Finite Group Databasebeta

← C32×#5053 #13162489

Subgroups — C32×#5053 #13162489

Order 1728.

1272 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1272 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
1C32×#1004C211yesmaximal, characteristic
2C33×#123C211yesmaximal, characteristic, complemented
3C32×#929C212yesmaximal, complemented
4C32×#834C211yesmaximal, characteristic, complemented
5C32×#806C212yesmaximal, complemented
6C32×#458—31nomaximal
7C3×#5053C314yesmaximal, direct product factor
8C2×C33×D4C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic, complemented
10C2×C32×#67—21no—
11C2×C32×#67C2212yes—
12C32×#237—22no—
13C32×D3×D4—22no—
14C32×#235—21no—
15C33×C4⋊C4—21no—
16C33×C22⋊C4—21no—
17C33×C22⋊C4C2211yescharacteristic, complemented
18C32×#217C2212yescomplemented
19C32×(C2×D3)⋊C4—22no—
20C32×Dic3⋊C4—22no—
21C32×H(5,𝔽2)—31no—
22C3×#1004C2×C314yes—
23C32×#123—24no—
24C32×#123—32no—
25C32×#123D311yescharacteristic, complemented
26C32×#123C2×C314yescomplemented
27C3×#929C2×C318yescomplemented
28C32×#99—31no—
29C32×#99—32no—
30C3×#834C2×C314yescomplemented
31C3×#806C2×C318yescomplemented
32C23×C33D411yescharacteristic
33C23×C33C2311yescharacteristic
34C22×C32×D3—21no—
35C22×C32×D3D412yes—
36C33×D4—22no—
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4—22no—
40C2×C33×C4D411yescharacteristic
41C32×#67—22no—
42C32×#67—22no—
43C32×#67—22no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3D412yes—
47C32×D12—21no—
48C32×C4×D3—22no—
49C32×Dic6—21no—
50C3×#458—34no—
51#5053C3211yescharacteristic, direct product factor
52C32×#41—62no—
53C32×#41—61no—
54C2×C32×D4—24no—
55C2×C32×D4—32no—
56C2×C32×D4—62no—
57C2×C32×D4—61no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C22×C314yes—
60C22×C32×C4—24no—
61C22×C32×C4C2×D311yescharacteristic, complemented
62C22×C32×C4C22×C314yescomplemented
63C2×C3×#67—24no—
64C2×C3×#67C22×C318yes—
65C3×#237—28no—
66C3×D3×D4—28no—
67C3×#235—24no—
68C32×C4⋊C4—28no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4—24no—
71C32×C4⋊C4—62no—
72C32×C22⋊C4—28no—
73C32×C22⋊C4—24no—
74C32×C22⋊C4—21no—
75C32×C22⋊C4—24no—
76C32×C22⋊C4—32no—
77C32×C22⋊C4—62no—
78C32×C22⋊C4C2×D311yescharacteristic, complemented
79C32×C22⋊C4C22×C314yescomplemented
80C3×#217C22×C318yescomplemented
81C3×(C2×D3)⋊C4—28no—
82C3×Dic3⋊C4—28no—
83C22×C33—21no—
84C22×C33—41no—
85C22×C33—41no—
86C22×C33C2×D412yes—
87C22×C33C2×D411yescharacteristic
88C2×C32×D3—22no—
89C2×C32×D3—42no—
90C2×C32×D3—81no—
91C33×C4—21no—
92C33×C4—41no—
93C33×C4—41no—
94C32×Dic3—22no—
95C32×Dic3—42no—
96C3×H(5,𝔽2)—34no—
97#1004C2×C3211yescharacteristic
98C3×#123—24no—
99C3×#123—38no—
100C3×#123C3×D314yescomplemented
101C3×#123C2×C3211yescharacteristic, complemented
102#929C2×C3212yescomplemented
103C3×#99—34no—
104C3×#99—38no—
105#834C2×C3211yescharacteristic, complemented
106#806C2×C3212yescomplemented
107C23×C32—24no—
108C23×C32—24no—
109C23×C32—32no—
110C23×C32—61no—
111C23×C32#6711yescharacteristic
112C23×C32C3×D414yes—
113C23×C32C22×D311yescharacteristic
114C23×C32C23×C314yes—
115C22×C3×D3—24no—
116C22×C3×D3C3×D418yes—
117C32×Q8—61no—
118C32×D4—22no—
119C32×D4—28no—
120C32×D4—48no—
121C32×D4—61no—
122C32×D4—62no—
123C32×D4—62no—
124C32×D4—62no—
125C2×C32×C4—28no—
126C2×C32×C4—28no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—28no—
134C2×C32×C4—32no—
135C2×C32×C4—48no—
136C2×C32×C4—62no—
137C2×C32×C4—61no—
138C2×C32×C4—62no—
139C2×C32×C4#6711yescharacteristic
140C2×C32×C4C3×D414yes—
141C3×#67—28no—
142C3×#67—28no—
143C3×#67—28no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—28no—
146C2×C3×Dic3C3×D418yes—
147C3×D12—24no—
148C3×C4×D3—28no—
149C3×Dic6—24no—
150#458—31noSylow
151C2×C33—21no—
152C2×C33—22no—
153C2×C33—41no—
154C2×C33#12011yescharacteristic
155C32×D3—42no—
156C3×#41—68no—
157C3×#41—64no—
158C2×C3×D4—24no—
159C2×C3×D4—38no—
160C2×C3×D4—68no—
161C2×C3×D4—64no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C22×C3211yescharacteristic
164C22×C3×C4—24no—
165C22×C3×C4C2×C3×D314yescomplemented
166C22×C3×C4C22×C3211yescharacteristic, complemented
167C2×#67—21no—
168C2×#67C22×C3212yes—
169#237—22no—
170D3×D4—22no—
171#235—21no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—68no—
