Finite Group Databasebeta

← #3757⋊A4 #13147306

Subgroups — #3757⋊A4 #13147306

Order 1728.

908 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C23×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊Dic3)⋊A4C212yesmaximal, complemented
3C22×#3757C311yesmaximal, characteristic, complemented
4A4×#217—31nomaximal
5C22×C22⋊H(3,𝔽3)—22no—
6C22×C22⋊H(3,𝔽3)C412yescomplemented
7C22×C22⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊Dic3)⋊A4—22no—
9#3757⋊C3—41nomaximal
10C25×C32C2×C311yescharacteristic
11C23×C3×A4—21no—
12C23×C3×A4D311yescharacteristic
13C23×C3⋊Dic3C2×C312yescomplemented
14C2×Dic3×A4—32no—
15C2×#3757—31no—
16C2×#3757—31no—
17C2×#3757—32no—
18C23×H(3,𝔽3)—41no—
19C2×C22⋊H(3,𝔽3)—22no—
20C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
21C2×C22⋊H(3,𝔽3)D412yes—
22C2×(C3⋊Dic3)⋊C3—42no—
23C22×#217—31no—
24C22×#217—91no—
25A4×C22⋊C4—91no—
26C24×C32—22no—
27C24×C32—31no—
28C24×C32—32no—
29C24×C32—31no—
30C24×C32—62no—
31C24×C32C3×C412yescomplemented
32C24×C32C22×C311yescharacteristic
33C22×C3×A4—24no—
34C22×C3×A4—22no—
35C22×C3×A4—21no—
36C22×C3×A4—22no—
37C22×C3×A4Dic312yescomplemented
38C22×C3×A4C2×D311yescharacteristic
39C22×C3⋊Dic3—22no—
40C22×C3⋊Dic3—32no—
41C22×C3⋊Dic3—32no—
42C22×C3⋊Dic3—62no—
43Dic3×A4—62no—
44#3757—32no—
45#3757—31no—
46#3757—32no—
47#3757A411yescharacteristic, complemented
48C3×#217—121no—
49C22×H(3,𝔽3)—41no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)—82no—
52C22⋊H(3,𝔽3)C22⋊C411yescharacteristic, complemented
53(C3⋊Dic3)⋊C3—82no—
54C25×C3—31no—
55C25×C3C3×D311yescharacteristic
56C23×A4—31no—
57C23×A4—61no—
58C23×Dic3—32no—
59C23×Dic3—92no—
60C2×C4×A4—92no—
61C2×#217—91no—
62C2×#217—93no—
63C2×#217—96no—
64C2×#217—93no—
65C2×#217—92no—
66C2×#217—91no—
67C23×C32—22no—
68C23×C32—31no—
69C23×C32—32no—
70C23×C32—31no—
71C23×C32—32no—
72C23×C32—32no—
73C23×C32—32no—
74C23×C32—32no—
75C23×C32—31no—
76C23×C32—41no—
77C23×C32—62no—
78C23×C32—62no—
79C23×C32—62no—
80C23×C32—64no—
81C23×C32—62no—
82C23×C32—62no—
83C23×C32—64no—
84C23×C32—81no—
85C23×C32C2×C3×C411yescharacteristic
86C23×C32C3×D412yes—
87C23×C32C2×A411yescharacteristic
88C2×C3×A4—24no—
89C2×C3×A4—22no—
90C2×C3×A4—21no—
91C2×C3×A4—22no—
92C2×C3×A4C2×Dic311yescharacteristic
93C2×C3×A4#6712yes—
94C2×C3⋊Dic3—32no—
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—62no—
97C2×C3⋊Dic3—62no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3C2×A412yescomplemented
100C2×C3×Dic3—122no—
101C22×C22⋊C4—91noSylow
102C2×H(3,𝔽3)—41no—
103C2×H(3,𝔽3)—42no—
104C2×H(3,𝔽3)—82no—
105C24×C3—22no—
106C24×C3—36no—
107C24×C3—33no—
108C24×C3—33no—
109C24×C3—32no—
110C24×C3—31no—
111C24×C3—31no—
112C24×C3—32no—
113C24×C3—31no—
114C24×C3—62no—
115C24×C3—62no—
116C24×C3—66no—
117C24×C3C3×Dic312yes—
118C24×C3C2×C3×D311yescharacteristic
119C22×A4—32no—
120C22×A4—31no—
121C22×A4—62no—
122C22×A4—64no—
