Finite Group Databasebeta

← D32×D28 #72246589

Subgroups — D32×D28 #72246589

Order 2016.

998 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 168 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×D32×D7C212yesmaximal, complemented
2#11759635C211yesmaximal, characteristic, complemented
3D3×D84C212yesmaximal, complemented
4C4×D32×C7C211yesmaximal, characteristic, complemented
5C3⋊D3×D28C211yesmaximal, characteristic, complemented
6C3×D3×D28C212yesmaximal, complemented
7#11759428C212yesmaximal, complemented
8D3×#4246C214yesmaximal, complemented
9C2×D3×D28—32nomaximal
10C2×(C3⋊D3)⋊D7C2212yescomplemented
11C2×D3×D21C2214yescomplemented
12C2×D32×C7C2211yescharacteristic
13C2×D7×C3⋊D3C2212yescomplemented
14C2×C3×D3×D7C2214yescomplemented
15D32×D7—28no—
16C4⋊(C3⋊D21)C2211yescharacteristic, complemented
17C4×C7×C3⋊D3C2211yescharacteristic, complemented
18C3×D84C2212yescomplemented
19C3×C4×D3×C7C2212yescomplemented
20C32×D28C2211yescharacteristic, complemented
21#92695C2212yescomplemented
22#92689C2214yescomplemented
23C7×#624C2211yescharacteristic, complemented
24D3×C7×Dic3C2212yescomplemented
25#92677C2212yescomplemented
26C3×#4246C2214yescomplemented
27C22×D3×D7—34no—
28C2×D84—32no—
29C2×C4×D3×C7—32no—
30C2×C3×D28—32no—
31C2×#4246—34no—
32D3×D28—34no—
33D3×D28—32no—
34D3×D28—32no—
35D3×D28—32no—
36D3×D28—34no—
37D3×D28—61no—
38D3×D28D312yesdirect product factor
39D32×D4—71nomaximal, Hall
40C2×C3⋊D21C2312yescomplemented
41C2×C7×C3⋊D3C2311yescharacteristic
42C2×C3×D21C2314yescomplemented
43C2×C3×D3×C7C2312yes—
44C2×C32×D7C2312yescomplemented
45(C3⋊D3)⋊D7—24no—
46D3×D21—28no—
47D32×C7D414yescomplemented
48D7×C3⋊D3—24no—
49C3×D3×D7—28no—
50C32×C4×C7C2311yescharacteristic, complemented
51C7×C3⋊Dic3C2311yescharacteristic, complemented
52C3×C7×Dic3C2312yescomplemented
53C22×D28—91noHall
54C22×D21—34no—
55C22×D3×C7—32no—
56C22×C3×D7—34no—
57C2×D3×D7—34no—
58C2×D3×D7—34no—
59C2×D3×D7—34no—
60C2×D3×D7—62no—
61C2×D3×D7—616no—
62C2×D3×D7C2×D314yescomplemented
63C2×C3×C4×C7—32no—
64D84—34no—
65D84—32no—
66D84—61no—
67D84C2×D312yescomplemented
68C2×C7×Dic3—32no—
69C4×D3×C7—32no—
70C4×D3×C7—32no—
71C4×D3×C7—32no—
72C4×D3×C7—61no—
73C4×D3×C7C2×D312yescomplemented
74C3×D28—21no—
75C3×D28—34no—
76C3×D28—32no—
77C3×D28C2×D312yescomplemented
78#4246—34no—
79#4246—34no—
80#4246—34no—
81#4246—62no—
82#4246C2×D314yescomplemented
83C22×D32—72no—
84D4×C3⋊D3—71no—
85C3×D3×D4—72no—
86#3811—72no—
87D3×#67—74no—
88#3802—71no—
89D3×D12—72no—
90C4×D32D711yescharacteristic, complemented
91C2×C32×C7C2411yescharacteristic
92C3⋊D21—22no—
93C7×C3⋊D3C2×D412yescomplemented
94C3×D21—24no—
95C3×D3×C7C2×D414yescomplemented
96C32×D7—22no—
97C23×D7—92no—
98C22×C4×C7—91no—
99C2×D28—32no—
100C2×D28—92no—
101C2×D28—91no—
102C2×D28—92no—
103C2×D28—94no—
104C2×D28—91no—
105C2×D3×D4—212no—
106C22×C3×C7—32no—
107C2×D21—34no—
108C2×D21—62no—
109C2×D21—68no—
110C2×D21C22×D314yescomplemented
111C2×D3×C7—32no—
112C2×D3×C7—38no—
113C2×D3×C7—61no—
114C2×D3×C7C22×D312yes—
115C2×C3×D7—22no—
116C2×C3×D7—34no—
117C2×C3×D7—68no—
118C2×C3×D7C22×D314yescomplemented
119D3×D7—28no—
120D3×D7—68no—
121D3×D7—68no—
122D3×D7—68no—
123D3×D7—124no—
