Finite Group Databasebeta

← C2×C3×#64266 #13181384

Subgroups — C2×C3×#64266 #13181384

Order 1728.

911 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×#64266C214yesmaximal, direct product factor
3C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
5C2×#64266C311yesmaximal, characteristic, direct product factor
6C2×C3×#985—33nomaximal
7C2×C3×#950—31nomaximal
8C2×C3×C4×C3⋊D3—41nomaximal
9C32×#41⋊C3C2212yescomplemented
10C2×C32×SL(2,𝔽3)C2211yescharacteristic
11C3×C3⋊GL(2,𝔽3)C2212yescomplemented
12C3×C3⋊#226C2212yescomplemented
13C2×C32×#41D311yescharacteristic, complemented
14C2×C3×#235—31no—
15C2×C3×#41⋊C3—21no—
16C2×C3×#41⋊C3—23no—
17C2×C3×#41⋊C3D313yescomplemented
18C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
19#64266C2×C314yesdirect product factor
20C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
21C2×C3⋊#226C2×C311yescharacteristic, complemented
22C3×#985—312no—
23C2×C3×GL(2,𝔽3)—33no—
24C2×C3×#226—33no—
25C3×#950—34no—
26C3×#950—34no—
27C2×C3×#216—31no—
28C2×C3×#215—31no—
29C2×C3×#214—31no—
30C2×C3×#213—31no—
31C22×C3×C3⋊C8—31no—
32C22×C3×C3⋊D3—41no—
33C2×C33×C4—41no—
34C2×C3×C3⋊Dic3—41no—
35C3×C4×C3⋊D3—44no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C2×#985—33no—
38C2×C3×#135—91no—
39C2×#950—31no—
40C32×#41—32no—
41C32×#41C2×D312yescomplemented
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×D4—31no—
44C22×C32×C4—31no—
45C2×C4×C3⋊D3—41no—
46C2×C3×#67—61no—
47C3×#235—34no—
48C3×#235—62no—
49C2×C3×D12—31no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—123no—
52C2×C3×Dic6—31no—
53C3×#41⋊C3—22no—
54C3×#41⋊C3—26no—
55C3×#41⋊C3C2×D316yescomplemented
56C3×#41⋊C3C22×C312yescomplemented
57C2×C3×SL(2,𝔽3)—21no—
58C2×C3×SL(2,𝔽3)—23no—
59C2×C3×SL(2,𝔽3)C2×D313yes—
60C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
61C3⋊GL(2,𝔽3)C22×C312yescomplemented
62C3⋊#226C22×C312yescomplemented
63C3×GL(2,𝔽3)—36no—
64C3×#226—36no—
65C3×#216—32no—
66C3×#216—32no—
67C3×#215—32no—
68C3×#215—32no—
69C3×#214—34no—
70C3×#213—34no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—34no—
74C22×C33—41no—
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—44no—
77C33×C4—42no—
78C3×C3⋊Dic3—42no—
79C2×C3×#41—21no—
80C2×C3×#41—91no—
81C2×C3×#41C3×D311yescharacteristic, complemented
82C2×C3×#41C3⋊D311yescharacteristic, complemented
83C2×#235—31no—
84C2×#41⋊C3—23no—
85C2×#41⋊C3C3×D313yescomplemented
86#985—312no—
87C2×GL(2,𝔽3)—33no—
88C2×#226—33no—
89C3×#135—94no—
90C3×#135—94no—
91C2×C3×Q16—91no—
92C2×C3×SD16—91no—
93C2×C3×SD16—91no—
94C2×C3×D8—91no—
95C22×C3×C8—91no—
96#950—34no—
97#950—34no—
98C2×#216—31no—
99C2×#215—31no—
100C2×#214—31no—
101C2×#213—31no—
102C22×C3⋊C8—31no—
103C23×C32—31no—
104C22×C3⋊D3—41no—
105C22×C3×D3—61no—
106C22×C3×D3—123no—
107C32×Q8—31no—
108C32×Q8C22×D311yescharacteristic
109C32×D4—32no—
110C32×D4—61no—
111C2×C32×C4—31no—
112C2×C32×C4—32no—
113C2×C32×C4—32no—
114C2×C32×C4—41no—
115C2×C32×C4—43no—
116C2×C32×C4—83no—
117C2×C32×C4—81no—
118C2×C32×C4S411yescharacteristic
119C2×C3⋊Dic3—41no—
120C4×C3⋊D3—44no—
121C3×#67—64no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—123no—
124C3×D12—32no—
125C3×D12—61no—
126C3×C4×D3—64no—
127C3×C4×D3—1212no—
