Finite Group Databasebeta

← C32×#5745 #13179122

Subgroups — C32×#5745 #13179122

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2C2×C33×M16C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4C32×#907C214yesmaximal, complemented
5C32×D3×M16C214yesmaximal, complemented
6C32×#901C214yesmaximal, complemented
7C32×#568—31nomaximal
8C3×#5745C314yesmaximal, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C32×#67C412yes—
11C32×#235C412yes—
12C32×#235C412yes—
13C32×#235C2212yes—
14C32×#235C2212yes—
15C2×C32×D12C411yescharacteristic
16C2×C32×C4×D3C2212yes—
17C2×C32×Dic6C411yescharacteristic
18C33×M16C2212yes—
19C33×M16C2212yescomplemented
20C2×C33×C8C2212yescomplemented
21C32×#208C2212yes—
22C32×#208C2212yescomplemented
23C2×C32×C3⋊C8C2212yescomplemented
24C32×ΓL(1,𝔽25)C2218yescomplemented
25C32×D3×C8C2218yescomplemented
26C2×C32×#41—31no—
27C2×C3×#235C2×C314yes—
28C32×#131—34no—
29C32×#131—34no—
30C2×C32×M16—24no—
31C2×C32×M16—31no—
32C2×C32×M16—34no—
33C2×C32×M16D311yescharacteristic, complemented
34C2×C32×M16C2×C314yescomplemented
35C2×C3×#208C2×C314yescomplemented
36C3×#907C2×C3116yescomplemented
37C3×D3×M16C2×C3116yescomplemented
38C3×#901C2×C3116yescomplemented
39C23×C33C2×C411yescharacteristic
40C22×C32×D3C2×C412yes—
41C2×C33×C4C2×C411yescharacteristic
42C2×C33×C4C2×C411yescharacteristic
43C2×C33×C4C2×C411yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C32×#67C2×C414yes—
48C32×#67C2×C414yes—
49C2×C32×Dic3C2×C412yes—
50C32×D12C2×C412yes—
51C32×D12C2×C412yes—
52C32×C4×D3C2×C414yes—
53C32×C4×D3C2314yes—
54C32×Dic6C2×C412yes—
55C32×Dic6C2×C412yes—
56C33×C8C2314yes—
57C32×C3⋊C8C2314yes—
58C3×#568—34no—
59#5745C3211yescharacteristic, direct product factor
60C32×#41—32no—
61C32×#41—32no—
62C32×#41—32no—
63C32×#41—32no—
64C2×C32×Q8—31no—
65C2×C32×D4—32no—
66C2×C32×D4—31no—
67C22×C32×C4—24no—
68C22×C32×C4—32no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C22×C314yes—
71C2×C3×#67C3×C418yes—
72C3×#235C3×C418yes—
73C3×#235C3×C418yes—
74C3×#235C22×C318yes—
75C3×#235C22×C318yes—
76C2×C3×D12C3×C414yes—
77C2×C3×C4×D3C22×C318yes—
78C2×C3×Dic6C3×C414yes—
79C32×M16—28no—
80C32×M16—28no—
81C32×M16—32no—
82C32×M16—32no—
83C32×M16—38no—
84C32×M16C2×D312yes—
85C32×M16C2×D312yescomplemented
86C32×M16C22×C318yes—
87C32×M16C22×C318yescomplemented
88C2×C32×C8—28no—
89C2×C32×C8—32no—
90C2×C32×C8—38no—
91C2×C32×C8C2×D312yescomplemented
92C2×C32×C8C22×C318yescomplemented
93C3×#208C22×C318yes—
94C3×#208C22×C318yescomplemented
95C2×C3×C3⋊C8C22×C318yescomplemented
96C3×ΓL(1,𝔽25)C22×C3132yescomplemented
97C3×D3×C8C22×C3132yescomplemented
98C22×C33—41no—
99C22×C33C22×C411yescharacteristic
100C22×C33C22×C411yescharacteristic
101C22×C33C22×C411yescharacteristic
102C2×C32×D3—42no—
103C2×C32×D3C22×C414yes—
104C33×C4C22×C411yescharacteristic
105C33×C4C22×C411yescharacteristic
106C33×C4C22×C411yescharacteristic
107C33×C4C2411yescharacteristic
