Finite Group Databasebeta

← C2×A4×#627 #13180914

Subgroups — C2×A4×#627 #13180914

Order 1728.

970 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 120 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×A4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×Dic3×A4C212yesmaximal
3A4×#627C214yesmaximal, direct product factor
4C23×#627C311yesmaximal, characteristic, complemented
5C2×A4×Dic6—32nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3×#627—41nomaximal
8A4×C3⋊Dic3C2212yes—
9C3×Dic3×A4C2214yes—
10C23×C3⋊Dic3C2×C311yescharacteristic
11C23×C3×Dic3C2×C312yes—
12C2×C3×C4×A4—32no—
13C22×#627—31no—
14C22×#627—31no—
15C22×#627—32no—
16C22×#627—34no—
17C22×#627C2×C314yescomplemented
18C2×Dic3×A4—22no—
19C2×Dic3×A4—32no—
20C2×Dic3×A4—61no—
21C2×Dic3×A4D312yes—
22A4×Dic6—38no—
23C2×C32×A4Q812yescomplemented
24C2×C32×A4C2311yescharacteristic
25C2×C3×C3⋊Dic3—41no—
26C2×C32×Dic3—42no—
27C3×#627—44no—
28C23×Dic6—32no—
29C2×Q8×A4—91no—
30C24×C32C22×C311yescharacteristic
31C22×C3×A4—22no—
32C22×C3×A4—21no—
33C22×C3×A4—41no—
34C22×C3×A4C2×D312yes—
35C22×C3⋊Dic3—31no—
36C22×C3⋊Dic3—31no—
37C22×C3⋊Dic3—32no—
38C22×C3⋊Dic3C22×C312yes—
39C22×C3×Dic3—32no—
40C22×C3×Dic3—32no—
41C22×C3×Dic3—34no—
42C22×C3×Dic3C22×C314yes—
43C2×C3×Dic6—122no—
44C3×C4×A4—34no—
45C2×#627—32no—
46C2×#627—34no—
47C2×#627—38no—
48C2×#627—31no—
49C2×#627—32no—
50C2×#627—34no—
51C2×#627—34no—
52C2×#627—34no—
53C2×#627—38no—
54C2×#627A411yescharacteristic, direct product factor
55Dic3×A4—24no—
56Dic3×A4—34no—
57Dic3×A4—62no—
58Dic3×A4C2×D314yes—
59C22×C33—41no—
60C32×A4C2×Q811yescharacteristic, complemented
61C3×C3⋊Dic3—42no—
62C32×Dic3—44no—
63C23×C3×C4—32no—
64C23×Dic3—32no—
65C23×Dic3—61no—
66C23×Dic3C3×D312yes—
67C22×Dic6—38no—
68C22×Dic6—92no—
69C22×Dic6—98no—
70C22×Dic6—92no—
71C22×Dic6—92no—
72C22×Dic6—92no—
73Q8×A4—94no—
74C2×C4×A4—32no—
75C2×C4×A4—62no—
76C2×C4×A4—91no—
77C23×C32—32no—
78C23×C32—31no—
79C23×C32—31no—
80C23×C32C3×Q812yescomplemented
81C23×C32C23×C311yescharacteristic
82C2×C3×A4—24no—
83C2×C3×A4—22no—
84C2×C3×A4—22no—
85C2×C3×A4—21no—
86C2×C3×A4—42no—
87C2×C3×A4—41no—
88C2×C3×A4Dic614yescomplemented
89C2×C3×A4C22×D312yes—
90C2×C32×C4—122no—
91C2×C3⋊Dic3—32no—
92C2×C3⋊Dic3—31no—
93C2×C3⋊Dic3—32no—
94C2×C3⋊Dic3—32no—
95C2×C3⋊Dic3—32no—
96C2×C3⋊Dic3C2×A411yescharacteristic
97C2×C3×Dic3—34no—
98C2×C3×Dic3—32no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—82no—
104C2×C3×Dic3—122no—
105C2×C3×Dic3—241no—
106C2×C3×Dic3C2×A412yes—
107C3×Dic6—128no—
108#627—38no—
109#627—34no—
110#627—38no—
111#627C2×A414yesdirect product factor
112C23×Q8—91noSylow
113C2×C33—42no—
114C2×C33—41no—
115C24×C3—21no—
116C24×C3C2×C3×D312yes—
117C22×A4—22no—
118C22×A4—41no—
119C22×A4D3211yescharacteristic