128C3×Dic6—32no—
129C3×Dic6—61no—
130C3×SL(2,𝔽3)—21no—
131C3×SL(2,𝔽3)—23no—
132C3×SL(2,𝔽3)C22×D313yes—
133C3×SL(2,𝔽3)C23×C311yescharacteristic
134C3×C3⋊C8—32no—
135C3×C3⋊C8—32no—
136C2×#135—91noSylow
137C2×C33—42no—
138C2×C33—41no—
139C3×C3⋊D3—44no—
140C3×#41—22no—
141C3×#41—32no—
142C3×#41—32no—
143C3×#41—62no—
144C3×#41—94no—
145C3×#41—182no—
146C3×#41C2×C3×D312yescomplemented
147C3×#41C2×C3⋊D312yescomplemented
148C2×C3×Q8—21no—
149C2×C3×Q8—91no—
150C2×C3×Q8C2×C3×D311yescharacteristic
151C2×C3×Q8C2×C3⋊D311yescharacteristic
152C2×C3×D4—31no—
153C2×C3×D4—31no—
154C2×C3×D4—61no—
155C2×C3×D4—91no—
156C2×C3×D4—181no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—61no—
160C22×C3×C4—181no—
161C2×#67—61no—
162#235—34no—
163#235—62no—
164C2×D12—31no—
165C2×C4×D3—61no—
166C2×C4×D3—123no—
167C2×Dic6—31no—
168#41⋊C3—26no—
169#41⋊C3C2×C3×D316yescomplemented
170C2×SL(2,𝔽3)—23no—
171C2×SL(2,𝔽3)C2×C3×D313yes—
172GL(2,𝔽3)—36no—
173#226—36no—
174C3×Q16—92no—
175C3×Q16—92no—
176C3×SD16—94no—
177C3×SD16—92no—
178C3×SD16—92no—
179C3×D8—94no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8—94no—
183#216—32no—
184#216—32no—
185#215—32no—
186#215—32no—
187#214—34no—
188#213—34no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—34no—
192C22×C32—32no—
193C22×C32—41no—
194C22×C32—43no—
195C22×C32—62no—
196C22×C32—83no—
197C22×C32—81no—
198C22×C32C2×S411yescharacteristic
199C2×C3⋊D3—42no—
200C2×C3⋊D3—44no—
201C2×C3×D3—62no—
202C2×C3×D3—126no—
203C2×C3×D3—1212no—
204C2×C3×D3—122no—
205C32×C4—31no—
206C32×C4—31no—
207C32×C4—42no—
208C32×C4—46no—
209C32×C4—86no—
210C32×C4—82no—
211C32×C4C2×S412yes—
212C3⋊Dic3—42no—
213C3×Dic3—62no—
214C3×Dic3—126no—
215C2×#41—91no—
216C2×#41C3×C3⋊D311yescharacteristic, complemented
217#135—94no—
218#135—94no—
219C2×Q16—91no—
220C2×SD16—91no—
221C2×SD16—91no—
222C2×D8—91no—
223C22×C8—91no—
224C33—41noSylow
225C23×C3—31no—
226C23×C3—31no—
227C23×C3—61no—
228C23×C3—181no—
229C22×D3—61no—
230C22×D3—123no—
231C3×Q8—21no—
232C3×Q8—31no—
233C3×Q8—31no—
234C3×Q8—61no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×Q8C22×C3×D311yescharacteristic
238C3×Q8C22×C3⋊D311yescharacteristic
239C3×D4—32no—
240C3×D4—32no—
241C3×D4—62no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—62no—
245C3×D4—92no—
246C3×D4—181no—
247C3×D4—184no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—32no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—43no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—61no—
259C2×C3×C4—83no—
260C2×C3×C4—181no—
261C2×C3×C4—184no—
262C2×C3×C4C3×S411yescharacteristic
263C2×C3×C4C3⋊S411yescharacteristic
264#67—64no—
265C2×Dic3—61no—
266C2×Dic3—123no—
267D12—32no—
268D12—61no—
269C4×D3—64no—
270C4×D3—1212no—
271Dic6—32no—
272Dic6—61no—
273SL(2,𝔽3)—23no—
274SL(2,𝔽3)C22×C3×D313yes—
275C3×C8—92no—
276C3×C8—92no—
277C3⋊C8—32no—
278C3⋊C8—32no—
279C2×C32—41no—
280C2×C32—42no—
281C2×C32—46no—
282C2×C32—43no—
283C2×C32—62no—
284C2×C32—83no—
285C2×C32—86no—
286C2×C32—81no—
287C2×C32—82no—
288C2×C32#98512yescomplemented
289C2×C32C22×S411yescharacteristic
290C3⋊D3—44no—
291C3×D3—1212no—
292C3×D3—122no—
293#41—32no—
294#41—94no—
295#41—182no—
296#41C2×C3×C3⋊D312yescomplemented
