Finite Group Databasebeta

← #72220529

Subgroups — #72220529

Order 2592.

669 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 37 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C3⋊(C3⋊S4)C311yesmaximal, characteristic, complemented
3A4×C3⋊S4—31nomaximal
4(C3⋊S4)⋊A4—32nomaximal
5#638⋊A4—31nomaximal
6C3⋊((C3⋊D3)⋊A4)—41nomaximal
7A4⋊((C3⋊D3)⋊C3)—41nomaximal
8C22×C32×A4C2×C311yescharacteristic, complemented
9C2×C3⋊(C3⋊S4)—31no—
10C2×C3⋊(C3⋊S4)—31no—
11C3×A42—21no—
12C3×A42D311yescharacteristic, complemented
13C32⋊(C22⋊A4)—21no—
14C22×C22⋊H(3,𝔽3)D311yescharacteristic, complemented
15C2×(C3⋊D3)⋊A4—31no—
16(C3×A4)⋊A4—22no—
17(C3×A4)⋊A4D312yescomplemented
18(C3⋊Dic3)⋊A4—31no—
19C3×C22⋊H(3,𝔽3)—41no—
20A4×H(3,𝔽3)—41no—
21C22×C3⋊S4—31no—
22C22×C3⋊S4—32no—
23C22×C3⋊S4—93no—
24A4×S4—93no—
25C22×#638—31no—
26A4×#67—91no—
27C22×C3⋊(C3⋊D3)—41no—
28C2×C32×A4—31no—
29A4×C3⋊D3—121no—
30C3⋊(C3⋊S4)—31no—
31C3⋊(C3⋊S4)A411yescharacteristic, complemented
32C3×C3⋊S4—121no—
33C2×C22⋊H(3,𝔽3)—31no—
34(C3⋊D3)⋊A4—61no—
35(C3⋊D3)⋊A4—122no—
36(C3⋊S4)⋊C3—122no—
37#638⋊C3—121no—
38C3⋊((C3⋊D3)⋊C3)—161no—
39C24×C32C3×D311yescharacteristic, complemented
40C23×C3⋊D3—31no—
41C3×C22⋊A4—22no—
42C3×C22⋊A4—21no—
43C22×C3×A4—21no—
44C22×C3×A4—33no—
45C22×C3×A4C3×D311yescharacteristic
46C22×C3×A4C3×D312yescomplemented
47C22×C3×A4C3⋊D311yescharacteristic, complemented
48C2×D3×A4—91no—
49C2×C3⋊S4—91no—
50C2×C3⋊S4—92no—
51C2×C3⋊S4—99no—
52C2×C3⋊S4—99no—
53C2×C3⋊S4—91no—
54C2×C3⋊S4—92no—
55A42—33no—
56A42—63no—
57C2×#638—91no—
58C2×#638—91no—
59C2×#638—91no—
60C2×#638—91no—
61C22×C3⋊Dic3—31no—
62Dic3×A4—91no—
63C22×C33—41no—
64C2×C3⋊(C3⋊D3)—121no—
65C2×C3⋊(C3⋊D3)—121no—
66C32×A4—41no—
67C32×A4—41no—
68C32×A4—81no—
69C32×A4—81no—
70C32×A4C2×A411yescharacteristic, complemented
71C22×H(3,𝔽3)—41no—
72C2×(C3⋊D3)⋊C3—121no—
73C22⋊H(3,𝔽3)—42no—
74C22⋊H(3,𝔽3)—42no—
75C22⋊H(3,𝔽3)—82no—
76C22⋊H(3,𝔽3)—81no—
77C22⋊H(3,𝔽3)—81no—
78C22⋊H(3,𝔽3)—82no—
79C22⋊H(3,𝔽3)—241no—
80C22⋊H(3,𝔽3)S411yescharacteristic, complemented
81(C3⋊Dic3)⋊C3—121no—
82C22×S4—93no—
83C22×S4—272no—
84C22×#67—91no—
85C22×#67—271no—
86D4×A4—271no—
87C3×H(3,𝔽3)—161noSylow
88C23×C32—31no—
89C23×C32—31no—
90C23×C32—91no—
91C22×C3⋊D3—61no—
92C22×C3⋊D3—91no—
93C22×C3⋊D3—91no—
94C22×C3⋊D3—121no—
95C22×C3⋊D3—122no—
96C22×C3⋊D3—181no—
97C22×C3⋊D3—363no—
98C2×C3×A4—39no—
99C2×C3×A4—31no—
100C2×C3×A4—32no—
101C2×C3×A4—31no—
102C2×C3×A4—61no—
103D3×A4—181no—
104D3×A4—363no—
105C3⋊S4—31no—
106C3⋊S4—32no—
107C3⋊S4—93no—
108C3⋊S4—99no—
109C3⋊S4—91no—
110C3⋊S4—92no—
111C3×S4—363no—
112#638—31no—
113#638—91no—
114#638—91no—
115#638—92no—
116#638—91no—
117C2×C3⋊Dic3—91no—
118C2×C3⋊Dic3—91no—
119C3×#67—361no—
120C2×C33—121no—
121C3⋊(C3⋊D3)—41no—
122C3⋊(C3⋊D3)—121no—
123C3×C3⋊D3—481no—
124C2×H(3,𝔽3)—121no—
125(C3⋊D3)⋊C3—241no—
126(C3⋊D3)⋊C3—482no—
127C24×C3—31no—
128C24×C3C3×C3⋊D311yescharacteristic
129C23×D3—91no—
130C23×D3—271no—
131C22⋊A4—61no—
132C22⋊A4—62no—
133C22×A4—32no—
134C22×A4—31no—
135C22×A4—61no—
136C22×A4—271no—
137C22×A4(C3⋊D3)⋊C313yescomplemented
138C2×S4—273no—
139C2×S4—276no—
140C2×S4—276no—
141C2×S4—273no—
142C2×#67—271no—
143C2×#67—273no—
144C2×#67—273no—
145C2×#67—273no—
146C2×#67—273no—
147C2×#67—271no—
148C2×#67—271no—
149C2×#67—271no—
150C22×Dic3—91no—
151C22×Dic3—271no—
152C4×A4—271no—
153C22×C32—41no—
154C22×C32—42no—
155C22×C32—41no—
156C22×C32—61no—
157C22×C32—81no—
158C22×C32—91no—
159C22×C32—91no—
160C22×C32—91no—
161C22×C32—123no—
162C22×C32C3×S411yescharacteristic, complemented
163C22×C32D3×A411yescharacteristic, complemented
164C2×C3⋊D3—31no—
165C2×C3⋊D3—91no—
166C2×C3⋊D3—181no—
167C2×C3⋊D3—181no—
168C2×C3⋊D3—181no—
169C2×C3⋊D3—181no—
170C2×C3⋊D3—361no—
171C2×C3⋊D3—362no—
172C2×C3⋊D3—369no—
