Finite Group Databasebeta

← C3×#10592886 #13163229

Subgroups — C3×#10592886 #13163229

Order 1728.

1152 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C2×C33×Q16C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63647C211yesmaximal, characteristic, complemented
8#10592886C311yesmaximal, characteristic, direct product factor
9C3×#5187—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×Q8C2211yescharacteristic
12C3×#3832—22no—
13C3×#3832—22no—
14C2×C3×C4⋊(C3⋊D3)—21no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C33×Q16—22no—
19C2×C33×C8C2211yescharacteristic
20C3×#3755—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750—21no—
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×#3832C2×C311yescharacteristic
26C2×C3×#239—34no—
27C2×C32×Q16—24no—
28C2×C32×Q16D314yescomplemented
29C2×C32×Q16C2×C311yescharacteristic, complemented
30#64155C2×C311yescharacteristic, complemented
31C2×#3755C2×C311yescharacteristic, complemented
32C3×#946—34no—
33C2×C3×#215—34no—
34#63662C2×C311yescharacteristic, complemented
35#63648C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#835—34no—
38C3×#821—34no—
39C3×(C2×D3)⋊C8—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8—21no—
44C33×Q8—21no—
45C33×Q8D412yes—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—24no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)D412yes—
54C3×C4×C3⋊D3—24no—
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#452—91no—
60#5187—34no—
61C2×C32×Q8—24no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D314yes—
65C2×C32×Q8C22×C311yescharacteristic
66C2×C32×Q8C22×C311yescharacteristic
67#3832—22no—
68#3832—22no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C3×#239—68no—
74C3×#239—68no—
75C2×C3×D12—34no—
76C2×C3×D12—64no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—64no—
79C32×Q16—216no—
80C32×Q16—28no—
81C32×Q16—22no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C22×C311yescharacteristic
85#3755—22no—
86#3752—21no—
87C4⋊(C3⋊Dic3)C22×C311yescharacteristic
88#3750—21no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#215—68no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—34no—
93C3×Dic3⋊C4—64no—
94C2×C3×C3⋊C8—34no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C33×C4—22no—
103C33×C4—41no—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—41no—
108C2×C3×#41—91no—
109C2×#239—34no—
110C2×C3×Q16—24no—
111C2×C3×Q16C3×D314yescomplemented
112C2×C3×Q16C3⋊D311yescharacteristic, complemented
113C2×C3×SD16—91no—
114C3×C22⋊Q8—91no—
115#946—34no—
116C2×#215—34no—
117C3×#103—91no—
118C3×#102—91no—
119C3×C22⋊C8—91no—
120#835—34no—
121#821—34no—
122(C2×D3)⋊C8—34no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3C3×D411yescharacteristic
125C22×C3×D3—34no—
126C22×C3×D3—64no—
127C32×Q8—21no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—28no—
133C32×Q8—24no—
134C32×Q8—24no—
135C32×Q8—44no—
136C32×Q8—44no—
137C32×Q8#6718yes—
138C32×Q8C3×D412yes—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—44no—
145C2×C32×C4—44no—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C23×C311yescharacteristic
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3C3×D411yescharacteristic
150C4⋊(C3⋊D3)—24no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)C3×D412yes—
153C4×C3⋊D3—24no—
154C4×C3⋊D3—21no—
155C4×C3⋊D3—21no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—64no—
158C3×D12—38no—
159C3×D12—616no—
160C3×D12—64no—
161C3×C4×D3—616no—
162C3×C4×D3—64no—
163C3×C4×D3—64no—
164C32×C8—28no—
165C32×C8—24no—
166C32×C8—21no—
167C3⋊(C3⋊C8)—21no—
168C3×C3⋊C8—64no—
169#452—91noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—41no—
174C3×C3⋊D3—42no—
175C3×#41—182no—
176C3×#41—182no—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8C2×C3×D314yes—
180C2×C3×Q8C2×C3×D314yes—
181C2×C3×Q8C2×C3⋊D311yescharacteristic
182C2×C3×Q8C2×C3⋊D311yescharacteristic
183C2×C3×D4—91no—
184C2×C3×D4—181no—
185C22×C3×C4—91no—
186C22×C3×C4—181no—
187#239—68no—
188#239—68no—
189C2×D12—34no—
190C2×D12—64no—
191C2×C4×D3—34no—
192C2×C4×D3—64no—
193C3×Q16—216no—
194C3×Q16—28no—
195C3×Q16—22no—
196C3×SD16—182no—
197C2×C3×C8—24no—
198C2×C3×C8—91no—
199C2×C3×C8C2×C3×D314yes—
200C2×C3×C8C2×C3⋊D311yescharacteristic
201C3×C4⋊C4—91no—
202C3×C4⋊C4—181no—
203C3×C22⋊C4—181no—
204#215—68no—
205(C2×D3)⋊C4—64no—
206C4⋊Dic3—34no—
207Dic3⋊C4—64no—
208C2×C3⋊C8—34no—
209C22×C32—24no—
210C22×C32D3×D414yes—
211C22×C32C2×C3×D411yescharacteristic
212C2×C3⋊D3—21no—
213C2×C3⋊D3—22no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—68no—