108C32×Dic3C22×C414yes—
109C2×C3×#41—34no—
110C2×#235C2×C3211yescharacteristic
111C3×#131—316no—
112C3×#131—316no—
113C2×C3×M16—24no—
114C2×C3×M16—34no—
115C2×C3×M16—316no—
116C2×C3×M16C3×D314yescomplemented
117C2×C3×M16C2×C3211yescharacteristic, complemented
118C2×#208C2×C3211yescharacteristic, complemented
119#907C2×C3214yescomplemented
120D3×M16C2×C3214yescomplemented
121#901C2×C3214yescomplemented
122C23×C32—24no—
123C23×C32—32no—
124C23×C32C4×D311yescharacteristic
125C23×C32C2×C3×C414yes—
126C22×C3×D3C2×C3×C418yes—
127C32×Q8—32no—
128C32×Q8—32no—
129C32×D4—34no—
130C32×D4—34no—
131C32×D4—32no—
132C32×D4—32no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—32no—
140C2×C32×C4—34no—
141C2×C32×C4—34no—
142C2×C32×C4C4×D311yescharacteristic
143C2×C32×C4C4×D311yescharacteristic
144C2×C32×C4C4×D311yescharacteristic
145C2×C32×C4C2×C3×C414yes—
146C2×C32×C4C2×C3×C414yes—
147C2×C32×C4C2×C3×C414yes—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C314yes—
152C2×C32×C4C23×C314yes—
153C2×C32×C4C23×C314yes—
154C3×#67C2×C3×C4116yes—
155C3×#67C2×C3×C4116yes—
156C2×C3×Dic3C2×C3×C418yes—
157C3×D12C2×C3×C418yes—
158C3×D12C2×C3×C418yes—
159C3×C4×D3C2×C3×C4116yes—
160C3×C4×D3C23×C3116yes—
161C3×Dic6C2×C3×C418yes—
162C3×Dic6C2×C3×C418yes—
163C32×C8—216no—
164C32×C8—34no—
165C32×C8C22×D314yes—
166C32×C8C23×C3116yes—
167C3×C3⋊C8C23×C3116yes—
168#568—31noSylow
169C2×C33—21no—
170C2×C33—21no—
171C2×C33—21no—
172C2×C33C23×C411yescharacteristic
173C32×D3—24no—
174C3×#41—38no—
175C3×#41—38no—
176C3×#41—38no—
177C3×#41—38no—
178C2×C3×Q8—34no—
179C2×C3×D4—38no—
180C2×C3×D4—34no—
181C22×C3×C4—24no—
182C22×C3×C4—38no—
183C22×C3×C4C2×C3×D314yes—
184C22×C3×C4C22×C3211yescharacteristic
185C2×#67C32×C412yes—
186#235C32×C412yes—
187#235C32×C412yes—
188#235C22×C3212yes—
189#235C22×C3212yes—
190C2×D12C32×C411yescharacteristic
191C2×C4×D3C22×C3212yes—
192C2×Dic6C32×C411yescharacteristic
193C3×M16—28no—
194C3×M16—28no—
195C3×M16—38no—
196C3×M16—38no—
197C3×M16—332no—
198C3×M16C2×C3×D318yes—
199C3×M16C2×C3×D318yescomplemented
200C3×M16C22×C3212yes—
201C3×M16C22×C3212yescomplemented
202C2×C3×C8—28no—
203C2×C3×C8—38no—
204C2×C3×C8—332no—
205C2×C3×C8C2×C3×D318yescomplemented
206C2×C3×C8C22×C3212yescomplemented
207#208C22×C3212yes—
208#208C22×C3212yescomplemented
209C2×C3⋊C8C22×C3212yescomplemented
210ΓL(1,𝔽25)C22×C3218yescomplemented
211D3×C8C22×C3218yescomplemented
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—24no—
215C22×C32—34no—
216C22×C32—41no—
217C22×C32—44no—
218C22×C32—48no—
219C22×C32—122no—
220C22×C32C2×C4×D311yescharacteristic
221C22×C32C2×C4×D311yescharacteristic
222C22×C32C2×C4×D311yescharacteristic
223C22×C32C22×C3×C414yes—
224C22×C32C22×C3×C414yes—
225C22×C32C22×C3×C414yes—
226C2×C3×D3—48no—
227C2×C3×D3C22×C3×C4116yes—
228C32×C4—24no—
229C32×C4—24no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—34no—
233C32×C4C2×C4×D311yescharacteristic