124C3×C4×C7—21no—
125C3×C4×C7—32no—
126C3×C4×C7C22×D312yescomplemented
127C7×Dic3—32no—
128C7×Dic3—61no—
129C7×Dic3C22×D312yescomplemented
130C22×C3⋊D3—72no—
131C22×C3×D3—74no—
132C2×D32—72no—
133C2×D32—74no—
134C2×D32—148no—
135C2×D32C2×D711yescharacteristic
136C32×D4—71no—
137#638—72no—
138C4⋊(C3⋊D3)—71no—
139C4×C3⋊D3C2×D711yescharacteristic, complemented
140C3×#67—74no—
141C3×D12—72no—
142C3×C4×D3C2×D712yescomplemented
143#626—74no—
144#625—72no—
145#624C2×D711yescharacteristic, complemented
146D3×Dic3C2×D712yescomplemented
147C32×C7C22×D411yescharacteristic, complemented, Hall
148C23×C7—91no—
149C22×D7—34no—
150C22×D7—92no—
151C22×D7—94no—
152C22×D7—92no—
153C22×D7—188no—
154C2×C4×C7—32no—
155C2×C4×C7—91no—
156C2×C4×C7—92no—
157C2×C4×C7—91no—
158D28—34no—
159D28—32no—
160D28—94no—
161D28—92no—
162D28—92no—
163D28—91no—
164D28D3211yescharacteristic, direct product factor
165C23×D3—214no—
166C2×C3×D4—212no—
167C2×#67—214no—
168D3×D4—72no—
169D3×D4—212no—
170D3×D4—214no—
171D3×D4—212no—
172D3×D4—214no—
173D3×D4—212no—
174D3×D4—421no—
175C2×D12—212no—
176C2×C4×D3—32no—
177C2×C3×C7—21no—
178C2×C3×C7—34no—
179C2×C3×C7C23×D312yes—
180D21—24no—
181D21—64no—
182D21—122no—
183C3×D7—24no—
184C3×D7—42no—
185C3×D7—64no—
186D3×C7—34no—
187D3×C7—62no—
188D3×C7D3×D414yescomplemented
189C22×C32—72no—
190C2×C3⋊D3—72no—
191C2×C3⋊D3—144no—
192C2×C3⋊D3C22×D711yescharacteristic
193C2×C3×D3—74no—
194C2×C3×D3—148no—
195C2×C3×D3C22×D712yes—
196D32—144no—
197D32—148no—
198D32D2814yesdirect product factor
199C32×C4C22×D711yescharacteristic, complemented
200C3⋊Dic3C22×D711yescharacteristic, complemented
201C3×Dic3C22×D712yescomplemented
202C22×D4—631noSylow
203C22×C7—32no—
204C22×C7—94no—
205C22×C7—91no—
206C2×D7—34no—
207C2×D7—68no—
208C2×D7—92no—
209C2×D7—184no—
210C2×D7—184no—
211C2×D7—188no—
212C2×D7C2×D3212yescomplemented
213C4×C7—32no—
214C4×C7—91no—
215C4×C7C2×D3211yescharacteristic, complemented
216C23×C3—214no—
217C22×D3—32no—
218C22×D3—74no—
219C22×D3—214no—
220C22×D3—214no—
221C22×D3—214no—
222C22×D3—214no—
223C22×D3—422no—
224C22×D3—4216no—
225C3×D4—72no—
226C3×D4—141no—
227C3×D4—214no—
228C3×D4—212no—
229C2×C3×C4—32no—
230#67—74no—
231#67—214no—
232#67—214no—
233#67—214no—
234#67—422no—
235C2×Dic3—32no—
236D12—72no—
237D12—214no—
238D12—212no—
239D12—421no—
240C4×D3—32no—
241C4×D3—32no—
242C4×D3—32no—
243C4×D3—61no—
244C4×D3D3×D712yescomplemented
245C3×C7—21no—
246C3×C7C2×D3×D412yescomplemented
247C2×C32—142no—
248C2×C32C23×D711yescharacteristic
249C3⋊D3—142no—
250C3⋊D3C2×D2812yescomplemented
251C3×D3—144no—
252C3×D3C2×D2814yescomplemented
253C24—632no—
254C2×D4—212no—
255C2×D4—632no—
256C2×D4—631no—
257C2×D4—632no—
258C2×D4—631no—
259C2×D4—634no—
260C22×C4—91no—
261C2×C7—34no—
262C2×C7—92no—
263C2×C7C22×D3211yescharacteristic
264D7—22no—
265D7—64no—
266D7—182no—
267C22×C3—32no—
268C22×C3—74no—
269C22×C3—142no—
270C22×C3—214no—
271C22×C3—428no—
272C2×D3—32no—
273C2×D3—38no—
274C2×D3—61no—
275C2×D3—74no—
276C2×D3—148no—
277C2×D3—214no—
278C2×D3—422no—
279C2×D3—428no—
280C2×D3—428no—
281C2×D3—428no—
282C2×D3—428no—
283C2×D3—844no—
284C2×D3C2×D3×D712yes—
285C3×C4—21no—
286C3×C4—32no—
