Finite Group Databasebeta

← C22⋊C9×Dic3⋊C4 #13147259

Subgroups — C22⋊C9×Dic3⋊C4 #13147259

Order 1728.

466 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 85 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×C4×C22⋊C9C211yesmaximal, characteristic
2C2×Dic3×C22⋊C9C211yesmaximal, characteristic
3C2×Dic3×C22⋊C9C211yesmaximal, characteristic
4C22×C3×Dic3⋊C4C311yesmaximal, characteristic
5C4⋊C4×C22⋊C9—31nomaximal
6C22×C3×C22⋊C9C2211yescharacteristic
7C3×C4×C22⋊C9—21no—
8Dic3×C22⋊C9—21no—
9Dic3×C22⋊C9C412yescomplemented
10C9×Dic3⋊C4—41nomaximal
11C23×C32×C4C2×C311yescharacteristic
12C23×C3×Dic3C2×C311yescharacteristic
13C23×C3×Dic3C2×C311yescharacteristic
14C2×C3×Dic3⋊C4—31no—
15C2×C3×Dic3⋊C4—31no—
16C2×C3×Dic3⋊C4—31no—
17C2×C3×Dic3⋊C4—31no—
18C2×C4×C22⋊C9—21no—
19C2×C4×C22⋊C9—31no—
20C2×C4×C22⋊C9—31no—
21C2×C4×C22⋊C9D311yescharacteristic
22C2×C3×C22⋊C9C2×C411yescharacteristic
23C2×C3×C22⋊C9D411yescharacteristic
24C2×C3×C22⋊C9Q811yescharacteristic
25C2×C3×C4×C9—41no—
26C2×C9×Dic3—41no—
27C2×C9×Dic3—41no—
28C22×C3×C4⋊C4—31no—
29C22×Dic3⋊C4C911yescharacteristic, complemented
30C24×C32C22×C311yescharacteristic
31C22×C32×C4—21no—
32C22×C32×C4—31no—
33C22×C32×C4—31no—
34C22×C32×C4—61no—
35C22×C3×Dic3—21no—
36C22×C3×Dic3—31no—
37C22×C3×Dic3—31no—
38C22×C3×Dic3—32no—
39C22×C3×Dic3—31no—
40C22×C3×Dic3—31no—
41C22×C3×Dic3—61no—
42C22×C3×Dic3C3×C412yes—
43C22×C22⋊C9—21no—
44C22×C22⋊C9C2×D311yescharacteristic
45C3×Dic3⋊C4—32no—
46C3×Dic3⋊C4—31no—
47C3×Dic3⋊C4—31no—
48C3×Dic3⋊C4—31no—
49C3×Dic3⋊C4A411yescharacteristic
50C4×C22⋊C9—22no—
51C4×C22⋊C9—21no—
52C4×C22⋊C9—32no—
53C4×C22⋊C9—61no—
54C9×C4⋊C4—121no—
55C22×C3×C9—41no—
56C3×C22⋊C9C4⋊C411yescharacteristic, complemented
57C3×C4×C9—81no—
58C9×Dic3—42no—
59C9×Dic3—81no—
60C23×C3×C4—21no—
61C23×C3×C4—31no—
62C23×C3×C4—31no—
63C23×C3×C4C2×C911yescharacteristic
64C23×C3×C4C3×D311yescharacteristic
65C23×Dic3C2×C911yescharacteristic
66C23×Dic3C2×C911yescharacteristic
67C2×C3×C4⋊C4—91no—
68C2×C3×C4⋊C4—91no—
69C2×C3×C4⋊C4—91no—
70C2×C3×C4⋊C4—91no—
71C2×Dic3⋊C4—31no—
72C2×Dic3⋊C4—31no—
73C2×Dic3⋊C4—31no—
74C2×Dic3⋊C4—31no—
75C23×C32—31no—
76C23×C32—31no—
77C23×C32—31no—
78C23×C32—31no—
79C23×C32C2×C3×C411yescharacteristic
80C23×C32C3×D411yescharacteristic
81C23×C32C3×Q811yescharacteristic
82C2×C32×C4—31no—
83C2×C32×C4—61no—
84C2×C32×C4—61no—
85C2×C32×C4—61no—
86C2×C32×C4—61no—
87C2×C32×C4C2×A411yescharacteristic
88C2×C3×Dic3—32no—
89C2×C3×Dic3—31no—
90C2×C3×Dic3—32no—
91C2×C3×Dic3—32no—
92C2×C3×Dic3—32no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3C2×A411yescharacteristic
99C2×C3×Dic3C2×A411yescharacteristic
100C2×C22⋊C9—21no—
101C2×C22⋊C9—21no—
102C2×C22⋊C9—21no—
103C2×C22⋊C9Dic611yescharacteristic
