Finite Group Databasebeta

← C3×#10592758 #13163094

Subgroups — C3×#10592758 #13163094

Order 1728.

1084 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C33×#124C211yesmaximal, characteristic, complemented
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5C2×C3×#3753C211yesmaximal, characteristic
6C3×#64084C211yesmaximal, characteristic, complemented
7C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592758C311yesmaximal, characteristic, direct product factor
9C3×#5059—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×C22⋊C4—22no—
15C33×C42C2211yescharacteristic
16C3×#3756—22no—
17C3×#3755—22no—
18C3×#3754—22no—
19C3×#3753—22no—
20C3×#3752—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#124—24no—
24C32×#124D314yescomplemented
25C32×#124C2×C311yescharacteristic, complemented
26C2×#3756C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28C2×#3754C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic
30#64084C2×C311yescharacteristic, complemented
31C2×C3×#216—34no—
32C2×C3×#215—34no—
33C2×C3×#214—34no—
34C2×C3×#213—34no—
35C3×#875—34no—
36C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
37C3×C4×C3⋊C8—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4—21no—
45C2×C33×C4D412yes—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×#634—21no—
51C3×#634—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53C3×C3⋊(C3⋊C8)D412yes—
54C3×#495—91no—
55#5059—34no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C22×C311yescharacteristic
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6—34no—
66C32×C22⋊C4—216no—
67C32×C22⋊C4—28no—
68C32×C22⋊C4—22no—
69C32×C42—24no—
70C32×C42C2×D314yes—
71C32×C42C22×C311yescharacteristic
72#3756—22no—
73#3755—22no—
74#3754—22no—
75#3753—22no—
76#3752—22no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#216—68no—
80C3×#215—68no—
81C3×#214—68no—
82C3×#213—68no—
83C3×(C2×D3)⋊C4—68no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33—41no—
87C22×C33—42no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—42no—
91C33×C4—22no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C3×C3⋊Dic3—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—91no—
98C2×C3×SD16—91no—
99C2×C3×D8—91no—
100C3×#124—24no—
101C3×#124—91no—
102C3×#124C3×D314yescomplemented
103C3×#124C3⋊D311yescharacteristic, complemented
104C2×#216—34no—
105C2×#215—34no—
106C2×#214—34no—
107C2×#213—34no—
108#875—34no—
109C3×C4×C8—91no—
110C4×C3⋊C8—34no—
111C23×C32—28no—
112C23×C32—24no—
113C23×C32—21no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—24no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8—44no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—24no—
125C32×D4—24no—
126C32×D4—44no—
127C32×D4—44no—
128C2×C32×C4—28no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—28no—
132C2×C32×C4—24no—
133C2×C32×C4#6718yes—
134C2×C32×C4C3×D412yes—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140#634—21no—
141#634—21no—
142C2×C3×Dic3—64no—
143C3×D12—64no—
144C3×D12—64no—
145C3×Dic6—64no—
146C3×Dic6—64no—
147C3⋊(C3⋊C8)C3×D412yes—
148C3⋊(C3⋊C8)C3×D412yes—
149C3×C3⋊C8—38no—
150C3×C3⋊C8—38no—
151#495—91noSylow
152C2×C33—81no—
153C2×C33#12711yescharacteristic
154C2×C33#13512yes—
155C3×C3⋊D3—81no—
156C2×C3×Q8—24no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×C3×D314yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—24no—
161C2×C3×D4—91no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C2×D12—34no—
165C2×Dic6—34no—
166C3×Q16—182no—
167C3×SD16—182no—
168C3×SD16—182no—
169C3×D8—182no—
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C3×C22⋊C4—216no—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—22no—
175C3×C22⋊C4—182no—
176C3×C42—24no—
177C3×C42C2×C3×D314yes—
178C3×C42C2×C3⋊D311yescharacteristic
179#216—68no—
180#215—68no—
181#214—68no—
182#213—68no—
183(C2×D3)⋊C4—68no—
184C2×C3⋊C8—34no—
185C2×C3⋊C8—34no—
186C22×C32—24no—
187C22×C32—48no—
188C22×C32—416no—
189C22×C32—44no—
190C22×C32—48no—
191C22×C32—41no—
192C22×C32—42no—
193C22×C32C2×#6714yes—
194C22×C32C2×C3×D411yescharacteristic
195C2×C3⋊D3—41no—
196C2×C3⋊D3—42no—
