Finite Group Databasebeta

← C3×#10592746 #13163089

Subgroups — C3×#10592746 #13163089

Order 1728.

1208 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C33×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#3756C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic, complemented
5C3×#63657C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592746C311yesmaximal, characteristic, direct product factor
9C3×#5047—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×#3828—22no—
14C3×#3828—22no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×#634C2211yescharacteristic
18C33×C4⋊C4—21no—
19C33×C4⋊C4C2211yescharacteristic
20C33×C22⋊C4—21no—
21C3×#3756—22no—
22C3×#3755—22no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×#3828C2×C311yescharacteristic
26C2×C3×#235—34no—
27C32×C22⋊Q8—24no—
28C32×C22⋊Q8D314yescomplemented
29C32×C22⋊Q8C2×C311yescharacteristic, complemented
30C2×#3756C2×C311yescharacteristic, complemented
31C2×#3755C2×C311yescharacteristic, complemented
32C2×C3×#216—34no—
33C2×C3×#215—34no—
34#63657C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36#63636C2×C311yescharacteristic, complemented
37C3×#830—34no—
38C3×#810—34no—
39C3×#809—34no—
40C23×C33D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C33×Q8—21no—
43C33×Q8—21no—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4D411yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C3×#638—24no—
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)D412yes—
54C3×C4×C3⋊D3—24no—
55C3×#634—21no—
56C3×#634D412yes—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#452—91no—
60#5047—34no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C22×C311yescharacteristic
64C22×C32×C4—24no—
65C22×C32×C4C2×D314yes—
66C22×C32×C4C22×C311yescharacteristic
67C2×#638—21no—
68#3828—22no—
69#3828—22no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×#634C22×C311yescharacteristic
73C2×C3×#67—64no—
74C3×#235—68no—
75C3×#235—68no—
76C2×C3×D12—34no—
77C2×C3×C4×D3—64no—
78C2×C3×Dic6—34no—
79C32×C4⋊C4—28no—
80C32×C4⋊C4—24no—
81C32×C4⋊C4—24no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D314yes—
84C32×C4⋊C4C22×C311yescharacteristic
85C32×C22⋊C4—28no—
86C32×C22⋊C4—24no—
87C32×C22⋊C4—21no—
88#3756—22no—
89#3755—22no—
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#216—68no—
93C3×#215—68no—
94C2×C3×C3⋊C8—34no—
95C2×C3×C3⋊C8—34no—
96C22×C33—21no—
97C22×C33—21no—
98C22×C33—21no—
99C22×C33C2×D411yescharacteristic
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C33×C4—21no—
104C33×C4—41no—
105C33×C4—41no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C2×C3×#41—91no—
110C2×#235—34no—
111C2×C3×Q16—91no—
112C2×C3×SD16—91no—
113C3×C22⋊Q8—24no—
114C3×C22⋊Q8C3×D314yescomplemented
115C3×C22⋊Q8C3⋊D311yescharacteristic, complemented
116C2×#216—34no—
117C2×#215—34no—
118C3×#103—91no—
119C3×#102—91no—
120C3×C22⋊C8—91no—
121#830—34no—
122#810—34no—
123#809—34no—
124C23×C32—24no—
125C23×C32#6714yes—
126C23×C32C3×D411yescharacteristic
127C22×C3⋊D3—21no—
128C22×C3×D3—64no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—24no—
132C32×Q8—24no—
133C32×Q8—44no—
