Finite Group Databasebeta

← C2×C3×D32⋊D3 #13049852

Subgroups — C2×C3×D32⋊D3 #13049852

Order 1296.

561 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 62 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×#1113C211yesmaximal, characteristic, complemented
2C2×C32×D32C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#157C211yesmaximal, characteristic, complemented
4C3×D32⋊D3C214yesmaximal, direct product factor
5C2×D32⋊D3C311yesmaximal, characteristic, direct product factor
6C2×C3×GO+(4,𝔽2)—31nomaximal
7C2×C32×C3⋊D3—21no—
8C2×C32×C3⋊D3C2211yescharacteristic
9C2×C33×D3—21no—
10C3×#1113—21no—
11C3×#1113C2212yescomplemented
12C32×D32—21no—
13C32×D32C2212yescomplemented
14C3×C3⋊#157C2212yescomplemented
15C22×C32×D3—61no—
16C2×#1113C2×C311yescharacteristic, complemented
17C2×C3×D32—21no—
18C2×C3×D32—31no—
19C2×C3×D32—61no—
20C2×C3×D32D311yescharacteristic, complemented
21C2×C3×D32C2×C311yescharacteristic, complemented
22C2×C3⋊#157C2×C311yescharacteristic, complemented
23C2×C3×#157—31no—
24D32⋊D3C2×C314yesdirect product factor
25C3×GO+(4,𝔽2)—34no—
26C2×C34D411yescharacteristic, complemented
27C32×C3⋊D3—22no—
28C32×C3⋊D3D411yescharacteristic, complemented
29C32×C3⋊D3C2311yescharacteristic
30C33×D3—22no—
31C2×GO+(4,𝔽2)—31no—
32C2×C3×#67—91nomaximal
33C22×C33—61no—
34C2×C3×C3⋊D3—21no—
35C2×C3×C3⋊D3—21no—
36C2×C3×C3⋊D3—21no—
37C2×C3×C3⋊D3—41no—
38C2×C3×C3⋊D3C2×D311yescharacteristic
39C2×C3×C3⋊D3C22×C311yescharacteristic
40C2×C32×D3—21no—
41C2×C32×D3—22no—
42C2×C32×D3—21no—
43C2×C32×D3—21no—
44C2×C32×D3—42no—
45C2×C32×D3—41no—
46C2×C32×D3—41no—
47C2×C32×D3—62no—
48C2×C32×D3—62no—
49C2×C32×D3—61no—
50C2×C32×D3—61no—
51C2×C32×D3—61no—
52C2×C32×D3—61no—
53C2×C32×D3—121no—
54#1113—21no—
55#1113C22×C312yescomplemented
56C3×D32—22no—
57C3×D32—22no—
58C3×D32—21no—
59C3×D32—21no—
60C3×D32—32no—
61C3×D32—62no—
62C3×D32—62no—
63C3×D32—61no—
64C3×D32C2×D312yescomplemented
65C3×D32C22×C312yescomplemented
66C3⋊#157C22×C312yescomplemented
67C3×#157—32no—
68C34C2×D411yescharacteristic, complemented, Sylow
69C23×C32—91no—
70C22×C3×D3—61no—
71C22×C3×D3—61no—
72C22×C3×D3—62no—
73C22×C3×D3—91no—
74C22×C3×D3—181no—
75C2×D32—31no—
76C2×D32—61no—
77C2×D32C3×D311yescharacteristic, complemented
78C2×#157—31no—
79GO+(4,𝔽2)—34no—
80C3×#67—94no—
81C2×C3×Dic3—91no—
82C2×C33—21no—
83C2×C33—21no—
84C2×C33—21no—
85C2×C33—42no—
86C2×C33—41no—
87C2×C33—41no—
88C2×C33—41no—
89C2×C33—41no—
90C2×C33—62no—
91C2×C33—81no—
92C2×C33#6711yescharacteristic, complemented
93C2×C33C3×D411yescharacteristic, complemented
94C3×C3⋊D3—22no—
95C3×C3⋊D3—22no—
96C3×C3⋊D3—21no—
97C3×C3⋊D3—21no—
98C3×C3⋊D3—42no—
99C3×C3⋊D3#6711yescharacteristic, complemented
100C3×C3⋊D3C3×D411yescharacteristic, complemented
101C3×C3⋊D3C22×D311yescharacteristic
102C3×C3⋊D3C23×C311yescharacteristic
103C32×D3—22no—
104C32×D3—24no—
105C32×D3—22no—
106C32×D3—22no—
107C32×D3—44no—
108C32×D3—42no—
109C32×D3—42no—
110C32×D3—61no—
111C32×D3—61no—
112C32×D3—61no—
113C32×D3—61no—