104C2×C22⋊C9C4×D311yescharacteristic
105C2×C22⋊C9#6711yescharacteristic
106C2×C4×C9—41no—
107C2×C4×C9—81no—
108C2×C4×C9—121no—
109C2×C4×C9—121no—
110C22×C4⋊C4—31noSylow
111C2×C3×C9—41no—
112C2×C3×C9—41no—
113C2×C3×C9—41no—
114C24×C3—21no—
115C24×C3C22×C911yescharacteristic
116C24×C3C2×C3×D311yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—32no—
125C22×C3×C4—61no—
126C22×C3×C4—61no—
127C22×C3×C4—62no—
128C22×C3×C4—61no—
129C22×C3×C4—61no—
130C22×C3×C4—61no—
131C22×C3×C4—91no—
132C22×C3×C4—91no—
133C22×C3×C4—91no—
134C22×C3×C4—92no—
135C22×C3×C4—91no—
136C22×C3×C4—181no—
137C22×Dic3—21no—
138C22×Dic3—31no—
139C22×Dic3—31no—
140C22×Dic3—32no—
141C22×Dic3—31no—
142C22×Dic3—31no—
143C22×Dic3—61no—
144C22×Dic3C4×C912yescomplemented
145C3×C4⋊C4—31no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—92no—
149C3×C4⋊C4—91no—
150Dic3⋊C4—32no—
151Dic3⋊C4—31no—
152Dic3⋊C4—31no—
153Dic3⋊C4—31no—
154Dic3⋊C4C22⋊C911yescharacteristic, direct product factor
155C22×C32—31no—
156C22×C32—31no—
157C22×C32—31no—
158C22×C32—32no—
159C22×C32—31no—
160C22×C32—31no—
161C22×C32—31no—
162C22×C32—31no—
163C22×C32—31no—
164C22×C32—31no—
165C22×C32C3×C4⋊C411yescharacteristic
166C22×C32C22×A411yescharacteristic
167C32×C4—21no—
168C32×C4—61no—
169C3×Dic3—21no—
170C3×Dic3—32no—
171C3×Dic3—61no—
172C3×Dic3C4×A412yes—
173C22×C9—41no—
174C22×C9—81no—
175C22⋊C9—21no—
176C22⋊C9Dic3⋊C411yescharacteristic, direct product factor
177C4×C9—81no—
178C4×C9—82no—
179C4×C9—122no—
180C4×C9—241no—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4D3×C911yescharacteristic
184C2×C4⋊C4—91no—
185C2×C4⋊C4—91no—
186C2×C4⋊C4—91no—
187C2×C4⋊C4—91no—
188C3×C9—41noSylow
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—31no—
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—31no—
199C23×C3—31no—
200C23×C3—61no—
201C23×C3—61no—
202C23×C3—61no—
203C23×C3—61no—
204C23×C3C2×C4×C911yescharacteristic
205C23×C3D4×C911yescharacteristic
206C23×C3Q8×C911yescharacteristic
207C23×C3C3×Dic611yescharacteristic
208C23×C3C3×C4×D311yescharacteristic
209C23×C3C3×#6711yescharacteristic
210C2×C3×C4—21no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—92no—
231C2×C3×C4—91no—
232C2×C3×C4—92no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4C2×C22⋊C911yescharacteristic
239C2×C3×C4D3×A411yescharacteristic
240C2×Dic3—32no—
241C2×Dic3—31no—
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—32no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3C2×C22⋊C911yescharacteristic
251C2×Dic3C2×C22⋊C911yescharacteristic
252C2×C32—31no—
253C2×C32—31no—
254C2×C32—31no—
255C2×C32—31no—
256C2×C32C2×C4×A411yescharacteristic
257C2×C32D4×A411yescharacteristic
258C2×C32Q8×A411yescharacteristic
259C2×C9—41no—
260C2×C9—41no—