123C22×A4—62no—
124C22×A4—61no—
125C22×Dic3—62no—
126C22×Dic3—92no—
127C22×Dic3—96no—
128C22×Dic3—96no—
129C22×Dic3—92no—
130C22×Dic3—182no—
131C22×Dic3—186no—
132C22×Dic3—182no—
133C4×A4—182no—
134C3×C22⋊C4—361no—
135#217—31no—
136#217—91no—
137#217—92no—
138#217—96no—
139#217—96no—
140#217—92no—
141#217—93no—
142#217—91no—
143C22×C32—22no—
144C22×C32—31no—
145C22×C32—32no—
146C22×C32—32no—
147C22×C32—31no—
148C22×C32—31no—
149C22×C32—32no—
150C22×C32—32no—
151C22×C32—32no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32—62no—
155C22×C32—64no—
156C22×C32—62no—
157C22×C32—62no—
158C22×C32—62no—
159C22×C32—64no—
160C22×C32—62no—
161C22×C32—82no—
162C22×C32—84no—
163C22×C32—81no—
164C22×C32—82no—
165C22×C32C3×C22⋊C411yescharacteristic, complemented
166C22×C32C4×A412yescomplemented
167C22×C32C22×A411yescharacteristic
168C3×A4—21no—
169C3×A4#21711yescharacteristic, complemented
170C3⋊Dic3—22no—
171C3⋊Dic3—62no—
172C3×Dic3—242no—
173C25(C3⋊D3)⋊C311yescharacteristic
174C23×C4—92no—
175C2×C22⋊C4—271no—
176C2×C22⋊C4—272no—
177C2×C22⋊C4—271no—
178H(3,𝔽3)—41noSylow
179C23×C3—22no—
180C23×C3—32no—
181C23×C3—31no—
182C23×C3—33no—
183C23×C3—36no—
184C23×C3—33no—
185C23×C3—31no—
186C23×C3—33no—
187C23×C3—36no—
188C23×C3—32no—
189C23×C3—36no—
190C23×C3—36no—
191C23×C3—36no—
192C23×C3—32no—
193C23×C3—31no—
194C23×C3—32no—
195C23×C3—32no—
196C23×C3—31no—
197C23×C3—32no—
198C23×C3—31no—
199C23×C3—62no—
200C23×C3—62no—
201C23×C3—62no—
202C23×C3—62no—
203C23×C3—64no—
204C23×C3—62no—
205C23×C3—62no—
206C23×C3—66no—
207C23×C3—66no—
208C23×C3—66no—
209C23×C3—66no—
210C23×C3—612no—
211C23×C3—66no—
212C23×C3—64no—
213C23×C3—612no—
214C23×C3—121no—
215C23×C3—241no—
216C23×C3C2×C3×Dic311yescharacteristic
217C23×C3C3×#6712yes—
218C23×C3D3×A411yescharacteristic
219C2×A4—31no—
220C2×A4—32no—
221C2×A4—62no—
222C2×A4—64no—
223C2×A4—61no—
224C2×A4—62no—
225C2×C3×C4—362no—
226C2×Dic3—32no—
227C2×Dic3—92no—
228C2×Dic3—96no—
229C2×Dic3—92no—
230C2×Dic3—182no—
231C2×Dic3—182no—
232C2×Dic3—186no—
233C2×Dic3—186no—
234C2×Dic3—186no—
235C2×Dic3—186no—
236C2×Dic3—182no—
237C2×Dic3—182no—
238C2×C32—22no—
239C2×C32—31no—
240C2×C32—32no—
241C2×C32—31no—
242C2×C32—41no—
243C2×C32—42no—
244C2×C32—62no—
245C2×C32—82no—
246C2×C32—84no—
247C2×C32—81no—
248C2×C32—82no—
249C2×C32C2×C4×A411yescharacteristic
250C2×C32D4×A412yes—
251C24—22no—
252C24—31no—
253C24—32no—
254C24—31no—
255C24—62no—
256C24(C3⋊Dic3)⋊C312yescomplemented
257C24C2×(C3⋊D3)⋊C311yescharacteristic
258C22×C4—182no—
259C22×C4—272no—
260C22×C4—272no—
261C22×C4—542no—
262C22⋊C4—91no—
263C22⋊C4—272no—
264C22⋊C4—272no—
265C22⋊C4—271no—
266C22×C3—22no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—36no—
270C22×C3—36no—
271C22×C3—33no—
272C22×C3—33no—
273C22×C3—33no—
274C22×C3—36no—
275C22×C3—36no—
276C22×C3—32no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—36no—
