Finite Group Databasebeta

← C4×(#41⋊C3)⋊A4 #72228363

Subgroups — C4×(#41⋊C3)⋊A4 #72228363

Order 2304.

1170 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 95 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×(#41⋊C3)⋊A4C211yesmaximal, characteristic
2C4×Ω+(4,𝔽3)C212yesmaximal, complemented
3C4×#5945C312yesmaximal, complemented
4C4×#5950C312yesmaximal, complemented
5(#41⋊C3)⋊A4C412yesdirect product factor
6C2×Ω+(4,𝔽3)C2211yescharacteristic
7C3×C4×#41⋊C3—42nomaximal
8C2×#5950C2×C312yes—
9C2×#5945C2×C312yes—
10C4×#997C2×C314yescomplemented
11C4×2−1+4⋊C3—34no—
12C4×#994C2×C314yescomplemented
13#3428⋊C3—32no—
14C2×C4×#41⋊C3—32no—
15C2×C3×#41⋊C3—42no—
16Ω+(4,𝔽3)C2×C411yescharacteristic, complemented
17C3×C4×SL(2,𝔽3)—44no—
18C4×#585C3211yescharacteristic, complemented, Sylow
19#5950C3×C414yescomplemented
20C2×#997C22×C312yes—
21#5945—34no—
22#5945C3×C414yescomplemented
23C2×2−1+4⋊C3—32no—
24C2×#994C22×C312yes—
25C22×#41⋊C3—32no—
26C3×C4×#41—42no—
27#520⋊C3—34no—
28#519⋊C3—34no—
29C4×#41⋊C3—32no—
30C4×#41⋊C3—34no—
31C4×#41⋊C3—44no—
32C4×#41⋊C3A412yescomplemented
33C2×C4×SL(2,𝔽3)—34no—
34C42×A4—122no—
35C3×#41⋊C3—44no—
36C2×C3×SL(2,𝔽3)—42no—
37C32×C42—161no—
38C2×#585C2×C3211yescharacteristic
39C4×2−1+4—34no—
40C4×H(5,𝔽2)—92no—
41C4×H(5,𝔽2)C2×C3212yes—
42#3428—32no—
43#3428—91no—
44C2×C4×#41—32no—
45C2×C4×#41—91no—
46C2×C3×#41—42no—
47#997C2×C3×C412yescomplemented
482−1+4⋊C3—34no—
492−1+4⋊C3—34no—
50#994—32no—
51#994C2×C3×C412yescomplemented
52C2×#41⋊C3—34no—
53C2×#41⋊C3—34no—
54C2×#41⋊C3—32no—
55C2×#41⋊C3—44no—
56C2×#41⋊C3C2×A412yes—
57C22×SL(2,𝔽3)—32no—
58C2×C4×A4—124no—
59C2×C4×A4—122no—
60C3×C4×Q8—44no—
61C3×C4×D4—124no—
62C3×#117—122no—
63C2×C3×C42—122no—
64C4×SL(2,𝔽3)—34no—
65C4×SL(2,𝔽3)—48no—
66C4×SL(2,𝔽3)C2×A414yes—
67C2×C32×C4—161no—
68C2×C32×C4—162no—
69C3×SL(2,𝔽3)—42no—
70#585—34no—
71#585—92no—
72#585C32×C412yescomplemented
73C2×2−1+4—32no—
74C2×H(5,𝔽2)—91no—
75C2×H(5,𝔽2)C22×C3211yescharacteristic
76C22×#41—32no—
77C22×#41—91no—
78#520—34no—
79#520—98no—
80#520—94no—
81#519—34no—
82#519—92no—
83#518—94no—
84#518—94no—
85#518—92no—
86C4×#41—32no—
87C4×#41—34no—
88C4×#41—92no—
89C4×#41—98no—
90C4×#41—94no—
91C4×#41—92no—
92C4×#41—94no—
93C4×#41C3×A412yes—
94C2×C4×Q8—34no—
95C2×C4×Q8—92no—
96C2×C4×D4—94no—
97C2×C4×D4—94no—
98C2×C4×D4—92no—
99C2×#117—92no—
100C2×#117—91no—
101C2×#117—92no—
102C22×C42—32no—
103C22×C42—91no—
104C22×A4—122no—
105C3×#41—44no—
106C3×#41—124no—
107C2×C3×Q8—42no—
108C2×C3×D4—122no—
