Finite Group Databasebeta

← C4×#88994 #13173407

Subgroups — C4×#88994 #13173407

Order 1728.

1096 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 145 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⋊D3C211yesmaximal, characteristic
2C2×#88994C212yesmaximal
3C2×C4×C3⋊#157C212yesmaximal, complemented
4#11723106C212yesmaximal, complemented
5C4×C22⋊#157—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157C2214yes—
8C22×C33×C4C411yescharacteristic, complemented
9C22×C3×C3⋊Dic3C411yescharacteristic, complemented
10C2×C3×C4×C3⋊D3—24no—
11C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
12C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15#88994C418yesdirect product factor
16C4×C3⋊#157—24no—
17C22×C4×C3⋊D3D311yescharacteristic
18C22×C3×C4×D3—62no—
19C2×C22⋊#157—32no—
20C2×C4×#157—32no—
21#63781—32no—
22C23×C33C2×C411yescharacteristic
23C22×C3×C3⋊D3—22no—
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C22×C3×C3⋊D3C2×C414yes—
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×C3⋊#157—24no—
29C2×C3⋊#157—24no—
30C2×C3⋊#157C2×C418yescomplemented
31C2×C33×C4—22no—
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C3×C4×C3⋊D3—22no—
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3D414yes—
41C4×#217—91nomaximal
42C23×C3⋊D3C2×D311yescharacteristic
43C23×C3×D3—62no—
44C22×#157—34no—
45C22×C32×C4—22no—
46C22×C32×C4—42no—
47C22×C32×C4Dic311yescharacteristic, complemented
48C22×C3⋊Dic3Dic311yescharacteristic, complemented
49C2×C4×C3⋊D3—24no—
50C2×C4×C3⋊D3Dic311yescharacteristic, complemented
51C2×C4×C3⋊D3Dic311yescharacteristic, complemented
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C22×C3×Dic3—62no—
55C2×C3×C4×D3—62no—
56C2×C3×C4×D3—62no—
57C2×C3×C4×D3—62no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—616no—
60C22⋊#157—38no—
61C4×#157—64no—
62C22×C33—21no—
63C22×C33—21no—
64C22×C33C4212yescomplemented
65C22×C33C22×C411yescharacteristic
66C2×C3×C3⋊D3—21no—
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—28no—
69C2×C3×C3⋊D3C4212yescomplemented
70C2×C3×C3⋊D3C22×C412yes—
71C2×C3×C3⋊D3C2×D412yes—
72C2×C3×C3⋊D3#4112yes—
73C3⋊#157—28no—
74C33×C4—21no—
75C33×C4C22⋊C412yescomplemented
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3C22⋊C412yescomplemented
78C23×C3×C4—91no—
79C22×C4×D3—62no—
80C2×#217—92no—
81C2×C4×Dic3—92no—
82C3⋊#95—92no—
83C23×C32—22no—
84C23×C32—42no—
85C23×C32C2×Dic311yescharacteristic
86C22×C3⋊D3—22no—
87C22×C3⋊D3—22no—
88C22×C3⋊D3C4×D314yes—
89C22×C3⋊D3C2×Dic311yescharacteristic
90C22×C3⋊D3C22×D311yescharacteristic
91C22×C3×D3—68no—
92C22×C3×D3—62no—
93C22×C3×D3—62no—
94C22×C3×D3—68no—
95C22×C3×D3—68no—
96C2×#157—38no—
97C2×#157—64no—
98C2×#157—64no—
99C2×C32×C4—22no—
100C2×C32×C4—28no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—48no—
104C2×C32×C4—42no—
105C2×C32×C4—42no—
106C2×C32×C4C2×Dic311yescharacteristic
107C2×C32×C4C2×Dic311yescharacteristic
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3C2×Dic311yescharacteristic
