Finite Group Databasebeta

← D4×A4×3−1+2 #72253039

Subgroups — D4×A4×3−1+2 #72253039

Order 2592.

880 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 234 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
1C22×A4×3−1+2C212yesmaximal, complemented
2C4×A4×3−1+2C211yesmaximal, characteristic, complemented
3C32×D4×A4C311yesmaximal, characteristic
4C22×D4×3−1+2C311yesmaximal, characteristic, complemented
5D4×#1094C312yesmaximal, complemented
6D4×C9⋊A4C316yesmaximal, complemented
7D4×C9×A4C313yesmaximal, complemented
8C2×A4×3−1+2—22no—
9C2×A4×3−1+2C2211yescharacteristic
10C3×D4×3−1+2—41nomaximal
11C22×C32×A4C2×C312yes—
12C32×C4×A4C2×C311yescharacteristic
13C24×3−1+2C2×C312yescomplemented
14C22×#1094C2×C314yescomplemented
15C22×C9⋊A4C2×C3112yescomplemented
16C22×C9×A4C2×C316yescomplemented
17C2×D4×3−1+2—31no—
18C2×D4×3−1+2—31no—
19C2×D4×3−1+2—32no—
20C22×C4×3−1+2C2×C311yescharacteristic, complemented
21C4×#1094C2×C312yescomplemented
22C4×C9⋊A4C2×C316yescomplemented
23C4×C9×A4C2×C313yescomplemented
24C22×C3×3−1+2—42no—
25A4×3−1+2D411yescharacteristic, direct product factor
26C3×C4×3−1+2—41no—
27C22×C32×D4C3211yescharacteristic
28C3×D4×A4—33no—
29C3×D4×A4C3212yes—
30C3×D4×A4C3211yescharacteristic
31C22×D4×C9C3213yescomplemented
32D4×C22⋊C9C3216yescomplemented
33C2×C32×A4—22no—
34C2×C32×A4C22×C311yescharacteristic
35C33×D4—41no—
36C23×3−1+2—22no—
37C23×3−1+2—32no—
38C23×3−1+2—32no—
39C23×3−1+2—62no—
40C23×3−1+2C22×C311yescharacteristic
41C2×#1094—24no—
42C2×#1094C22×C312yes—
43C2×C9⋊A4—212no—
44C2×C9⋊A4C22×C316yes—
45C2×C9×A4—26no—
46C2×C9×A4C22×C313yes—
47D4×3−1+2—32no—
48D4×3−1+2—32no—
49D4×3−1+2—31no—
50D4×3−1+2—42no—
51D4×3−1+2—46no—
52D4×3−1+2A411yescharacteristic, direct product factor
53C3×D4×C9—43no—
54C2×C4×3−1+2—31no—
55C2×C4×3−1+2—31no—
56C2×C3×3−1+2—41no—
57C2×C3×3−1+2—82no—
58C24×C32C2×C3212yes—
59C22×C3×A4—36no—
60C22×C3×A4C2×C3214yes—
61C22×C3×A4C2×C3212yes—
62C2×C32×D4—31no—
63C2×C32×D4—31no—
64C2×C32×D4—32no—
65C22×C32×C4C2×C3211yescharacteristic
66C3×C4×A4—33no—
67C3×C4×A4C2×C3212yes—
68C3×C4×A4C2×C3211yescharacteristic
69C24×C9C2×C3216yescomplemented
70C22×C22⋊C9C2×C32112yescomplemented
71C2×D4×C9—33no—
72C2×D4×C9—33no—
73C2×D4×C9—36no—
74C22×C4×C9C2×C3213yescomplemented
75C4×C22⋊C9C2×C3216yescomplemented
76C22×C33—42no—
77C32×A4C3×D411yescharacteristic
78C33×C4—41no—
79C22×3−1+2—31no—
80C22×3−1+2—31no—
81C22×3−1+2—32no—
82C22×3−1+2—44no—
83C22×3−1+2—412no—
84C22×3−1+2—62no—
85C22×3−1+2—62no—
86C22×3−1+2—62no—
87C22×3−1+2—62no—
88C22×3−1+2C3×D411yescharacteristic, complemented
89C22×3−1+2C2×A412yescomplemented
90C22×C3×C9—46no—
91#1094C3×D412yescomplemented
92C9⋊A4C3×D416yescomplemented
93C9×A4C3×D413yescomplemented
94C4×3−1+2—31no—
95C4×3−1+2—42no—
96C4×3−1+2—46no—
97C4×3−1+2C2×A411yescharacteristic, complemented
98C3×C4×C9—43no—
99C22×C3×D4—31no—
100C22×C3×D4C3311yescharacteristic
101D4×A4—32no—
102D4×A43−1+213yesdirect product factor
103C3×3−1+2—41noSylow
104C23×C32—22no—
105C23×C32—32no—
106C23×C32—32no—
107C23×C32—62no—
108C23×C32C22×C3211yescharacteristic
109C2×C3×A4—22no—
110C2×C3×A4—24no—
111C2×C3×A4—33no—
112C2×C3×A4—66no—
113C2×C3×A4C22×C3212yes—
114C2×C3×A4C22×C3211yescharacteristic
115C32×D4—32no—
116C32×D4—32no—
117C32×D4—31no—
118C32×D4—41no—
119C32×D4—42no—
120C32×D4—123no—
121C32×D4C3×A411yescharacteristic
122C2×C32×C4—31no—
123C2×C32×C4—31no—
124C23×C9—26no—
125C23×C9—36no—
126C23×C9—36no—
127C23×C9—66no—
128C23×C9C22×C3213yes—
129C2×C22⋊C9—212no—
130C2×C22⋊C9C22×C3216yes—
131D4×C9—36no—
132D4×C9—36no—
133D4×C9—33no—
134D4×C9—46no—
135D4×C9C3×A413yescomplemented
136C2×C4×C9—33no—
137C2×C4×C9—33no—
138C2×C33—41no—
139C2×C33—82no—
140C2×3−1+2—22no—
141C2×3−1+2—31no—
142C2×3−1+2—31no—
143C2×3−1+2—42no—
144C2×3−1+2—46no—
145C2×3−1+2—62no—
146C2×3−1+2—812no—
147C2×3−1+2—84no—
148C2×3−1+2C22×A411yescharacteristic
149C2×C3×C9—43no—
150C2×C3×C9—86no—
