Finite Group Databasebeta

← C3×#10590374 #13157400

Subgroups — C3×#10590374 #13157400

Order 1728.

710 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C3×#64084C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C2×C3×#3719C212yesmaximal, complemented
4C2×C3×#3717C212yesmaximal
5C32×#803C211yesmaximal, characteristic, complemented
6C3×#5086—31nomaximal
7C3×#4715—31nomaximal
8#10590374C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C33×C42C2211yescharacteristic
13C3×#3752—22no—
14C32×C4⋊Dic3—22no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3719—24no—
17C3×#3717—24no—
18C32×C4⋊Q8—31no—
19#64084C2×C311yescharacteristic
20C2×C3×#216—32no—
21C2×C3×#215—32no—
22C2×C3×Dic12—32no—
23C2×C3×#204—32no—
24C3×#875—31no—
25C3×#875—31no—
26C3×#875—61no—
27C3×#869—21no—
28C3×#869D311yescharacteristic, complemented
29C3×#869C2×C311yescharacteristic, complemented
30C2×#3719C2×C312yescomplemented
31C2×#3717C2×C312yes—
32C32×#97—31no—
33C3×#803—21no—
34C3×#803D311yescharacteristic, complemented
35C3×#803C2×C311yescharacteristic, complemented
36C22×C3×C3⋊D3—21no—
37C2×C33×C4D412yes—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C3×C4⋊(C3⋊D3)—22no—
41C3×#634—22no—
42C2×C32×Dic3—22no—
43C32×Dic6—22no—
44C32×Dic6—22no—
45C32×C3⋊C8D414yes—
46C3×#497—91no—
47#5086—31no—
48#4715—31no—
49C2×C32×Q8—32no—
50C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
51C2×#634C22×C311yescharacteristic
52C2×C3×D12—31no—
53C2×C3×D12—31no—
54C2×C3×D12—61no—
55C2×C3×Dic6—22no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—31no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6C2×D312yes—
60C2×C3×Dic6C22×C312yes—
61C2×C32×C8—32no—
62C32×C4⋊C4—62no—
63C32×C42—21no—
64C32×C42—21no—
65C32×C42—21no—
66C32×C42—41no—
67C32×C42C2×D311yescharacteristic
68C32×C42C2×D311yescharacteristic
69C32×C42C22×C311yescharacteristic
70#3752—22no—
71C3×#216—64no—
72C3×#215—64no—
73C3×(C2×D3)⋊C4—64no—
74C3×(C2×D3)⋊C4—62no—
75C3×(C2×D3)⋊C4—62no—
76C3×C4⋊Dic3—24no—
77C3×C4⋊Dic3—22no—
78C3×C4⋊Dic3—22no—
79C2×C3×C3⋊C8—22no—
80C2×C3×C3⋊C8C2×D312yes—
81C2×C3×C3⋊C8C22×C312yes—
82C3×Dic12—64no—
83C3×#204—64no—
84#3719—24no—
85#3717—24no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—42no—
89C33×C4—22no—
90C33×C4C2×D412yes—
91C3×C3⋊Dic3—41no—
92C32×Dic3—42no—
93C2×C3×Q16—92no—
94C2×C3×SD16—92no—
95C3×C4⋊Q8—31no—
96C3×C4⋊Q8—31no—
97C3×C4⋊Q8—61no—
98C3×#124—91no—
99C2×#216—32no—
100C2×#215—32no—
101C2×Dic12—32no—
102C2×#204—32no—
103#875—31no—
104#875—31no—
105#875—61no—
106#869C3×D311yescharacteristic, complemented
107C3×#97—31no—
108C3×#97—31no—
109C3×#97—61no—
110#803C3×D311yescharacteristic, complemented
111C22×C3⋊D3—21no—
112C22×C3×D3—62no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C32×Q8—62no—
116C32×Q8—62no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—42no—
124C2×C32×C4—41no—