261C2×C9—41no—
262C2×C9—81no—
263C2×C9—81no—
264C2×C9—81no—
265C24C2×D3×C911yescharacteristic
266C22×C4—21no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—32no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—91no—
273C22×C4—91no—
274C22×C4—91no—
275C22×C4—92no—
276C22×C4—91no—
277C22×C4—181no—
278C4⋊C4—31no—
279C4⋊C4—91no—
280C4⋊C4—91no—
281C4⋊C4—92no—
282C4⋊C4—91no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—62no—
315C22×C3C9×C4⋊C411yescharacteristic, complemented
316C22×C3C3×Dic3⋊C411yescharacteristic
317C22×C3C22×C22⋊C911yescharacteristic
318C22×C3C2×D3×A411yescharacteristic
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—32no—
323C3×C4—62no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—61no—
327C3×C4—92no—
328C3×C4—181no—
329Dic3—21no—
330Dic3—32no—
331Dic3—61no—
332Dic3C4×C22⋊C912yescomplemented
333C32A4×C4⋊C411yescharacteristic
334C9—41no—
335C9—81no—
336C23—31no—
337C23—31no—
338C23—31no—
339C23—31no—
340C23C9×Dic611yescharacteristic
341C23C4×D3×C911yescharacteristic
342C23C9×#6711yescharacteristic
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—92no—
351C2×C4—92no—
352C2×C4—92no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—91no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4D3×C22⋊C911yescharacteristic
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3C2×C4×C22⋊C911yescharacteristic
377C2×C3D4×C22⋊C911yescharacteristic
378C2×C3Q8×C22⋊C911yescharacteristic
379C2×C3A4×Dic611yescharacteristic
380C2×C3C4×D3×A411yescharacteristic
381C2×C3A4×#6711yescharacteristic
382C22—31no—
383C22—31no—
384C22—31no—
385C22—32no—
386C22—31no—
387C22—31no—
388C22—31no—
389C22—31no—
390C22—31no—
391C22—31no—
392C22C9×Dic3⋊C411yescharacteristic, complemented
393C22C2×D3×C22⋊C911yescharacteristic
394C4—21no—
395C4—32no—
396C4—61no—
397C4—61no—
398C4—92no—
399C4—181no—
400C3—21no—
401C3C4⋊C4×C22⋊C911yescharacteristic, complemented
402C3A4×Dic3⋊C411yescharacteristic
403C2—31no—
404C2—31no—
405C2—31no—
406C2—31no—
407C2Dic6×C22⋊C911yescharacteristic
408C2C4×D3×C22⋊C911yescharacteristic
409C2#67×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC23×C3C2×C4×C9
FittingC23×C32×C4C2×C3
socle, prime radical, cosocleC24×C32C22×C3
hypercenterC2×C3×C4D3×A4
hypocenterC22×C3C9×C4⋊C4
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
9. Dic3×C22⋊C9C4semidirect
29. C22×Dic3⋊C4C9semidirect
56. C3×C22⋊C9C4⋊C4semidirect
144. C22×Dic3C4×C9semidirect
154. Dic3⋊C4C22⋊C9direct (N × H)
176. C22⋊C9Dic3⋊C4direct (N × H)
315. C22×C3C9×C4⋊C4semidirect
332. Dic3C4×C22⋊C9semidirect
392. C22C9×Dic3⋊C4semidirect
401. C3C4⋊C4×C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
C2×Dic3×C22⋊C92
C22×C3×Dic3⋊C43
C4⋊C4×C22⋊C93
C9×Dic3⋊C44