120C2×C3×Q8—361no—
121C22×C3×C4—34no—
122C22×C3×C4—92no—
123C22×C3×C4—94no—
124C22×C3×C4—92no—
125C22×Dic3—32no—
126C22×Dic3—32no—
127C22×Dic3—34no—
128C22×Dic3—34no—
129C22×Dic3—62no—
130C22×Dic3—92no—
131C22×Dic3—94no—
132C22×Dic3—92no—
133C22×Dic3—181no—
134C22×Dic3—182no—
135C22×Dic3—181no—
136C22×Dic3C2×C3×D314yes—
137C2×Dic6—32no—
138C2×Dic6—98no—
139C2×Dic6—98no—
140C2×Dic6—92no—
141C2×Dic6—92no—
142C2×Dic6—98no—
143C2×Dic6—98no—
144C2×Dic6—94no—
145C2×Dic6—98no—
146C2×Dic6—92no—
147C2×Dic6—98no—
148C2×Dic6—98no—
149C2×Dic6—98no—
150C4×A4—34no—
151C4×A4—64no—
152C4×A4—92no—
153C22×C32—32no—
154C22×C32—31no—
155C22×C32—32no—
156C22×C32—32no—
157C22×C32—32no—
158C22×C32—31no—
159C22×C32—31no—
160C22×C32—42no—
161C22×C32—81no—
162C22×C32—82no—
163C22×C32—161no—
164C22×C32C2×C3×Q811yescharacteristic, complemented
165C22×C32C22×A411yescharacteristic
166C3×A4—22no—
167C3×A4—21no—
168C3×A4—41no—
169C3×A4C2×Dic612yescomplemented
170C32×C4—124no—
171C3⋊Dic3—32no—
172C3⋊Dic3C22×A412yes—
173C3×Dic3—34no—
174C3×Dic3—44no—
175C3×Dic3—84no—
176C3×Dic3—124no—
177C3×Dic3—242no—
178C3×Dic3C22×A414yes—
179C22×Q8—94no—
180C22×Q8—271no—
181C22×Q8—272no—
182C22×Q8—271no—
183C22×Q8—274no—
184C23×C4—32no—
185C23×C4—91no—
186C33—41noSylow
187C23×C3—22no—
188C23×C3—21no—
189C23×C3—32no—
190C23×C3—34no—
191C23×C3—32no—
192C23×C3—61no—
193C23×C3—61no—
194C23×C3—62no—
195C23×C3C3×Dic614yescomplemented
196C23×C3C22×C3×D312yes—
197C2×A4—24no—
198C2×A4—22no—
199C2×A4—42no—
200C2×A4—41no—
201C2×A4#62712yesdirect product factor
202C2×A4C2×D3211yescharacteristic
203C3×Q8—364no—
204C2×C3×C4—32no—
205C2×C3×C4—94no—
206C2×C3×C4—94no—
207C2×C3×C4—94no—
208C2×C3×C4—92no—
209C2×C3×C4—94no—
210C2×C3×C4—122no—
211C2×C3×C4—242no—
212C2×C3×C4—361no—
213C2×Dic3—34no—
214C2×Dic3—32no—
215C2×Dic3—32no—
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—34no—
219C2×Dic3—61no—
220C2×Dic3—94no—
221C2×Dic3—94no—
222C2×Dic3—94no—
223C2×Dic3—92no—
224C2×Dic3—94no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C2×Dic3—181no—
228C2×Dic3—182no—
229C2×Dic3—182no—
230C2×Dic3D3×A412yes—
231Dic6—38no—
232Dic6—98no—
233Dic6—98no—
234Dic6—916no—
235Dic6—98no—
236C2×C32—32no—
237C2×C32—31no—
238C2×C32—31no—
239C2×C32—44no—
240C2×C32—42no—
241C2×C32—81no—
242C2×C32—82no—
243C2×C32—84no—
244C2×C32—82no—
245C2×C32—161no—
246C2×C32—162no—
247C2×C32Q8×A412yescomplemented
248C2×C32C23×A411yescharacteristic
249C24C3×D3211yescharacteristic
250C2×Q8—91no—
251C2×Q8—272no—
252C2×Q8—271no—
253C2×Q8—274no—
254C2×Q8—274no—
255C2×Q8—278no—
256C2×Q8—274no—
257C2×Q8—278no—
258C2×Q8—274no—
259C2×Q8—272no—
260C22×C4—34no—
261C22×C4—92no—
262C22×C4—94no—
263C22×C4—92no—
264C22×C4—92no—
265C22×C4—271no—
266C22×C4—272no—