297C2×Q8—91no—
298C2×Q8C2×C3×C3⋊D311yescharacteristic
299C2×D4—31no—
300C2×D4—91no—
301C2×D4—181no—
302C22×C4—31no—
303C22×C4—181no—
304Q16—92no—
305Q16—92no—
306SD16—94no—
307SD16—92no—
308SD16—92no—
309D8—94no—
310C2×C8—91no—
311C2×C8—91no—
312C2×C8—94no—
313C22×C3—21no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—43no—
317C22×C3—62no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—83no—
321C22×C3—122no—
322C22×C3—182no—
323C22×C3—362no—
324C22×C3C2×C3×S411yescharacteristic
325C22×C3C2×C3⋊S411yescharacteristic
326C2×D3—62no—
327C2×D3—126no—
328C2×D3—1212no—
329C2×D3—122no—
330C3×C4—22no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—46no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—86no—
339C3×C4—182no—
340C3×C4C2×C3×S412yes—
341C3×C4C2×C3⋊S412yes—
342Dic3—62no—
343Dic3—126no—
344C32—41no—
345C32—43no—
346C32—83no—
347C32—81no—
348C32C2×#98511yescharacteristic, complemented
349C23—31no—
350C23—181no—
351Q8—31no—
352Q8—92no—
353Q8—181no—
354Q8C22×C3×C3⋊D311yescharacteristic
355D4—32no—
356D4—61no—
357D4—92no—
358D4—181no—
359D4—184no—
360C2×C4—31no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—181no—
364C2×C4—184no—
365C2×C4C3×C3⋊S411yescharacteristic
366C8—92no—
367C8—92no—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—46no—
371C2×C3—43no—
372C2×C3—62no—
373C2×C3—62no—
374C2×C3—86no—
375C2×C3—83no—
376C2×C3—122no—
377C2×C3—362no—
378C2×C3C3×#98512yescomplemented
379C2×C3#6426612yesdirect product factor
380C2×C3C22×C3×S411yescharacteristic
381C2×C3C22×C3⋊S411yescharacteristic
382D3—1212no—
383D3—122no—
384C22—32no—
385C22—62no—
386C22—182no—
387C22—362no—
388C22C2×C3×C3⋊S411yescharacteristic
389C4—31no—
390C4—31no—
391C4—182no—
392C4C2×C3×C3⋊S412yes—
393C3—21no—
394C3—43no—
395C3—83no—
396C3C2×C3×#98511yescharacteristic, complemented
397C3C2×#6426611yescharacteristic, direct product factor
398C2—62no—
399C2—362no—
400C2C3×#6426612yesdirect product factor
401C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3×#64266C2direct (N × H)
3. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
4. C2×C3×C3⋊#226C2semidirect
5. C2×#64266C3direct (N × H)
9. C32×#41⋊C3C22semidirect
11. C3×C3⋊GL(2,𝔽3)C22semidirect
12. C3×C3⋊#226C22semidirect
13. C2×C32×#41D3semidirect
17. C2×C3×#41⋊C3D3semidirect
18. C2×C3×#41⋊C3C2×C3semidirect
19. #64266C2×C3direct (N × H)
20. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
21. C2×C3⋊#226C2×C3semidirect
41. C32×#41C2×D3semidirect
55. C3×#41⋊C3C2×D3semidirect
56. C3×#41⋊C3C22×C3semidirect
61. C3⋊GL(2,𝔽3)C22×C3semidirect
62. C3⋊#226C22×C3semidirect
81. C2×C3×#41C3×D3semidirect
82. C2×C3×#41C3⋊D3semidirect
85. C2×#41⋊C3C3×D3semidirect
146. C3×#41C2×C3×D3semidirect
147. C3×#41C2×C3⋊D3semidirect
169. #41⋊C3C2×C3×D3semidirect
216. C2×#41C3×C3⋊D3semidirect
288. C2×C32#985semidirect
296. #41C2×C3×C3⋊D3semidirect
348. C32C2×#985semidirect
378. C2×C3C3×#985semidirect
379. C2×C3#64266direct (N × H)
396. C3C2×C3×#985semidirect
397. C3C2×#64266direct (N × H)
400. C2C3×#64266direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×#642662
C2×C3×C3⋊GL(2,𝔽3)2
C2×C3×C3⋊#2262
C2×#642663
C2×C3×#9853
C2×C3×#9503
C2×C3×C4×C3⋊D34