151C24×C3—32no—
152C24×C3C2×C3312yes—
153C22×A4—34no—
154C22×A4C2×3−1+216yescomplemented
155C2×C3×D4—31no—
156C2×C3×D4—31no—
157C2×C3×D4—32no—
158C2×C3×D4—91no—
159C2×C3×D4—91no—
160C2×C3×D4—92no—
161C22×C3×C4—31no—
162C22×C3×C4C2×C3311yescharacteristic
163C4×A4—32no—
164C4×A4C2×3−1+213yescomplemented
165C22×C32—31no—
166C22×C32—31no—
167C22×C32—32no—
168C22×C32—42no—
169C22×C32—44no—
170C22×C32—62no—
171C22×C32—62no—
172C22×C32—62no—
173C22×C32—62no—
174C22×C32—126no—
175C22×C32C32×D411yescharacteristic
176C22×C32C2×C3×A412yes—
177C3×A4—33no—
178C3×A4C32×D412yes—
179C3×A4C32×D411yescharacteristic
180C32×C4—31no—
181C32×C4—41no—
182C32×C4—42no—
183C32×C4—123no—
184C32×C4C2×C3×A411yescharacteristic
185C22×C9—33no—
186C22×C9—33no—
187C22×C9—36no—
188C22×C9—412no—
189C22×C9—66no—
190C22×C9—66no—
191C22×C9—66no—
192C22×C9—66no—
193C22×C9C32×D413yescomplemented
194C22×C9C2×C3×A416yescomplemented
195C22⋊C9C32×D416yescomplemented
196C4×C9—33no—
197C4×C9—46no—
198C4×C9C2×C3×A413yescomplemented
199C22×D4C3×3−1+211yescharacteristic, complemented, Sylow
200C33—41no—
2013−1+2—42no—
2023−1+2—46no—
2033−1+2D4×A411yescharacteristic, direct product factor
204C3×C9—43no—
205C23×C3—22no—
206C23×C3—32no—
207C23×C3—32no—
208C23×C3—31no—
209C23×C3—62no—
210C23×C3—62no—
211C23×C3—92no—
212C23×C3—92no—
213C23×C3—182no—
214C23×C3C22×C3311yescharacteristic
215C2×A4—26no—
216C2×A4—32no—
217C2×A4—64no—
218C2×A4C22×3−1+213yes—
219C3×D4—32no—
220C3×D4—32no—
221C3×D4—31no—
222C3×D4—31no—
223C3×D4—43no—
224C3×D4—91no—
225C3×D4—92no—
226C3×D4—92no—
227C3×D4—122no—
228C3×D4C32×A411yescharacteristic
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C32—22no—
234C2×C32—31no—
235C2×C32—31no—
236C2×C32—41no—
237C2×C32—42no—
238C2×C32—62no—
239C2×C32—82no—
240C2×C32—84no—
241C2×C32—123no—
242C2×C32—246no—
243C2×C32C22×C3×A411yescharacteristic
244C2×C9—26no—
245C2×C9—33no—
246C2×C9—33no—
247C2×C9—46no—
248C2×C9—66no—
249C2×C9—812no—
250C2×C9C22×C3×A413yes—
251C24C2×C3×3−1+212yescomplemented
252C2×D4—31no—
253C2×D4—31no—
254C2×D4—32no—
255C22×C4C2×C3×3−1+211yescharacteristic, complemented
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—32no—
259C22×C3—31no—
260C22×C3—32no—
261C22×C3—46no—
262C22×C3—62no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—62no—
266C22×C3—91no—
267C22×C3—92no—
268C22×C3—91no—
269C22×C3—124no—
270C22×C3—182no—
271C22×C3—182no—
272C22×C3—182no—
273C22×C3—182no—
274C22×C3C33×D411yescharacteristic
275C22×C3C2×C32×A412yes—
276A4—32no—
277A4D4×3−1+213yesdirect product factor
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—43no—
281C3×C4—91no—
282C3×C4—122no—
283C3×C4C2×C32×A411yescharacteristic
284C32—41no—
285C32—42no—
286C32—123no—
287C32C3×D4×A411yescharacteristic
288C9—46no—
289C9C3×D4×A413yescomplemented
290C23—22no—
291C23—32no—
292C23—32no—
293C23—62no—
294C23C22×C3×3−1+211yescharacteristic
295D4—32no—
296D4—32no—
297D4—31no—
298D4A4×3−1+211yescharacteristic, direct product factor
299C2×C4—31no—
300C2×C4—31no—
301C2×C3—22no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—43no—
306C2×C3—62no—
307C2×C3—62no—
308C2×C3—86no—
309C2×C3—91no—
310C2×C3—91no—
311C2×C3—122no—
312C2×C3—182no—
313C2×C3—244no—
314C2×C3C22×C32×A411yescharacteristic
315C22—31no—
316C22—31no—
317C22—32no—
318C22—62no—
319C22—62no—
320C22—62no—
321C22—62no—
322C22C3×D4×3−1+211yescharacteristic, complemented
323C22C2×A4×3−1+212yescomplemented
324C4—31no—
325C4C2×A4×3−1+211yescharacteristic, complemented
326C3—31no—
327C3—43no—
328C3—122no—
329C3C32×D4×A411yescharacteristic
330C2—22no—
331C2—31no—
332C2—31no—
333C2—62no—
334C2C22×A4×3−1+211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×C32×A4
commutator, socle, cosocleC23×C3C22×C33
FittingC22×D4×3−1+2C3
prime radicalC32×D4×A4C3
hypercenterD4×3−1+2A4
hypocenterC22C3×D4×3−1+2
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C2×C3 ◁ D4×3−1+2