173C2×C3⋊D3—369no—
174C2×C3⋊D3—362no—
175C2×C3⋊D3—361no—
176C2×C3×D3—361no—
177C3×A4—21no—
178C3×A4—33no—
179C3×A4—82no—
180C3×A4—81no—
181C3×A4—82no—
182C3×A4—81no—
183C3×A4—123no—
184C3×A4—123no—
185C3×A4—242no—
186C3×A4—241no—
187C3×A4—243no—
188C3×A4—243no—
189C3×A4C3⋊S411yescharacteristic, complemented
190C3×A4D3×A411yescharacteristic
191C3×A4D3×A412yescomplemented
192C3⋊Dic3—31no—
193C3⋊Dic3—91no—
194C3×Dic3—361no—
195C22×D4—271noSylow
196C33—41no—
197C33—161no—
198C33—321no—
199H(3,𝔽3)—41no—
200H(3,𝔽3)—162no—
201H(3,𝔽3)—322no—
202H(3,𝔽3)—321no—
203C23×C3—33no—
204C23×C3—31no—
205C23×C3—31no—
206C23×C3—91no—
207C23×C3—91no—
208C23×C3—93no—
209C22×D3—181no—
210C22×D3—271no—
211C22×D3—273no—
212C22×D3—273no—
213C22×D3—271no—
214C22×D3—363no—
215C22×D3—541no—
216C22×D3—543no—
217C22×D3—541no—
218C22×D3—1082no—
219C2×A4—36no—
220C2×A4—33no—
221C2×A4—91no—
222C2×A4—181no—
223C2×A4—541no—
224S4—93no—
225S4—272no—
226S4—276no—
227S4—273no—
228C3×D4—1081no—
229#67—91no—
230#67—271no—
231#67—271no—
232#67—271no—
233#67—273no—
234#67—273no—
235#67—273no—
236#67—272no—
237#67—276no—
238#67—271no—
239C2×Dic3—271no—
240C2×Dic3—273no—
241C2×Dic3—273no—
242C2×Dic3—271no—
243C2×C32—31no—
244C2×C32—31no—
245C2×C32—91no—
246C2×C32—121no—
247C2×C32—122no—
248C2×C32—129no—
249C2×C32—121no—
250C2×C32—241no—
251C3⋊D3—61no—
252C3⋊D3—121no—
253C3⋊D3—122no—
254C3⋊D3—181no—
255C3⋊D3—363no—
256C3⋊D3—361no—
257C3⋊D3—362no—
258C3⋊D3—369no—
259C3×D3—721no—
260C3×D3—1443no—
261C24—271no—
262C24C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
263C2×D4—811no—
264C2×D4—811no—
265C2×D4—811no—
266C2×D4—811no—
267C22×C4—271no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—43no—
271C22×C3—61no—
272C22×C3—91no—
273C22×C3—91no—
274C22×C3—91no—
275C22×C3—93no—
276C22×C3—93no—
277C22×C3—93no—
278C22×C3—122no—
279C22×C3—121no—
280C22×C3—181no—
281C22×C3—241no—
282C22×C3—1081no—
283C22×C3C3×C3⋊S411yescharacteristic
284C22×C3A4×C3⋊D311yescharacteristic
285C2×D3—91no—
286C2×D3—271no—
287C2×D3—271no—
288C2×D3—273no—
289C2×D3—541no—
290C2×D3—541no—
291C2×D3—543no—
292C2×D3—543no—
293C2×D3—543no—
294C2×D3—543no—
295C2×D3—541no—
296C2×D3—541no—
297C2×D3—1083no—
298C2×D3—1086no—
299C2×D3—1086no—
300C2×D3—1083no—
301A4—31no—
302A4—32no—
303A4—61no—
304A4—241no—
305A4—242no—
306A4—241no—
307A4—242no—
308A4—721no—
309A4—722no—
310A4(C3⋊D3)⋊A413yescomplemented
311C3×C4—1081no—
312Dic3—91no—
313Dic3—271no—
314Dic3—271no—
315Dic3—273no—
316C32—41no—
317C32—41no—
318C32—42no—
319C32—81no—
320C32—123no—
321C32—322no—
322C32—321no—
323C32—483no—
324C32—963no—
325C32A4×S411yescharacteristic, complemented
326C23—31no—
327C23—31no—
328C23—91no—
329C23—541no—
330C23—811no—
331C23—811no—
332C23—1621no—
333D4—271no—
334D4—811no—
335D4—811no—
336D4—812no—
337D4—811no—
338C2×C4—811no—
339C2×C4—811no—
340C2×C3—33no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—91no—
344C2×C3—93no—
345C2×C3—91no—
346C2×C3—123no—
347C2×C3—126no—
348C2×C3—361no—
349C2×C3—721no—
350C2×C3—2161no—
351D3—181no—
352D3—363no—
353D3—541no—
354D3—543no—
355D3—541no—
356D3—1082no—
357D3—1083no—
358D3—1086no—
359C22—61no—
360C22—91no—
361C22—91no—
362C22—91no—
363C22—271no—
364C22—811no—
365C22—1621no—
366C22—1621no—
367C22—1621no—
368C22—1621no—
369C22A4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
370C22C3⋊((C3⋊D3)⋊A4)11yescharacteristic, complemented
371C4—271no—
372C4—811no—
373C3—31no—
374C3—43no—
375C3—122no—
376C3—121no—
377C3—241no—
378C3—961no—
379C3—962no—
380C3A4×C3⋊S411yescharacteristic
381C2—31no—
382C2—31no—
383C2—91no—
384C2—541no—
385C2—1621no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×C32×A4C2×C3
FrattiniC3A4×C3⋊S4
FittingC24×C32C3×D3
socleC24×C3C3×C3⋊D3
LCSG ▷ C22×C32×A4
UCSC1

