Finite Group Databasebeta

← C3⋊D3×A4⋊Q8 #13180737

Subgroups — C3⋊D3×A4⋊Q8 #13180737

Order 1728.

1086 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×A4×C3⋊D3C211yesmaximal, characteristic
2#11724463C211yesmaximal, characteristic, complemented
3C32×A4⋊Q8C211yesmaximal, characteristic, complemented
4C3⋊D3×SL(2,ℤ/4)C212yesmaximal
5#11724184C212yesmaximal, complemented
6D3×A4⋊Q8—34nomaximal
7C3⋊D3×C22⋊Q8—31nomaximal
8C2×A4×C3⋊D3C2211yescharacteristic
9C32×C4×A4C2211yescharacteristic
10C3⋊D3×Dic6—41nomaximal
11A4×C3⋊Dic3C2211yescharacteristic
12C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
13C32×SL(2,ℤ/4)C2212yes—
14C2×Q8×C3⋊D3—31no—
15C22×C4×C3⋊D3D311yescharacteristic
16C4×D3×A4—34no—
17#64259—34no—
18#64259—64no—
19C3×A4⋊Q8D314yescomplemented
20D3×SL(2,ℤ/4)—38no—
21#64204—38no—
22C32×C22⋊Q8—31no—
23#64155—31no—
24#64133—31no—
25#64106—31no—
26#64105—32no—
27C4⋊C4×C3⋊D3—31no—
28C4⋊C4×C3⋊D3—32no—
29C22⋊C4×C3⋊D3—32no—
30#64087—32no—
31C2×C32×A4C2311yescharacteristic
32A4×C3⋊D3Q812yescomplemented
33#6308—41no—
34C3×C4×C3⋊D3—41no—
35C32×Dic6—41no—
36#6293—42no—
37Dic3×C3⋊D3—42no—
38C2×A4⋊Q8—91no—
39D3×C22⋊Q8—94no—
40C23×C3⋊D3C2×D311yescharacteristic
41C2×D3×A4—34no—
42C2×C32×Q8—31no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3C2×D311yescharacteristic
45Q8×C3⋊D3—61no—
46Q8×C3⋊D3—61no—
47Q8×C3⋊D3—61no—
48Q8×C3⋊D3—61no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—32no—
54C2×C4×C3⋊D3—32no—
55C2×#634—32no—
56C2×#634—31no—
57C3×C4×A4—24no—
58C3×C4×A4C2×D314yes—
59D3×Dic6—124no—
60Dic3×A4—34no—
61C3⋊SL(2,ℤ/4)—38no—
62C3⋊SL(2,ℤ/4)—68no—
63C3×SL(2,ℤ/4)C2×D318yes—
64C32×C4⋊C4—32no—
65C32×C4⋊C4—31no—
66C32×C22⋊C4—32no—
67#3757—32no—
68#3752—32no—
69#3752—32no—
70C4⋊(C3⋊Dic3)—32no—
71C4⋊(C3⋊Dic3)—31no—
72#3750—32no—
73#3750—32no—
74#3750—32no—
75C2×C3×C3⋊D3—41no—
76C32×A4C2×Q811yescharacteristic, complemented
77C33×C4—41no—
78C3⋊(C3⋊Dic3)—42no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—42no—
81C2×D3×Q8—94no—
82C22×C4×D3—34no—
83C2×C4×A4—91no—
84C2×SL(2,ℤ/4)—92no—
85A4⋊Q8—91no—
86A4⋊Q8—92no—
87A4⋊Q8—184no—
88A4⋊Q8C3⋊D311yescharacteristic, direct product factor
89C3×C22⋊Q8—34no—
90#946—94no—
91#924—94no—
92#897—94no—
93#896—98no—
94D3×C4⋊C4—94no—
95D3×C4⋊C4—98no—
96D3×C22⋊C4—98no—
97#878—98no—
98C23×C32C22×D311yescharacteristic
99C22×C3⋊D3—31no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—31no—
102C22×C3⋊D3Dic612yescomplemented
103C2×C3×A4—24no—
104C2×C3×A4C22×D314yes—
105D3×A4—38no—
106C32×Q8—61no—
107C32×Q8—61no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—32no—
111C2×C32×C4—32no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—32no—
