Finite Group Databasebeta

← #13155711

Subgroups — #13155711

Order 1728.

690 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C3×Dic3×D12C212yesmaximal, complemented
2C2×C3×#3713C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic, complemented
4#11722031C212yesmaximal, complemented
5#11722019C211yesmaximal, characteristic, complemented
6#10591430—32nomaximal
7#10590344—31nomaximal
8C2×C32×D12C2212yes—
9C2×C3×D3×Dic3—22no—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×C4×Dic3C2211yescharacteristic
14C3×(C2×D3)⋊Dic3—22no—
15C3×#3713C414yescomplemented
16C3×D4×Dic3—32no—
17C2×C3×#213—32no—
18C3×C4×D12—32no—
19Dic3×D12D312yescomplemented
20C2×#3713D311yescharacteristic
21#63659—32no—
22#63630—31no—
23#63574—31no—
24#63573—32no—
25#63562—32no—
26#63562—32no—
27#63562—62no—
28#63554—32no—
29#63554—61no—
30C22×C32×D3—22no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3D412yes—
34C32×D12—22no—
35C32×D12C2×C414yescomplemented
36C3×D3×Dic3—44no—
37C3⋊(C3⋊(C3⋊C8))—21no—
38C3×C3⋊(C3⋊C8)C2×C412yescomplemented
39#5152—91no—
40#5014—92no—
41C2×C32×D4—32no—
42C22×C3×Dic3—62no—
43C2×C3×D12—22no—
44C2×C3×D12C2×D312yes—
45C2×C3×D12C2×D312yes—
46C2×C3×C4×D3—62no—
47C2×D3×Dic3—22no—
48C32×C42—31no—
49C4⋊(C3⋊Dic3)C2×D312yes—
50C2×C3⋊(C3⋊C8)—32no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—61no—
53C2×C3⋊(C3⋊C8)—62no—
54C2×C3⋊(C3⋊C8)—121no—
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56C3×#217—62no—
57C3×#213—38no—
58C3×(C2×D3)⋊C4—62no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—62no—
62C3×C4×Dic3—21no—
63C3×C4×Dic3C2×D312yes—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66(C2×D3)⋊Dic3—22no—
67#3713Dic314yescomplemented
68C22×C33C2×D411yescharacteristic
69C2×C32×D3—24no—
70C2×C32×D3—42no—
71C2×C32×D3—42no—
72C33×C4C22×C411yescharacteristic
73C33×C4#4111yescharacteristic
74C3×C3⋊Dic3—42no—
75C32×Dic3—22no—
76C2×C3×D8—91no—
77C3×C4×D4—92no—
78D4×Dic3—32no—
79C2×#213—32no—
80C4×D12—32no—
81#832—92no—
82#832—92no—
83#832—182no—
84#817—91no—
85#815—91no—
86#809—92no—
87#808—92no—
88#803—92no—
89#803—181no—
90C23×C32—62no—
91C22×C3×D3—22no—
92C22×C3×D3—22no—
93C22×C3×D3—42no—
94C32×D4—34no—
95C32×D4—62no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—32no—
99C2×C32×C4—41no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—122no—
107C2×C3×Dic3—124no—
108C2×C3×Dic3#6712yes—
109C2×C3×Dic3#6712yes—
110C3×D12—22no—
111C3×D12—22no—
112C3×D12—24no—
113C3×D12—42no—
114C3×D12C4×D314yescomplemented
115C3×D12C2×Dic314yescomplemented
116C3×C4×D3—124no—
117D3×Dic3—44no—
118C3⋊(C3⋊C8)—61no—
119C3⋊(C3⋊C8)—62no—
120C3⋊(C3⋊C8)—62no—
121C3⋊(C3⋊C8)—122no—
122C3⋊(C3⋊C8)—122no—
123C3⋊(C3⋊C8)C2×Dic312yescomplemented
124C3×C3⋊C8—34no—
125C3×C3⋊C8—62no—
126#442—271noSylow
127C2×C33C4×D411yescharacteristic
128C2×C33AGL(1,ℤ/8)12yes—
129C32×D3—44no—
130C2×C3×D4—32no—
131C2×C3×D4—32no—
132C2×C3×D4—62no—
133C22×C3×C4—182no—
134C22×Dic3—62no—
135C2×D12D3212yes—
136C2×C4×D3—62no—
137C3×D8—94no—
138C2×C3×C8—91no—
139C3×C4⋊C4—92no—
140C3×C22⋊C4—182no—
141C3×C42—32no—
142C3×C42—61no—
143#217—62no—
144#213—38no—
145(C2×D3)⋊C4—62no—
146C4⋊Dic3—32no—
147C4⋊Dic3—32no—
148C4⋊Dic3—62no—
149C4×Dic3D3211yescharacteristic
150C2×C3⋊C8—32no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—92no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—181no—
155C2×C3⋊C8—182no—
156C2×C3⋊C8—361no—
157C22×C32—21no—
158C22×C32—22no—
159C22×C32—41no—
160C22×C32—64no—
161C22×C32—122no—
162C22×C32—122no—
163C22×C32D3×D411yescharacteristic
164C22×C32C2×#6712yes—
165C2×C3×D3—24no—
166C2×C3×D3—24no—
167C2×C3×D3—44no—
