Finite Group Databasebeta

← A4×#3719 #13166134

Subgroups — A4×#3719 #13166134

Order 1728.

390 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 54 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
1A4×#634C211yesmaximal, characteristic
2C3×A4×Dic6C211yesmaximal, characteristic
3C3×A4×C3⋊C8C211yesmaximal, characteristic
4C22×#3719C311yesmaximal, characteristic, complemented
5A4×#216—31nomaximal
6A4×Dic12—31nomaximal
7C32×C4×A4C2211yescharacteristic
8A4×C3⋊Dic3—21no—
9C3×Dic3×A4—21no—
10C3×#3719—41nomaximal
11C22×#634C2×C311yescharacteristic
12C22×C3×Dic6C2×C311yescharacteristic
13C3×Q8×A4—31no—
14A4×Dic6—21no—
15A4×Dic6—31no—
16A4×Dic6—31no—
17A4×Dic6—61no—
18A4×Dic6D311yescharacteristic
19C22×C3×C3⋊C8C2×C311yescharacteristic
20C3×C8×A4—31no—
21C2×#3719—31no—
22C2×#3719—31no—
23C2×#3719—31no—
24C2×#3719—31no—
25A4×C3⋊C8—21no—
26A4×C3⋊C8D311yescharacteristic
27C2×C32×A4D411yescharacteristic
28C3×#634—41no—
29C32×Dic6—41no—
30C32×C3⋊C8—41no—
31C22×#216—31no—
32C22×Dic12—31no—
33A4×Q16—91no—
34C22×C32×C4C22×C311yescharacteristic
35C22×C3⋊Dic3—21no—
36C2×#634—31no—
37C2×#634—31no—
38C2×#634—61no—
39C22×C3×Dic3—21no—
40C2×C3×Dic6—31no—
41C2×C3×Dic6—31no—
42C2×C3×Dic6—61no—
43C3×C4×A4—21no—
44C3×C4×A4—21no—
45C3×C4×A4—21no—
46C3×C4×A4—41no—
47C3×C4×A4—61no—
48C3×C4×A4C2×D311yescharacteristic
49C3×C4×A4C2×D311yescharacteristic
50Dic3×A4—22no—
51Dic3×A4—21no—
52Dic3×A4—62no—
53Dic3×A4—61no—
54Dic3×A4—61no—
55C3×#216—121no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C3×Dic12—121no—
59#3719—32no—
60#3719—31no—
61#3719—31no—
62#3719—31no—
63#3719A411yescharacteristic, direct product factor
64C32×A4Q1611yescharacteristic, complemented
65C33×C4—41no—
66C3×C3⋊Dic3—81no—
67C32×Dic3—81no—
68C22×C3×Q8—31no—
69C22×Dic6—31no—
70C22×Dic6—31no—
71C22×Dic6—61no—
72C22×Dic6C3×D311yescharacteristic
73Q8×A4—31no—
74Q8×A4—61no—
75Q8×A4—91no—
76C22×C3×C8—31no—
77C2×#216—91no—
78C2×#216—91no—
79C2×#216—91no—
80C2×#216—91no—
81C22×C3⋊C8C3×D311yescharacteristic
82C2×Dic12—91no—
83C2×Dic12—91no—
84C2×Dic12—91no—
85C2×Dic12—91no—
86C8×A4—31no—
87C8×A4—61no—
88C23×C32C3×D411yescharacteristic
89C2×C3×A4—21no—
90C2×C3×A4—21no—
91C2×C3×A4—21no—
92C2×C3×A4—41no—
93C2×C3×A4D1211yescharacteristic
94C2×C3×A4#6711yescharacteristic
95C32×Q8—121no—
96C2×C32×C4—31no—
97C2×C32×C4—31no—
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—61no—
100#634—31no—
101#634—61no—
102#634—61no—
103#634C2×A411yescharacteristic
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×Dic6—31no—
107C3×Dic6—41no—
108C3×Dic6—61no—
109C3×Dic6—61no—
110C3×Dic6—81no—
111C3×Dic6—121no—
112C3×Dic6—121no—
113C3×Dic6—241no—
114C3×Dic6C2×A411yescharacteristic
115C32×C8—121no—
116C3×C3⋊C8—31no—
117C3×C3⋊C8—41no—
118C3×C3⋊C8—81no—
119C3×C3⋊C8C2×A411yescharacteristic
120C22×Q16—91noSylow
121C2×C33—41no—
122C2×C3×Q8—91no—
123C2×C3×Q8—91no—
124C2×C3×Q8—181no—
125C22×C3×C4—21no—
126C22×C3×C4—61no—
127C22×C3×C4C2×C3×D311yescharacteristic
128C22×C3×C4C2×C3×D311yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—62no—
131C22×Dic3—61no—
132C22×Dic3—61no—
133C2×Dic6—31no—
134C2×Dic6—31no—
135C2×Dic6—61no—
136C2×Dic6—91no—
137C2×Dic6—91no—
138C2×Dic6—91no—
139C2×Dic6—91no—
140C2×Dic6—181no—
141C2×Dic6—181no—
142C2×Dic6—182no—
143C2×Dic6—181no—
144C2×Dic6—181no—
145C4×A4—21no—
146C4×A4—21no—
147C4×A4—41no—
148C4×A4—62no—
149C4×A4—61no—
150C4×A4—181no—
151C4×A4D3211yescharacteristic
152C3×Q16—361no—
153C2×C3×C8—91no—
154C2×C3×C8—91no—
155#216—31no—
156#216—91no—
157#216—91no—
158#216—92no—
159#216—91no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—31no—
162Dic12—31no—
163Dic12—91no—
164Dic12—91no—
165Dic12—92no—
166Dic12—91no—
167C22×C32—31no—
168C22×C32—31no—
169C22×C32C3×Q1611yescharacteristic, complemented
170C3×A4—21no—
171C3×A4—21no—
172C3×A4—21no—
173C3×A4—41no—
174C3×A4Dic1211yescharacteristic, complemented
175C3×A4#21611yescharacteristic, complemented
176C32×C4—31no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—81no—