Complements

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

NH ≅ G/Ntype
1. A4×C22⋊H(3,𝔽3)C2semidirect
2. C22×C3⋊(C3⋊S4)C3semidirect
8. C22×C32×A4C2×C3semidirect
12. C3×A42D3semidirect
14. C22×C22⋊H(3,𝔽3)D3semidirect
17. (C3×A4)⋊A4D3semidirect
31. C3⋊(C3⋊S4)A4semidirect
39. C24×C32C3×D3semidirect
46. C22×C3×A4C3×D3semidirect
47. C22×C3×A4C3⋊D3semidirect
70. C32×A4C2×A4semidirect
80. C22⋊H(3,𝔽3)S4semidirect
137. C22×A4(C3⋊D3)⋊C3semidirect
162. C22×C32C3×S4semidirect
163. C22×C32D3×A4semidirect
189. C3×A4C3⋊S4semidirect
191. C3×A4D3×A4semidirect
262. C24C3⋊((C3⋊D3)⋊C3)semidirect
310. A4(C3⋊D3)⋊A4semidirect
325. C32A4×S4semidirect
369. C22A4⋊((C3⋊D3)⋊C3)semidirect
370. C22C3⋊((C3⋊D3)⋊A4)semidirect

Maximal subgroups

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

subgroupindex
A4×C22⋊H(3,𝔽3)2
C22×C3⋊(C3⋊S4)3
A4×C3⋊S43
(C3⋊S4)⋊A43
#638⋊A43
C3⋊((C3⋊D3)⋊A4)4
A4⋊((C3⋊D3)⋊C3)4