Finite Group Databasebeta

← D3×A4×C3⋊C8 #13166124

Subgroups — D3×A4×C3⋊C8 #13166124

Order 1728.

458 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C3×C4×D3×A4C211yesmaximal, characteristic
2A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
4C22×D3×C3⋊C8C311yesmaximal, characteristic, complemented
5C2×A4×C3⋊C8—31nomaximal
6D3×C8×A4—31nomaximal
7C2×C3×D3×A4C411yescharacteristic
8C32×C4×A4C2211yescharacteristic
9C3×Dic3×A4C411yescharacteristic
10C3×D3×C3⋊C8—41nomaximal
11C22×C3×C4×D3C2×C311yescharacteristic
12C2×C3×C4×A4—31no—
13C4×D3×A4—21no—
14C4×D3×A4D311yescharacteristic
15C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
16C22×C3×C3⋊C8C2×C311yescharacteristic, complemented
17C3×C8×A4—31no—
18C2×D3×C3⋊C8—31no—
19C2×D3×C3⋊C8—31no—
20C2×D3×C3⋊C8—31no—
21C2×D3×C3⋊C8—31no—
22A4×C3⋊C8—21no—
23A4×C3⋊C8—31no—
24A4×C3⋊C8—31no—
25A4×C3⋊C8—61no—
26A4×C3⋊C8D311yescharacteristic, direct product factor
27C2×C32×A4C2×C411yescharacteristic
28C3×D3×A4C812yescomplemented
29C32×C4×D3—41no—
30C3×C3⋊(C3⋊C8)—41no—
31C32×C3⋊C8—41no—
32C23×C3⋊C8—31no—
33C22×D3×C8—31no—
34C2×C8×A4—91no—
35C23×C3×D3C3×C411yescharacteristic
36C22×C3×A4—31no—
37C2×D3×A4—21no—
38C2×D3×A4Dic311yescharacteristic
39C22×C32×C4C22×C311yescharacteristic
40C22×C3×Dic3C3×C411yescharacteristic
41C2×C3×C4×D3—31no—
42C2×C3×C4×D3—31no—
43C2×C3×C4×D3—31no—
44C2×C3×C4×D3—31no—
45C3×C4×A4—21no—
46C3×C4×A4—21no—
47C3×C4×A4—21no—
48C3×C4×A4—31no—
49C3×C4×A4—41no—
50C3×C4×A4C2×D311yescharacteristic
51C3×C4×A4C2×D311yescharacteristic
52Dic3×A4—21no—
53Dic3×A4Dic311yescharacteristic
54C2×C3⋊(C3⋊C8)—31no—
55C2×C3⋊(C3⋊C8)—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—121no—
59C3×D3×C8—121no—
60D3×C3⋊C8—32no—
61D3×C3⋊C8—31no—
62D3×C3⋊C8—31no—
63D3×C3⋊C8—31no—
64D3×C3⋊C8A411yescharacteristic, direct product factor
65C2×C32×D3—41no—
66C32×A4C2×C811yescharacteristic, complemented
67C33×C4—41no—
68C32×Dic3—41no—
69C23×C3×C4—31no—
70C22×C4×D3C3×D311yescharacteristic
71C2×C4×A4—31no—
72C2×C4×A4—61no—
73C22×C3×C8—31no—
74C22×C3⋊C8—31no—
75C22×C3⋊C8—31no—
76C22×C3⋊C8—61no—
77C22×C3⋊C8—91no—
78C22×C3⋊C8—91no—
79C22×C3⋊C8—91no—
80C22×C3⋊C8—91no—
81C22×C3⋊C8C3×D311yescharacteristic, complemented
82C2×D3×C8—91no—
83C2×D3×C8—91no—
84C2×D3×C8—91no—
85C2×D3×C8—91no—
86C8×A4—31no—
87C8×A4—61no—
88C8×A4—91no—
89C23×C32C2×C3×C411yescharacteristic
90C22×C3×D3—31no—
91C22×C3×D3—32no—
92C22×C3×D3—31no—
93C22×C3×D3C3×C812yescomplemented
94C2×C3×A4—21no—
95C2×C3×A4—21no—
96C2×C3×A4—21no—
97C2×C3×A4—32no—
98C2×C3×A4—41no—
99C2×C3×A4C4×D311yescharacteristic
100C2×C3×A4C2×Dic311yescharacteristic
101D3×A4—22no—
102D3×A4C3⋊C812yesdirect product factor
103C2×C32×C4—31no—
104C2×C32×C4—31no—
105C2×C32×C4—121no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108C3×C4×D3—32no—
109C3×C4×D3—31no—
110C3×C4×D3—31no—
111C3×C4×D3—31no—
112C3×C4×D3—41no—
113C3×C4×D3—81no—
114C3×C4×D3C2×A411yescharacteristic
115C32×C8—121no—
116C3⋊(C3⋊C8)—31no—
117C3⋊(C3⋊C8)C2×A411yescharacteristic, complemented
118C3×C3⋊C8—31no—
119C3×C3⋊C8—41no—
120C3×C3⋊C8—81no—
121C3×C3⋊C8—121no—
122C3×C3⋊C8—121no—
123C3×C3⋊C8—241no—