287C3×C4C2×D3×D712yescomplemented
288Dic3—32no—
289Dic3—61no—
290Dic3C2×D3×D712yescomplemented
291C32C22×D2811yescharacteristic, complemented, Sylow
292C23—91no—
293C23—214no—
294C23—634no—
295C23—632no—
296C23—632no—
297C23—1268no—
298D4—71no—
299D4—214no—
300D4—212no—
301D4—634no—
302D4—632no—
303D4—632no—
304D4—631no—
305C2×C4—32no—
306C2×C4—91no—
307C2×C4—92no—
308C2×C4—91no—
309C7D32×D411yescharacteristic, complemented, Sylow
310C2×C3—21no—
311C2×C3—34no—
312C2×C3—144no—
313C2×C3—282no—
314C2×C3—424no—
315C2×C3C22×D3×D712yes—
316D3—34no—
317D3—62no—
318D3—144no—
319D3—424no—
320D3—842no—
321D3D3×D2814yesdirect product factor
322C22—32no—
323C22—72no—
324C22—94no—
325C22—91no—
326C22—214no—
327C22—428no—
328C22—632no—
329C22—1268no—
330C22—1264no—
331C22—1264no—
332C4—32no—
333C4—91no—
334C4D32×D711yescharacteristic, complemented
335C3—21no—
336C3C2×D3×D2812yescomplemented
337C2—34no—
338C2—92no—
339C2—142no—
340C2—424no—
341C2—1262no—
342C2C2×D32×D711yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D7
commutator, socle, cosocleC2×C32×C7C24
FittingC32×C4×C7C23
hypercenterC4D32×D7
hypocenterC32×C7C22×D4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D32×D7C2semidirect
2. #11759635C2semidirect
3. D3×D84C2semidirect
4. C4×D32×C7C2semidirect
5. C3⋊D3×D28C2semidirect
6. C3×D3×D28C2semidirect
7. #11759428C2semidirect
8. D3×#4246C2semidirect
10. C2×(C3⋊D3)⋊D7C22semidirect
11. C2×D3×D21C22semidirect
13. C2×D7×C3⋊D3C22semidirect
14. C2×C3×D3×D7C22semidirect
16. C4⋊(C3⋊D21)C22semidirect
17. C4×C7×C3⋊D3C22semidirect
18. C3×D84C22semidirect
19. C3×C4×D3×C7C22semidirect
20. C32×D28C22semidirect
21. #92695C22semidirect
22. #92689C22semidirect
23. C7×#624C22semidirect
24. D3×C7×Dic3C22semidirect
25. #92677C22semidirect
26. C3×#4246C22semidirect
38. D3×D28D3direct (N × H)
40. C2×C3⋊D21C23semidirect
42. C2×C3×D21C23semidirect
44. C2×C32×D7C23semidirect
47. D32×C7D4semidirect
50. C32×C4×C7C23semidirect
51. C7×C3⋊Dic3C23semidirect
52. C3×C7×Dic3C23semidirect
62. C2×D3×D7C2×D3semidirect
67. D84C2×D3semidirect
73. C4×D3×C7C2×D3semidirect
77. C3×D28C2×D3semidirect
82. #4246C2×D3semidirect
90. C4×D32D7semidirect
93. C7×C3⋊D3C2×D4semidirect
95. C3×D3×C7C2×D4semidirect
110. C2×D21C22×D3semidirect
118. C2×C3×D7C22×D3semidirect
126. C3×C4×C7C22×D3semidirect
129. C7×Dic3C22×D3semidirect
139. C4×C3⋊D3C2×D7semidirect
142. C3×C4×D3C2×D7semidirect
145. #624C2×D7semidirect
146. D3×Dic3C2×D7semidirect
147. C32×C7C22×D4semidirect
164. D28D32direct (N × H)
188. D3×C7D3×D4semidirect
198. D32D28direct (N × H)
199. C32×C4C22×D7semidirect
200. C3⋊Dic3C22×D7semidirect
201. C3×Dic3C22×D7semidirect
212. C2×D7C2×D32semidirect
215. C4×C7C2×D32semidirect
244. C4×D3D3×D7semidirect
246. C3×C7C2×D3×D4semidirect
250. C3⋊D3C2×D28semidirect
252. C3×D3C2×D28semidirect
287. C3×C4C2×D3×D7semidirect
290. Dic3C2×D3×D7semidirect
291. C32C22×D28semidirect
309. C7D32×D4semidirect
321. D3D3×D28direct (N × H)
334. C4D32×D7semidirect
336. C3C2×D3×D28semidirect

Maximal subgroups

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

subgroupindex
C2×D32×D72
#117596352
D3×D842
C4×D32×C72
C3⋊D3×D282
C3×D3×D282
#117594282
D3×#42462
C2×D3×D283
D32×D47