134C32×Q8—44no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—28no—
138C2×C32×C4—28no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—24no—
146C2×C32×C4—24no—
147C2×C32×C4—44no—
148C2×C32×C4—44no—
149C2×C32×C4#6714yes—
150C2×C32×C4C3×D411yescharacteristic
151C2×C32×C4C22×D314yes—
152C2×C32×C4C23×C311yescharacteristic
153#638—24no—
154C2×C3⋊Dic3—21no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)C3×D412yes—
157C4×C3⋊D3—24no—
158#634—21no—
159#634C3×D412yes—
160C3×#67—616no—
161C2×C3×Dic3—64no—
162C3×D12—38no—
163C3×D12—64no—
164C3×C4×D3—616no—
165C3×Dic6—38no—
166C3×Dic6—64no—
167C3⋊(C3⋊C8)—21no—
168C3⋊(C3⋊C8)—21no—
169C3×C3⋊C8—64no—
170C3×C3⋊C8—64no—
171#452—91noSylow
172C2×C33—41no—
173C2×C33#12011yescharacteristic
174C2×C33#13511yescharacteristic
175C2×C33#13711yescharacteristic
176C3×C3⋊D3—42no—
177C3×#41—182no—
178C3×#41—182no—
179C2×C3×Q8—24no—
180C2×C3×Q8—91no—
181C2×C3×Q8C2×C3×D314yes—
182C2×C3×Q8C2×C3⋊D311yescharacteristic
183C2×C3×D4—91no—
184C2×C3×D4—181no—
185C22×C3×C4—24no—
186C22×C3×C4—181no—
187C22×C3×C4C2×C3×D314yes—
188C22×C3×C4C2×C3⋊D311yescharacteristic
189C2×#67—64no—
190#235—68no—
191#235—68no—
192C2×D12—34no—
193C2×C4×D3—64no—
194C2×Dic6—34no—
195C3×Q16—182no—
196C3×SD16—182no—
197C2×C3×C8—91no—
198C2×C3×C8—91no—
199C3×C4⋊C4—28no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4C2×C3×D314yes—
204C3×C4⋊C4C2×C3⋊D311yescharacteristic
205C3×C22⋊C4—28no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—21no—
208#216—68no—
209#215—68no—
210C2×C3⋊C8—34no—
211C2×C3⋊C8—34no—
212C22×C32—28no—
213C22×C32—24no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—24no—
218C22×C32—24no—
219C22×C32—24no—
220C22×C32—44no—
221C22×C32—44no—
222C22×C32C2×#6714yes—
223C22×C32C2×C3×D411yescharacteristic
224C2×C3⋊D3—22no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3×D3—68no—
228C2×C3×D3—124no—
229C2×C3×D3—124no—
230C32×C4—21no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4—44no—
235C32×C4—48no—
236C32×C4—48no—
237C32×C4—44no—
238C32×C4—44no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4D3×D414yes—
242C32×C4D3×D414yes—
243C32×C4C2×C3×D411yescharacteristic
244C32×C4C2×C3×D411yescharacteristic
245C3⋊Dic3—22no—
246C3×Dic3—68no—
247C2×#41—91no—
248C2×Q16—91no—
249C2×SD16—91no—
250C22⋊Q8C3×C3⋊D311yescharacteristic, complemented
251#103—91no—
252#102—91no—
253C22⋊C8—91no—
254C33#45211yescharacteristic, complemented, Sylow
255C23×C3—24no—
256C23×C3—181no—
257C23×C3C3×#6714yes—
258C23×C3#63811yescharacteristic
259C22×D3—64no—
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—44no—
265C3×Q8—44no—
266C3×Q8—92no—
267C3×Q8—181no—
268C3×D4—92no—
269C3×D4—184no—
270C3×D4—181no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—28no—
276C2×C3×C4—28no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4—184no—
286C2×C3×C4—181no—
287C2×C3×C4C3×#6714yes—
288C2×C3×C4#63811yescharacteristic
289C2×C3×C4C22×C3×D314yes—
290C2×C3×C4C22×C3⋊D311yescharacteristic
291#67—616no—
292C2×Dic3—64no—
293D12—38no—
294D12—64no—
295C4×D3—616no—
296Dic6—38no—
297Dic6—64no—
298C3×C8—181no—
299C3×C8—181no—
300C3⋊C8—64no—