124C3×C3⋊C8C2×A411yescharacteristic, complemented
125C23×C8—91noSylow
126C2×C33—41no—
127C32×D3—42no—
128C24×C3—31no—
129C23×D3C3×Dic311yescharacteristic
130C22×A4—31no—
131C22×A4—61no—
132C22×C3×C4—21no—
133C22×C3×C4—31no—
134C22×C3×C4—91no—
135C22×C3×C4—91no—
136C22×C3×C4—91no—
137C22×C3×C4—91no—
138C22×C3×C4C2×C3×D311yescharacteristic
139C22×C3×C4C2×C3×D311yescharacteristic
140C22×Dic3C3×Dic311yescharacteristic
141C2×C4×D3—31no—
142C2×C4×D3—31no—
143C2×C4×D3—31no—
144C2×C4×D3—31no—
145C4×A4—21no—
146C4×A4—21no—
147C4×A4—31no—
148C4×A4—41no—
149C4×A4—61no—
150C4×A4D3211yescharacteristic
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C2×C3×C8—361no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—92no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—181no—
166C2×C3⋊C8—181no—
167D3×C8—31no—
168D3×C8—91no—
169D3×C8—91no—
170D3×C8—92no—
171D3×C8—91no—
172C22×C32—31no—
173C22×C32—31no—
174C22×C32—121no—
175C22×C32C2×C3×C811yescharacteristic, complemented
176C2×C3×D3—32no—
177C2×C3×D3—31no—
178C2×C3×D3—32no—
179C2×C3×D3—32no—
180C2×C3×D3—32no—
181C2×C3×D3—41no—
182C2×C3×D3—81no—
183C2×C3×D3C4×A411yescharacteristic
184C3×A4—21no—
185C3×A4—21no—
186C3×A4—21no—
187C3×A4—41no—
188C3×A4D3×C811yescharacteristic, complemented
189C3×A4C2×C3⋊C811yescharacteristic, complemented
190C32×C4—31no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—81no—
194C32×C4—81no—
195C32×C4—81no—
196C32×C4—121no—
197C32×C4—161no—
198C32×C4C22×A411yescharacteristic
199C3×Dic3—31no—
200C3×Dic3—41no—
201C3×Dic3—81no—
202C3×Dic3C4×A411yescharacteristic
203C23×C4—31no—
204C22×C8—31no—
205C22×C8—91no—
206C22×C8—271no—
207C22×C8—271no—
208C22×C8—271no—
209C22×C8—271no—
210C33—41noSylow
211C23×C3—21no—
212C23×C3—32no—
213C23×C3—91no—
214C23×C3—91no—
215C23×C3—92no—
216C23×C3C3×C4×D311yescharacteristic
217C23×C3C2×C3×Dic311yescharacteristic
218C22×D3—31no—
219C22×D3—31no—
220C22×D3—32no—
221C22×D3C3×C3⋊C812yescomplemented
222C2×A4—21no—
223C2×A4—21no—
224C2×A4—32no—
225C2×A4—41no—
226C2×A4—62no—
227C2×A4D3×Dic311yescharacteristic
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—92no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—121no—
242C2×C3×C4—241no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C4×D3—32no—
246C4×D3—31no—
247C4×D3—31no—
248C4×D3—31no—
249C4×D3D3×A411yescharacteristic
250C3×C8—31no—
251C3×C8—91no—
252C3×C8—121no—
253C3×C8—241no—
254C3×C8—361no—
255C3⋊C8—31no—
256C3⋊C8—31no—
257C3⋊C8—31no—
258C3⋊C8—61no—
259C3⋊C8—91no—
260C3⋊C8—91no—
261C3⋊C8—181no—
262C3⋊C8D3×A411yescharacteristic, direct product factor
263C2×C32—31no—
264C2×C32—31no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—81no—
268C2×C32—81no—
269C2×C32—81no—
270C2×C32—122no—
271C2×C32—161no—
272C2×C32C2×C4×A411yescharacteristic
273C3×D3—32no—
274C3×D3—42no—
275C3×D3—82no—
276C3×D3C8×A412yescomplemented
277C24—31no—
278C22×C4—31no—
279C22×C4—91no—
280C22×C4—91no—
281C22×C4—91no—
282C22×C4—91no—
283C22×C4C3×D3211yescharacteristic
284C2×C8—91no—
285C2×C8—91no—
286C2×C8—91no—
287C2×C8—271no—
288C2×C8—271no—
289C2×C8—271no—
290C2×C8—271no—
291C2×C8—272no—
292C2×C8—271no—
