Finite Group Databasebeta

← #13173389

Subgroups — #13173389

Order 1728.

752 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 85 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×C3×#3828C211yesmaximal, characteristic
2#11724514C212yesmaximal, complemented
3#11723111C212yesmaximal, complemented
4#11723099C212yesmaximal, complemented
5#10593792—31nomaximal
6C22×C33×C4C411yescharacteristic, complemented
7C2×C3×#638C411yescharacteristic, complemented
8C2×C3×#638C2211yescharacteristic
9C3×#3828—22no—
10C3×#3828—22no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C411yescharacteristic, complemented
15#88994C2214yes—
16C4⋊(C3⋊#157)—22no—
17C4×C3⋊#157—22no—
18C2×#3828D311yescharacteristic
19C2×C3×#235—62no—
20#64286—32no—
21#63786—32no—
22#63774—32no—
23C23×C33C2×C411yescharacteristic
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C22×C3×C3⋊D3C2311yescharacteristic
26C2×C3⋊#157—24no—
27C2×C33×C4—21no—
28C2×C33×C4—21no—
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C3×#638—22no—
32C3×#638—22no—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4×C3⋊D3—21no—
37C3×C4×C3⋊D3—21no—
38C3×C4×C3⋊D3D412yes—
39C3×C4×C3⋊D3D412yes—
40C3×#634—22no—
41#5240—91nomaximal
42C22×C32×C4—22no—
43C22×C32×C4—42no—
44C22×C32×C4Dic311yescharacteristic, complemented
45C2×#638Dic311yescharacteristic, complemented
46C2×#638C2×D311yescharacteristic
47#3828—22no—
48#3828—22no—
49C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
50C2×C4×C3⋊D3Dic311yescharacteristic, complemented
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×#634Dic311yescharacteristic, complemented
53C2×C3×#67—62no—
54C2×C3×#67—62no—
55C3×#235—68no—
56C3×#235—68no—
57C2×C3×D12—62no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—62no—
61C22⋊#157—34no—
62C4⋊#157—62no—
63C4×#157—62no—
64C22×C33—21no—
65C22×C33—41no—
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—41no—
69C2×C3×C3⋊D3—41no—
70C2×C3×C3⋊D3C2×D412yes—
71C3⋊#157—44no—
72C33×C4—21no—
73C33×C4C22⋊C411yescharacteristic
74C33×C4C22⋊C411yescharacteristic
75C3×C3⋊Dic3—21no—
76C3×C3⋊Dic3C22⋊C412yes—
77C2×C3×#41—91no—
78C2×#235—62no—
79#926—92no—
80#834—92no—
81#834—92no—
82C23×C32—22no—
83C23×C32—42no—
84C23×C32C2×Dic311yescharacteristic
85C22×C3⋊D3C2×Dic311yescharacteristic
86C22×C3⋊D3C22×D311yescharacteristic
87C22×C3×D3—62no—
88C22×C3×D3—62no—
89C2×#157—64no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—44no—
97C2×C32×C4—44no—
98C2×C32×C4—42no—
99C2×C32×C4—42no—
100C2×C32×C4C2×Dic311yescharacteristic
101C2×C32×C4C2×Dic311yescharacteristic
102#638—22no—
103#638—22no—
104C2×C3⋊Dic3C2×Dic311yescharacteristic
105C2×C3⋊Dic3C2×Dic311yescharacteristic
106C4⋊(C3⋊D3)—22no—
107C4×C3⋊D3—21no—
108C4×C3⋊D3—21no—
109C4×C3⋊D3#6712yes—
110C4×C3⋊D3#6712yes—
111#634—22no—
112C3×#67—68no—
113C3×#67—68no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C3×D12—68no—
117C3×C4×D3—64no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×C4×D3—64no—
121C3×Dic6—68no—
122#410—271noSylow
123C2×C33—21no—
124C2×C33—41no—
125C2×C33C2×C22⋊C411yescharacteristic
126C3×C3⋊D3—22no—
127C3×C3⋊D3—41no—
128C3×#41—182no—
129C3×#41—182no—
130C2×C3×Q8—91no—
131C2×C3×D4—91no—
132C2×C3×D4—91no—
133C2×C3×D4—91no—
134C22×C3×C4—22no—
135C22×C3×C4—42no—
136C22×C3×C4—91no—
137C22×C3×C4—91no—
138C22×C3×C4#15711yescharacteristic, complemented
139C2×#67—62no—
140C2×#67—62no—
141#235—68no—
142#235—68no—
143C2×D12—62no—
144C2×C4×D3—62no—
145C2×C4×D3—62no—
146C2×Dic6—62no—
147#217—94no—
148C4⋊Dic3—182no—
149C4×Dic3—182no—
150C22×C32—21no—
151C22×C32—24no—
152C22×C32—22no—
153C22×C32—44no—
154C22×C32—42no—
155C22×C32—41no—
156C22×C32—44no—
157C22×C32—48no—
158C22×C32C22×Dic311yescharacteristic
159C2×C3⋊D3—21no—
160C2×C3⋊D3—41no—
161C2×C3⋊D3—41no—
162C2×C3⋊D3C2×#6712yes—
163C2×C3×D3—64no—