109C22×C3×C4—122no—
110C22×C3×C4—124no—
111#41⋊C3—32no—
112#41⋊C3—32no—
113#41⋊C3—48no—
114#41⋊C3C4×A414yescomplemented
115C2×SL(2,𝔽3)—34no—
116C2×SL(2,𝔽3)—44no—
117C2×SL(2,𝔽3)C22×A412yes—
118C4×A4—124no—
119C4×A4—128no—
120C3×C4⋊C4—124no—
121C3×C22⋊C4—124no—
122C3×C42—42no—
123C3×C42—122no—
124C3×C42—124no—
125C3×C42—162no—
126C22×C32—161no—
127C32×C4—164no—
128C32×C4—162no—
1292−1+4—34no—
1302−1+4—34no—
1312−1+4—94no—
1322−1+4—94no—
133H(5,𝔽2)—32no—
134H(5,𝔽2)—94no—
135H(5,𝔽2)—98no—
136H(5,𝔽2)—91no—
137H(5,𝔽2)—92no—
138H(5,𝔽2)—92no—
139H(5,𝔽2)C2×C32×C411yescharacteristic
140C2×#41—34no—
141C2×#41—34no—
142C2×#41—32no—
143C2×#41—92no—
144C2×#41—94no—
145C2×#41—92no—
146C2×#41—92no—
147C2×#41—98no—
148C2×#41—94no—
149C2×#41—92no—
150C2×#41—94no—
151C2×#41—94no—
152C2×#41C2×C3×A412yes—
153C22×Q8—32no—
154C22×Q8—91no—
155C22×D4—92no—
156C22×D4—92no—
157C22×D4—91no—
158C23×C4—32no—
159C23×C4—34no—
160C23×C4—91no—
161C23×C4—92no—
162C4×Q8—34no—
163C4×Q8—94no—
164C4×Q8—98no—
165C4×Q8—94no—
166C4×Q8C2×C3×A414yes—
167C4×D4—34no—
168C4×D4—94no—
169C4×D4—98no—
170C4×D4—94no—
171C4×D4—98no—
172C4×D4—916no—
173C4×D4—94no—
174C4×D4—98no—
175#117—32no—
176#117—94no—
177#117—94no—
178#117—98no—
179#117—94no—
180#117—94no—
181#117—92no—
182C2×C4⋊C4—94no—
183C2×C4⋊C4—94no—
184C2×C4⋊C4—92no—
185C2×C22⋊C4—94no—
186C2×C22⋊C4—94no—
187C2×C22⋊C4—92no—
188C2×C42—32no—
189C2×C42—91no—
190C2×C42—92no—
191C2×C42—92no—
192C2×C42—94no—
193C2×C42—91no—
194C2×C42—92no—
195C2×C42—94no—
196C23×C3—122no—
197C2×A4—122no—
198C2×A4—124no—
199C3×Q8—42no—
200C3×Q8—122no—
201C3×D4—124no—
202C3×D4—124no—
203C2×C3×C4—44no—
204C2×C3×C4—42no—
205C2×C3×C4—124no—
206C2×C3×C4—124no—
207C2×C3×C4—122no—
208C2×C3×C4—124no—
209C2×C3×C4—164no—
210C2×C3×C4—162no—
211C2×C3×C4—248no—
212SL(2,𝔽3)—44no—
213SL(2,𝔽3)C2×C4×A412yes—
214C2×C32—161no—
215C2×C32—162no—
216C24—32no—
217C24—91no—
218#41—34no—
219#41—32no—
220#41—32no—
221#41—98no—
222#41—94no—
223#41—94no—
224#41—94no—
225#41—94no—
226#41—916no—
227#41—98no—
228#41—98no—
229#41—94no—
230#41—94no—
231#41—92no—
232#41—92no—
233#41C3×C4×A414yes—
234C2×Q8—34no—
235C2×Q8—92no—
236C2×Q8—92no—
237C2×Q8—94no—
238C2×Q8—92no—
239C2×Q8—94no—
240C2×Q8C22×C3×A412yes—
241C2×D4—32no—
242C2×D4—92no—
243C2×D4—94no—
244C2×D4—94no—
245C2×D4—98no—
246C2×D4—94no—
247C2×D4—92no—
248C2×D4—94no—
249C2×D4—98no—
250C2×D4—94no—
251C2×D4—94no—
252C2×D4—91no—
253C2×D4—91no—
254C22×C4—32no—
255C22×C4—34no—
256C22×C4—34no—
257C22×C4—91no—