221C2×C3×D3—64no—
222C2×C3×D3—124no—
223C2×C3×D3—124no—
224C32×C4—22no—
225C32×C4—24no—
226C32×C4—24no—
227C32×C4—28no—
228C32×C4—48no—
229C32×C4—48no—
230C32×C4—44no—
231C32×C4—41no—
232C32×C4D3×D414yes—
233C32×C4C2×#6714yes—
234C32×C4C2×C3×D411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C3⋊Dic3—21no—
237C3⋊Dic3—41no—
238C3×Dic3—64no—
239C3×Dic3—124no—
240C2×#41—91no—
241C2×Q16C3×C3⋊D311yescharacteristic, complemented
242C2×SD16—91no—
243C22⋊Q8—91no—
244#103—91no—
245#102—91no—
246C22⋊C8—91no—
247C33#45211yescharacteristic, complemented, Sylow
248C23×C3—91no—
249C23×C3—181no—
250C22×D3—34no—
251C22×D3—64no—
252C3×Q8—24no—
253C3×Q8—21no—
254C3×Q8—28no—
255C3×Q8—24no—
256C3×Q8—24no—
257C3×Q8—21no—
258C3×Q8—21no—
259C3×Q8—28no—
260C3×Q8—44no—
261C3×Q8—44no—
262C3×Q8C3×#6718yes—
263C3×Q8#63812yes—
264C3×D4—92no—
265C3×D4—184no—
266C3×D4—181no—
267C2×C3×C4—24no—
268C2×C3×C4—21no—
269C2×C3×C4—24no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—91no—
275C2×C3×C4—184no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C22×C3×D314yes—
280C2×C3×C4C22×C3⋊D311yescharacteristic
281C2×Dic3—34no—
282C2×Dic3—64no—
283D12—38no—
284D12—616no—
285D12—64no—
286C4×D3—616no—
287C4×D3—64no—
288C4×D3—64no—
289C3×C8—28no—
290C3×C8—24no—
291C3×C8—21no—
292C3×C8—181no—
293C3⋊C8—64no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32#91814yes—
298C2×C32#91914yes—
299C2×C32#93714yes—
300C2×C32C3×#12011yescharacteristic
301C2×C32C3×#13511yescharacteristic
302C2×C32C3×#13711yescharacteristic
303C3⋊D3—41no—
304C3⋊D3—42no—
305C3×D3—124no—
306C3×D3—128no—
307#41—182no—
308#41—182no—
309C2×Q8C2×C3×C3⋊D311yescharacteristic
310C2×Q8C2×C3×C3⋊D311yescharacteristic
311C2×D4—91no—
312C2×D4—181no—
313C22×C4—91no—
314C22×C4—181no—
315Q16—22no—
316SD16—182no—
317C2×C8—91no—
318C2×C8C2×C3×C3⋊D311yescharacteristic
319C4⋊C4—91no—
320C4⋊C4—181no—
321C22⋊C4—181no—
322C22×C3—24no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—182no—
326C22×C3—181no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3C3×D3×D414yes—
330C22×C3D4×C3⋊D311yescharacteristic
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—68no—
334C2×D3—64no—
335C2×D3—124no—
336C2×D3—124no—
337C3×C4—28no—
338C3×C4—22no—
339C3×C4—24no—
340C3×C4—24no—
341C3×C4—48no—
342C3×C4—48no—
343C3×C4—44no—
344C3×C4—41no—
345C3×C4—181no—
346C3×C4—361no—
347C3×C4C3×D3×D414yes—
348C3×C4C2×C3×#6714yes—
349C3×C4D4×C3⋊D311yescharacteristic
350C3×C4C2×#63811yescharacteristic
351Dic3—64no—
352Dic3—124no—
353C32—24no—
354C32#518714yescomplemented
355C32C3×#45211yescharacteristic, complemented
356C23—91no—
357C23—181no—
358Q8—21no—
359Q8—21no—
360Q8—21no—
361Q8C3×#63812yes—
362D4—92no—
363D4—184no—
364D4—181no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—91no—
368C2×C4—184no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4C22×C3×C3⋊D311yescharacteristic
373C8—21no—
374C8—181no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—361no—
379C2×C3—362no—
380C2×C3C3×#91814yes—
381C2×C3C3×#91914yes—
382C2×C3C3×#93714yes—
383C2×C3#6412711yescharacteristic
384C2×C3#6412811yescharacteristic
385C2×C3#6414611yescharacteristic
386D3—124no—
387D3—128no—
388C22—181no—
389C22—181no—
390C22—182no—
391C22—181no—
392C22—361no—
393C22—361no—
394C22C3×D4×C3⋊D311yescharacteristic
395C4—22no—
396C4—41no—
397C4—181no—
398C4—361no—
399C4C3×D4×C3⋊D311yescharacteristic
400C4C2×C3×#63811yescharacteristic
401C3—24no—
402C3C3×#518714yescomplemented
403C3#1059288611yescharacteristic, direct product factor
404C2—361no—
405C2—362no—
406C2C3×#6412711yescharacteristic
407C2C3×#6412811yescharacteristic
408C2C3×#6414611yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×Q16C2
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
hypercenterC2×C3×Q16C3⋊D3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C2×C33×Q16C2semidirect
3. C3×#64155C2semidirect
4. C2×C3×#3755C2semidirect
5. C3×#63662C2semidirect
6. C3×#63648C2semidirect
7. C3×#63647C2semidirect
8. #10592886C3direct (N × H)
28. C2×C32×Q16D3semidirect
29. C2×C32×Q16C2×C3semidirect
30. #64155C2×C3semidirect
31. C2×#3755C2×C3semidirect
34. #63662C2×C3semidirect
35. #63648C2×C3semidirect
36. #63647C2×C3semidirect
111. C2×C3×Q16C3×D3semidirect
112. C2×C3×Q16C3⋊D3semidirect
241. C2×Q16C3×C3⋊D3semidirect
247. C33#452semidirect
354. C32#5187semidirect
355. C32C3×#452semidirect
402. C3C3×#5187semidirect
403. C3#10592886direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C33×Q162
C3×#641552
C2×C3×#37552
C3×#636622
C3×#636482
C3×#636472
#105928863
C3×#51873