Finite Group Databasebeta

← C2×C3×#64188 #13180086

Subgroups — C2×C3×#64188 #13180086

Order 1728.

1046 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 50 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×C22⋊(C22⋊C9)C211yesmaximal, characteristic, complemented
2C3×#64188C212yesmaximal, direct product factor
3C2×#64188C311yesmaximal, characteristic, direct product factor
4C2×C3×#953—31nomaximal
5C3×C22⋊(C22⋊C9)C2211yescharacteristic, complemented
6C2×C3×(C22⋊C9)⋊C2—43nomaximal
7C25×C32D311yescharacteristic
8C22×C3×#67—33no—
9C2×C22⋊(C22⋊C9)—21no—
10C2×C22⋊(C22⋊C9)C2×C311yescharacteristic, complemented
11#64188C2×C312yesdirect product factor
12C3×#953—36no—
13C3×#953—32no—
14C2×C3×#217—33no—
15C2×C3×C22⋊C9—43no—
16C2×C3×C22⋊C9—81no—
17C3×(C22⋊C9)⋊C2—46no—
18C2×C3×#120—91no—
19C2×#953—31no—
20C24×C32—33no—
21C24×C32—33no—
22C24×C32—61no—
23C24×C32—61no—
24C24×C32C2×D311yescharacteristic
25C23×C3×D3—31no—
26C2×C3×#67—36no—
27C2×C3×#67—36no—
28C2×C3×#67—63no—
29C2×C3×#67—63no—
30C2×C3×#67—63no—
31C2×C3×#67—63no—
32C22×C3×Dic3—33no—
33C22⋊(C22⋊C9)—21no—
34C22⋊(C22⋊C9)C22×C311yescharacteristic, complemented
35C2×(C22⋊C9)⋊C2—43no—
36C3×#217—36no—
37C3×#217—36no—
38C2×C3×D9—161no—
39C3×C22⋊C9—43no—
40C3×C22⋊C9—81no—
41C25×C3—21no—
42C25×C3D911yescharacteristic, complemented
43C25×C3C3×D311yescharacteristic
44C22×C3×D4—93no—
45C22×#67—33no—
46C3×#120—96no—
47C3×#120—92no—
48C2×C3×C22⋊C4—93no—
49#953—36no—
50#953—32no—
51C2×#217—33no—
52C23×C32—21no—
53C23×C32—31no—
54C23×C32—33no—
55C23×C32—33no—
56C23×C32—33no—
57C23×C32—33no—
58C23×C32—33no—
59C23×C32—61no—
60C23×C32—61no—
61C23×C32—66no—
62C23×C32—63no—
63C23×C32—63no—
64C23×C32—63no—
65C23×C32S413yes—
66C22×C3×D3—32no—
67C22×C3×D3—63no—
68C22×C3×D3—63no—
69C3×#67—66no—
70C3×#67—66no—
71C3×#67—66no—
72C3×#67—66no—
73C2×C3×Dic3—36no—
74C2×C3×Dic3—63no—
75C2×C3×Dic3—63no—
76C2×C22⋊C9—43no—
77C2×C22⋊C9—81no—
78C2×C22⋊C9—82no—
79C2×C22⋊C9—83no—
80(C22⋊C9)⋊C2—46no—
81C2×#120—91noSylow
82C2×C3×C9—161no—
83C3×D9—162no—
84C24×C3—21no—
85C24×C3—33no—
86C24×C3—33no—
87C24×C3—33no—
88C24×C3—33no—
89C24×C3—63no—
90C24×C3—63no—
91C24×C3—61no—
92C24×C3—61no—
93C24×C3—62no—
94C24×C3—61no—
95C24×C3—61no—
96C24×C3—62no—
97C24×C3—91no—
98C24×C3C2×D911yescharacteristic, complemented
99C24×C3C2×C3×D311yescharacteristic
100C23×D3—31no—
101C2×C3×D4—96no—
102C2×C3×D4—96no—
103C2×C3×D4—183no—
104C2×C3×D4—183no—
105C2×C3×D4—183no—
106C2×C3×D4—183no—
107C22×C3×C4—93no—
108C2×#67—36no—
109C2×#67—36no—
110C2×#67—63no—
