Finite Group Databasebeta

← #13165728

Subgroups — #13165728

Order 1728.

648 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 60 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×#88941C211yesmaximal, characteristic, complemented
2C2×#88935C212yesmaximal, complemented
3C2×C3×#3774C211yesmaximal, characteristic, complemented
4#11723029C212yesmaximal, complemented
5Dic3×#622C211yesmaximal, characteristic, complemented
6#10593418—31nomaximal
7C2×#6295C2212yes—
8C2×#6290—22no—
9C2×C3×#626—22no—
10C2×C3×#625C2212yes—
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88941—22no—
14#88935—24no—
15C3×#3774—22no—
16Dic3×C3⋊Dic3C2211yescharacteristic
17C2×#3774—32no—
18C2×#3774—31no—
19C2×#3774D311yescharacteristic, complemented
20#63979—62no—
21#63914—32no—
22C4×#622—31no—
23C22×C3×C3⋊D3—22no—
24C22×C32×D3—22no—
25C2×C3⋊(C3⋊Dic3)D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3D411yescharacteristic
28#6295—22no—
29#6295—22no—
30#6290—44no—
31C3×#626—44no—
32C3×#625—22no—
33C3×#625—22no—
34C3⋊#622D412yescomplemented
35C3×#622D412yescomplemented
36#5151—91nomaximal
37C2×#638—62no—
38C2×C3×#67—62no—
39C2×C3×#67—62no—
40C2×C3×D12—62no—
41C2×#626—22no—
42C2×#626—62no—
43C2×#625—32no—
44C2×#625—62no—
45C2×#625—62no—
46C2×#625—122no—
47C2×#625C2×D312yes—
48C2×#622—31no—
49C2×#622C2×D311yescharacteristic
50#3774—22no—
51#3774—64no—
52#3774—62no—
53C4×C3⋊Dic3—31no—
54Dic32—62no—
55C22×C33C2×D411yescharacteristic
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3—42no—
59C2×C32×D3—42no—
60C2×C32×D3—42no—
61C2×C32×D3—42no—
62C3⋊(C3⋊Dic3)D812yescomplemented
63C3×C3⋊Dic3C2×D411yescharacteristic
64C3×C3⋊Dic3C2×D411yescharacteristic
65C32×Dic3D812yescomplemented
66C2×C3×D8—91no—
67#940—92no—
68#940—182no—
69C2×#213—92no—
70C2×D24—91no—
71C8×Dic3—91no—
72C23×C32—62no—
73C22×C3⋊D3—22no—
74C22×C3×D3—22no—
75C22×C3×D3—22no—
76C22×C3×D3—42no—
77C22×C3×D3—62no—
78C22×C3×D3—62no—
79C22×C3×D3—122no—
80C2×C32×C4—31no—
81#638—124no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—62no—
84C2×C3⋊Dic3—122no—
85C2×C3⋊Dic3C22×D311yescharacteristic
86C3×#67—122no—
87C3×#67—122no—
88C3×#67—124no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—62no—
91C3×D12—124no—
92#626—44no—
93#626—122no—
94#626—122no—
95#625—22no—
96#625—22no—
97#625—62no—
98#625—62no—
99#625—124no—
100#625—124no—
101#625—128no—
102#622—32no—
103#622#6712yescomplemented
104#493—271noSylow
105C2×C33#12711yescharacteristic
106C2×C33C2×D811yescharacteristic
107C2×C33C2×D811yescharacteristic
108C3×C3⋊D3—82no—
109C32×D3—82no—
110C2×C3×D4—92no—
111C2×C3×D4—182no—
112C2×#67—62no—
113C2×#67—62no—
114C2×#67—182no—
115C2×#67—182no—
116C2×#67—182no—
117C2×#67—182no—
118C2×#67—362no—
119C2×D12—62no—
120C2×D12—92no—
121C3×D8—182no—
122C2×C3×C8—91no—
123#213—184no—
124C4×Dic3—91no—
125C4×Dic3—182no—
126C2×C3⋊C8—91no—
127D24—182no—
128C22×C32—22no—
129C22×C32—42no—
130C22×C32—122no—
131C22×C32—122no—
132C22×C32—122no—
133C22×C32D3×D411yescharacteristic
134C2×C3⋊D3—42no—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C2×C3×D3—42no—
140C2×C3×D3—44no—
141C2×C3×D3—42no—
142C2×C3×D3—42no—
143C2×C3×D3—42no—
144C2×C3×D3—82no—
145C2×C3×D3—82no—
146C2×C3×D3—122no—
147C2×C3×D3—122no—
148C2×C3×D3—122no—
149C2×C3×D3—124no—
150C2×C3×D3—122no—
151C2×C3×D3—122no—
152C2×C3×D3—122no—
153C2×C3×D3—242no—
154C2×C3×D3—242no—
155C32×C4—32no—
156C3⋊Dic3—32no—
157C3⋊Dic3—64no—