234C32×C4C2×C4×D311yescharacteristic
235C32×C4C2×C4×D311yescharacteristic
236C32×C4C22×C3×C414yes—
237C32×C4C22×C3×C414yes—
238C32×C4C22×C3×C414yes—
239C32×C4C23×D311yescharacteristic
240C32×C4C24×C314yes—
241C3×Dic3C22×C3×C4116yes—
242C2×#41—31no—
243#131—34no—
244#131—34no—
245C2×M16—31no—
246C2×M16—34no—
247C2×M16C32×D311yescharacteristic, complemented
248C33#56811yescharacteristic, complemented, Sylow
249C23×C3—24no—
250C23×C3—38no—
251C23×C3C3×C4×D314yes—
252C23×C3C2×C32×C411yescharacteristic
253C22×D3C2×C32×C412yes—
254C3×Q8—38no—
255C3×Q8—38no—
256C3×D4—316no—
257C3×D4—316no—
258C3×D4—38no—
259C3×D4—38no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—38no—
267C2×C3×C4—316no—
268C2×C3×C4—316no—
269C2×C3×C4C3×C4×D314yes—
270C2×C3×C4C3×C4×D314yes—
271C2×C3×C4C3×C4×D314yes—
272C2×C3×C4C2×C32×C411yescharacteristic
273C2×C3×C4C2×C32×C411yescharacteristic
274C2×C3×C4C2×C32×C411yescharacteristic
275C2×C3×C4C22×C3×D314yes—
276C2×C3×C4C22×C3×D314yes—
277C2×C3×C4C22×C3×D314yes—
278C2×C3×C4C23×C3211yescharacteristic
279C2×C3×C4C23×C3211yescharacteristic
280C2×C3×C4C23×C3211yescharacteristic
281#67C2×C32×C414yes—
282#67C2×C32×C414yes—
283C2×Dic3C2×C32×C412yes—
284D12C2×C32×C412yes—
285D12C2×C32×C412yes—
286C4×D3C2×C32×C414yes—
287C4×D3C23×C3214yes—
288Dic6C2×C32×C412yes—
289Dic6C2×C32×C412yes—
290C3×C8—216no—
291C3×C8—316no—
292C3×C8C22×C3×D3116yes—
293C3×C8C23×C3214yes—
294C3⋊C8C23×C3214yes—
295C2×C32—28no—
296C2×C32—24no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—44no—
304C2×C32—44no—
305C2×C32—64no—
306C2×C32C22×C4×D311yescharacteristic
307C2×C32C23×C3×C414yes—
308C3×D3—216no—
309#41—32no—
310#41—32no—
311#41—32no—
312#41—32no—
313C2×Q8—31no—
314C2×D4—32no—
315C2×D4—31no—
316C22×C4—32no—
317C22×C4C2×C32×D311yescharacteristic
318M16—32no—
319M16—32no—
320M16—38no—
321M16C2×C32×D312yes—
322M16C2×C32×D312yescomplemented
323C2×C8—32no—
324C2×C8—38no—
325C2×C8C2×C32×D312yescomplemented
326C22×C3—24no—
327C22×C3—24no—
328C22×C3—24no—
329C22×C3—316no—
330C22×C3—44no—
331C22×C3—41no—
332C22×C3—48no—
333C22×C3—128no—
334C22×C3C2×C3×C4×D314yes—
335C22×C3C2×C3×C4×D314yes—
336C22×C3C2×C3×C4×D314yes—
337C22×C3C22×C32×C411yescharacteristic
338C22×C3C22×C32×C411yescharacteristic
339C22×C3C22×C32×C411yescharacteristic
340C2×D3—42no—
341C2×D3C22×C32×C414yes—
342C3×C4—24no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—24no—
346C3×C4—316no—
347C3×C4C2×C3×C4×D314yes—
348C3×C4C2×C3×C4×D314yes—
349C3×C4C2×C3×C4×D314yes—
350C3×C4C22×C32×C411yescharacteristic
351C3×C4C22×C32×C411yescharacteristic
352C3×C4C22×C32×C411yescharacteristic
353C3×C4C23×C3×D314yes—
354C3×C4C24×C3211yescharacteristic
355Dic3C22×C32×C414yes—
356C32—24no—
357C32#574511yescharacteristic, direct product factor
358C32C3×#56814yescomplemented
359C23—32no—
360C23C32×C4×D311yescharacteristic
361Q8—32no—
362Q8—32no—
363D4—34no—
364D4—34no—
365D4—32no—
366D4—32no—