111C2×#67—63no—
112C2×#67—63no—
113C2×#67—63no—
114C22×Dic3—33no—
115C3×C22⋊C4—96no—
116C3×C22⋊C4—96no—
117#217—36no—
118#217—36no—
119C22×C32—21no—
120C22×C32—33no—
121C22×C32—31no—
122C22×C32—33no—
123C22×C32—33no—
124C22×C32—33no—
125C22×C32—33no—
126C22×C32—63no—
127C22×C32—66no—
128C22×C32—63no—
129C22×C32—63no—
130C22×C32—61no—
131C22×C32—61no—
132C22×C32C2×S413yes—
133C2×C3×D3—66no—
134C2×C3×D3—123no—
135C2×C3×D3—121no—
136C3×Dic3—66no—
137C2×D9—161no—
138C22⋊C9—43no—
139C22⋊C9—81no—
140C22⋊C9—82no—
141C22⋊C9—83no—
142C25C3×D911yescharacteristic, complemented
143C22×D4—93no—
144#120—96no—
145#120—92no—
146C2×C22⋊C4—93no—
147C3×C9—161noSylow
148C23×C3—21no—
149C23×C3—22no—
150C23×C3—21no—
151C23×C3—23no—
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—33no—
155C23×C3—33no—
156C23×C3—33no—
157C23×C3—33no—
158C23×C3—33no—
159C23×C3—33no—
160C23×C3—33no—
161C23×C3—33no—
162C23×C3—33no—
163C23×C3—33no—
164C23×C3—63no—
165C23×C3—63no—
166C23×C3—61no—
167C23×C3—63no—
168C23×C3—63no—
169C23×C3—63no—
170C23×C3—61no—
171C23×C3—61no—
172C23×C3—62no—
173C23×C3—61no—
174C23×C3—61no—
175C23×C3—62no—
176C23×C3—66no—
177C23×C3—66no—
178C23×C3—63no—
179C23×C3—63no—
180C23×C3—63no—
181C23×C3—63no—
182C23×C3—63no—
183C23×C3—63no—
184C23×C3—66no—
185C23×C3—612no—
186C23×C3—66no—
187C23×C3—66no—
188C23×C3—92no—
189C23×C3—183no—
190C23×C3—183no—
191C23×C3(C22⋊C9)⋊C213yescomplemented
192C23×C3C3×S413yes—
193C22×D3—32no—
194C22×D3—63no—
195C22×D3—63no—
196C3×D4—186no—
197C3×D4—186no—
198C3×D4—186no—
199C3×D4—186no—
200C2×C3×C4—96no—
201C2×C3×C4—183no—
202C2×C3×C4—183no—
203#67—66no—
204#67—66no—
205#67—66no—
206#67—66no—
207C2×Dic3—36no—
208C2×Dic3—63no—
209C2×Dic3—63no—
210C2×C32—33no—
211C2×C32—33no—
212C2×C32—61no—
213C2×C32—61no—
214C2×C32#102011yescharacteristic
215C3×D3—122no—
216C2×C9—161no—
217C2×C9—321no—
218D9—162no—
219C24—33no—
220C24—33no—
221C24—61no—
222C24—61no—
223C24—91no—
224C24C2×C3×D911yescharacteristic, complemented
225C2×D4—96no—
226C2×D4—96no—
227C2×D4—183no—
228C2×D4—183no—
229C2×D4—183no—
230C2×D4—183no—
231C22×C4—93no—
232C22⋊C4—96no—
233C22⋊C4—96no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—23no—
238C22×C3—33no—
239C22×C3—33no—
240C22×C3—31no—
241C22×C3—33no—
242C22×C3—31no—
243C22×C3—33no—
244C22×C3—33no—
245C22×C3—33no—
246C22×C3—33no—
247C22×C3—33no—
248C22×C3—33no—
249C22×C3—33no—
250C22×C3—63no—
251C22×C3—63no—
252C22×C3—61no—
253C22×C3—63no—
254C22×C3—63no—
255C22×C3—63no—
256C22×C3—62no—
257C22×C3—63no—
258C22×C3—66no—
