Finite Group Databasebeta

← #13173405

Subgroups — #13173405

Order 1728.

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

normal subgroups only

This group has 736 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#11724511C212yesmaximal, complemented
3#11723113C212yesmaximal, complemented
4#11723105C212yesmaximal, complemented
5#10593808—31nomaximal
6C22×C33×C4C411yescharacteristic
7C2×C3×#638C411yescharacteristic
8C2×C3×#638C2211yescharacteristic
9C3×#3828—22no—
10C3×#3828—22no—
11C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
12C2×C3×C4×C3⋊D3C411yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15#88993C2214yes—
16#88989—22no—
17#88988—22no—
18C2×#3828D311yescharacteristic
19C2×C3×#235—62no—
20#64283—32no—
21#63788—32no—
22#63780—32no—
23C23×C33C2×C411yescharacteristic
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C33×C4—21no—
27C2×C33×C4—21no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C3×#638—22no—
31C3×#638—22no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C3×C4×C3⋊D3D412yes—
38C3×C4×C3⋊D3D412yes—
39C3×#634—22no—
40C3⋊#622—24no—
41#5237—91nomaximal
42C22×C32×C4—22no—
43C22×C32×C4—42no—
44C22×C32×C4Dic311yescharacteristic
45C2×#638Dic311yescharacteristic
46C2×#638C2×D311yescharacteristic
47#3828—22no—
48#3828—22no—
49C2×C4⋊(C3⋊D3)Dic311yescharacteristic
50C2×C4×C3⋊D3Dic311yescharacteristic
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×#634C2×D311yescharacteristic
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—
61#3792—34no—
62#3788—62no—
63#3787—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⋊D3C22⋊C412yes—
71C33×C4—21no—
72C33×C4C22⋊C411yescharacteristic
73C33×C4C22⋊C411yescharacteristic
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3C2×D412yes—
76C2×C3×#41—91no—
77C2×#235—62no—
78C2×#208—92no—
79#836—92no—
80#833—92no—
81C23×C32—22no—
82C23×C32—42no—
83C23×C32C2×Dic311yescharacteristic
84C22×C3⋊D3C2×Dic311yescharacteristic
85C22×C3⋊D3C2×Dic311yescharacteristic
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—44no—
95C2×C32×C4—44no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4C2×Dic311yescharacteristic
99C2×C32×C4C2×Dic311yescharacteristic
100#638—22no—
101#638—22no—
102C2×C3⋊Dic3C2×Dic311yescharacteristic
103C2×C3⋊Dic3C22×D311yescharacteristic
104C4⋊(C3⋊D3)—22no—
105C4×C3⋊D3—21no—
106C4×C3⋊D3—21no—
107C4×C3⋊D3#6712yes—
108C4×C3⋊D3#6712yes—
109#634—22no—
110C3×#67—68no—
111C3×#67—68no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C3×D12—68no—
115C3×C4×D3—64no—
116C3×C4×D3—64no—
117C3×C4×D3—64no—
118C3×C4×D3—64no—
119C3×Dic6—68no—
120#622—64no—
121#413—271noSylow
122C2×C33—21no—
123C2×C33—41no—
124C2×C33C2×C22⋊C411yescharacteristic
125C3×C3⋊D3—22no—
126C3×C3⋊D3—41no—
127C3×#41—182no—
128C3×#41—182no—
129C2×C3×Q8—91no—
130C2×C3×D4—91no—
131C2×C3×D4—91no—
132C2×C3×D4—91no—
133C22×C3×C4—22no—
134C22×C3×C4—42no—
135C22×C3×C4—91no—
136C22×C3×C4—91no—
137C22×C3×C4#15711yescharacteristic
138C2×#67—62no—
139C2×#67—62no—
140#235—68no—
141#235—68no—
142C2×D12—62no—
143C2×C4×D3—62no—
144C2×C4×D3—62no—
145C2×Dic6—62no—
146#208—94no—
147#208—182no—
148C2×C3⋊C8—182no—
149C22×C32—21no—
150C22×C32—24no—
151C22×C32—22no—
152C22×C32—44no—
153C22×C32—42no—
154C22×C32—41no—
155C22×C32—44no—
156C22×C32—48no—
157C22×C32C22×Dic311yescharacteristic
158C2×C3⋊D3—21no—
159C2×C3⋊D3—41no—
160C2×C3⋊D3—41no—
161C2×C3⋊D3#21712yes—
162C2×C3×D3—64no—
163C2×C3×D3—64no—
