Finite Group Databasebeta

← C2×#11723111 #13173440

Subgroups — C2×#11723111 #13173440

Order 1728.

1160 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×C3×#638C211yesmaximal, characteristic
2C2×#88994C212yesmaximal, complemented
3#11723111C214yesmaximal, direct product factor
4C2×#63786—31nomaximal
5C24×C33C411yescharacteristic, complemented
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157—22no—
8C2×C3×#638—22no—
9C2×C3×#638—22no—
10C2×C3×#638C412yescomplemented
11C2×C3×#638C2212yes—
12C22×C3×C3⋊Dic3C411yescharacteristic, complemented
13#88994—22no—
14#88994C2214yescomplemented
15C22×#638D311yescharacteristic
16C22×C3×#67—62no—
17C2×C22⋊#157—32no—
18#63786—34no—
19C23×C33—21no—
20C23×C33—21no—
21C23×C33—42no—
22C23×C33C2×C411yescharacteristic
23C23×C33C2×C412yescomplemented
24C22×C3×C3⋊D3—42no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C22×C3×C3⋊D3D412yes—
27C22×C3×C3⋊D3D412yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157—24no—
30C2×C3⋊#157—44no—
31C3×#638—24no—
32C3×#638—24no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2×C412yescomplemented
36C2×#834—91nomaximal
37C24×C32—22no—
38C24×C32—42no—
39C24×C32Dic311yescharacteristic, complemented
40C23×C3⋊D3C2×D311yescharacteristic
41C23×C3×D3—62no—
42C22×#157—62no—
43C2×#638—22no—
44C2×#638—22no—
45C2×#638Dic312yescomplemented
46C2×#638C2×D312yes—
47C22×C3⋊Dic3Dic311yescharacteristic, complemented
48C2×C3×#67—64no—
49C2×C3×#67—64no—
50C2×C3×#67—68no—
51C2×C3×#67—68no—
52C22×C3×Dic3—62no—
53C22⋊#157—34no—
54C22⋊#157—62no—
55C22×C33—22no—
56C22×C33—22no—
57C22×C33—42no—
58C22×C33—42no—
59C22×C33—42no—
60C22×C33C22⋊C411yescharacteristic
61C22×C33C22⋊C411yescharacteristic
62C22×C33C22×C411yescharacteristic
63C2×C3×C3⋊D3—24no—
64C2×C3×C3⋊D3—41no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3—41no—
67C2×C3×C3⋊D3C22⋊C412yes—
68C2×C3×C3⋊D3C2×D412yes—
69C3⋊#157—44no—
70C3×C3⋊Dic3—22no—
71C22×C3×D4—91no—
72C22×#67—62no—
73C2×#217—92no—
74#834—94no—
75C23×C32—21no—
76C23×C32—21no—
77C23×C32—24no—
78C23×C32—24no—
79C23×C32—22no—
80C23×C32—24no—
81C23×C32—44no—
82C23×C32—44no—
83C23×C32—42no—
84C23×C32—44no—
85C23×C32—42no—
86C23×C32—48no—
87C23×C32—416no—
88C23×C32C2×Dic311yescharacteristic
89C23×C32C2×Dic312yescomplemented
90C22×C3⋊D3—42no—
91C22×C3⋊D3C2×Dic311yescharacteristic
92C22×C3⋊D3#6712yes—
93C22×C3⋊D3#6712yes—
94C22×C3⋊D3C22×D311yescharacteristic
95C22×C3×D3—62no—
96C22×C3×D3—64no—
97C22×C3×D3—64no—
98C22×C3×D3—62no—
99C22×C3×D3—128no—
100C2×#157—64no—
101C2×#157—124no—
102#638—24no—
103#638—24no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3C2×Dic312yescomplemented
107C3×#67—616no—
108C3×#67—616no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3—64no—
112C2×#99—271noSylow
113C2×C33—21no—
114C2×C33—21no—
115C2×C33—42no—
116C2×C33#9912yescomplemented
117C2×C33C2×C22⋊C411yescharacteristic
118C3×C3⋊D3—22no—
119C3×C3⋊D3—22no—
120C24×C3—22no—
121C24×C3—42no—
122C24×C3—91no—
123C24×C3#15711yescharacteristic, complemented
124C23×D3—62no—
125C2×C3×D4—92no—
126C2×C3×D4—92no—
127C2×C3×D4—182no—
128C2×C3×D4—182no—
129C22×C3×C4—91no—
130C2×#67—64no—
131C2×#67—64no—
132C2×#67—68no—
133C2×#67—68no—
134C22×Dic3—62no—
135C22×Dic3—182no—
136#217—94no—
137#217—182no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—28no—
141C22×C32—28no—
142C22×C32—22no—
143C22×C32—22no—
144C22×C32—22no—
145C22×C32—48no—
146C22×C32—48no—
147C22×C32—42no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32—42no—
151C22×C32—48no—
152C22×C32—42no—
153C22×C32—48no—
154C22×C32—48no—
155C22×C32—416no—
156C22×C32—416no—
157C22×C32—416no—
158C22×C32—42no—
159C22×C32#21711yescharacteristic
160C22×C32#21711yescharacteristic
161C22×C32C22×Dic311yescharacteristic
162C2×C3⋊D3—24no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—41no—
166C2×C3⋊D3#21712yes—
167C2×C3⋊D3C2×#6712yes—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C2×C3×D3—128no—
171C2×C3×D3—124no—
172C2×C3×D3—128no—
173C2×C3×D3—124no—
174#157—124no—
175C3⋊Dic3—22no—
176C3×Dic3—68no—
177C22×D4—91no—