267C22×C4—271no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—34no—
271C22×C3—32no—
272C22×C3—34no—
273C22×C3—34no—
274C22×C3—34no—
275C22×C3—32no—
276C22×C3—32no—
277C22×C3—41no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—62no—
281C22×C3—62no—
282C22×C3—62no—
283C22×C3—61no—
284C22×C3—62no—
285C22×C3—82no—
286C22×C3—161no—
287C22×C3C2×C3×Dic612yescomplemented
288C22×C3C2×D3×A412yes—
289A4—22no—
290A4—41no—
291A4C2×#62711yescharacteristic, direct product factor
292C3×C4—34no—
293C3×C4—94no—
294C3×C4—124no—
295C3×C4—244no—
296C3×C4—362no—
297Dic3—34no—
298Dic3—34no—
299Dic3—62no—
300Dic3—94no—
301Dic3—182no—
302Dic3C2×D3×A414yes—
303C32—42no—
304C32—81no—
305C32—82no—
306C32—161no—
307C32C2×Q8×A411yescharacteristic, complemented
308C23—31no—
309C23—32no—
310C23—31no—
311C23C3×#62712yescomplemented
312C23C2×C3×D3211yescharacteristic
313Q8—94no—
314Q8—278no—
315Q8—274no—
316Q8—278no—
317C2×C4—32no—
318C2×C4—94no—
319C2×C4—94no—
320C2×C4—94no—
321C2×C4—92no—
322C2×C4—91no—
323C2×C4—94no—
324C2×C4—272no—
325C2×C4—272no—
326C2×C4—272no—
327C2×C4—271no—
328C2×C4—272no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—32no—
332C2×C3—34no—
333C2×C3—32no—
334C2×C3—41no—
335C2×C3—42no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—62no—
339C2×C3—82no—
340C2×C3—84no—
341C2×C3—161no—
342C2×C3—162no—
343C2×C3A4×Dic614yescomplemented
344C2×C3C22×D3×A412yes—
345C22—32no—
346C22—31no—
347C22—32no—
348C22—31no—
349C22—31no—
350C22—32no—
351C22—32no—
352C22C2×C3×#62711yescharacteristic, complemented
353C22D32×A411yescharacteristic
354C4—34no—
355C4—94no—
356C4—92no—
357C4—272no—
358C3—21no—
359C3—41no—
360C3—82no—
361C3—161no—
362C3C2×A4×Dic612yescomplemented
363C2—31no—
364C2—32no—
365C2—31no—
366C2A4×#62712yesdirect product factor
367C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, cosocleC23×C32C23×C3
FrattiniC2C2×D32×A4
Fitting, socleC24×C32C22×C3
prime radicalC22×C32×A4C22
hypocenterC22×C32C2×C3×Q8
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. A4×#627C2direct (N × H)
4. C23×#627C3semidirect
17. C22×#627C2×C3semidirect
23. C2×C32×A4Q8semidirect
54. C2×#627A4direct (N × H)
60. C32×A4C2×Q8semidirect
80. C23×C32C3×Q8semidirect
88. C2×C3×A4Dic6semidirect
111. #627C2×A4direct (N × H)
164. C22×C32C2×C3×Q8semidirect
169. C3×A4C2×Dic6semidirect
195. C23×C3C3×Dic6semidirect
201. C2×A4#627direct (N × H)
247. C2×C32Q8×A4semidirect
287. C22×C3C2×C3×Dic6semidirect
291. A4C2×#627direct (N × H)
307. C32C2×Q8×A4semidirect
311. C23C3×#627semidirect
343. C2×C3A4×Dic6semidirect
352. C22C2×C3×#627semidirect
362. C3C2×A4×Dic6semidirect
366. C2A4×#627direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
C2×C3×Dic3×A42
A4×#6272
C23×#6273
C2×A4×Dic63
C2×C3×#6274