125C2×C32×C4—62no—
126C2×C32×C4D1212yes—
127C2×C32×C4#6712yes—
128C2×C32×C4C3×D412yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C23×C311yescharacteristic
132C2×C3⋊Dic3—21no—
133C4⋊(C3⋊D3)—22no—
134#634—22no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—62no—
142C3×D12—62no—
143C3×D12—122no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—22no—
147C3×Dic6—22no—
148C3×Dic6—42no—
149C3×Dic6—42no—
150C3×Dic6—62no—
151C3×Dic6—62no—
152C3×Dic6—122no—
153C32×C8—34no—
154C3×C3⋊C8—24no—
155C3×C3⋊C8D1214yes—
156C3×C3⋊C8C3×D414yes—
157#497—91noSylow
158C2×C33#12711yescharacteristic
159C2×C33#13712yes—
160C3×C3⋊D3—81no—
161C2×C3×Q8—32no—
162C2×C3×Q8—32no—
163C2×C3×Q8—62no—
164C2×C3×Q8—91no—
165C2×C3×D4—91no—
166C2×D12—31no—
167C2×D12—31no—
168C2×D12—61no—
169C2×Dic6—31no—
170C2×Dic6—31no—
171C2×Dic6—61no—
172C2×Dic6C2×C3×D312yes—
173C3×Q16—184no—
174C3×SD16—184no—
175C2×C3×C8—32no—
176C2×C3×C8—32no—
177C2×C3×C8—62no—
178C3×C4⋊C4—64no—
179C3×C4⋊C4—62no—
180C3×C4⋊C4—62no—
181C3×C22⋊C4—182no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—41no—
186C3×C42D3211yescharacteristic
187C3×C42C2×C3×D311yescharacteristic
188C3×C42C2×C3×D311yescharacteristic
189#216—64no—
190#215—64no—
191(C2×D3)⋊C4—64no—
192(C2×D3)⋊C4—62no—
193(C2×D3)⋊C4—62no—
194C4⋊Dic3—22no—
195C2×C3⋊C8C2×C3×D312yes—
196Dic12—64no—
197#204—64no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—41no—
202C22×C32C2×D1211yescharacteristic
203C22×C32C2×#6711yescharacteristic
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—41no—
206C2×C3⋊D3—42no—
207C2×C3×D3—122no—
208C2×C3×D3—124no—
209C2×C3×D3—121no—
210C2×C3×D3—122no—
211C2×C3×D3—121no—
212C2×C3×D3—122no—
213C32×C4—22no—
214C32×C4—24no—
215C32×C4—22no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—42no—
221C32×C4—42no—
222C32×C4—42no—
223C32×C4—44no—
224C32×C4—122no—
225C32×C4C2×D1212yes—
226C32×C4D3×D412yes—
227C32×C4C2×C3×D412yes—
228C3⋊Dic3—41no—
229C3×Dic3—44no—
230C3×Dic3—42no—
231C3×Dic3—42no—
232C3×Dic3—122no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C2×Q16—92no—
236C2×SD16—92no—
237C4⋊Q8—31no—
238#124—91no—
239#97—31no—
240C33#49711yescharacteristic, complemented, Sylow
241C23×C3—181no—
242C22×D3—62no—
243C22×D3—61no—
244C22×D3—61no—
245C3×Q8—62no—
246C3×Q8—62no—
247C3×Q8—62no—
248C3×Q8—62no—
249C3×Q8—122no—
250C3×Q8—122no—
251C3×Q8—182no—
252C3×D4—182no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—64no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—181no—
265C2×C3×C4#62612yes—
266C2×C3×C4C3×D1212yes—
267C2×C3×C4C3×#6712yes—
268C2×C3×C4C2×D3211yescharacteristic
269C2×C3×C4C22×C3×D311yescharacteristic
270C2×C3×C4C22×C3×D311yescharacteristic
271C2×Dic3—22no—
272C2×Dic3—62no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275D12—62no—
276D12—62no—
277D12—122no—
278Dic6—22no—
279Dic6—22no—
280Dic6—62no—
