Finite Group Databasebeta

← C3×#10596519 #13181038

Subgroups — C3×#10596519 #13181038

Order 1728.

992 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64312C212yesmaximal, complemented
2C3×D4×#157C211yesmaximal, characteristic, complemented
3C3×#64242C212yesmaximal, complemented
4C2×C3×#3773C212yesmaximal, complemented
5C3×#64233C214yesmaximal, complemented
6C3×C4×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
7C3×#64222C211yesmaximal, characteristic, complemented
8C3×D32⋊Q8C212yesmaximal, complemented
9#10596519C311yesmaximal, characteristic, direct product factor
10C22×C3×#157C2212yes—
11C2×C3×GO+(4,𝔽2)C2211yescharacteristic
12C3×D4×C3⋊D3C2211yescharacteristic
13C32×#237—22no—
14C3×#3811C2214yes—
15C2×C3×#624C2214yes—
16C3×#3804—24no—
17C3×C4×D32C2212yes—
18C3×#3799—22no—
19C3×#3797C2212yes—
20C3×C22⋊#157C2212yes—
21C3×C4⋊#157C2211yescharacteristic
22C3×C4×#157C2211yescharacteristic
23C3×#3773C2212yes—
24C3×#3773C2214yes—
25C3×#3773C2211yescharacteristic
26C3×#3773C2212yes—
27C3×D32⋊C4C2212yes—
28C3×D32⋊C4C2214yes—
29C3×D3×#41—62no—
30#64312C2×C312yescomplemented
31D4×#157C2×C311yescharacteristic, complemented
32#64242C2×C312yescomplemented
33C2×#3773C2×C312yescomplemented
34#64233C2×C314yescomplemented
35C4×GO+(4,𝔽2)C2×C311yescharacteristic, complemented
36#64222C2×C311yescharacteristic, complemented
37D32⋊Q8C2×C312yescomplemented
38C22×C3×C3⋊D3C2312yes—
39C2×C3×D32C2312yes—
40C2×C3×#157—24no—
41C2×C3×#157C2312yes—
42C2×C3×#157C2311yescharacteristic
43C2×C3×#157C2311yescharacteristic
44C3×GO+(4,𝔽2)—22no—
45C33×D4D411yescharacteristic, complemented
46C3×#638D412yescomplemented
47C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
48C3×C4×C3⋊D3C2311yescharacteristic
49C32×#67—24no—
50C2×C32×Dic3—24no—
51C32×C4×D3—22no—
52C32×Dic6—22no—
53C3×#627—22no—
54C3×#626—24no—
55C3×#624—24no—
56C3×#624C2314yes—
57C3×#624C2312yes—
58C3×D3×Dic3—22no—
59C3×#547—91nomaximal
60C22×#157C22×C312yes—
61C2×GO+(4,𝔽2)C22×C311yescharacteristic
62C32×#41—62no—
63D4×C3⋊D3C22×C311yescharacteristic
64C3×#239—62no—
65C3×D3×Q8—62no—
66C3×#237—22no—
67C3×#237—22no—
68C3×#237—42no—
69C3×#237—64no—
70C3×D3×D4—64no—
71C3×D3×D4—62no—
72C3×#235—62no—
73C3×#235—64no—
74C2×C3×C4×D3—62no—
75C2×C3×C4×D3—64no—
76#3811C22×C314yes—
77C2×#624C22×C314yes—
78#3804—24no—
79C4×D32C22×C312yes—
80#3799—22no—
81#3797C22×C312yes—
82C22⋊#157C22×C312yes—
83C4⋊#157C22×C311yescharacteristic
84C4×#157C22×C311yescharacteristic
85#3773C22×C314yes—
86#3773C22×C312yes—
87#3773C22×C312yes—
88#3773C22×C311yescharacteristic
89D32⋊C4C22×C314yes—
90D32⋊C4C22×C312yes—
91C22×C33C2×D412yes—
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3C2×D412yes—
95C2×C3×C3⋊D3C2411yescharacteristic
96C2×C32×D3—22no—
97C3×D32—22no—
98C3×D32—22no—
99C3×#157—22no—
100C3×#157—21no—
101C3×#157#4112yescomplemented
102C33×C4C2×D411yescharacteristic
103C3×C3⋊Dic3C2×D411yescharacteristic
104C32×Dic3—22no—
105C32×Dic3—24no—
106C2×C3×#41—92no—
107D3×#41—62no—
108C3×C4⋊Q8—91no—
109C3×#123—94no—
110C3×C22⋊Q8—92no—
111C3×#121—92no—
112C3×C4×D4—91no—
113C3×C4×D4—91no—
114C2×C3×C4⋊C4—92no—
115C22×C3⋊D3C23×C312yes—
116C22×C3×D3—62no—
117C22×C3×D3—64no—
118C2×D32C23×C312yes—
119C2×#157—24no—
120C2×#157C23×C312yes—
121C2×#157C23×C311yescharacteristic
122C2×#157C23×C311yescharacteristic
123GO+(4,𝔽2)—22no—
