Finite Group Databasebeta

← C2×D3×#41⋊C9 #13164626

Subgroups — C2×D3×#41⋊C9 #13164626

Order 1728.

843 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 133 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×#41⋊C9C211yesmaximal, characteristic, complemented
2D3×#41⋊C9C214yesmaximal, direct product factor
3C2×D3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C2×C3⋊(#41⋊C9)C211yesmaximal, characteristic, complemented
5C2×C3×D3×#41C311yesmaximal, characteristic
6C22×#41⋊C9—31nomaximal
7C2×C4×D3×C9—41nomaximal
8C3×#41⋊C9C2212yescomplemented
9C2×C3×Q8⋊C9C2211yescharacteristic, complemented
10D3×Q8⋊C9C2212yescomplemented
11C3⋊(#41⋊C9)C2212yescomplemented
12C2×C32×#41C2×C311yescharacteristic
13C3×D3×#41—34no—
14C3×D3×#41C2×C314yes—
15C2×C3×#239C2×C311yescharacteristic
16C2×C3×D3×Q8C2×C311yescharacteristic
17C2×C3×#237—31no—
18C2×C3×D3×D4—31no—
19C2×C3×#235—31no—
20C22×C3×C4×D3—31no—
21C2×#41⋊C9—21no—
22C2×#41⋊C9—34no—
23C2×#41⋊C9—31no—
24C2×#41⋊C9D311yescharacteristic, direct product factor
25C22×Q8⋊C9—31no—
26C22×D3×C9—41no—
27C2×C3×C4×C9—41no—
28C2×C9×Dic3—41no—
29C4×D3×C9—44no—
30C3×Q8⋊C9C2311yescharacteristic, complemented
31C22×C3×#41—31no—
32C2×D3×#41C911yescharacteristic, complemented
33C23×C3×D3—31no—
34C32×#41—32no—
35C32×#41C22×C312yes—
36C2×C32×Q8C22×C311yescharacteristic
37C2×C32×D4—31no—
38C22×C32×C4—31no—
39C2×C3×#67—32no—
40C22×C3×Dic3—31no—
41C3×#239—32no—
42C3×#239C22×C312yes—
43C3×D3×Q8—32no—
44C3×D3×Q8C22×C312yes—
45C3×#237—34no—
46C3×#237—34no—
47C3×D3×D4—34no—
48C3×D3×D4—34no—
49C3×#235—34no—
50C3×#235—34no—
51C2×C3×D12—31no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—34no—
57C2×C3×C4×D3A411yescharacteristic
58C2×C3×Dic6—31no—
59C22×C4×C9—121no—
60#41⋊C9—22no—
61#41⋊C9—32no—
62#41⋊C9C2×D312yesdirect product factor
63C2×Q8⋊C9—21no—
64C2×Q8⋊C9—32no—
65C2×Q8⋊C9C2×D311yescharacteristic, complemented
66C22×C3×C9—41no—
67C2×D3×C9—44no—
68C2×D3×C9—42no—
69C3×C4×C9—42no—
70C9×Dic3—42no—
71C2×C3×#41—21no—
72C2×C3×#41—34no—
73C2×C3×#41—31no—
74C2×C3×#41—91no—
75C2×C3×#41—91no—
76C2×C3×#41—94no—
77C2×C3×#41C2×C911yescharacteristic, complemented
78C2×C3×#41C3×D311yescharacteristic
79C22×C3×Q8—31no—
80C22×C3×D4—91no—
81C23×C3×C4—91no—
82D3×#41—34no—
83D3×#41C2×C914yescomplemented
84C2×#239C2×C911yescharacteristic, complemented
85C2×D3×Q8C2×C911yescharacteristic
86C2×#237—31no—
87C2×D3×D4—31no—
88C2×#235—31no—
89C22×C4×D3—31no—
90C23×C32—31no—
91C22×C3×D3—31no—
92C22×C3×D3—34no—
93C22×C3×D3—64no—
94C22×C3×D3C2×A411yescharacteristic
95C32×Q8—31no—
96C32×Q8C23×C311yescharacteristic
97C32×D4—32no—
98C32×D4—32no—
99C2×C32×C4—31no—
100C2×C32×C4—32no—
101C2×C32×C4—32no—
102C2×C32×C4C2×A411yescharacteristic
103C3×#67—38no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—34no—
106C2×C3×Dic3C2×A411yescharacteristic
107C3×D12—32no—
108C3×D12—32no—
109C3×C4×D3—32no—
110C3×C4×D3—32no—
111C3×C4×D3—34no—
112C3×C4×D3—32no—
113C3×C4×D3—32no—
114C3×C4×D3C2×A414yes—
115C3×Dic6—32no—
116C3×Dic6—32no—
117C23×C9—121no—
118C2×C4×C9—41no—
119C2×C4×C9—81no—
120C2×C4×C9—124no—
121C2×C4×C9—121no—
122Q8⋊C9—21no—
123Q8⋊C9C22×D311yescharacteristic, complemented
124C22×#41—31noSylow
