Finite Group Databasebeta

← C2×#11723099 #13173388

Subgroups — C2×#11723099 #13173388

Order 1728.

1000 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 1000 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×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×#88994C212yesmaximal, complemented
3#11723099C214yesmaximal, direct product factor
4C2×#63774—31nomaximal
5C23×C3×C3⋊D3C411yescharacteristic, complemented
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157—22no—
8C22×C33×C4C411yescharacteristic, complemented
9C2×C3×C4⋊(C3⋊D3)—22no—
10C2×C3×C4⋊(C3⋊D3)—22no—
11C2×C3×C4⋊(C3⋊D3)C412yescomplemented
12C2×C3×C4⋊(C3⋊D3)C2212yes—
13#88994—22no—
14#88994C2214yescomplemented
15C22×C4⋊(C3⋊D3)D311yescharacteristic
16C22×C3×D12—62no—
17C2×C22⋊#157—32no—
18#63774—34no—
19C23×C33C2×C411yescharacteristic
20C22×C3×C3⋊D3—21no—
21C22×C3×C3⋊D3—21no—
22C22×C3×C3⋊D3—42no—
23C22×C3×C3⋊D3—42no—
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C22×C3×C3⋊D3C2×C412yescomplemented
26C22×C3×C3⋊D3D412yes—
27C22×C3×C3⋊D3D412yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157—24no—
30C2×C3⋊#157—44no—
31C2×C33×C4—21no—
32C2×C33×C4—21no—
33C2×C33×C4C2×C412yescomplemented
34C3×C4⋊(C3⋊D3)—24no—
35C3×C4⋊(C3⋊D3)—24no—
36C2×#834—91nomaximal
37C23×C3⋊D3Dic311yescharacteristic, complemented
38C23×C3⋊D3C2×D311yescharacteristic
39C23×C3×D3—62no—
40C23×C3×D3—62no—
41C22×#157—62no—
42C22×C32×C4—22no—
43C22×C32×C4—42no—
44C22×C32×C4Dic311yescharacteristic, complemented
45C2×C4⋊(C3⋊D3)—22no—
46C2×C4⋊(C3⋊D3)—22no—
47C2×C4⋊(C3⋊D3)Dic312yescomplemented
48C2×C4⋊(C3⋊D3)C2×D312yes—
49C2×C3×D12—64no—
50C2×C3×D12—64no—
51C2×C3×D12—68no—
52C2×C3×D12—68no—
53C22⋊#157—34no—
54C22⋊#157—62no—
55C22×C33—22no—
56C22×C33C22⋊C411yescharacteristic
57C22×C33C22⋊C411yescharacteristic
58C22×C33C22×C411yescharacteristic
59C2×C3×C3⋊D3—24no—
60C2×C3×C3⋊D3—22no—
61C2×C3×C3⋊D3—41no—
62C2×C3×C3⋊D3—42no—
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3—42no—
67C2×C3×C3⋊D3C22⋊C412yes—
68C2×C3×C3⋊D3C2×D412yes—
69C3⋊#157—44no—
70C33×C4—22no—
71C22×C3×D4—91no—
72C22×D12—62no—
73C2×#217—92no—
74#834—94no—
75C23×C32—22no—
76C23×C32—42no—
77C23×C32C2×Dic311yescharacteristic
78C22×C3⋊D3—21no—
79C22×C3⋊D3—21no—
80C22×C3⋊D3—42no—
81C22×C3⋊D3—42no—
82C22×C3⋊D3C2×Dic311yescharacteristic
83C22×C3⋊D3C2×Dic312yescomplemented
84C22×C3⋊D3#6712yes—
85C22×C3⋊D3#6712yes—
86C22×C3⋊D3C22×D311yescharacteristic
87C22×C3×D3—62no—
88C22×C3×D3—64no—
89C22×C3×D3—64no—
90C22×C3×D3—64no—
91C22×C3×D3—62no—
92C22×C3×D3—64no—
93C22×C3×D3—64no—
94C22×C3×D3—128no—
95C22×C3×D3—128no—
96C2×#157—64no—
97C2×#157—124no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—44no—
104C2×C32×C4—44no—
105C2×C32×C4—44no—
106C2×C32×C4C2×Dic312yescomplemented
107C4⋊(C3⋊D3)—24no—
108C4⋊(C3⋊D3)—24no—
109C3×D12—616no—
110C3×D12—616no—
111C2×#99—271noSylow
112C2×C33—21no—
113C2×C33—21no—
114C2×C33#9912yescomplemented
115C2×C33C2×C22⋊C411yescharacteristic
116C3×C3⋊D3—22no—
117C3×C3⋊D3—22no—
118C3×C3⋊D3—42no—
119C24×C3—91no—
120C24×C3—91no—
121C23×D3—62no—
122C23×D3—62no—
123C2×C3×D4—92no—
124C2×C3×D4—92no—
125C2×C3×D4—182no—
126C2×C3×D4—182no—
127C22×C3×C4—22no—
128C22×C3×C4—42no—
129C22×C3×C4#15711yescharacteristic, complemented
130C22×Dic3—182no—
131C2×D12—64no—
132C2×D12—64no—
133C2×D12—68no—
134C2×D12—68no—
135#217—94no—
136#217—182no—
137C22×C32—22no—
138C22×C32—28no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—48no—
143C22×C32—42no—
144C22×C32—42no—
145C22×C32—42no—
146C22×C32#21711yescharacteristic
147C22×C32#21711yescharacteristic
148C22×C32C22×Dic311yescharacteristic
149C2×C3⋊D3—22no—
150C2×C3⋊D3—24no—
151C2×C3⋊D3—41no—
152C2×C3⋊D3—42no—
153C2×C3⋊D3—41no—
154C2×C3⋊D3—42no—
155C2×C3⋊D3—42no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3#21712yes—
158C2×C3⋊D3C2×#6712yes—
159C2×C3×D3—64no—
160C2×C3×D3—64no—
161C2×C3×D3—68no—
162C2×C3×D3—128no—
163C2×C3×D3—124no—
164C2×C3×D3—128no—
165C2×C3×D3—124no—
166C2×C3×D3—128no—
167C2×C3×D3—128no—
