Finite Group Databasebeta

← A4×#3797 #13180883

Subgroups — A4×#3797 #13180883

Order 1728.

946 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 102 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
2C3×A4×Dic6C212yesmaximal, complemented
3A4×#627C212yesmaximal, complemented
4A4×#624C212yesmaximal
5C22×#3797C311yesmaximal, characteristic, complemented
6D3×Q8×A4—32nomaximal
7C2×A4×C3⋊D3C2211yescharacteristic
8C32×C4×A4C2211yescharacteristic
9C3×#3797—41nomaximal
10A4×C3⋊Dic3C2211yescharacteristic
11C3×Dic3×A4C2214yes—
12C22×C4×C3⋊D3C2×C311yescharacteristic
13C22×C3×Dic6C2×C312yescomplemented
14C3×Q8×A4—32no—
15C22×#627C2×C312yescomplemented
16C22×#624C2×C312yes—
17C2×#3797—31no—
18C2×#3797—31no—
19C2×#3797—32no—
20C2×#3797—32no—
21C2×#3797—32no—
22C4×D3×A4—32no—
23C4×D3×A4—34no—
24C4×D3×A4—61no—
25A4×Dic6—22no—
26A4×Dic6—34no—
27A4×Dic6D312yescomplemented
28C2×C32×A4C2311yescharacteristic
29A4×C3⋊D3Q812yescomplemented
30C3×C4×C3⋊D3—41no—
31C32×Dic6—42no—
32C3×#627—42no—
33C3×#624—42no—
34C22×D3×Q8—32no—
35C2×Q8×A4—91no—
36C23×C3⋊D3C22×C311yescharacteristic
37C2×D3×A4—32no—
38C2×D3×A4—61no—
39C22×C32×C4C22×C311yescharacteristic
40C22×C3⋊Dic3C22×C311yescharacteristic
41C2×C4×C3⋊D3—31no—
42C2×C4×C3⋊D3—31no—
43C2×C4×C3⋊D3—31no—
44C2×C4×C3⋊D3—31no—
45C22×C3×Dic3C22×C314yes—
46C3×D3×Q8—122no—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—34no—
50C3×C4×A4—22no—
51C3×C4×A4—21no—
52C3×C4×A4—34no—
53C3×C4×A4—41no—
54C3×C4×A4C2×D312yes—
55C2×#627—32no—
56C2×#627—32no—
57C2×#627—34no—
58C2×#624—32no—
59C2×#624—32no—
60C2×#624—34no—
61#3797—32no—
62#3797—32no—
63#3797—38no—
64#3797—32no—
65#3797—31no—
66#3797—32no—
67#3797—32no—
68#3797—32no—
69#3797A411yescharacteristic, direct product factor
70Dic3×A4—24no—
71Dic3×A4—32no—
72Dic3×A4—61no—
73Dic3×A4C2×D314yes—
74C2×C3×C3⋊D3—41no—
75C32×A4C2×Q811yescharacteristic, complemented
76C33×C4—41no—
77C3×C3⋊Dic3—41no—
78C32×Dic3—44no—
79C22×C3×Q8—32no—
80C2×D3×Q8—92no—
81C2×D3×Q8—92no—
82C2×D3×Q8—94no—
83C2×D3×Q8—94no—
84C2×D3×Q8—92no—
85C2×D3×Q8—92no—
86C22×C4×D3—32no—
87C22×C4×D3—34no—
88C22×C4×D3—61no—
89C22×Dic6—34no—
90C22×Dic6C3×D312yescomplemented
91Q8×A4—32no—
92Q8×A4—62no—
93Q8×A4—92no—
94C2×C4×A4—92no—
95C2×C4×A4—91no—
96C23×C32C23×C311yescharacteristic
97C22×C3⋊D3—31no—
98C22×C3⋊D3—32no—
99C22×C3⋊D3—31no—
100C22×C3⋊D3C3×Q812yescomplemented
101C2×C3×A4—22no—
102C2×C3×A4—21no—
103C2×C3×A4—41no—
104C2×C3×A4C22×D312yes—
105D3×A4—34no—
106D3×A4—62no—
107C32×Q8—122no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C4×C3⋊D3—32no—
113C4×C3⋊D3—31no—
114C4×C3⋊D3—31no—
115C4×C3⋊D3—31no—
116C4×C3⋊D3C2×A411yescharacteristic
117C2×C3×Dic3—34no—
118C2×C3×Dic3—34no—
119C3×C4×D3—124no—
120C3×C4×D3—122no—
121C3×C4×D3—241no—
122C3×Dic6—34no—
123C3×Dic6—32no—
124C3×Dic6—34no—
125C3×Dic6—42no—
126C3×Dic6—82no—
127C3×Dic6—124no—
128C3×Dic6C2×A412yescomplemented
129#627—34no—
130#627—32no—
131#627—34no—
132#627C2×A412yescomplemented
133#624—34no—
134#624—32no—
135#624—34no—
136#624C2×A412yes—
137C23×Q8—91noSylow
138C2×C33—41no—
139C3×C3⋊D3—42no—