124C32×Q8—62no—
125C32×D4—22no—
126C32×D4—42no—
127C32×D4—64no—
128C32×D4C3×D411yescharacteristic, complemented
129C2×C32×C4—62no—
130C2×C32×C4—64no—
131#638C3×D412yescomplemented
132C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
133C4×C3⋊D3C23×C311yescharacteristic
134C3×#67—24no—
135C3×#67—24no—
136C3×#67—44no—
137C3×#67—64no—
138C3×#67—64no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—44no—
142C2×C3×Dic3—62no—
143C3×D12—62no—
144C3×D12—64no—
145C3×C4×D3—22no—
146C3×C4×D3—22no—
147C3×C4×D3—42no—
148C3×C4×D3—62no—
149C3×C4×D3—64no—
150C3×C4×D3—62no—
151C3×C4×D3—64no—
152C3×C4×D3—64no—
153C3×C4×D3—62no—
154C3×Dic6—22no—
155C3×Dic6—22no—
156C3×Dic6—42no—
157C3×Dic6—64no—
158#627—22no—
159#626—24no—
160#624—24no—
161#624C23×C314yes—
162#624C23×C312yes—
163D3×Dic3—22no—
164#547—91noSylow
165C2×C33—22no—
166C2×C33C22×D411yescharacteristic
167C3×C3⋊D3—22no—
168C3×C3⋊D3C2×#4111yescharacteristic
169C3×C3⋊D32−1+411yescharacteristic
170C32×D3—42no—
171C3×#41—62no—
172C3×#41—62no—
173C3×#41—122no—
174C3×#41—184no—
175C3×#41—182no—
176C2×C3×Q8—92no—
177C2×C3×D4—94no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C22×C3×C4—94no—
181C22×C3×C4—92no—
182C22×C3×C4—92no—
183#239—62no—
184D3×Q8—62no—
185#237—22no—
186#237—64no—
187D3×D4—64no—
188D3×D4—62no—
189#235—62no—
190#235—64no—
191C2×C4×D3—62no—
192C2×C4×D3—64no—
193C3×C4⋊C4—91no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—94no—
196C3×C4⋊C4—92no—
197C3×C4⋊C4—91no—
198C3×C22⋊C4—92no—
199C3×C22⋊C4—94no—
200C3×C22⋊C4—92no—
201C3×C42—91no—
202C22×C32—24no—
203C22×C32—44no—
204C22×C32—62no—
205C22×C32C2×C3×D412yes—
206C2×C3⋊D3—22no—
207C2×C3⋊D3—22no—
208C2×C3⋊D3C2×C3×D412yes—
209C2×C3⋊D3C24×C311yescharacteristic
210C2×C3×D3—22no—
211C2×C3×D3—22no—
212C2×C3×D3—42no—
213C2×C3×D3—62no—
214C2×C3×D3—64no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219D32—22no—
220D32—22no—
221#157—22no—
222#157—21no—
223#157C3×#4112yescomplemented
224C32×C4—22no—
225C32×C4—42no—
226C32×C4—64no—
227C32×C4—62no—
228C32×C4C2×C3×D411yescharacteristic
229C3⋊Dic3C2×C3×D411yescharacteristic
230C3×Dic3—24no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—24no—
234C3×Dic3—44no—
235C3×Dic3—42no—
236C3×Dic3—62no—
237C2×#41—92no—
238C4⋊Q8—91no—
239#123—94no—
240C22⋊Q8—92no—
241#121—92no—
242C4×D4—91no—
243C4×D4—91no—
244C2×C4⋊C4—92no—
245C33#54711yescharacteristic, complemented, Sylow
246C23×C3—92no—
247C23×C3—92no—
248C22×D3—62no—
249C22×D3—64no—
250C3×Q8—62no—
251C3×Q8—62no—
252C3×Q8—122no—
253C3×Q8—182no—
254C3×D4—22no—
255C3×D4—42no—
256C3×D4—64no—
257C3×D4—64no—
258C3×D4—92no—
259C3×D4—91no—
260C3×D4—124no—
261C3×D4—184no—
262C3×D4—182no—
263C3×D4GO+(4,𝔽2)11yescharacteristic, complemented
264C2×C3×C4—62no—
265C2×C3×C4—64no—
266C2×C3×C4—64no—
267C2×C3×C4—62no—
268C2×C3×C4—92no—
269C2×C3×C4—94no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—92no—
274C2×C3×C4—122no—
275C2×C3×C4—124no—
276C2×C3×C4—184no—
277C2×C3×C4—182no—
278C2×C3×C4—184no—
279#67—24no—
280#67—64no—
281#67—64no—
282C2×Dic3—24no—
283C2×Dic3—62no—
284D12—62no—
285D12—64no—
286C4×D3—22no—
287C4×D3—62no—
288C4×D3—64no—
289C4×D3—62no—
290C4×D3—64no—
291C4×D3—64no—
292C4×D3—62no—
293Dic6—22no—
294Dic6—64no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—42no—
298C2×C32—44no—
299C2×C32—84no—
300C2×C32—122no—