168C2×C3×D3—128no—
169#157—124no—
170C32×C4—22no—
171C32×C4—28no—
172C32×C4—48no—
173C22×D4—91no—
174C2×C22⋊C4—272no—
175#99—274no—
176C33C2×#9911yescharacteristic, complemented, Sylow
177C23×C3—22no—
178C23×C3—42no—
179C23×C3—92no—
180C23×C3—91no—
181C23×C3—91no—
182C23×C3—92no—
183C23×C3—92no—
184C23×C3—181no—
185C23×C3—181no—
186C23×C3—362no—
187C23×C3—362no—
188C23×C3C2×#15711yescharacteristic
189C22×D3—62no—
190C22×D3—64no—
191C22×D3—64no—
192C22×D3—64no—
193C22×D3—62no—
194C22×D3—64no—
195C22×D3—64no—
196C22×D3—128no—
197C22×D3—128no—
198C3×D4—184no—
199C3×D4—184no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—24no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4C2×#15712yescomplemented
209C2×Dic3—184no—
210C2×Dic3—364no—
211D12—616no—
212D12—616no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32—22no—
219C2×C32—44no—
220C2×C32—44no—
221C2×C32—44no—
222C2×C32—42no—
223C2×C32#83412yescomplemented
224C2×C32C2×#21711yescharacteristic
225C3⋊D3—22no—
226C3⋊D3—22no—
227C3⋊D3—42no—
228C3×D3—124no—
229C3×D3—124no—
230C3×D3—128no—
231C24—91no—
232C24—91no—
233C2×D4—92no—
234C2×D4—92no—
235C2×D4—182no—
236C2×D4—182no—
237C22×C4—542no—
238C22×C4C3⋊#15711yescharacteristic, complemented
239C22⋊C4—274no—
240C22⋊C4—542no—
241C22×C3—28no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—48no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—42no—
250C22×C3—92no—
251C22×C3—92no—
252C22×C3—184no—
253C22×C3—182no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3—362no—
257C22×C3—362no—
258C22×C3—362no—
259C22×C3—362no—
260C22×C3C22⋊#15711yescharacteristic
261C22×C3C22⋊#15711yescharacteristic
262C22×C3C22×#15711yescharacteristic
263C2×D3—64no—
264C2×D3—64no—
265C2×D3—68no—
266C2×D3—128no—
267C2×D3—124no—
268C2×D3—128no—
269C2×D3—124no—
270C2×D3—128no—
271C2×D3—128no—
272C2×D3—128no—
273C3×C4—28no—
274C3×C4—22no—
275C3×C4—48no—
276Dic3—364no—
277C32—22no—
278C32—42no—
279C32C2×#83411yescharacteristic, complemented
280C23—92no—
281C23—91no—
282C23—91no—
283C23—92no—
284C23—92no—
285C23—181no—
286C23—181no—
287C23—362no—
288C23—362no—
289C23C2×C3⋊#15711yescharacteristic
290D4—184no—
291D4—184no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—544no—
295C2×C4—1084no—
296C2×C4C2×C3⋊#15712yescomplemented
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—24no—
302C2×C3—22no—
303C2×C3—44no—
304C2×C3—44no—
305C2×C3—44no—
306C2×C3—42no—
307C2×C3—182no—
308C2×C3—182no—
309C2×C3—362no—
310C2×C3#6377412yescomplemented
311C2×C3C2×C22⋊#15711yescharacteristic
312D3—124no—
313D3—124no—
314D3—128no—
315C22—22no—
316C22—92no—
317C22—92no—
318C22—184no—
319C22—182no—
320C22—361no—
321C22—361no—
322C22—362no—
323C22—362no—
324C22—362no—
325C22—362no—
326C22#8899411yescharacteristic
327C22#8899411yescharacteristic
328C22C22×C3⋊#15711yescharacteristic
329C4—22no—
330C4—1084no—
331C3—22no—
332C3—42no—
333C3C2×#6377411yescharacteristic, complemented
334C2—21no—
335C2—21no—
336C2—182no—
337C2—182no—
338C2—362no—
339C2#1172309912yesdirect product factor
340C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
cosocleC22×C3×C3⋊D3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×#99
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
3. #11723099C2direct (N × H)
5. C23×C3×C3⋊D3C4semidirect
8. C22×C33×C4C4semidirect
11. C2×C3×C4⋊(C3⋊D3)C4semidirect
14. #88994C22semidirect
25. C22×C3×C3⋊D3C2×C4semidirect
33. C2×C33×C4C2×C4semidirect
37. C23×C3⋊D3Dic3semidirect
44. C22×C32×C4Dic3semidirect
47. C2×C4⋊(C3⋊D3)Dic3semidirect
83. C22×C3⋊D3C2×Dic3semidirect
106. C2×C32×C4C2×Dic3semidirect
114. C2×C33#99semidirect
129. C22×C3×C4#157semidirect
176. C33C2×#99semidirect
208. C2×C3×C4C2×#157semidirect
223. C2×C32#834semidirect
238. C22×C4C3⋊#157semidirect
279. C32C2×#834semidirect
296. C2×C4C2×C3⋊#157semidirect
310. C2×C3#63774semidirect
333. C3C2×#63774semidirect
339. C2#11723099direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C2×#889942
#117230992
C2×#637743
C2×#8349