140C23×D3—32no—
141C23×D3—61no—
142C22×A4—91no—
143C2×C3×Q8—92no—
144C2×C3×Q8—92no—
145C2×C3×Q8—94no—
146C2×C3×Q8—361no—
147C22×C3×C4—21no—
148C22×C3×C4—34no—
149C22×C3×C4C2×C3×D312yes—
150C22×Dic3—32no—
151C22×Dic3—61no—
152C22×Dic3C2×C3×D314yes—
153D3×Q8—32no—
154D3×Q8—94no—
155D3×Q8—92no—
156D3×Q8—94no—
157D3×Q8—92no—
158D3×Q8—94no—
159D3×Q8—98no—
160D3×Q8—94no—
161D3×Q8—94no—
162D3×Q8—98no—
163D3×Q8—92no—
164C2×C4×D3—94no—
165C2×C4×D3—92no—
166C2×C4×D3—94no—
167C2×C4×D3—94no—
168C2×C4×D3—94no—
169C2×C4×D3—92no—
170C2×C4×D3—92no—
171C2×C4×D3—92no—
172C2×C4×D3—181no—
173C2×C4×D3—181no—
174C2×C4×D3—181no—
175C2×C4×D3—181no—
176C2×Dic6—32no—
177C2×Dic6—32no—
178C2×Dic6—34no—
179C2×Dic6—94no—
180C2×Dic6—94no—
181C2×Dic6—94no—
182C2×Dic6—94no—
183C4×A4—22no—
184C4×A4—34no—
185C4×A4—41no—
186C4×A4—64no—
187C4×A4—91no—
188C4×A4D3211yescharacteristic
189C22×C32—31no—
190C22×C32—31no—
191C22×C32C2×C3×Q811yescharacteristic, complemented
192C2×C3⋊D3—32no—
193C2×C3⋊D3—31no—
194C2×C3⋊D3—32no—
195C2×C3⋊D3—32no—
196C2×C3⋊D3—32no—
197C2×C3⋊D3C22×A411yescharacteristic
198C2×C3×D3—122no—
199C2×C3×D3—241no—
200C3×A4—22no—
201C3×A4—21no—
202C3×A4—41no—
203C3×A4D3×Q812yescomplemented
204C32×C4—31no—
205C32×C4—42no—
206C32×C4—81no—
207C32×C4—82no—
208C32×C4—124no—
209C32×C4—161no—
210C32×C4C22×A411yescharacteristic
211C3⋊Dic3—31no—
212C3⋊Dic3C22×A411yescharacteristic
213C3×Dic3—34no—
214C3×Dic3—44no—
215C3×Dic3—84no—
216C3×Dic3—122no—
217C3×Dic3—241no—
218C3×Dic3C22×A414yes—
219C22×Q8—32no—
220C22×Q8—92no—
221C22×Q8—271no—
222C22×Q8—271no—
223C22×Q8—272no—
224C22×Q8—272no—
225C22×Q8—272no—
226C23×C4—92no—
227C23×C4—91no—
228C33—41noSylow
229C23×C3—21no—
230C23×C3C22×C3×D312yes—
231C22×D3—34no—
232C22×D3—62no—
233C22×D3—92no—
234C22×D3—92no—
235C22×D3—94no—
236C22×D3—181no—
237C22×D3—181no—
238C22×D3—182no—
239C2×A4—22no—
240C2×A4—41no—
241C2×A4—92no—
242C2×A4C2×D3211yescharacteristic
243C3×Q8—32no—
244C3×Q8—94no—
245C3×Q8—94no—
246C3×Q8—92no—
247C3×Q8—122no—
248C3×Q8—242no—
249C3×Q8—362no—
250C2×C3×C4—32no—
251C2×C3×C4—32no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—94no—
255C2×C3×C4—94no—
256C2×C3×C4—361no—
257C2×C3×C4—362no—
258C2×Dic3—34no—
259C2×Dic3—34no—
260C2×Dic3—92no—
261C2×Dic3—92no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C4×D3—32no—
265C4×D3—34no—
266C4×D3—61no—
267C4×D3—94no—
268C4×D3—92no—
269C4×D3—94no—
270C4×D3—98no—
271C4×D3—92no—
272C4×D3—94no—
273C4×D3—92no—
274C4×D3—94no—
275C4×D3—181no—
276C4×D3—181no—
277C4×D3—181no—
278C4×D3—182no—
279Dic6—34no—
280Dic6—32no—
281Dic6—34no—
282Dic6—34no—
283Dic6—94no—
284Dic6—94no—
285Dic6—98no—
286Dic6—94no—
287Dic6D3×A412yescomplemented
288C2×C32—31no—
289C2×C32—31no—
290C2×C32—42no—
291C2×C32—81no—
292C2×C32—82no—
293C2×C32—161no—
294C2×C32C23×A411yescharacteristic
295C3⋊D3—32no—
296C3⋊D3Q8×A412yescomplemented
297C3×D3—124no—
298C3×D3—242no—
299C24—91no—
300C2×Q8—92no—
301C2×Q8—92no—
302C2×Q8—94no—
303C2×Q8—91no—
304C2×Q8—272no—
305C2×Q8—274no—
306C2×Q8—272no—
307C2×Q8—272no—
308C2×Q8—271no—
309C2×Q8—272no—
310C2×Q8—272no—
311C2×Q8—278no—