114C32×D3—62no—
115C32×D3—62no—
116C32×D3—122no—
117C2×C3×D4—271no—
118C2×#67—91no—
119C22×C32—61no—
120C22×C32—62no—
121C22×C32—61no—
122C22×C32—61no—
123C22×C32—92no—
124C22×C32—91no—
125C22×C32—122no—
126C22×C32—121no—
127C22×C32—121no—
128C22×C32—181no—
129C22×C32—181no—
130C2×C3⋊D3—21no—
131C2×C3⋊D3C2×C3×D311yescharacteristic
132C2×C3×D3—22no—
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3—21no—
136C2×C3×D3—41no—
137C2×C3×D3—41no—
138C2×C3×D3—42no—
139C2×C3×D3—62no—
140C2×C3×D3—62no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C2×C3×D3—61no—
144C2×C3×D3—61no—
145C2×C3×D3—64no—
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—64no—
149C2×C3×D3—61no—
150C2×C3×D3—62no—
151C2×C3×D3—61no—
152C2×C3×D3—61no—
153C2×C3×D3—61no—
154C2×C3×D3—92no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C2×C3×D3—121no—
160C2×C3×D3—122no—
161C2×C3×D3—182no—
162C2×C3×D3—181no—
163C2×C3×D3—182no—
164D32—21no—
165D32—32no—
166D32—62no—
167D32—62no—
168D32—61no—
169D32C2×C3×D312yescomplemented
170#157—32no—
171C3×Dic3—92no—
172C33—21no—
173C33—21no—
174C33—21no—
175C33—42no—
176C33—41no—
177C33—41no—
178C33—41no—
179C33—41no—
180C33—81no—
181C33C2×#6711yescharacteristic, complemented
182C33C2×C3×D411yescharacteristic, complemented
183C23×C3—91no—
184C23×C3—91no—
185C23×C3—181no—
186C23×C3—271no—
187C22×D3—61no—
188C22×D3—91no—
189C22×D3—181no—
190C3×D4—274no—
191C2×C3×C4—271no—
192#67—94no—
193C2×Dic3—91no—
194C2×C32—22no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—41no—
200C2×C32—42no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—42no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—42no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—62no—
212C2×C32—64no—
213C2×C32—62no—
214C2×C32—62no—
215C2×C32—81no—
216C2×C32—81no—
217C2×C32—82no—
218C2×C32—82no—
219C2×C32—81no—
220C2×C32—91no—
221C2×C32—91no—
222C2×C32—124no—
223C2×C32—122no—
224C2×C32—122no—
225C2×C32—182no—
226C2×C32C3×#6711yescharacteristic, complemented
227C2×C32GO+(4,𝔽2)11yescharacteristic, complemented
228C3⋊D3—22no—
229C3⋊D3C3×#6711yescharacteristic, complemented
230C3⋊D3C22×C3×D311yescharacteristic
231C3×D3—24no—
232C3×D3—22no—
233C3×D3—22no—
234C3×D3—22no—
235C3×D3—42no—
236C3×D3—42no—
237C3×D3—44no—
238C3×D3—61no—
239C3×D3—61no—
240C3×D3—61no—
241C3×D3—61no—
242C3×D3—61no—
243C3×D3—61no—
244C3×D3—61no—
245C3×D3—61no—
246C3×D3—62no—
247C3×D3—62no—
248C3×D3—62no—
249C3×D3—62no—
250C3×D3—62no—
251C3×D3—62no—
252C3×D3—121no—
253C3×D3—121no—
254C3×D3—122no—
255C3×D3—122no—
256C3×D3—122no—
257C3×D3—122no—
258C3×D3—124no—
259C2×D4—271noSylow
260C22×C3—61no—
261C22×C3—62no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—92no—
265C22×C3—92no—
266C22×C3—91no—
267C22×C3—91no—
268C22×C3—122no—
269C22×C3—121no—
270C22×C3—121no—
271C22×C3—181no—
272C22×C3—182no—
273C22×C3—181no—
274C22×C3—181no—
275C22×C3—181no—
276C22×C3—182no—
277C22×C3—181no—
278C22×C3—272no—
279C22×C3—361no—
280C22×C3—541no—
281C2×D3—21no—
282C2×D3—62no—
283C2×D3—61no—