197C2×C3×D3—124no—
198C2×C3×D3—128no—
199C32×C4—22no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—28no—
203C32×C4—48no—
204C32×C4—48no—
205C32×C4—44no—
206C32×C4—41no—
207C32×C4D3×D414yes—
208C32×C4D3×D414yes—
209C32×C4C2×C3×D411yescharacteristic
210C32×C4C2×C3×D411yescharacteristic
211C3⋊Dic3—41no—
212C3×Dic3—124no—
213C2×Q16—91no—
214C2×SD16—91no—
215C2×SD16—91no—
216C2×D8—91no—
217#124—91no—
218#124C3×C3⋊D311yescharacteristic, complemented
219C4×C8—91no—
220C33#49511yescharacteristic, complemented, Sylow
221C23×C3—28no—
222C23×C3—24no—
223C23×C3—21no—
224C23×C3—181no—
225C22×D3—64no—
226C3×Q8—24no—
227C3×Q8—24no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—44no—
231C3×Q8—44no—
232C3×Q8—181no—
233C3×Q8—181no—
234C3×D4—24no—
235C3×D4—24no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—44no—
239C3×D4—44no—
240C3×D4—181no—
241C3×D4—181no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—28no—
246C2×C3×C4—24no—
247C2×C3×C4—181no—
248C2×C3×C4C3×#6718yes—
249C2×C3×C4#63812yes—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252C2×Dic3—64no—
253D12—64no—
254D12—64no—
255Dic6—64no—
256Dic6—64no—
257C3×C8—92no—
258C3×C8—92no—
259C3⋊C8—38no—
260C3⋊C8—38no—
261C2×C32—24no—
262C2×C32—28no—
263C2×C32—88no—
264C2×C32—84no—
265C2×C32—81no—
266C2×C32#94014yes—
267C2×C32#95018yes—
268C2×C32C3×#12711yescharacteristic
269C2×C32C3×#13512yes—
270C3⋊D3—81no—
271C3×D3—244no—
272C2×Q8—91no—
273C2×Q8C2×C3×C3⋊D311yescharacteristic
274C2×D4—91no—
275C2×D4C2×C3×C3⋊D311yescharacteristic
276Q16—182no—
277SD16—182no—
278SD16—182no—
279D8—182no—
280C2×C8—91no—
281C2×C8—91no—
282C22⋊C4—22no—
283C22⋊C4—182no—
284C42C2×C3×C3⋊D311yescharacteristic
285C22×C3—24no—
286C22×C3—48no—
287C22×C3—416no—
288C22×C3—44no—
289C22×C3—48no—
290C22×C3—41no—
291C22×C3—42no—
292C22×C3—361no—
293C22×C3—362no—
294C22×C3C2×C3×#6714yes—
295C22×C3C2×#63811yescharacteristic
296C2×D3—124no—
297C2×D3—128no—
298C3×C4—28no—
299C3×C4—22no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—48no—
303C3×C4—48no—
304C3×C4—44no—
305C3×C4—41no—
306C3×C4—361no—
307C3×C4C3×D3×D414yes—
308C3×C4C3×D3×D414yes—
309C3×C4D4×C3⋊D311yescharacteristic
310C3×C4D4×C3⋊D311yescharacteristic
311Dic3—124no—
312C32—24no—
313C32#505914yescomplemented
314C32C3×#49511yescharacteristic, complemented
315C23—21no—
316C23—181no—
317Q8—21no—
318Q8—21no—
319Q8—181no—
320Q8—181no—
321D4—21no—
322D4—21no—
323D4—181no—
324D4—181no—
325C2×C4—21no—
326C2×C4—181no—
327C2×C4C3×#63812yes—
328C2×C4C22×C3×C3⋊D311yescharacteristic
329C8—92no—
330C8—92no—
331C2×C3—24no—
332C2×C3—28no—
333C2×C3—88no—
334C2×C3—84no—
335C2×C3—81no—
336C2×C3—721no—
337C2×C3C3×#94014yes—
338C2×C3C3×#95018yes—
339C2×C3#6414911yescharacteristic
340C2×C3#6415912yes—
341D3—244no—
342C22—41no—
343C22—42no—
344C22—361no—
345C22—362no—
346C22C2×C3×#63811yescharacteristic
347C4—22no—
348C4—41no—
349C4—361no—
350C4C3×D4×C3⋊D311yescharacteristic
351C4C3×D4×C3⋊D311yescharacteristic
352C3—24no—
353C3C3×#505914yescomplemented
354C3#1059275811yescharacteristic, direct product factor
355C2—81no—
356C2—721no—
357C2C3×#6414911yescharacteristic
358C2C3×#6415912yes—
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#124C2
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC3×#124C3⋊D3
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C33×#124C2semidirect
2. C2×C3×#3756C2semidirect
3. C2×C3×#3755C2semidirect
4. C2×C3×#3754C2semidirect
6. C3×#64084C2semidirect
7. C3×C4×C3⋊(C3⋊C8)C2semidirect
8. #10592758C3direct (N × H)
24. C32×#124D3semidirect
25. C32×#124C2×C3semidirect
26. C2×#3756C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. C2×#3754C2×C3semidirect
30. #64084C2×C3semidirect
36. C4×C3⋊(C3⋊C8)C2×C3semidirect
102. C3×#124C3×D3semidirect
103. C3×#124C3⋊D3semidirect
218. #124C3×C3⋊D3semidirect
220. C33#495semidirect
313. C32#5059semidirect
314. C32C3×#495semidirect
353. C3C3×#5059semidirect
354. C3#10592758direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1242
C2×C3×#37562
C2×C3×#37552
C2×C3×#37542
C2×C3×#37532
C3×#640842
C3×C4×C3⋊(C3⋊C8)2
#105927583
C3×#50593