110C2×C3⋊Dic3C2×Dic311yescharacteristic
111C4×C3⋊D3—22no—
112C4×C3⋊D3—22no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3#6714yes—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—68no—
118C3×C4×D3—68no—
119C3×C4×D3—68no—
120C3×C4×D3—68no—
121C3×C4×D3—68no—
122C4×C22⋊C4—271noSylow
123C2×C33—22no—
124C2×C33C2×C4211yescharacteristic
125C2×C33C2×C22⋊C411yescharacteristic
126C2×C33#11711yescharacteristic
127C3×C3⋊D3—22no—
128C3×C3⋊D3C4×D414yes—
129C24×C3—91no—
130C23×D3—62no—
131C22×C3×C4—22no—
132C22×C3×C4—42no—
133C22×C3×C4—91no—
134C22×C3×C4—91no—
135C22×C3×C4—91no—
136C22×C3×C4—91no—
137C22×C3×C4—91no—
138C22×C3×C4—184no—
139C22×C3×C4#15711yescharacteristic, complemented
140C22×Dic3—62no—
141C22×Dic3—94no—
142C2×C4×D3—62no—
143C2×C4×D3—62no—
144C2×C4×D3—62no—
145C2×C4×D3—62no—
146C2×C4×D3—616no—
147#217—98no—
148C4×Dic3—184no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—24no—
152C22×C32—24no—
153C22×C32—22no—
154C22×C32—24no—
155C22×C32—44no—
156C22×C32—44no—
157C22×C32—44no—
158C22×C32—42no—
159C22×C32C4×Dic312yescomplemented
160C22×C32C22×Dic311yescharacteristic
161C2×C3⋊D3—21no—
162C2×C3⋊D3—21no—
163C2×C3⋊D3—28no—
164C2×C3⋊D3C4×Dic312yescomplemented
165C2×C3⋊D3C2×C4×D312yes—
166C2×C3⋊D3#23512yes—
167C2×C3⋊D3C2×#6712yes—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C2×C3×D3—632no—
174C2×C3×D3—64no—
175#157—68no—
176C32×C4—21no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4—44no—
181C32×C4#21712yescomplemented
182C3⋊Dic3—21no—
183C3⋊Dic3#21712yescomplemented
184C3×Dic3—64no—
185C3×Dic3—64no—
186C23×C4—91no—
187C2×C22⋊C4—272no—
188C2×C42—272no—
189#95—272no—
190C33C4×C22⋊C411yescharacteristic, complemented, Sylow
191C23×C3—22no—
192C23×C3—42no—
193C23×C3—91no—
194C23×C3—91no—
195C23×C3—94no—
196C23×C3—182no—
197C23×C3—182no—
198C23×C3C2×#15711yescharacteristic
199C22×D3—68no—
200C22×D3—62no—
201C22×D3—62no—
202C22×D3—68no—
203C22×D3—68no—
204C2×C3×C4—28no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—48no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—91no—
212C2×C3×C4—94no—
213C2×C3×C4—91no—
214C2×C3×C4—182no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4C2×#15711yescharacteristic
219C2×C3×C4C2×#15711yescharacteristic
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—68no—
223C2×Dic3—98no—
224C2×Dic3—184no—
225C2×Dic3—184no—
226C4×D3—68no—
227C4×D3—68no—
228C4×D3—68no—
229C4×D3—68no—
230C2×C32—22no—
231C2×C32—28no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—22no—
235C2×C32—48no—
236C2×C32—42no—
237C2×C32—42no—
238C2×C32—42no—
239C2×C32C2×C4×Dic311yescharacteristic
240C2×C32#92611yescharacteristic
241C2×C32C2×#21711yescharacteristic
242C3⋊D3—22no—
243C3⋊D3C4×#6714yes—
244C3×D3—68no—
245C3×D3—68no—
246C24—91no—
247C22×C4—91no—
248C22×C4—91no—
249C22×C4—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—184no—
253C22×C4—274no—
254C22×C4C3⋊#15711yescharacteristic, complemented
255C22⋊C4—278no—
256C42—544no—
257C22×C3—24no—
