33#12111yescharacteristic
114C3×C3⋊D3—21no—
115C3×C3⋊D3—41no—
116C3×C3⋊D3—82no—
117D3×D4—182no—
118#235—91no—
119#235—181no—
120C2×D12—92no—
121C3×D8—181no—
122C3×M16—92no—
123C2×C3×C8—91no—
124C2×C3×C8—181no—
125#213—182no—
126#208—91no—
127#208—181no—
128D24—61no—
129D24—61no—
130D24—121no—
131D24—181no—
132D24—182no—
133#204—182no—
134ΓL(1,𝔽25)—181no—
135ΓL(1,𝔽25)—181no—
136ΓL(1,𝔽25)—181no—
137D3×C8—61no—
138D3×C8—61no—
139D3×C8—61no—
140D3×C8—61no—
141D3×C8—121no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3—42no—
144C2×C3⋊D3#23911yescharacteristic
145C2×C3×D3—61no—
146C2×C3×D3—61no—
147C2×C3×D3—61no—
148C2×C3×D3—61no—
149C2×C3×D3—121no—
150C2×C3×D3—122no—
151C2×C3×D3—122no—
152C2×C3×D3—242no—
153D32—244no—
154D32—242no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4C2×D1211yescharacteristic
160C3⋊Dic3—21no—
161C3⋊Dic3D3×D411yescharacteristic
162C3×Dic3—61no—
163C3×Dic3—61no—
164C3×Dic3—61no—
165C3×Dic3—61no—
166C3×Dic3—121no—
167AGL(1,ℤ/8)—272no—
168C2×D8—271no—
169#131—271no—
170#108—91no—
171#100—272no—
172C33#47211yescharacteristic, complemented, Sylow
173C22×D3—182no—
174C22×D3—362no—
175C3×D4—182no—
176C2×C3×C4—91no—
177C2×C3×C4—181no—
178#67—181no—
179#67—181no—
180#67—181no—
181#67—362no—
182D12—62no—
183D12—62no—
184D12—91no—
185D12—122no—
186D12—182no—
187D12—181no—
188C4×D3—61no—
189C4×D3—61no—
190C4×D3—61no—
191C4×D3—61no—
192C4×D3—121no—
193Dic6—91no—
194Dic6—181no—
195C3×C8—21no—
196C3×C8—21no—
197C3×C8—41no—
198C3×C8—41no—
199C3×C8—91no—
200C3×C8—181no—
201C3×C8—182no—
202C3×C8GO+(4,𝔽2)11yescharacteristic
203C3⋊C8—61no—
204C3⋊C8—61no—
205C3⋊C8—61no—
206C3⋊C8—61no—
207C3⋊C8—121no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32#89511yescharacteristic
213C3⋊D3—21no—
214C3⋊D3—41no—
215C3⋊D3—82no—
216C3×D3—62no—
217C3×D3—121no—
218C3×D3—121no—
219C3×D3—122no—
220C3×D3—121no—
221C3×D3—242no—
222C3×D3—242no—
223C3×D3—482no—
224#41—271no—
225C2×D4—272no—
226SD16—542no—
227D8—181no—
228D8—541no—
229D8—542no—
230M16—92no—
231M16—271no—
232M16—541no—
233C2×C8—91no—
234C2×C8—181no—
235C22×C3—91no—
236C22×C3—181no—
237C22×C3—362no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—61no—
242C2×D3—121no—
243C2×D3—122no—
244C2×D3—122no—
245C2×D3—242no—
246C2×D3—724no—
247C2×D3—722no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—91no—
253C3×C4—181no—
254C3×C4C2×GO+(4,𝔽2)11yescharacteristic
255Dic3—61no—
256Dic3—61no—
257Dic3—61no—
258Dic3—61no—
259Dic3—121no—
260C32—21no—
261C32—21no—
262C32—41no—
263C32—41no—
264C32#489711yescharacteristic, complemented
265C23—542no—
266Q8—271no—
267D4—182no—
268D4—271no—
269D4—542no—
270D4—541no—
271C2×C4—91no—
272C2×C4—181no—
273C8—91no—
274C8—182no—
275C8—181no—
276C8#157⋊D311yescharacteristic
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—181no—
282C2×C3—361no—
283C2×C3—722no—
284C2×C3#6423011yescharacteristic
285D3—62no—
286D3—121no—
287D3—121no—
288D3—122no—
289D3—121no—
290D3—242no—
291D3—242no—
292D3—482no—
293C22—91no—
294C22—181no—
295C22—362no—
296C22—2162no—
297C4—91no—
298C4—181no—
299C4C2×#157⋊D311yescharacteristic
300C3—21no—
301C3—21no—
302C3—41no—
303C3—41no—
304C3#1059330111yescharacteristic, complemented
305C2—181no—
306C2—361no—
307C2—722no—
308C2#1172434311yescharacteristic
subgroupG/H
centerC2#11724343
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#121
hypercenterC8#157⋊D3
hypocenterC33#472
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724340C2semidirect
2. #11723651C2semidirect
3. #11723647C2semidirect
4. C3×#63771C2semidirect
5. #11722781C2semidirect
23. #63771D3semidirect
172. C33#472semidirect
264. C32#4897semidirect
304. C3#10593301semidirect

Maximal subgroups

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

subgroupindex
#117243402
#117236512
#117236472
C3×#637712
#117227812
#105933013
#48979