301C2×C32C22×C3×D411yescharacteristic
302C3⋊D3—22no—
303C3⋊D3C2×C3×#4111yescharacteristic
304C3⋊D3C3×2−1+411yescharacteristic
305C3×D3—42no—
306C3×D3—42no—
307C3×D3—62no—
308C3×D3—62no—
309C3×D3—82no—
310C3×D3—124no—
311#41—62no—
312#41—184no—
313#41—182no—
314C2×Q8—92no—
315C2×D4—94no—
316C2×D4—91no—
317C2×D4—91no—
318C22×C4—94no—
319C22×C4—92no—
320C22×C4—92no—
321C4⋊C4—91no—
322C4⋊C4—92no—
323C4⋊C4—94no—
324C4⋊C4—91no—
325C4⋊C4—92no—
326C22⋊C4—92no—
327C22⋊C4—92no—
328C22⋊C4—94no—
329C42—91no—
330C22×C3—24no—
331C22×C3—44no—
332C22×C3—62no—
333C22×C3—62no—
334C22×C3—91no—
335C22×C3—92no—
336C22×C3—122no—
337C22×C3—182no—
338C22×C3—182no—
339C22×C3—182no—
340C22×C3—182no—
341C22×C3C2×GO+(4,𝔽2)12yes—
342C2×D3—22no—
343C2×D3—62no—
344C2×D3—64no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—124no—
348C2×D3—124no—
349C3×C4—22no—
350C3×C4—42no—
351C3×C4—64no—
352C3×C4—62no—
353C3×C4—62no—
354C3×C4—64no—
355C3×C4—92no—
356C3×C4—91no—
357C3×C4—122no—
358C3×C4—124no—
359C3×C4—182no—
360C3×C4—181no—
361C3×C4C2×GO+(4,𝔽2)11yescharacteristic
362Dic3—24no—
363Dic3—22no—
364Dic3—62no—
365C32—22no—
366C32—42no—
367C32C3×#54711yescharacteristic, complemented
368C23—92no—
369C23—92no—
370Q8—62no—
371Q8—182no—
372D4—64no—
373D4—92no—
374D4—91no—
375D4—184no—
376D4—182no—
377D4C3×GO+(4,𝔽2)11yescharacteristic, complemented
378C2×C4—64no—
379C2×C4—62no—
380C2×C4—92no—
381C2×C4—94no—
382C2×C4—91no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—92no—
386C2×C4—184no—
387C2×C4—182no—
388C2×C4—184no—
389C2×C3—22no—
390C2×C3—22no—
391C2×C3—44no—
392C2×C3—42no—
393C2×C3—84no—
394C2×C3—91no—
395C2×C3—91no—
396C2×C3—122no—
397C2×C3—122no—
398C2×C3—182no—
399C2×C3—242no—
400C2×C3C22×GO+(4,𝔽2)11yescharacteristic
401D3—42no—
402D3—62no—
403D3—62no—
404D3—124no—
405C22—62no—
406C22—91no—
407C22—92no—
408C22—182no—
409C22—182no—
410C22—182no—
411C22—182no—
412C22C2×C3×GO+(4,𝔽2)12yes—
413C4—62no—
414C4—64no—
415C4—92no—
416C4—91no—
417C4—182no—
418C4—181no—
419C4C2×C3×GO+(4,𝔽2)11yescharacteristic
420C3—22no—
421C3—42no—
422C3#1059651911yescharacteristic, direct product factor
423C2—22no—
424C2—91no—
425C2—91no—
426C2—122no—
427C2—182no—
428C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC33×D4D4
socleC2×C33C22×D4
hypercenterC3×D4GO+(4,𝔽2)
hypocenterC32C3×#547
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64312C2semidirect
2. C3×D4×#157C2semidirect
3. C3×#64242C2semidirect
4. C2×C3×#3773C2semidirect
5. C3×#64233C2semidirect
6. C3×C4×GO+(4,𝔽2)C2semidirect
7. C3×#64222C2semidirect
8. C3×D32⋊Q8C2semidirect
9. #10596519C3direct (N × H)
30. #64312C2×C3semidirect
31. D4×#157C2×C3semidirect
32. #64242C2×C3semidirect
33. C2×#3773C2×C3semidirect
34. #64233C2×C3semidirect
35. C4×GO+(4,𝔽2)C2×C3semidirect
36. #64222C2×C3semidirect
37. D32⋊Q8C2×C3semidirect
45. C33×D4D4semidirect
46. C3×#638D4semidirect
47. C3×C4⋊(C3⋊D3)D4semidirect
101. C3×#157#41semidirect
128. C32×D4C3×D4semidirect
131. #638C3×D4semidirect
132. C4⋊(C3⋊D3)C3×D4semidirect
223. #157C3×#41semidirect
245. C33#547semidirect
263. C3×D4GO+(4,𝔽2)semidirect
367. C32C3×#547semidirect
377. D4C3×GO+(4,𝔽2)semidirect
422. C3#10596519direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643122
C3×D4×#1572
C3×#642422
C2×C3×#37732
C3×#642332
C3×C4×GO+(4,𝔽2)2
C3×#642222
C3×D32⋊Q82
#105965193
C3×#5479