259C22×C3—63no—
260C22×C3—63no—
261C22×C3—66no—
262C22×C3—63no—
263C22×C3—63no—
264C22×C3—63no—
265C22×C3—62no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—66no—
269C22×C3—612no—
270C22×C3—66no—
271C22×C3—66no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—186no—
275C22×C3—363no—
276C22×C3—361no—
277C22×C3C2×(C22⋊C9)⋊C213yescomplemented
278C22×C3C2×C3×S413yes—
279C2×D3—66no—
280C2×D3—123no—
281C2×D3—121no—
282C3×C4—186no—
283Dic3—66no—
284C32C2×#102011yescharacteristic
285C9—161no—
286C9—321no—
287C23—21no—
288C23—31no—
289C23—33no—
290C23—33no—
291C23—33no—
292C23—33no—
293C23—33no—
294C23—61no—
295C23—61no—
296C23—66no—
297C23—63no—
298C23—63no—
299C23—63no—
300C23—92no—
301C23—183no—
302C23—183no—
303C23C3×(C22⋊C9)⋊C213yescomplemented
304D4—186no—
305D4—186no—
306D4—186no—
307D4—186no—
308C2×C4—96no—
309C2×C4—183no—
310C2×C4—183no—
311C2×C3—21no—
312C2×C3—33no—
313C2×C3—33no—
314C2×C3—33no—
315C2×C3—33no—
316C2×C3—63no—
317C2×C3—63no—
318C2×C3—62no—
319C2×C3—62no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—362no—
325C2×C3#6418811yescharacteristic, direct product factor
326C2×C3C3×#102011yescharacteristic
327D3—122no—
328C22—21no—
329C22—33no—
330C22—31no—
331C22—33no—
332C22—33no—
333C22—33no—
334C22—33no—
335C22—63no—
336C22—66no—
337C22—63no—
338C22—63no—
339C22—61no—
340C22—61no—
341C22—186no—
342C22—363no—
343C22—361no—
344C22C2×C3×(C22⋊C9)⋊C213yescomplemented
345C4—186no—
346C3—21no—
347C3C2×#6418811yescharacteristic, direct product factor
348C3C2×C3×#102011yescharacteristic
349C2—33no—
350C2—33no—
351C2—61no—
352C2—61no—
353C2—362no—
354C2C3×#6418811yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#64188
commutator, cosocle, hypocenterC22⋊(C22⋊C9)C22×C3
FrattiniC3C2×C3×#1020
Fitting, socleC25×C32D3
LCSG ▷ C22⋊(C22⋊C9)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊(C22⋊C9)C2semidirect
2. C3×#64188C2direct (N × H)
3. C2×#64188C3direct (N × H)
5. C3×C22⋊(C22⋊C9)C22semidirect
10. C2×C22⋊(C22⋊C9)C2×C3semidirect
11. #64188C2×C3direct (N × H)
34. C22⋊(C22⋊C9)C22×C3semidirect
42. C25×C3D9semidirect
98. C24×C3C2×D9semidirect
142. C25C3×D9semidirect
191. C23×C3(C22⋊C9)⋊C2semidirect
224. C24C2×C3×D9semidirect
277. C22×C3C2×(C22⋊C9)⋊C2semidirect
303. C23C3×(C22⋊C9)⋊C2semidirect
325. C2×C3#64188direct (N × H)
344. C22C2×C3×(C22⋊C9)⋊C2semidirect
347. C3C2×#64188direct (N × H)
354. C2C3×#64188direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊(C22⋊C9)2
C3×#641882
C2×#641883
C2×C3×#9533
C2×C3×(C22⋊C9)⋊C24