115C2×C3⋊Dic3—32no—
116C4×C3⋊D3—31no—
117C4×C3⋊D3—31no—
118C4×C3⋊D3—31no—
119C4×C3⋊D3—61no—
120C4×C3⋊D3—64no—
121C4×C3⋊D3—62no—
122C4×C3⋊D3—62no—
123C4×C3⋊D3S411yescharacteristic
124#634—62no—
125#634—61no—
126#634—61no—
127#634—62no—
128#634—124no—
129#634—244no—
130C3×C4×D3—124no—
131C3×Dic6—44no—
132#627—128no—
133D3×Dic3—128no—
134C2×C22⋊Q8—271noSylow
135C2×C33—41no—
136C3×C3⋊D3—42no—
137C23×D3—34no—
138C22×A4—91no—
139C2×C3×Q8—34no—
140C22×C3×C4D3214yes—
141C22×Dic3—34no—
142D3×Q8—184no—
143D3×Q8—184no—
144D3×Q8—184no—
145D3×Q8—184no—
146C2×C4×D3—94no—
147C2×C4×D3—94no—
148C2×C4×D3—94no—
149C2×C4×D3—94no—
150C2×C4×D3—98no—
151C2×C4×D3—98no—
152C2×Dic6—98no—
153C2×Dic6—94no—
154C2×Dic6—361no—
155C4×A4—24no—
156C4×A4—91no—
157C4×A4C2×C3⋊D311yescharacteristic
158SL(2,ℤ/4)—92no—
159SL(2,ℤ/4)—188no—
160SL(2,ℤ/4)C2×C3⋊D312yes—
161C3×C4⋊C4—38no—
162C3×C4⋊C4—34no—
163C3×C22⋊C4—38no—
164#217—98no—
165(C2×D3)⋊C4—98no—
166(C2×D3)⋊C4—98no—
167C4⋊Dic3—98no—
168C4⋊Dic3—94no—
169Dic3⋊C4—98no—
170Dic3⋊C4—98no—
171Dic3⋊C4—98no—
172C22×C32—31no—
173C22×C32—31no—
174C22×C32C2×Dic611yescharacteristic, complemented
175C2×C3⋊D3—31no—
176C2×C3⋊D3—32no—
177C2×C3⋊D3—32no—
178C2×C3⋊D3—32no—
179C2×C3⋊D3—61no—
180C2×C3⋊D3C2×S411yescharacteristic
181C2×C3×D3—124no—
182C3×A4—24no—
183C3×A4D3×Q814yescomplemented
184C32×C4—31no—
185C32×C4—44no—
186C32×C4—62no—
187C32×C4—62no—
188C32×C4—84no—
189C32×C4C2×S411yescharacteristic
190C3⋊Dic3—31no—
191C3⋊Dic3—62no—
192C3⋊Dic3—62no—
193C3⋊Dic3—128no—
194C3⋊Dic3—248no—
195C3⋊Dic3C2×S411yescharacteristic
196C3×Dic3—48no—
197C3×Dic3—124no—
198C22×Q8—271no—
199C23×C4—91no—
200C22⋊Q8—31no—
201C22⋊Q8—272no—
202C22⋊Q8—272no—
203C22⋊Q8—271no—
204C22⋊Q8—271no—
205C22⋊Q8—271no—
206C2×C4⋊C4—271no—
207C2×C4⋊C4—272no—
208C2×C22⋊C4—272no—
209C33—41noSylow
210C23×C3C2×D3214yes—
211C22×D3—38no—
212C22×D3—94no—
213C22×D3—94no—
214C22×D3—98no—
215C2×A4—24no—
216C2×A4—92no—
217C2×A4C22×C3⋊D311yescharacteristic
218C3×Q8—64no—
219C3×Q8—64no—
220C2×C3×C4—34no—
221C2×C3×C4—34no—
222C2×C3×C4—38no—
223C2×C3×C4—38no—
224C2×C3×C4—361no—
225C2×Dic3—94no—
226C2×Dic3—94no—
227C2×Dic3—98no—
228C2×Dic3—98no—
229C2×Dic3—362no—
230C4×D3—34no—
231C4×D3—94no—
232C4×D3—94no—
233C4×D3—94no—
234C4×D3—184no—
235C4×D3—188no—
236C4×D3—1816no—
237C4×D3—188no—
238Dic6—41no—
239Dic6—188no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—362no—
244Dic6—361no—
245Dic6—724no—
246C2×C32—31no—
247C2×C32—31no—
248C2×C32—44no—
249C2×C32—84no—
250C2×C32C22×S411yescharacteristic
251C3⋊D3—32no—
252C3⋊D3A4⋊Q812yesdirect product factor
253C3×D3—128no—
254C24—91no—
255C2×Q8—31no—
256C2×Q8—271no—
257C2×Q8—272no—
258C2×Q8—541no—