281Dic6—62no—
282Dic6—122no—
283C3×C8—34no—
284C3×C8—34no—
285C3×C8—64no—
286C3⋊C8C3×D1214yes—
287C2×C32—22no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—42no—
295C2×C32#87411yescharacteristic
296C2×C32#90912yes—
297C2×C32#94011yescharacteristic
298C2×C32#94212yes—
299C2×C32C3×#12711yescharacteristic
300C2×C32C3×#13712yes—
301C3⋊D3—81no—
302C3×D3—242no—
303C3×D3—241no—
304C3×D3—241no—
305C2×Q8—32no—
306C2×Q8—91no—
307C2×D4—91no—
308Q16—184no—
309SD16—184no—
310C2×C8—32no—
311C4⋊C4—62no—
312C22⋊C4—182no—
313C42C3×D3211yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—362no—
319C22×C3—361no—
320C22×C3C2×#62611yescharacteristic
321C22×C3C2×C3×D1211yescharacteristic
322C22×C3C2×C3×#6711yescharacteristic
323C2×D3—122no—
324C2×D3—124no—
325C2×D3—122no—
326C2×D3—121no—
327C2×D3—122no—
328C2×D3—121no—
329C3×C4—24no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—42no—
338C3×C4—44no—
339C3×C4—42no—
340C3×C4—124no—
341C3×C4—122no—
342C3×C4—122no—
343C3×C4—361no—
344C3×C4D3×D1212yes—
345C3×C4C2×C3×D1212yes—
346C3×C4C3×D3×D412yes—
347Dic3—42no—
348Dic3—122no—
349Dic3—121no—
350Dic3—121no—
351C32—21no—
352C32—21no—
353C32—21no—
354C32—41no—
355C32#471511yescharacteristic, complemented
356C32#508611yescharacteristic, complemented
357C32C3×#49711yescharacteristic, complemented
358C23—181no—
359Q8—62no—
360Q8—62no—
361Q8—182no—
362D4—182no—
363C2×C4—62no—
364C2×C4—181no—
365C2×C4C3×#62612yes—
366C2×C4C2×C3×D3211yescharacteristic
367C8—34no—
368C2×C3—22no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—42no—
375C2×C3—41no—
376C2×C3—721no—
377C2×C3#6383312yes—
378C2×C3#6391111yescharacteristic
379C2×C3C3×#87411yescharacteristic
380C2×C3C3×#90912yes—
381C2×C3C3×#94011yescharacteristic
382C2×C3C3×#94212yes—
383D3—242no—
384D3—241no—
385D3—241no—
386C22—362no—
387C22—361no—
388C22C2×C3×#62611yescharacteristic
389C4—22no—
390C4—122no—
391C4—361no—
392C4C3×D3×D1212yes—
393C3—21no—
394C3—21no—
395C3—21no—
396C3—41no—
397C3#1059037411yescharacteristic, direct product factor
398C3C3×#471511yescharacteristic, complemented
399C3C3×#508611yescharacteristic, complemented
400C2—721no—
401C2C3×#6383312yes—
402C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C42D32
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#869C2semidirect
3. C2×C3×#3719C2semidirect
5. C32×#803C2semidirect
8. #10590374C3direct (N × H)
28. C3×#869D3semidirect
29. C3×#869C2×C3semidirect
30. C2×#3719C2×C3semidirect
34. C3×#803D3semidirect
35. C3×#803C2×C3semidirect
106. #869C3×D3semidirect
110. #803C3×D3semidirect
240. C33#497semidirect
355. C32#4715semidirect
356. C32#5086semidirect
357. C32C3×#497semidirect
397. C3#10590374direct (N × H)
398. C3C3×#4715semidirect
399. C3C3×#5086semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×#8692
C2×C3×#37192
C2×C3×#37172
C32×#8032
C3×#50863
C3×#47153
#105903743