284C2×D3—61no—
285C2×D3—62no—
286C2×D3—61no—
287C2×D3—61no—
288C2×D3—92no—
289C2×D3—121no—
290C2×D3—182no—
291C2×D3—181no—
292C2×D3—182no—
293C3×C4—272no—
294Dic3—92no—
295C32—22no—
296C32—21no—
297C32—21no—
298C32—21no—
299C32—21no—
300C32—41no—
301C32—41no—
302C32—41no—
303C32—41no—
304C32—41no—
305C32—42no—
306C32—42no—
307C32—42no—
308C32—41no—
309C32—41no—
310C32—41no—
311C32—41no—
312C32—81no—
313C32—81no—
314C32—82no—
315C32—82no—
316C32—81no—
317C32C2×C3×#6711yescharacteristic, complemented
318C32C2×GO+(4,𝔽2)11yescharacteristic, complemented
319C23—91no—
320C23—271no—
321D4—274no—
322C2×C4—271no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—41no—
327C2×C3—42no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—62no—
332C2×C3—64no—
333C2×C3—62no—
334C2×C3—62no—
335C2×C3—81no—
336C2×C3—91no—
337C2×C3—91no—
338C2×C3—91no—
339C2×C3—91no—
340C2×C3—124no—
341C2×C3—122no—
342C2×C3—122no—
343C2×C3—181no—
344C2×C3—181no—
345C2×C3—182no—
346C2×C3—182no—
347C2×C3—362no—
348C2×C3C3×GO+(4,𝔽2)11yescharacteristic, complemented
349C2×C3D32⋊D311yescharacteristic, direct product factor
350D3—22no—
351D3—61no—
352D3—61no—
353D3—61no—
354D3—61no—
355D3—62no—
356D3—62no—
357D3—122no—
358C22—61no—
359C22—92no—
360C22—91no—
361C22—181no—
362C22—181no—
363C22—272no—
364C22—541no—
365C4—272no—
366C3—21no—
367C3—21no—
368C3—21no—
369C3—41no—
370C3—42no—
371C3—41no—
372C3—41no—
373C3—41no—
374C3—81no—
375C3C2×C3×GO+(4,𝔽2)11yescharacteristic, complemented
376C3C2×D32⋊D311yescharacteristic, direct product factor
377C2—62no—
378C2—91no—
379C2—91no—
380C2—182no—
381C2C3×D32⋊D311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3D32⋊D3
commutator, cosocleC3×C3⋊D3C23×C3
FrattiniC1G
Fitting, socleC2×C34D4
hypocenterC33C2×C3×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1113C2semidirect
2. C2×C32×D32C2semidirect
3. C2×C3×C3⋊#157C2semidirect
4. C3×D32⋊D3C2direct (N × H)
5. C2×D32⋊D3C3direct (N × H)
11. C3×#1113C22semidirect
13. C32×D32C22semidirect
14. C3×C3⋊#157C22semidirect
16. C2×#1113C2×C3semidirect
20. C2×C3×D32D3semidirect
21. C2×C3×D32C2×C3semidirect
22. C2×C3⋊#157C2×C3semidirect
24. D32⋊D3C2×C3direct (N × H)
26. C2×C34D4semidirect
28. C32×C3⋊D3D4semidirect
55. #1113C22×C3semidirect
64. C3×D32C2×D3semidirect
65. C3×D32C22×C3semidirect
66. C3⋊#157C22×C3semidirect
68. C34C2×D4semidirect
77. C2×D32C3×D3semidirect
92. C2×C33#67semidirect
93. C2×C33C3×D4semidirect
99. C3×C3⋊D3#67semidirect
100. C3×C3⋊D3C3×D4semidirect
169. D32C2×C3×D3semidirect
181. C33C2×#67semidirect
182. C33C2×C3×D4semidirect
226. C2×C32C3×#67semidirect
227. C2×C32GO+(4,𝔽2)semidirect
229. C3⋊D3C3×#67semidirect
317. C32C2×C3×#67semidirect
318. C32C2×GO+(4,𝔽2)semidirect
348. C2×C3C3×GO+(4,𝔽2)semidirect
349. C2×C3D32⋊D3direct (N × H)
375. C3C2×C3×GO+(4,𝔽2)semidirect
376. C3C2×D32⋊D3direct (N × H)
381. C2C3×D32⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11132
C2×C32×D322
C2×C3×C3⋊#1572
C3×D32⋊D32
C2×D32⋊D33
C2×C3×GO+(4,𝔽2)3
C2×C3×#679