125C2×C3×C9—42no—
126C2×C3×C9—41no—
127D3×C9—44no—
128C24×C3—91no—
129C23×D3—31no—
130C3×#41—22no—
131C3×#41—32no—
132C3×#41—32no—
133C3×#41—32no—
134C3×#41—62no—
135C3×#41—94no—
136C3×#41—94no—
137C3×#41—94no—
138C3×#41—94no—
139C3×#41—92no—
140C3×#41C22×C912yescomplemented
141C3×#41C2×C3×D312yes—
142C2×C3×Q8—21no—
143C2×C3×Q8—32no—
144C2×C3×Q8—91no—
145C2×C3×Q8—92no—
146C2×C3×Q8C22×C911yescharacteristic
147C2×C3×Q8C2×C3×D311yescharacteristic
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4—91no—
152C2×C3×D4—94no—
153C2×C3×D4—94no—
154C2×C3×D4—92no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—61no—
159C22×C3×C4—91no—
160C22×C3×C4—94no—
161C22×C3×C4—94no—
162C22×C3×C4—91no—
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C2×#67—32no—
166C22×Dic3—31no—
167#239—32no—
168#239C22×C912yescomplemented
169D3×Q8—32no—
170D3×Q8C22×C912yes—
171#237—34no—
172#237—34no—
173D3×D4—34no—
174D3×D4—34no—
175#235—34no—
176#235—34no—
177C2×D12—31no—
178C2×C4×D3—31no—
179C2×C4×D3—31no—
180C2×C4×D3—31no—
181C2×C4×D3—34no—
182C2×C4×D3—34no—
183C2×C4×D3C22⋊C911yescharacteristic
184C2×Dic6—31no—
185C22×C32—32no—
186C22×C32—62no—
187C22×C32C22×A411yescharacteristic
188C2×C3×D3—32no—
189C2×C3×D3—62no—
190C2×C3×D3—68no—
191C2×C3×D3#41⋊C314yes—
192C2×C3×D3C22×A412yes—
193C32×C4—31no—
194C32×C4—31no—
195C32×C4C22×A412yes—
196C3×Dic3—32no—
197C3×Dic3C22×A412yes—
198C22×C9—41no—
199C22×C9—81no—
200C22×C9—122no—
201C22×C9—124no—
202C4×C9—42no—
203C4×C9—82no—
204C4×C9—122no—
205C2×#41—34no—
206C2×#41—31no—
207C2×#41—91no—
208C2×#41—91no—
209C2×#41—94no—
210C2×#41D3×C911yescharacteristic, complemented
211C22×Q8—31no—
212C22×D4—91no—
213C23×C4—91no—
214C3×C9—41noSylow
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—31no—
218C23×C3—61no—
219C23×C3—94no—
220C23×C3—91no—
221C23×C3—184no—
222C22×D3—31no—
223C22×D3—34no—
224C22×D3—64no—
225C22×D3C2×C22⋊C911yescharacteristic
226C3×Q8—21no—
227C3×Q8—31no—
228C3×Q8—31no—
229C3×Q8—61no—
230C3×Q8—92no—
231C3×Q8—92no—
232C3×Q8C23×C911yescharacteristic
233C3×Q8C22×C3×D311yescharacteristic
234C3×D4—32no—
235C3×D4—32no—
236C3×D4—32no—
237C3×D4—32no—
238C3×D4—62no—
239C3×D4—62no—
240C3×D4—92no—
241C3×D4—92no—
242C3×D4—98no—
243C2×C3×C4—21no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—34no—
248C2×C3×C4—32no—
249C2×C3×C4—32no—
250C2×C3×C4—32no—
251C2×C3×C4—32no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—61no—
255C2×C3×C4—92no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—94no—
260C2×C3×C4—91no—
261C2×C3×C4—94no—
262C2×C3×C4C2×C22⋊C911yescharacteristic
263C2×C3×C4D3×A411yescharacteristic
264#67—38no—
265C2×Dic3—31no—
266C2×Dic3—34no—
267C2×Dic3C2×C22⋊C911yescharacteristic
268D12—32no—
269D12—32no—
270C4×D3—32no—
271C4×D3—32no—
272C4×D3—34no—
273C4×D3—32no—
274C4×D3—32no—
275C4×D3C2×C22⋊C914yes—
276Dic6—32no—
277Dic6—32no—
278C2×C32—62no—
279C2×C32C2×#41⋊C312yes—
280C2×C32C23×A411yescharacteristic
281C3×D3—62no—
282C3×D3C2×#41⋊C314yes—
283C2×C9—42no—
284C2×C9—41no—
285C2×C9—82no—
286C2×C9—81no—
287C2×C9—124no—
288C24—91no—
289#41—32no—
290#41—32no—
291#41—94no—
292#41—94no—
293#41—94no—
294#41—94no—
295#41—92no—
296#41C2×D3×C912yescomplemented