176C3×C22⋊C4—28no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—38no—
181C3×C22⋊C4—68no—
182C3×C22⋊C4C2×C3×D314yescomplemented
183C3×C22⋊C4C22×C3211yescharacteristic, complemented
184#217C22×C3212yescomplemented
185(C2×D3)⋊C4—22no—
186Dic3⋊C4—22no—
187C22×C32—21no—
188C22×C32—28no—
189C22×C32—24no—
190C22×C32—24no—
191C22×C32—44no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—48no—
197C22×C32—48no—
198C22×C32—62no—
199C22×C32—122no—
200C22×C32—241no—
201C22×C32D3×D412yes—
202C22×C32C2×#6711yescharacteristic
203C22×C32C2×C3×D418yes—
204C22×C32C2×C3×D414yes—
205C2×C3×D3—28no—
206C2×C3×D3—48no—
207C2×C3×D3—84no—
208C32×C4—21no—
209C32×C4—24no—
210C32×C4—44no—
211C32×C4—48no—
212C32×C4—41no—
213C32×C4—44no—
214C32×C4—48no—
215C32×C4—41no—
216C32×C4—44no—
217C32×C4—62no—
218C32×C4—122no—
219C3×Dic3—28no—
220C3×Dic3—48no—
221H(5,𝔽2)—31no—
222#123—32no—
223#123C32×D311yescharacteristic, complemented
224#99—31no—
225#99—32no—
226C33#45811yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—38no—
230C23×C3—64no—
231C23×C3C3×#6714yes—
232C23×C3C32×D411yescharacteristic
233C23×C3C22×C3×D314yes—
234C23×C3C23×C3211yescharacteristic
235C22×D3—21no—
236C22×D3C32×D412yes—
237C3×Q8—64no—
238C3×D4—28no—
239C3×D4—22no—
240C3×D4—48no—
241C3×D4—64no—
242C3×D4—68no—
243C3×D4—68no—
244C3×D4—68no—
245C2×C3×C4—28no—
246C2×C3×C4—28no—
247C2×C3×C4—28no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—38no—
255C2×C3×C4—48no—
256C2×C3×C4—68no—
257C2×C3×C4—64no—
258C2×C3×C4—68no—
259C2×C3×C4C3×#6714yes—
260C2×C3×C4C32×D411yescharacteristic
261#67—22no—
262#67—22no—
263#67—22no—
264C2×Dic3—21no—
265C2×Dic3—22no—
266C2×Dic3C32×D412yes—
267D12—21no—
268C4×D3—22no—
269Dic6—21no—
270C2×C32—28no—
271C2×C32—21no—
272C2×C32—22no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—28no—
276C2×C32—48no—
277C2×C32—41no—
278C2×C32—44no—
279C2×C32—84no—
280C2×C32—122no—
281C2×C32#93711yescharacteristic
282C2×C32C3×#12014yes—
283C3×D3—48no—
284#41—62no—
285#41—61no—
286C2×D4—32no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4C2×C32×D311yescharacteristic
290C22×C4C2×C32×D311yescharacteristic, complemented
291C4⋊C4—21no—
292C4⋊C4—62no—
293C22⋊C4—21no—
294C22⋊C4—32no—
295C22⋊C4—62no—
296C22⋊C4C2×C32×D311yescharacteristic, complemented
297C22×C3—24no—
298C22×C3—21no—
299C22×C3—28no—
300C22×C3—24no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—41no—
304C22×C3—44no—
305C22×C3—41no—
306C22×C3—48no—
307C22×C3—48no—
308C22×C3—68no—
309C22×C3—128no—
310C22×C3—244no—
311C22×C3C3×D3×D418yes—
312C22×C3C2×C3×#6714yes—
313C22×C3C2×C32×D412yes—
314C22×C3C2×C32×D411yescharacteristic
315C2×D3—22no—
316C2×D3—42no—
317C2×D3—81no—
318C3×C4—24no—
319C3×C4—21no—
320C3×C4—44no—
321C3×C4—48no—
322C3×C4—48no—
323C3×C4—41no—
324C3×C4—44no—
325C3×C4—44no—
326C3×C4—41no—
327C3×C4—68no—
328C3×C4—128no—
329Dic3—22no—
330Dic3—42no—
331C32—24no—
332C32#505311yescharacteristic, direct product factor
333C32C3×#45814yescomplemented
334C23—32no—
335C23—61no—
336C23C32×#6711yescharacteristic
337C23C22×C32×D311yescharacteristic
338Q8—61no—
339D4—22no—
340D4—61no—
341D4—62no—
342D4—62no—
343D4—62no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—32no—
348C2×C4—62no—
349C2×C4—61no—
350C2×C4—62no—
351C2×C4C32×#6711yescharacteristic
352C2×C3—28no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—24no—
356C2×C3—28no—
357C2×C3—24no—
358C2×C3—48no—
359C2×C3—41no—
360C2×C3—44no—
361C2×C3—84no—
362C2×C3—128no—
363C2×C3C3×#93714yes—
364C2×C3C32×#12011yescharacteristic
365D3—42no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—62no—
370C22—122no—
371C22—241no—
372C22C32×D3×D412yes—
373C22C2×C32×#6711yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—62no—
378C4—122no—
379C3—24no—
380C3C3×#505314yesdirect product factor
381C3C32×#45811yescharacteristic, complemented
382C2—21no—
383C2—22no—
384C2—41no—
385C2—122no—
386C2C32×#93711yescharacteristic
subgroupG/H
centerC2×C32#937
commutator, cosocleC23×C3C23×C32
FrattiniC23C22×C32×D3
FittingC33×#123C2
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC32×#123D3
hypocenterC3C32×#458
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C23×C32 ◁ C32×#123