178C2×C22⋊C4—272no—
179#99—274no—
180C33C2×#9911yescharacteristic, complemented, Sylow
181C23×C3—24no—
182C23×C3—24no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—22no—
186C23×C3—24no—
187C23×C3—44no—
188C23×C3—44no—
189C23×C3—42no—
190C23×C3—48no—
191C23×C3—416no—
192C23×C3—42no—
193C23×C3—44no—
194C23×C3—92no—
195C23×C3—91no—
196C23×C3—91no—
197C23×C3—92no—
198C23×C3—362no—
199C23×C3C2×#15711yescharacteristic
200C23×C3C2×#15712yescomplemented
201C22×D3—62no—
202C22×D3—64no—
203C22×D3—64no—
204C22×D3—62no—
205C22×D3—128no—
206C3×D4—184no—
207C3×D4—184no—
208C2×C3×C4—92no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211#67—616no—
212#67—616no—
213C2×Dic3—64no—
214C2×Dic3—64no—
215C2×Dic3—64no—
216C2×Dic3—184no—
217C2×Dic3—364no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32—22no—
223C2×C32—24no—
224C2×C32—44no—
225C2×C32—44no—
226C2×C32—44no—
227C2×C32—42no—
228C2×C32—48no—
229C2×C32—416no—
230C2×C32—42no—
231C2×C32#83412yescomplemented
232C2×C32C2×#21711yescharacteristic
233C3⋊D3—22no—
234C3⋊D3—22no—
235C3×D3—124no—
236C3×D3—124no—
237C24—91no—
238C24C3⋊#15711yescharacteristic, complemented
239C2×D4—92no—
240C2×D4—92no—
241C2×D4—182no—
242C2×D4—182no—
243C22×C4—91no—
244C22×C4—542no—
245C22⋊C4—274no—
246C22⋊C4—542no—
247C22×C3—28no—
248C22×C3—22no—
249C22×C3—28no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—48no—
255C22×C3—48no—
256C22×C3—42no—
257C22×C3—48no—
258C22×C3—48no—
259C22×C3—42no—
260C22×C3—48no—
261C22×C3—416no—
262C22×C3—416no—
263C22×C3—416no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—92no—
269C22×C3—92no—
270C22×C3—184no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3—362no—
274C22×C3C22⋊#15711yescharacteristic
275C22×C3C22⋊#15711yescharacteristic
276C22×C3C22×#15711yescharacteristic
277C2×D3—64no—
278C2×D3—64no—
279C2×D3—128no—
280C2×D3—124no—
281C2×D3—128no—
282C2×D3—124no—
283C3×C4—182no—
284Dic3—68no—
285Dic3—364no—
286C32—22no—
287C32—42no—
288C32C2×#83411yescharacteristic, complemented
289C23—21no—
290C23—21no—
291C23—42no—
292C23—92no—
293C23—91no—
294C23—91no—
295C23—92no—
296C23—362no—
297C23C2×C3⋊#15711yescharacteristic
298C23C2×C3⋊#15712yescomplemented
299D4—184no—
300D4—184no—
301C2×C4—92no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C4—544no—
305C2×C4—1084no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—24no—
311C2×C3—22no—
312C2×C3—44no—
313C2×C3—44no—
314C2×C3—48no—
315C2×C3—416no—
316C2×C3—42no—
317C2×C3—44no—
318C2×C3—42no—
319C2×C3—182no—
320C2×C3—182no—
321C2×C3#6378612yescomplemented
322C2×C3C2×C22⋊#15711yescharacteristic
323D3—124no—
324D3—124no—
325C22—22no—
326C22—22no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—92no—
331C22—92no—
332C22—184no—
333C22—361no—
334C22—361no—
335C22—362no—
336C22#8899411yescharacteristic
337C22#8899411yescharacteristic
338C22C22×C3⋊#15711yescharacteristic
339C4—182no—
340C4—1084no—
341C3—22no—
342C3—42no—
343C3C2×#6378611yescharacteristic, complemented
344C2—21no—
345C2—21no—
346C2—42no—
347C2—182no—
348C2—182no—
349C2#1172311112yesdirect product factor
350C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC24×C33C4
socleC22×C33C22⋊C4
prime radicalC22×C3×#638C2
cosocleC22×C3×C3⋊D3C23
hypercenterC24C3⋊#157
hypocenterC33C2×#99
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
3. #11723111C2direct (N × H)
5. C24×C33C4semidirect
10. C2×C3×#638C4semidirect
12. C22×C3×C3⋊Dic3C4semidirect
14. #88994C22semidirect
23. C23×C33C2×C4semidirect
35. C2×C3×C3⋊Dic3C2×C4semidirect
39. C24×C32Dic3semidirect
45. C2×#638Dic3semidirect
47. C22×C3⋊Dic3Dic3semidirect
89. C23×C32C2×Dic3semidirect
106. C2×C3⋊Dic3C2×Dic3semidirect
116. C2×C33#99semidirect
123. C24×C3#157semidirect
180. C33C2×#99semidirect
200. C23×C3C2×#157semidirect
231. C2×C32#834semidirect
238. C24C3⋊#157semidirect
288. C32C2×#834semidirect
298. C23C2×C3⋊#157semidirect
321. C2×C3#63786semidirect
343. C3C2×#63786semidirect
349. C2#11723111direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×#889942
#117231112
C2×#637863
C2×#8349