180C32×C4—81no—
181C32×C4—81no—
182C32×C4—161no—
183C32×C4—241no—
184C32×C4C22×A411yescharacteristic
185C3⋊Dic3—21no—
186C3⋊Dic3—61no—
187C3×Dic3—21no—
188C3×Dic3—61no—
189C3×Dic3—81no—
190C3×Dic3—82no—
191C3×Dic3—241no—
192C3×Dic3—241no—
193C3×Dic3—242no—
194C22×Q8—31no—
195C22×Q8—91no—
196C2×Q16—271no—
197C2×Q16—271no—
198C2×Q16—271no—
199C2×Q16—271no—
200C22×C8—31no—
201C33—41noSylow
202C23×C3—21no—
203C23×C3C3×D1211yescharacteristic
204C23×C3C3×#6711yescharacteristic
205C2×A4—21no—
206C2×A4—21no—
207C2×A4—41no—
208C2×A4#62611yescharacteristic
209C3×Q8—31no—
210C3×Q8—91no—
211C3×Q8—121no—
212C3×Q8—181no—
213C3×Q8—181no—
214C3×Q8—241no—
215C3×Q8—361no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—182no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—182no—
232Dic6—31no—
233Dic6—31no—
234Dic6—31no—
235Dic6—61no—
236Dic6—61no—
237Dic6—61no—
238Dic6—91no—
239Dic6—91no—
240Dic6—181no—
241Dic6—181no—
242Dic6—181no—
243Dic6—182no—
244Dic6—182no—
245Dic6—181no—
246Dic6—181no—
247Dic6D3×A411yescharacteristic
248C3×C8—31no—
249C3×C8—91no—
250C3×C8—121no—
251C3×C8—241no—
252C3⋊C8—31no—
253C3⋊C8D3×A411yescharacteristic
254C2×C32—31no—
255C2×C32—31no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—81no—
259C2×C32—81no—
260C2×C32—81no—
261C2×C32—161no—
262C2×C32D4×A411yescharacteristic
263C2×Q8—91no—
264C2×Q8—91no—
265C2×Q8—181no—
266C2×Q8—271no—
267C2×Q8—271no—
268C2×Q8—541no—
269C22×C4—61no—
270C22×C4—181no—
271C22×C4C3×D3211yescharacteristic
272Q16—91no—
273Q16—271no—
274Q16—271no—
275Q16—271no—
276Q16—272no—
277C2×C8—91no—
278C2×C8—91no—
279C22×C3—21no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3C3×Dic1211yescharacteristic, complemented
287C22×C3C3×#21611yescharacteristic, complemented
288A4—21no—
289A4—21no—
290A4—41no—
291A4#371911yescharacteristic, direct product factor
292C3×C4—21no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—41no—
296C3×C4—61no—
297C3×C4—61no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—161no—
301C3×C4—181no—
302C3×C4—241no—
303C3×C4—242no—
304C3×C4—721no—
305C3×C4C2×D3×A411yescharacteristic
306C3×C4C2×D3×A411yescharacteristic
307Dic3—21no—
308Dic3—62no—
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—182no—
313Dic3—181no—
314Dic3—181no—
315C32—41no—
316C32—41no—
317C32—81no—
318C32—81no—
319C32—81no—
320C32—161no—
321C32A4×Q1611yescharacteristic, complemented
322C23C3×#62611yescharacteristic
323Q8—31no—
324Q8—91no—
325Q8—91no—
326Q8—181no—
327Q8—181no—
328Q8—271no—
329Q8—541no—
330Q8—541no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—541no—
336C2×C4—541no—
337C8—31no—
338C8—91no—
339C2×C3—21no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—41no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—81no—
348C2×C3—81no—
349C2×C3—161no—
350C2×C3A4×D1211yescharacteristic
351C2×C3A4×#6711yescharacteristic
352C22—31no—
353C22—31no—
354C22C3×#371911yescharacteristic, complemented
355C4—31no—
356C4—61no—
357C4—181no—
358C4—181no—
359C4—541no—
360C4D32×A411yescharacteristic
361C3—21no—
362C3—41no—
363C3—81no—
364C3—81no—
365C3—161no—
366C3A4×Dic1211yescharacteristic, complemented
367C3A4×#21611yescharacteristic, complemented
368C2—31no—
369C2—31no—
370C2A4×#62611yescharacteristic
subgroupG/H
centerC2A4×#626
commutator, Fitting, cosocleC22×C32×C4C22×C3
Frattini, hypercenterC4D32×A4
socleC23×C32C3×D4
prime radicalC2×C32×A4D4
hypocenterC22×C32C3×Q16
LCSG ▷ C22×C32×C4 ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
4. C22×#3719C3semidirect
63. #3719A4direct (N × H)
64. C32×A4Q16semidirect
169. C22×C32C3×Q16semidirect
174. C3×A4Dic12semidirect
175. C3×A4#216semidirect
286. C22×C3C3×Dic12semidirect
287. C22×C3C3×#216semidirect
291. A4#3719direct (N × H)
321. C32A4×Q16semidirect
354. C22C3×#3719semidirect
366. C3A4×Dic12semidirect
367. C3A4×#216semidirect

Maximal subgroups

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

subgroupindex
A4×#6342
C3×A4×Dic62
C3×A4×C3⋊C82
C22×#37193
A4×#2163
A4×Dic123
C3×#37194