282C22×C3—32no—
283C22×C3—32no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—62no—
287C22×C3—62no—
288C22×C3—64no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—62no—
292C22×C3—66no—
293C22×C3—66no—
294C22×C3—612no—
295C22×C3—66no—
296C22×C3—66no—
297C22×C3—612no—
298C22×C3—64no—
299C22×C3—66no—
300C22×C3—62no—
301C22×C3—122no—
302C22×C3—121no—
303C22×C3—242no—
304C22×C3—244no—
305C22×C3—242no—
306C22×C3—241no—
307C22×C3C3×#21711yescharacteristic
308C22×C3Dic3×A412yes—
309C22×C3C2×D3×A411yescharacteristic
310A4—31no—
311A4—61no—
312C3×C4—722no—
313Dic3—62no—
314Dic3—182no—
315Dic3—186no—
316Dic3—182no—
317C32—41no—
318C32—81no—
319C32A4×C22⋊C411yescharacteristic, complemented
320C23—22no—
321C23—32no—
322C23—31no—
323C23—31no—
324C23—32no—
325C23—32no—
326C23—32no—
327C23—32no—
328C23—31no—
329C23—62no—
330C23—62no—
331C23—62no—
332C23—64no—
333C23—62no—
334C23—62no—
335C23—64no—
336C23C2×(C3⋊Dic3)⋊C311yescharacteristic
337C23#638⋊C312yes—
338C23(C3⋊D3)⋊A411yescharacteristic
339C2×C4—92no—
340C2×C4—272no—
341C2×C4—542no—
342C2×C4—542no—
343C2×C4—542no—
344C2×C4—542no—
345C2×C3—22no—
346C2×C3—33no—
347C2×C3—33no—
348C2×C3—36no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—32no—
352C2×C3—31no—
353C2×C3—32no—
354C2×C3—62no—
355C2×C3—66no—
356C2×C3—62no—
357C2×C3—121no—
358C2×C3—122no—
359C2×C3—242no—
360C2×C3—244no—
361C2×C3—241no—
362C2×C3—242no—
363C2×C3C2×Dic3×A411yescharacteristic
364C2×C3A4×#6712yes—
365C22—22no—
366C22—31no—
367C22—32no—
368C22—32no—
369C22—31no—
370C22—31no—
371C22—32no—
372C22—32no—
373C22—32no—
374C22—62no—
375C22—62no—
376C22—64no—
377C22—62no—
378C22—62no—
379C22—64no—
380C22—62no—
381C22#3757⋊C311yescharacteristic, complemented
382C22(C3⋊Dic3)⋊A412yescomplemented
383C22C2×(C3⋊D3)⋊A411yescharacteristic
384C4—182no—
385C4—542no—
386C3—31no—
387C3—121no—
388C3—241no—
389C3A4×#21711yescharacteristic
390C2—22no—
391C2—31no—
392C2—31no—
393C2—32no—
394C2—62no—
395C2C2×(C3⋊Dic3)⋊A411yescharacteristic
396C2#638⋊A412yes—
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutatorC23×C32C2×C3×C4
FrattiniC22×C3C2×D3×A4
FittingC25×C32C2×C3
socleC24×C3C2×C3×D3
prime radicalC23×C22⋊H(3,𝔽3)C2
cosocleC24×C32C22×C3
hypercenterC23(C3⋊D3)⋊A4
hypocenterC22×C32C3×C22⋊C4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊Dic3)⋊A4C2semidirect
3. C22×#3757C3semidirect
6. C22×C22⋊H(3,𝔽3)C4semidirect
13. C23×C3⋊Dic3C2×C3semidirect
31. C24×C32C3×C4semidirect
37. C22×C3×A4Dic3semidirect
47. #3757A4semidirect
52. C22⋊H(3,𝔽3)C22⋊C4semidirect
99. C2×C3⋊Dic3C2×A4semidirect
165. C22×C32C3×C22⋊C4semidirect
166. C22×C32C4×A4semidirect
169. C3×A4#217semidirect
256. C24(C3⋊Dic3)⋊C3semidirect
319. C32A4×C22⋊C4semidirect
381. C22#3757⋊C3semidirect
382. C22(C3⋊Dic3)⋊A4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C2×(C3⋊Dic3)⋊A42
C22×#37573
A4×#2173
#3757⋊C34