258C22×C3—24no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—22no—
262C22×C3—24no—
263C22×C3—44no—
264C22×C3—44no—
265C22×C3—42no—
266C22×C3—44no—
267C22×C3—92no—
268C22×C3—92no—
269C22×C3—92no—
270C22×C3—92no—
271C22×C3—188no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3C4×#15712yescomplemented
275C22×C3C22×#15711yescharacteristic
276C2×D3—64no—
277C2×D3—64no—
278C2×D3—64no—
279C2×D3—64no—
280C2×D3—64no—
281C2×D3—632no—
282C2×D3—64no—
283C3×C4—24no—
284C3×C4—21no—
285C3×C4—24no—
286C3×C4—44no—
287C3×C4—44no—
288C3×C4—92no—
289C3×C4—181no—
290C3×C4C22⋊#15712yescomplemented
291Dic3—64no—
292Dic3—64no—
293Dic3—188no—
294C32—22no—
295C32—42no—
296C32C4×#21711yescharacteristic, complemented
297C23—91no—
298C23—91no—
299C23—94no—
300C23—182no—
301C23—182no—
302C23C2×C3⋊#15711yescharacteristic
303C2×C4—22no—
304C2×C4—91no—
305C2×C4—94no—
306C2×C4—91no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4—182no—
311C2×C4—278no—
312C2×C4—544no—
313C2×C4—544no—
314C2×C4C2×C3⋊#15711yescharacteristic
315C2×C4C2×C3⋊#15711yescharacteristic
316C2×C3—28no—
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—48no—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3—42no—
325C2×C3—94no—
326C2×C3—182no—
327C2×C3C2×C4×#15711yescharacteristic
328C2×C3#6428611yescharacteristic
329C2×C3C2×C22⋊#15711yescharacteristic
330D3—68no—
331D3—68no—
332C22—21no—
333C22—21no—
334C22—92no—
335C22—92no—
336C22—92no—
337C22—92no—
338C22—188no—
339C22—181no—
340C22—181no—
341C22C4×C3⋊#15712yescomplemented
342C22C22×C3⋊#15711yescharacteristic
343C4—21no—
344C4—92no—
345C4—181no—
346C4—548no—
347C4#8899412yesdirect product factor
348C3—22no—
349C3—42no—
350C3C4×C22⋊#15711yescharacteristic, complemented
351C2—22no—
352C2—94no—
353C2—182no—
354C2C2×C4×C3⋊#15711yescharacteristic
355C2#1172451411yescharacteristic
356C2C2×#8899411yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊#157
commutatorC2×C33C2×C42
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C4×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×C4×C3⋊#157C2semidirect
4. #11723106C2semidirect
8. C22×C33×C4C4semidirect
9. C22×C3×C3⋊Dic3C4semidirect
11. C2×C3×C4×C3⋊D3C4semidirect
12. C2×C3×C4×C3⋊D3C4semidirect
15. #88994C4direct (N × H)
30. C2×C3⋊#157C2×C4semidirect
47. C22×C32×C4Dic3semidirect
48. C22×C3⋊Dic3Dic3semidirect
50. C2×C4×C3⋊D3Dic3semidirect
51. C2×C4×C3⋊D3Dic3semidirect
64. C22×C33C42semidirect
69. C2×C3×C3⋊D3C42semidirect
75. C33×C4C22⋊C4semidirect
77. C3×C3⋊Dic3C22⋊C4semidirect
139. C22×C3×C4#157semidirect
159. C22×C32C4×Dic3semidirect
164. C2×C3⋊D3C4×Dic3semidirect
181. C32×C4#217semidirect
183. C3⋊Dic3#217semidirect
190. C33C4×C22⋊C4semidirect
254. C22×C4C3⋊#157semidirect
274. C22×C3C4×#157semidirect
290. C3×C4C22⋊#157semidirect
296. C32C4×#217semidirect
341. C22C4×C3⋊#157semidirect
347. C4#88994direct (N × H)
350. C3C4×C22⋊#157semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×#889942
C2×C4×C3⋊#1572
#117231062
C4×C22⋊#1573
C4×#2179