312C2×Q8—272no—
313C2×Q8—272no—
314C2×Q8—272no—
315C22×C4—34no—
316C22×C4—91no—
317C22×C4—271no—
318C22×C4—271no—
319C22×C4—271no—
320C22×C4—271no—
321C22×C4—272no—
322C22×C4—274no—
323C22×C4—272no—
324C22×C4C3×D3211yescharacteristic
325C22×C3—21no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—361no—
331C22×C3C3×D3×Q812yescomplemented
332C2×D3—32no—
333C2×D3—61no—
334C2×D3—94no—
335C2×D3—94no—
336C2×D3—94no—
337C2×D3—92no—
338C2×D3—94no—
339C2×D3—182no—
340C2×D3—182no—
341C2×D3—181no—
342C2×D3—182no—
343C2×D3—182no—
344A4—22no—
345A4—41no—
346A4#379711yescharacteristic, direct product factor
347C3×C4—21no—
348C3×C4—32no—
349C3×C4—34no—
350C3×C4—41no—
351C3×C4—61no—
352C3×C4—82no—
353C3×C4—94no—
354C3×C4—124no—
355C3×C4—161no—
356C3×C4—244no—
357C3×C4—361no—
358C3×C4C2×D3×A412yes—
359Dic3—34no—
360Dic3—32no—
361Dic3—61no—
362Dic3—92no—
363Dic3—181no—
364Dic3C2×D3×A414yes—
365C32—42no—
366C32—81no—
367C32—82no—
368C32—161no—
369C32C2×Q8×A411yescharacteristic, complemented
370C23—92no—
371C23—271no—
372C23—271no—
373C23—272no—
374C23C2×C3×D3211yescharacteristic
375Q8—32no—
376Q8—94no—
377Q8—92no—
378Q8—94no—
379Q8—92no—
380Q8—274no—
381Q8—272no—
382Q8—274no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—92no—
386C2×C4—94no—
387C2×C4—91no—
388C2×C4—94no—
389C2×C4—274no—
390C2×C4—271no—
391C2×C4—271no—
392C2×C4—272no—
393C2×C4—272no—
394C2×C4—274no—
395C2×C4—271no—
396C2×C4—271no—
397C2×C4—271no—
398C2×C3—21no—
399C2×C3—32no—
400C2×C3—32no—
401C2×C3—41no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—82no—
405C2×C3—161no—
406C2×C3—362no—
407C2×C3C22×D3×A412yes—
408D3—34no—
409D3—62no—
410D3—94no—
411D3—182no—
412C22—31no—
413C22—31no—
414C22—91no—
415C22—272no—
416C22—272no—
417C22—272no—
418C22—272no—
419C22—271no—
420C22C3×#379711yescharacteristic, complemented
421C4—31no—
422C4—34no—
423C4—94no—
424C4—91no—
425C4—271no—
426C4D32×A411yescharacteristic
427C3—21no—
428C3—41no—
429C3—82no—
430C3—161no—
431C3D3×Q8×A412yescomplemented
432C2—31no—
433C2—31no—
434C2—92no—
435C2—272no—
436C2C2×D32×A411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×A4
commutator, socle, cosocleC23×C32C23×C3
FittingC22×C32×C4C22×C3
prime radicalC2×A4×C3⋊D3C22
hypercenterC4D32×A4
hypocenterC22×C32C2×C3×Q8
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×A4×Dic6C2semidirect
3. A4×#627C2semidirect
5. C22×#3797C3semidirect
13. C22×C3×Dic6C2×C3semidirect
15. C22×#627C2×C3semidirect
27. A4×Dic6D3semidirect
29. A4×C3⋊D3Q8semidirect
69. #3797A4direct (N × H)
75. C32×A4C2×Q8semidirect
90. C22×Dic6C3×D3semidirect
100. C22×C3⋊D3C3×Q8semidirect
128. C3×Dic6C2×A4semidirect
132. #627C2×A4semidirect
191. C22×C32C2×C3×Q8semidirect
203. C3×A4D3×Q8semidirect
287. Dic6D3×A4semidirect
296. C3⋊D3Q8×A4semidirect
331. C22×C3C3×D3×Q8semidirect
346. A4#3797direct (N × H)
369. C32C2×Q8×A4semidirect
420. C22C3×#3797semidirect
431. C3D3×Q8×A4semidirect

Maximal subgroups

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

subgroupindex
C4×A4×C3⋊D32
C3×A4×Dic62
A4×#6272
A4×#6242
C22×#37973
D3×Q8×A43
C3×#37974