293C22×C3—21no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—92no—
302C22×C3—92no—
303C22×C3—92no—
304C22×C3—92no—
305C22×C3—91no—
306C22×C3—121no—
307C22×C3—241no—
308C22×C3C3×D3×C811yescharacteristic, complemented
309C22×C3C2×C3×C3⋊C811yescharacteristic, complemented
310C2×D3—32no—
311C2×D3—31no—
312C2×D3—32no—
313C2×D3—32no—
314C2×D3—32no—
315C2×D3Dic3×A411yescharacteristic
316A4—21no—
317A4—21no—
318A4—41no—
319A4D3×C3⋊C811yescharacteristic, direct product factor
320C3×C4—21no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—41no—
325C3×C4—61no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4—91no—
329C3×C4—121no—
330C3×C4—161no—
331C3×C4—241no—
332C3×C4C2×D3×A411yescharacteristic
333C3×C4C2×D3×A411yescharacteristic
334Dic3—31no—
335Dic3Dic3×A411yescharacteristic
336C32—41no—
337C32—41no—
338C32—81no—
339C32—81no—
340C32—81no—
341C32—161no—
342C32C2×C8×A411yescharacteristic, complemented
343C23—32no—
344C23—91no—
345C23—91no—
346C23—92no—
347C23C3×D3×Dic311yescharacteristic
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—91no—
356C2×C4—91no—
357C8—31no—
358C8—91no—
359C8—91no—
360C8—271no—
361C2×C3—21no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—32no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—41no—
368C2×C3—61no—
369C2×C3—61no—
370C2×C3—81no—
371C2×C3—81no—
372C2×C3—92no—
373C2×C3—122no—
374C2×C3—161no—
375C2×C3—242no—
376C2×C3C4×D3×A411yescharacteristic
377C2×C3C2×Dic3×A411yescharacteristic
378D3—32no—
379D3A4×C3⋊C812yesdirect product factor
380C22—31no—
381C22—31no—
382C22—31no—
383C22—92no—
384C22—92no—
385C22—92no—
386C22—92no—
387C22—91no—
388C22C3×D3×C3⋊C811yescharacteristic, complemented
389C4—31no—
390C4—31no—
391C4—91no—
392C4D32×A411yescharacteristic
393C3—21no—
394C3—41no—
395C3—81no—
396C3—81no—
397C3—161no—
398C3D3×C8×A411yescharacteristic, complemented
399C3C2×A4×C3⋊C811yescharacteristic, complemented
400C2—31no—
401C2—31no—
402C2—32no—
403C2—92no—
404C2D3×Dic3×A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D32×A4
commutator, hypocenterC22×C32C2×C3×C8
Fitting, cosocleC22×C32×C4C22×C3
socleC23×C32C2×C3×C4
prime radicalC2×C3×D3×A4C4
LCSG ▷ C22×C32
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. A4×C3⋊(C3⋊C8)C2semidirect
3. C3×A4×C3⋊C8C2semidirect
4. C22×D3×C3⋊C8C3semidirect
15. C22×C3⋊(C3⋊C8)C2×C3semidirect
16. C22×C3×C3⋊C8C2×C3semidirect
26. A4×C3⋊C8D3direct (N × H)
28. C3×D3×A4C8semidirect
64. D3×C3⋊C8A4direct (N × H)
66. C32×A4C2×C8semidirect
81. C22×C3⋊C8C3×D3semidirect
93. C22×C3×D3C3×C8semidirect
102. D3×A4C3⋊C8direct (N × H)
117. C3⋊(C3⋊C8)C2×A4semidirect
124. C3×C3⋊C8C2×A4semidirect
175. C22×C32C2×C3×C8semidirect
188. C3×A4D3×C8semidirect
189. C3×A4C2×C3⋊C8semidirect
221. C22×D3C3×C3⋊C8semidirect
262. C3⋊C8D3×A4direct (N × H)
276. C3×D3C8×A4semidirect
308. C22×C3C3×D3×C8semidirect
309. C22×C3C2×C3×C3⋊C8semidirect
319. A4D3×C3⋊C8direct (N × H)
342. C32C2×C8×A4semidirect
379. D3A4×C3⋊C8direct (N × H)
388. C22C3×D3×C3⋊C8semidirect
398. C3D3×C8×A4semidirect
399. C3C2×A4×C3⋊C8semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×A42
A4×C3⋊(C3⋊C8)2
C3×A4×C3⋊C82
C22×D3×C3⋊C83
C2×A4×C3⋊C83
D3×C8×A43
C3×D3×C3⋊C84