301C3⋊C8—64no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32—48no—
306C2×C32—44no—
307C2×C32—41no—
308C2×C32#93714yes—
309C2×C32#94214yes—
310C2×C32#95014yes—
311C2×C32C3×#12011yescharacteristic
312C2×C32C3×#13511yescharacteristic
313C2×C32C3×#13711yescharacteristic
314C3⋊D3—42no—
315C3×D3—128no—
316#41—182no—
317#41—182no—
318C2×Q8—91no—
319C2×Q8C2×C3×C3⋊D311yescharacteristic
320C2×D4—91no—
321C2×D4—181no—
322C22×C4—181no—
323C22×C4C2×C3×C3⋊D311yescharacteristic
324Q16—182no—
325SD16—182no—
326C2×C8—91no—
327C2×C8—91no—
328C4⋊C4—21no—
329C4⋊C4C2×C3×C3⋊D311yescharacteristic
330C22⋊C4—21no—
331C22×C3—28no—
332C22×C3—24no—
333C22×C3—24no—
334C22×C3—24no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—24no—
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3C2×C3×#6714yes—
345C22×C3C2×#63811yescharacteristic
346C2×D3—68no—
347C2×D3—124no—
348C2×D3—124no—
349C3×C4—24no—
350C3×C4—21no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—44no—
354C3×C4—48no—
355C3×C4—48no—
356C3×C4—44no—
357C3×C4—44no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—182no—
361C3×C4C3×D3×D414yes—
362C3×C4C3×D3×D414yes—
363C3×C4D4×C3⋊D311yescharacteristic
364C3×C4D4×C3⋊D311yescharacteristic
365Dic3—68no—
366C32—24no—
367C32#504714yescomplemented
368C32C3×#45211yescharacteristic, complemented
369C23—181no—
370C23C3×#63811yescharacteristic
371Q8—21no—
372Q8—21no—
373Q8—92no—
374Q8—181no—
375D4—92no—
376D4—184no—
377D4—181no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—184no—
383C2×C4—181no—
384C2×C4C3×#63811yescharacteristic
385C2×C4C22×C3×C3⋊D311yescharacteristic
386C8—181no—
387C8—181no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—48no—
392C2×C3—44no—
393C2×C3—41no—
394C2×C3—362no—
395C2×C3C3×#93714yes—
396C2×C3C3×#94214yes—
397C2×C3C3×#95014yes—
398C2×C3#6414611yescharacteristic
399C2×C3#6415111yescharacteristic
400C2×C3#6415911yescharacteristic
401D3—128no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—182no—
406C22—361no—
407C22—361no—
408C22C2×C3×#63811yescharacteristic
409C4—21no—
410C4—41no—
411C4—41no—
412C4—182no—
413C4C3×D4×C3⋊D311yescharacteristic
414C4C3×D4×C3⋊D311yescharacteristic
415C3—24no—
416C3C3×#504714yescomplemented
417C3#1059274611yescharacteristic, direct product factor
418C2—41no—
419C2—362no—
420C2C3×#6414611yescharacteristic
421C2C3×#6415111yescharacteristic
422C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C33×C22⋊Q8C2semidirect
3. C2×C3×#3756C2semidirect
4. C2×C3×#3755C2semidirect
5. C3×#63657C2semidirect
6. C3×#63637C2semidirect
7. C3×#63636C2semidirect
8. #10592746C3direct (N × H)
28. C32×C22⋊Q8D3semidirect
29. C32×C22⋊Q8C2×C3semidirect
30. C2×#3756C2×C3semidirect
31. C2×#3755C2×C3semidirect
34. #63657C2×C3semidirect
35. #63637C2×C3semidirect
36. #63636C2×C3semidirect
114. C3×C22⋊Q8C3×D3semidirect
115. C3×C22⋊Q8C3⋊D3semidirect
250. C22⋊Q8C3×C3⋊D3semidirect
254. C33#452semidirect
367. C32#5047semidirect
368. C32C3×#452semidirect
416. C3C3×#5047semidirect
417. C3#10592746direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C33×C22⋊Q82
C2×C3×#37562
C2×C3×#37552
C3×#636572
C3×#636372
C3×#636362
#105927463
C3×#50473