297C2×Q8—32no—
298C2×Q8—91no—
299C2×Q8—92no—
300C2×Q8C2×D3×C911yescharacteristic
301C2×D4—31no—
302C2×D4—91no—
303C2×D4—94no—
304C2×D4—94no—
305C2×D4—92no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—91no—
309C22×C4—94no—
310C22×C4—94no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—91no—
314C22×C3—21no—
315C22×C3—34no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—32no—
319C22×C3—62no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—92no—
323C22×C3—122no—
324C22×C3—188no—
325C22×C3—182no—
326C22×C3C22×C22⋊C911yescharacteristic
327C22×C3C2×D3×A411yescharacteristic
328C2×D3—32no—
329C2×D3—62no—
330C2×D3—68no—
331C2×D3#41⋊C914yesdirect product factor
332C2×D3C22×C22⋊C912yes—
333C3×C4—22no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—32no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—92no—
342C3×C4C22×C22⋊C912yes—
343C3×C4C2×D3×A412yes—
344Dic3—32no—
345Dic3C22×C22⋊C912yes—
346C32C22×#41⋊C311yescharacteristic
347C9—41no—
348C9—81no—
349C23—31no—
350C23—31no—
351C23—94no—
352C23—91no—
353C23—184no—
354Q8—31no—
355Q8—92no—
356Q8—92no—
357Q8C22×D3×C911yescharacteristic
358D4—32no—
359D4—32no—
360D4—92no—
361D4—92no—
362D4—98no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—32no—
366C2×C4—32no—
367C2×C4—34no—
368C2×C4—92no—
369C2×C4—92no—
370C2×C4—92no—
371C2×C4—92no—
372C2×C4—94no—
373C2×C4—91no—
374C2×C4—94no—
375C2×C4D3×C22⋊C911yescharacteristic
376C2×C3—21no—
377C2×C3—22no—
378C2×C3—34no—
379C2×C3—62no—
380C2×C3—62no—
381C2×C3—122no—
382C2×C3—182no—
383C2×C3C2×#41⋊C912yescomplemented
384C2×C3C23×C22⋊C911yescharacteristic
385C2×C3D3×#41⋊C312yes—
386C2×C3C22×D3×A411yescharacteristic
387D3—62no—
388D3C2×#41⋊C914yesdirect product factor
389C22—32no—
390C22—34no—
391C22—32no—
392C22—62no—
393C22—92no—
394C22—188no—
395C22—182no—
396C22C2×D3×C22⋊C911yescharacteristic
397C4—31no—
398C4—31no—
399C4—32no—
400C4—92no—
401C4C2×D3×C22⋊C912yes—
402C3—21no—
403C3C22×#41⋊C911yescharacteristic, complemented
404C3C2×D3×#41⋊C311yescharacteristic
405C2—34no—
406C2—62no—
407C2—182no—
408C2D3×#41⋊C912yesdirect product factor
409C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC3×Q8C23×C9
FrattiniC2×C3C22×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
prime radicalC2×C3×D3×#41C3
cosocleC32×Q8C23×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. D3×#41⋊C9C2direct (N × H)
3. C2×D3×Q8⋊C9C2semidirect
4. C2×C3⋊(#41⋊C9)C2semidirect
8. C3×#41⋊C9C22semidirect
9. C2×C3×Q8⋊C9C22semidirect
10. D3×Q8⋊C9C22semidirect
11. C3⋊(#41⋊C9)C22semidirect
24. C2×#41⋊C9D3direct (N × H)
30. C3×Q8⋊C9C23semidirect
32. C2×D3×#41C9semidirect
62. #41⋊C9C2×D3direct (N × H)
65. C2×Q8⋊C9C2×D3semidirect
77. C2×C3×#41C2×C9semidirect
83. D3×#41C2×C9semidirect
84. C2×#239C2×C9semidirect
123. Q8⋊C9C22×D3semidirect
140. C3×#41C22×C9semidirect
168. #239C22×C9semidirect
210. C2×#41D3×C9semidirect
296. #41C2×D3×C9semidirect
331. C2×D3#41⋊C9direct (N × H)
383. C2×C3C2×#41⋊C9semidirect
388. D3C2×#41⋊C9direct (N × H)
403. C3C22×#41⋊C9semidirect
408. C2D3×#41⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
D3×#41⋊C92
C2×D3×Q8⋊C92
C2×C3⋊(#41⋊C9)2
C2×C3×D3×#413
C22×#41⋊C93
C2×C4×D3×C94