168C2×C3×D3—42no—
169C2×C3×D3—42no—
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C2×C3×D3—82no—
173C2×C3×D3—82no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—62no—
181C32×C4C2×C4×D312yes—
182C32×C4#23512yes—
183C32×C4#23711yescharacteristic
184C32×C4C22×Dic311yescharacteristic
185C3⋊Dic3—42no—
186C3×Dic3—22no—
187C3×Dic3—22no—
188C3×Dic3—42no—
189C3×Dic3—122no—
190C3×Dic3—122no—
191C3×Dic3—242no—
192C2×D8—91no—
193C4×D4—92no—
194#106—271no—
195#102—272no—
196#97—271no—
197C33#44211yescharacteristic, complemented, Sylow
198C23×C3—62no—
199C23×C3—62no—
200C23×C3—122no—
201C22×D3—22no—
202C3×D4—34no—
203C3×D4—34no—
204C3×D4—64no—
205C3×D4—62no—
206C3×D4—62no—
207C3×D4—122no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4—41no—
213C2×C3×C4—62no—
214C2×C3×C4—182no—
215C2×C3×C4—364no—
216C2×C3×C4C2×D3211yescharacteristic
217C2×C3×C4C2×D3212yes—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—122no—
221C2×Dic3—124no—
222C2×Dic3#62512yes—
223D12—22no—
224D12D3×Dic314yescomplemented
225C4×D3—124no—
226C3×C8—92no—
227C3⋊C8—34no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C3⋊C8—181no—
231C3⋊C8—182no—
232C3⋊C8—362no—
233C3⋊C8—362no—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—41no—
240C2×C32—42no—
241C2×C32—124no—
242C2×C32#91112yes—
243C2×C32C4×#6712yes—
244C2×C32#93212yes—
245C2×C32D4×Dic311yescharacteristic
246C2×C32#94912yes—
247C3×D3—44no—
248C3×D3—44no—
249C3×D3—84no—
250C2×D4—32no—
251C22×C4—182no—
252D8—94no—
253C2×C8—91no—
254C2×C8—271no—
255C4⋊C4—92no—
256C22⋊C4—182no—
257C42—31no—
258C22×C3—22no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—64no—
262C22×C3—64no—
263C22×C3—124no—
264C22×C3—122no—
265C22×C3—122no—
266C22×C3—122no—
267C22×C3—122no—
268C22×C3—242no—
269C22×C3—242no—
270C22×C3C2×#62511yescharacteristic
271C22×C3D3×#6712yes—
272C2×D3—24no—
273C2×D3—42no—
274C2×D3—42no—
275C3×C4—21no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—122no—
284C3×C4—362no—
285C3×C4#379811yescharacteristic
286C3×C4C4×D3211yescharacteristic
287C3×C4C2×D3×Dic312yes—
288C3×C4#380412yes—
289Dic3—22no—
290Dic3—122no—
291Dic3—122no—
292Dic3—242no—
293C32—21no—
294C32—22no—
295C32—41no—
296C32#501412yescomplemented
297C32#515211yescharacteristic, complemented
298C23—62no—
299D4—34no—
300D4—62no—
301C2×C4—32no—
302C2×C4—182no—
303C2×C4—364no—
304C2×C4D3311yescharacteristic
305C8—92no—
306C8—541no—
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—124no—
315C2×C3—124no—
316C2×C3—244no—
317C2×C3#6382312yes—
318C2×C3C4×#62511yescharacteristic
319C2×C3#6392512yes—
320C2×C3#6393212yes—
321C2×C3Dic3×#6712yes—
322D3—44no—
323C22—64no—
324C22—122no—
325C22—122no—
326C22D3×#62511yescharacteristic
327C4—62no—
328C4—362no—
329C4D32×Dic311yescharacteristic
330C4#8895911yescharacteristic
331C3—22no—
332C3—21no—
333C3—41no—
334C3#1059034411yescharacteristic, complemented
335C3#1059143012yescomplemented
336C2—124no—
337C2#1172286612yes—
338C2Dic3×#62511yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C3×Dic3×D12C2semidirect
3. #11722079C2semidirect
4. #11722031C2semidirect
5. #11722019C2semidirect
15. C3×#3713C4semidirect
19. Dic3×D12D3semidirect
35. C32×D12C2×C4semidirect
38. C3×C3⋊(C3⋊C8)C2×C4semidirect
67. #3713Dic3semidirect
114. C3×D12C4×D3semidirect
115. C3×D12C2×Dic3semidirect
123. C3⋊(C3⋊C8)C2×Dic3semidirect
197. C33#442semidirect
224. D12D3×Dic3semidirect
296. C32#5014semidirect
297. C32#5152semidirect
334. C3#10590344semidirect
335. C3#10591430semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×D122
C2×C3×#37132
#117220792
#117220312
#117220192
#105914303
#105903443