Complements

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

NH ≅ G/Ntype
1. C22×A4×3−1+2C2semidirect
2. C4×A4×3−1+2C2semidirect
4. C22×D4×3−1+2C3semidirect
5. D4×#1094C3semidirect
6. D4×C9⋊A4C3semidirect
7. D4×C9×A4C3semidirect
13. C24×3−1+2C2×C3semidirect
14. C22×#1094C2×C3semidirect
15. C22×C9⋊A4C2×C3semidirect
16. C22×C9×A4C2×C3semidirect
20. C22×C4×3−1+2C2×C3semidirect
21. C4×#1094C2×C3semidirect
22. C4×C9⋊A4C2×C3semidirect
23. C4×C9×A4C2×C3semidirect
25. A4×3−1+2D4direct (N × H)
31. C22×D4×C9C32semidirect
32. D4×C22⋊C9C32semidirect
52. D4×3−1+2A4direct (N × H)
69. C24×C9C2×C32semidirect
70. C22×C22⋊C9C2×C32semidirect
74. C22×C4×C9C2×C32semidirect
75. C4×C22⋊C9C2×C32semidirect
88. C22×3−1+2C3×D4semidirect
89. C22×3−1+2C2×A4semidirect
91. #1094C3×D4semidirect
92. C9⋊A4C3×D4semidirect
93. C9×A4C3×D4semidirect
97. C4×3−1+2C2×A4semidirect
102. D4×A43−1+2direct (N × H)
135. D4×C9C3×A4semidirect
154. C22×A4C2×3−1+2semidirect
164. C4×A4C2×3−1+2semidirect
193. C22×C9C32×D4semidirect
194. C22×C9C2×C3×A4semidirect
195. C22⋊C9C32×D4semidirect
198. C4×C9C2×C3×A4semidirect
199. C22×D4C3×3−1+2semidirect
203. 3−1+2D4×A4direct (N × H)
251. C24C2×C3×3−1+2semidirect
255. C22×C4C2×C3×3−1+2semidirect
277. A4D4×3−1+2direct (N × H)
289. C9C3×D4×A4semidirect
298. D4A4×3−1+2direct (N × H)
322. C22C3×D4×3−1+2semidirect
323. C22C2×A4×3−1+2semidirect
325. C4C2×A4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C22×A4×3−1+22
C4×A4×3−1+22
C32×D4×A43
C22×D4×3−1+23
D4×#10943
D4×C9⋊A43
D4×C9×A43
C3×D4×3−1+24