164C2×C3×D3—124no—
165C2×C3×D3—124no—
166C32×C4—21no—
167C32×C4—24no—
168C32×C4—22no—
169C32×C4—22no—
170C32×C4—44no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4#21711yescharacteristic
174C32×C4#21711yescharacteristic
175C3⋊Dic3—21no—
176C3⋊Dic3C2×#6712yes—
177C3×Dic3—64no—
178C3×Dic3—64no—
179C2×#41—91no—
180C2×M16—272no—
181#101—272no—
182#100—272no—
183C33#41311yescharacteristic, complemented, Sylow
184C23×C3—22no—
185C23×C3—42no—
186C23×C3—91no—
187C23×C3—91no—
188C23×C3C2×#15711yescharacteristic
189C22×D3—62no—
190C22×D3—62no—
191C3×Q8—182no—
192C3×D4—182no—
193C3×D4—182no—
194C3×D4—182no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—91no—
206C2×C3×C4—92no—
207C2×C3×C4—92no—
208C2×C3×C4—91no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4C2×#15711yescharacteristic
212C2×C3×C4C2×#15711yescharacteristic
213#67—68no—
214#67—68no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217D12—68no—
218C4×D3—64no—
219C4×D3—64no—
220C4×D3—64no—
221C4×D3—64no—
222Dic6—68no—
223C3⋊C8—184no—
224C2×C32—21no—
225C2×C32—24no—
226C2×C32—22no—
227C2×C32—44no—
228C2×C32—42no—
229C2×C32—44no—
230C2×C32—48no—
231C2×C32—41no—
232C2×C32C2×#21711yescharacteristic
233C3⋊D3—22no—
234C3⋊D3—41no—
235C3×D3—124no—
236C3×D3—124no—
237#41—182no—
238#41—182no—
239C2×Q8—91no—
240C2×D4—91no—
241C2×D4—91no—
242C2×D4—91no—
243C22×C4—91no—
244C22×C4—91no—
245C22×C4C3⋊#15711yescharacteristic
246M16—274no—
247M16—542no—
248C2×C8—542no—
249C22×C3—24no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—44no—
253C22×C3—41no—
254C22×C3—44no—
255C22×C3—48no—
256C22×C3—42no—
257C22×C3—92no—
258C22×C3—181no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3C22×#15711yescharacteristic
262C2×D3—64no—
263C2×D3—64no—
264C2×D3—124no—
265C2×D3—124no—
266C3×C4—24no—
267C3×C4—21no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—44no—
271C3×C4—42no—
272C3×C4—42no—
273C3×C4—92no—
274C3×C4—181no—
275C3×C4C22⋊#15711yescharacteristic
276C3×C4C22⋊#15711yescharacteristic
277Dic3—64no—
278Dic3—64no—
279C32—22no—
280C32—42no—
281C32#523711yescharacteristic, complemented
282C23—91no—
283C23—91no—
284C23C2×C3⋊#15711yescharacteristic
285Q8—182no—
286D4—182no—
287D4—182no—
288D4—182no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—91no—
292C2×C4—92no—
293C2×C4—92no—
294C2×C4—91no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4C2×C3⋊#15711yescharacteristic
298C2×C4C2×C3⋊#15711yescharacteristic
299C8—544no—
300C2×C3—24no—
301C2×C3—21no—
302C2×C3—22no—
303C2×C3—44no—
304C2×C3—44no—
305C2×C3—48no—
306C2×C3—41no—
307C2×C3—42no—
308C2×C3—182no—
309C2×C3—361no—
310C2×C3C2×C22⋊#15711yescharacteristic
311D3—124no—
312D3—124no—
313C22—21no—
314C22—41no—
315C22—92no—
316C22—181no—
317C22—361no—
318C22—361no—
319C22C22×C3⋊#15711yescharacteristic
320C4—21no—
321C4—92no—
322C4—181no—
323C4#8899411yescharacteristic
324C4#8899411yescharacteristic
325C3—22no—
326C3—42no—
327C3#1059380811yescharacteristic, complemented
328C2—21no—
329C2—41no—
330C2—182no—
331C2—361no—
332C2C2×#8899411yescharacteristic
subgroupG/H
centerC4#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#413
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. #11724511C2semidirect
3. #11723113C2semidirect
4. #11723105C2semidirect
183. C33#413semidirect
281. C32#5237semidirect
327. C3#10593808semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
#117245112
#117231132
#117231052
#105938083
#52379