367C2×C4—32no—
368C2×C4—34no—
369C2×C4—34no—
370C2×C4C32×C4×D311yescharacteristic
371C2×C4C32×C4×D311yescharacteristic
372C2×C4C32×C4×D311yescharacteristic
373C2×C4C22×C32×D311yescharacteristic
374C2×C4C22×C32×D311yescharacteristic
375C2×C4C22×C32×D311yescharacteristic
376C8—34no—
377C8C22×C32×D314yes—
378C2×C3—28no—
379C2×C3—24no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—24no—
386C2×C3—44no—
387C2×C3—44no—
388C2×C3—616no—
389C2×C3C22×C3×C4×D314yes—
390C2×C3C23×C32×C411yescharacteristic
391D3—24no—
392C22—34no—
393C22—41no—
394C22—122no—
395C22C2×C32×C4×D311yescharacteristic
396C22C2×C32×C4×D311yescharacteristic
397C22C2×C32×C4×D311yescharacteristic
398C4—34no—
399C4C2×C32×C4×D311yescharacteristic
400C4C2×C32×C4×D311yescharacteristic
401C4C2×C32×C4×D311yescharacteristic
402C4C23×C32×D311yescharacteristic
403C3—24no—
404C3C3×#574514yesdirect product factor
405C3C32×#56811yescharacteristic, complemented
406C2—21no—
407C2—21no—
408C2—21no—
409C2—64no—
410C2C22×C32×C4×D311yescharacteristic
subgroupG/H
centerC32×C4C23×D3
commutatorC2×C3C23×C32×C4
FrattiniC4C23×C32×D3
FittingC2×C33×M16C2
socleC2×C33C23×C4
prime radicalC2×C32×#235C2
cosocleC3×C4C24×C32
hypercenterC2×C32×M16D3
hypocenterC3C32×#568
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C32×C4 ◁ C2×C32×M16

Complements

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

NH ≅ G/Ntype
2. C2×C33×M16C2semidirect
3. C2×C32×#208C2semidirect
4. C32×#907C2semidirect
5. C32×D3×M16C2semidirect
6. C32×#901C2semidirect
8. C3×#5745C3direct (N × H)
19. C33×M16C22semidirect
20. C2×C33×C8C22semidirect
22. C32×#208C22semidirect
23. C2×C32×C3⋊C8C22semidirect
24. C32×ΓL(1,𝔽25)C22semidirect
25. C32×D3×C8C22semidirect
33. C2×C32×M16D3semidirect
34. C2×C32×M16C2×C3semidirect
35. C2×C3×#208C2×C3semidirect
36. C3×#907C2×C3semidirect
37. C3×D3×M16C2×C3semidirect
38. C3×#901C2×C3semidirect
59. #5745C32direct (N × H)
85. C32×M16C2×D3semidirect
87. C32×M16C22×C3semidirect
91. C2×C32×C8C2×D3semidirect
92. C2×C32×C8C22×C3semidirect
94. C3×#208C22×C3semidirect
95. C2×C3×C3⋊C8C22×C3semidirect
96. C3×ΓL(1,𝔽25)C22×C3semidirect
97. C3×D3×C8C22×C3semidirect
116. C2×C3×M16C3×D3semidirect
117. C2×C3×M16C2×C32semidirect
118. C2×#208C2×C32semidirect
119. #907C2×C32semidirect
120. D3×M16C2×C32semidirect
121. #901C2×C32semidirect
199. C3×M16C2×C3×D3semidirect
201. C3×M16C22×C32semidirect
205. C2×C3×C8C2×C3×D3semidirect
206. C2×C3×C8C22×C32semidirect
208. #208C22×C32semidirect
209. C2×C3⋊C8C22×C32semidirect
210. ΓL(1,𝔽25)C22×C32semidirect
211. D3×C8C22×C32semidirect
247. C2×M16C32×D3semidirect
248. C33#568semidirect
322. M16C2×C32×D3semidirect
325. C2×C8C2×C32×D3semidirect
357. C32#5745direct (N × H)
358. C32C3×#568semidirect
404. C3C3×#5745direct (N × H)
405. C3C32×#568semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C33×M162
C2×C32×#2082
C32×#9072
C32×D3×M162
C32×#9012
C32×#5683
C3×#57453