Complements

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

NH ≅ G/Ntype
2. C33×#123C2semidirect
3. C32×#929C2semidirect
4. C32×#834C2semidirect
5. C32×#806C2semidirect
7. C3×#5053C3direct (N × H)
9. C22×C33×C4C22semidirect
17. C33×C22⋊C4C22semidirect
18. C32×#217C22semidirect
25. C32×#123D3semidirect
26. C32×#123C2×C3semidirect
27. C3×#929C2×C3semidirect
30. C3×#834C2×C3semidirect
31. C3×#806C2×C3semidirect
51. #5053C32direct (N × H)
61. C22×C32×C4C2×D3semidirect
62. C22×C32×C4C22×C3semidirect
78. C32×C22⋊C4C2×D3semidirect
79. C32×C22⋊C4C22×C3semidirect
80. C3×#217C22×C3semidirect
100. C3×#123C3×D3semidirect
101. C3×#123C2×C32semidirect
102. #929C2×C32semidirect
105. #834C2×C32semidirect
106. #806C2×C32semidirect
165. C22×C3×C4C2×C3×D3semidirect
166. C22×C3×C4C22×C32semidirect
182. C3×C22⋊C4C2×C3×D3semidirect
183. C3×C22⋊C4C22×C32semidirect
184. #217C22×C32semidirect
223. #123C32×D3semidirect
226. C33#458semidirect
290. C22×C4C2×C32×D3semidirect
296. C22⋊C4C2×C32×D3semidirect
332. C32#5053direct (N × H)
333. C32C3×#458semidirect
380. C3C3×#5053direct (N × H)
381. C3C32×#458semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C33×#1232
C32×#9292
C32×#8342
C32×#8062
C32×#4583
C3×#50533