164C2×C3×D3—64no—
165C2×C3×D3—124no—
166C2×C3×D3—124no—
167#157—124no—
168C32×C4—21no—
169C32×C4—24no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—44no—
173C32×C4—42no—
174C32×C4—42no—
175C32×C4#21711yescharacteristic
176C32×C4#21711yescharacteristic
177C3⋊Dic3—21no—
178C3⋊Dic3#21712yes—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C2×#41—91no—
182#117—272no—
183#99—272no—
184#99—272no—
185C33#41011yescharacteristic, complemented, Sylow
186C23×C3—22no—
187C23×C3—42no—
188C23×C3—91no—
189C23×C3—91no—
190C23×C3C2×#15711yescharacteristic
191C22×D3—62no—
192C22×D3—62no—
193C3×Q8—182no—
194C3×D4—182no—
195C3×D4—182no—
196C3×D4—182no—
197C2×C3×C4—24no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—91no—
208C2×C3×C4—92no—
209C2×C3×C4—92no—
210C2×C3×C4—91no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4C2×#15711yescharacteristic
214C2×C3×C4C2×#15711yescharacteristic
215#67—68no—
216#67—68no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—184no—
220D12—68no—
221C4×D3—64no—
222C4×D3—64no—
223C4×D3—64no—
224C4×D3—64no—
225Dic6—68no—
226C2×C32—21no—
227C2×C32—24no—
228C2×C32—22no—
229C2×C32—44no—
230C2×C32—42no—
231C2×C32—44no—
232C2×C32—48no—
233C2×C32—41no—
234C2×C32C2×#21711yescharacteristic
235C3⋊D3—22no—
236C3⋊D3—41no—
237C3×D3—124no—
238C3×D3—124no—
239#41—182no—
240#41—182no—
241C2×Q8—91no—
242C2×D4—91no—
243C2×D4—91no—
244C2×D4—91no—
245C22×C4—91no—
246C22×C4—91no—
247C22×C4C3⋊#15711yescharacteristic, complemented
248C4⋊C4—542no—
249C22⋊C4—274no—
250C42—542no—
251C22×C3—24no—
252C22×C3—21no—
253C22×C3—22no—
254C22×C3—44no—
255C22×C3—41no—
256C22×C3—44no—
257C22×C3—48no—
258C22×C3—42no—
259C22×C3—92no—
260C22×C3—181no—
261C22×C3—361no—
262C22×C3—361no—
263C22×C3C22×#15711yescharacteristic
264C2×D3—64no—
265C2×D3—64no—
266C2×D3—124no—
267C2×D3—124no—
268C3×C4—24no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—44no—
273C3×C4—42no—
274C3×C4—42no—
275C3×C4—92no—
276C3×C4—181no—
277C3×C4C22⋊#15711yescharacteristic
278C3×C4C22⋊#15711yescharacteristic
279Dic3—64no—
280Dic3—64no—
281Dic3—364no—
282C32—22no—
283C32—42no—
284C32#524011yescharacteristic, complemented
285C23—91no—
286C23—91no—
287C23C2×C3⋊#15711yescharacteristic
288Q8—182no—
289D4—182no—
290D4—182no—
291D4—182no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—91no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—91no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—544no—
301C2×C4C2×C3⋊#15711yescharacteristic
302C2×C4C2×C3⋊#15711yescharacteristic
303C2×C3—24no—
304C2×C3—21no—
305C2×C3—22no—
306C2×C3—44no—
307C2×C3—44no—
308C2×C3—48no—
309C2×C3—41no—
310C2×C3—42no—
311C2×C3—182no—
312C2×C3—361no—
313C2×C3C2×C22⋊#15711yescharacteristic
314D3—124no—
315D3—124no—
316C22—21no—
317C22—41no—
318C22—92no—
319C22—181no—
320C22—361no—
321C22—361no—
322C22C22×C3⋊#15711yescharacteristic
323C4—21no—
324C4—92no—
325C4—181no—
326C4—1084no—
327C4#8899411yescharacteristic
328C4#8899411yescharacteristic
329C3—22no—
330C3—42no—
331C3#1059379211yescharacteristic, complemented
332C2—21no—
333C2—41no—
334C2—182no—
335C2—361no—
336C2C2×#8899411yescharacteristic
subgroupG/H
centerC4#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC22×C3×C3⋊D3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. #11723111C2semidirect
4. #11723099C2semidirect
6. C22×C33×C4C4semidirect
7. C2×C3×#638C4semidirect
12. C2×C3×C4×C3⋊D3C4semidirect
14. C2×C3×#634C4semidirect
44. C22×C32×C4Dic3semidirect
45. C2×#638Dic3semidirect
50. C2×C4×C3⋊D3Dic3semidirect
52. C2×#634Dic3semidirect
138. C22×C3×C4#157semidirect
185. C33#410semidirect
247. C22×C4C3⋊#157semidirect
284. C32#5240semidirect
331. C3#10593792semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
#117245142
#117231112
#117230992
#105937923
#52409