259C2×Q8—541no—
260C2×Q8—541no—
261C2×Q8—541no—
262C22×C4—91no—
263C22×C4—271no—
264C22×C4—271no—
265C22×C4—271no—
266C22×C4—271no—
267C22×C4—272no—
268C22×C4—272no—
269C22×C4D3×C3⋊D311yescharacteristic
270C4⋊C4—31no—
271C4⋊C4—32no—
272C4⋊C4—272no—
273C4⋊C4—272no—
274C4⋊C4—272no—
275C4⋊C4—271no—
276C4⋊C4—272no—
277C22⋊C4—32no—
278C22⋊C4—272no—
279C22⋊C4—272no—
280C22⋊C4—272no—
281C22×C3—34no—
282C22×C3—34no—
283C22×C3—361no—
284C22×C3D3×Dic614yescomplemented
285C2×D3—34no—
286C2×D3—98no—
287C2×D3—98no—
288C2×D3—94no—
289C2×D3—98no—
290C2×D3—184no—
291A4—24no—
292A4Q8×C3⋊D311yescharacteristic, complemented
293C3×C4—34no—
294C3×C4—41no—
295C3×C4—68no—
296C3×C4—68no—
297C3×C4—84no—
298C3×C4—361no—
299C3×C4D3×S414yes—
300Dic3—34no—
301Dic3—42no—
302Dic3—94no—
303Dic3—188no—
304Dic3—188no—
305Dic3—362no—
306Dic3—728no—
307C32—44no—
308C32—84no—
309C32C2×A4⋊Q811yescharacteristic, complemented
310C23—92no—
311C23—271no—
312C23—271no—
313C23—272no—
314C23C2×D3×C3⋊D311yescharacteristic
315Q8—61no—
316Q8—61no—
317Q8—541no—
318Q8—542no—
319Q8—542no—
320Q8—541no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—91no—
326C2×C4—271no—
327C2×C4—271no—
328C2×C4—271no—
329C2×C4—271no—
330C2×C4—271no—
331C2×C4—272no—
332C2×C4—272no—
333C2×C4—541no—
334C2×C4—542no—
335C2×C4—542no—
336C2×C4—544no—
337C2×C3—34no—
338C2×C3—34no—
339C2×C3—41no—
340C2×C3—84no—
341C2×C3—362no—
342C2×C3C2×D3×S414yes—
343D3—38no—
344D3—98no—
345C22—31no—
346C22—31no—
347C22—91no—
348C22—272no—
349C22—272no—
350C22—272no—
351C22—271no—
352C22—541no—
353C22C3⋊D3×Dic611yescharacteristic, complemented
354C4—31no—
355C4—62no—
356C4—62no—
357C4—91no—
358C4—271no—
359C4—542no—
360C4—542no—
361C4C3⋊D3×S411yescharacteristic
362C3—41no—
363C3—84no—
364C3D3×A4⋊Q814yescomplemented
365C2—31no—
366C2—31no—
367C2—92no—
368C2—272no—
369C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×S4
commutator, cosocleC2×C32×A4C23
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
prime radicalC2×A4×C3⋊D3C22
hypercenterC4C3⋊D3×S4
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11724463C2semidirect
3. C32×A4⋊Q8C2semidirect
5. #11724184C2semidirect
19. C3×A4⋊Q8D3semidirect
32. A4×C3⋊D3Q8semidirect
76. C32×A4C2×Q8semidirect
88. A4⋊Q8C3⋊D3direct (N × H)
102. C22×C3⋊D3Dic6semidirect
174. C22×C32C2×Dic6semidirect
183. C3×A4D3×Q8semidirect
252. C3⋊D3A4⋊Q8direct (N × H)
284. C22×C3D3×Dic6semidirect
292. A4Q8×C3⋊D3semidirect
309. C32C2×A4⋊Q8semidirect
353. C22C3⋊D3×Dic6semidirect
364. C3D3×A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C4×A4×C3⋊D32
#117244632
C32×A4⋊Q82
C3⋊D3×SL(2,ℤ/4)2
#117241842
D3×A4⋊Q83
C3⋊D3×C22⋊Q83
C3⋊D3×Dic64