258C22×C4—92no—
259C22×C4—94no—
260C22×C4—92no—
261C22×C4—92no—
262C22×C4—92no—
263C22×C4—94no—
264C22×C4—91no—
265C22×C4—92no—
266C22×C4—94no—
267C22×C4—92no—
268C22×C4—94no—
269C22×C4—128no—
270C22×C4—364no—
271C4⋊C4—34no—
272C4⋊C4—94no—
273C4⋊C4—98no—
274C22⋊C4—34no—
275C22⋊C4—98no—
276C22⋊C4—94no—
277C42—32no—
278C42—34no—
279C42—92no—
280C42—94no—
281C42—94no—
282C42—91no—
283C42—92no—
284C42A4211yescharacteristic
285C22×C3—42no—
286C22×C3—124no—
287C22×C3—162no—
288C22×C3—244no—
289A4—122no—
290C3×C4—44no—
291C3×C4—48no—
292C3×C4—122no—
293C3×C4—122no—
294C3×C4—164no—
295C3×C4—168no—
296C3×C4—244no—
297C32—161noSylow
298C23—32no—
299C23—32no—
300C23—92no—
301C23—92no—
302C23—91no—
303C23—92no—
304C23—124no—
305C23—362no—
306Q8—32no—
307Q8—94no—
308Q8—94no—
309Q8C2×C3×C4×A412yes—
310D4—34no—
311D4—34no—
312D4—92no—
313D4—92no—
314D4—94no—
315D4—98no—
316D4—98no—
317C2×C4—32no—
318C2×C4—34no—
319C2×C4—34no—
320C2×C4—34no—
321C2×C4—68no—
322C2×C4—91no—
323C2×C4—92no—
324C2×C4—91no—
325C2×C4—94no—
326C2×C4—94no—
327C2×C4—94no—
328C2×C4—91no—
329C2×C4—92no—
330C2×C4—92no—
331C2×C4—94no—
332C2×C4—92no—
333C2×C4—184no—
334C2×C4—362no—
335C2×C4—364no—
336C2×C4—364no—
337C2×C4C2×A4211yescharacteristic
338C2×C4C2×A4212yes—
339C2×C3—42no—
340C2×C3—44no—
341C2×C3—162no—
342C2×C3—164no—
343C2×C3—244no—
344C22—34no—
345C22—64no—
346C22—91no—
347C22—91no—
348C22—122no—
349C22—182no—
350C22—362no—
351C22—362no—
352C22—362no—
353C22C22×A4211yescharacteristic
354C4—32no—
355C4—32no—
356C4—64no—
357C4—92no—
358C4—182no—
359C4C4×A4212yes—
360C4(#41⋊C3)⋊A414yesdirect product factor
361C3—42no—
362C3—162no—
363C2—64no—
364C2—181no—
365C2—181no—
366C2C2×C4×A4211yescharacteristic
367C2C2×(#41⋊C3)⋊A412yes—
subgroupG/H
center, hypercenterC42A42
commutator, hypocenterH(5,𝔽2)C2×C32×C4
Frattini, socleC22C22×A42
FittingC4×#585C32
prime radicalC2×(#41⋊C3)⋊A4C2
cosocleC2×H(5,𝔽2)C22×C32
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×Ω+(4,𝔽3)C2semidirect
3. C4×#5945C3semidirect
4. C4×#5950C3semidirect
5. (#41⋊C3)⋊A4C4direct (N × H)
10. C4×#997C2×C3semidirect
12. C4×#994C2×C3semidirect
16. Ω+(4,𝔽3)C2×C4semidirect
18. C4×#585C32semidirect
19. #5950C3×C4semidirect
22. #5945C3×C4semidirect
32. C4×#41⋊C3A4semidirect
47. #997C2×C3×C4semidirect
51. #994C2×C3×C4semidirect
72. #585C32×C4semidirect
114. #41⋊C3C4×A4semidirect
360. C4(#41⋊C3)⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×(#41⋊C3)⋊A42
C4×Ω+(4,𝔽3)2
C4×#59453
C4×#59503
C3×C4×#41⋊C34