158C3⋊Dic3—124no—
159C3⋊Dic3D3×D411yescharacteristic
160C3⋊Dic3C2×#6711yescharacteristic
161C3×Dic3—24no—
162C3×Dic3—62no—
163C3×Dic3—62no—
164C2×D8—91no—
165C2×D8—271no—
166C2×D8—272no—
167#127—272no—
168C4×C8—271no—
169C33#49311yescharacteristic, complemented, Sylow
170C23×C3—62no—
171C23×C3—62no—
172C23×C3—122no—
173C23×C3—182no—
174C22×D3—22no—
175C22×D3—62no—
176C22×D3—62no—
177C22×D3—122no—
178C3×D4—182no—
179C3×D4—182no—
180C3×D4—364no—
181C2×C3×C4—62no—
182C2×C3×C4—91no—
183#67—122no—
184#67—122no—
185#67—124no—
186#67—362no—
187#67—362no—
188#67—364no—
189#67—364no—
190#67—364no—
191#67—368no—
192C2×Dic3—62no—
193C2×Dic3—91no—
194C2×Dic3—182no—
195C2×Dic3—362no—
196C2×Dic3GO+(4,𝔽2)11yescharacteristic
197D12—124no—
198D12—182no—
199D12—182no—
200C3×C8—92no—
201C3⋊C8—92no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—42no—
206C2×C32—42no—
207C2×C32—42no—
208C2×C32—242no—
209C2×C32D3×D811yescharacteristic
210C2×C32D3×D811yescharacteristic
211C2×C32#94011yescharacteristic
212C3⋊D3—82no—
213C3×D3—82no—
214C3×D3—84no—
215C3×D3—82no—
216C3×D3—242no—
217C3×D3—244no—
218C3×D3—242no—
219C2×D4—92no—
220C2×D4—182no—
221C2×D4—272no—
222C2×D4—542no—
223D8—182no—
224D8—542no—
225D8—544no—
226C2×C8—91no—
227C2×C8—271no—
228C42—271no—
229C22×C3—22no—
230C22×C3—42no—
231C22×C3—122no—
232C22×C3—122no—
233C22×C3—122no—
234C22×C3—124no—
235C22×C3—122no—
236C22×C3—122no—
237C22×C3—122no—
238C22×C3—242no—
239C22×C3—242no—
240C22×C3—362no—
241C22×C3—362no—
242C22×C3—362no—
243C22×C3C2×GO+(4,𝔽2)11yescharacteristic
244C2×D3—42no—
245C2×D3—42no—
246C2×D3—42no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—124no—
251C2×D3—122no—
252C2×D3—122no—
253C2×D3—122no—
254C2×D3—242no—
255C2×D3—242no—
256C3×C4—64no—
257C3×C4—91no—
258C3×C4—91no—
259Dic3—62no—
260Dic3—62no—
261Dic3—92no—
262Dic3—184no—
263Dic3—364no—
264Dic3#377412yescomplemented
265C32—22no—
266C32—42no—
267C32#515111yescharacteristic, complemented
268C23—62no—
269C23—182no—
270D4—182no—
271D4—182no—
272D4—364no—
273D4—542no—
274D4—542no—
275D4—1084no—
276C2×C4—31no—
277C2×C4—91no—
278C2×C4—271no—
279C8—92no—
280C8—272no—
281C2×C3—22no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—42no—
285C2×C3—42no—
286C2×C3—42no—
287C2×C3—242no—
288C2×C3—244no—
289C2×C3—242no—
290C2×C3—722no—
291C2×C3#6422511yescharacteristic
292C2×C3#6423111yescharacteristic
293C2×C3C2×#377411yescharacteristic
294D3—82no—
295D3—242no—
296D3—244no—
297D3—242no—
298C22—122no—
299C22—122no—
300C22—122no—
301C22—362no—
302C22—362no—
303C22—362no—
304C22D3×GO+(4,𝔽2)11yescharacteristic
305C4—32no—
306C4—91no—
307C4—91no—
308C4—272no—
309C3—22no—
310C3—42no—
311C3#1059341811yescharacteristic, complemented
312C2—242no—
313C2—722no—
314C2#1172412411yescharacteristic
315C2D3×#377411yescharacteristic
316C2#1172412711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#493
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88941C2semidirect
2. C2×#88935C2semidirect
3. C2×C3×#3774C2semidirect
4. #11723029C2semidirect
5. Dic3×#622C2semidirect
19. C2×#3774D3semidirect
34. C3⋊#622D4semidirect
35. C3×#622D4semidirect
62. C3⋊(C3⋊Dic3)D8semidirect
65. C32×Dic3D8semidirect
103. #622#67semidirect
169. C33#493semidirect
264. Dic3#3774semidirect
267. C32#5151semidirect
311. C3#10593418semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
C2×#889352
C2×C3×#37742
#117230292
Dic3×#6222
#105934183
#51519