Finite Group Databasebeta

← C2×D3×C3⋊GL(2,𝔽3) #13180812

Subgroups — C2×D3×C3⋊GL(2,𝔽3) #13180812

Order 1728.

1361 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1361 conjugacy classes of subgroups — too many to draw. Only its 98 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×C3×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4D3×C3⋊GL(2,𝔽3)C214yesmaximal, direct product factor
5C22×C3⋊GL(2,𝔽3)—31nomaximal
6C2×D3×GL(2,𝔽3)—33nomaximal
7C2×D3×#215—31nomaximal
8C22×D3×C3⋊D3—41nomaximal
9C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
10C3×D3×SL(2,𝔽3)C2212yescomplemented
11C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
12C3×C3⋊GL(2,𝔽3)C2212yescomplemented
13C2×C3×D3×Q8D311yescharacteristic, complemented
14C22×C3×SL(2,𝔽3)—31no—
15C2×D3×D12—31no—
16C2×D3×SL(2,𝔽3)D313yescomplemented
17C2×C3⋊GL(2,𝔽3)—31no—
18C2×C3⋊GL(2,𝔽3)—33no—
19C2×C3⋊GL(2,𝔽3)—34no—
20C2×C3⋊GL(2,𝔽3)—61no—
21C2×C3⋊GL(2,𝔽3)—63no—
22C2×C3⋊GL(2,𝔽3)D311yescharacteristic, direct product factor
23C2×C3×GL(2,𝔽3)—33no—
24D3×GL(2,𝔽3)—312no—
25C2×#3755—31no—
26C2×C3×#215—31no—
27D3×#215—34no—
28D3×#215—34no—
29C2×#3717—31no—
30C2×#3715—31no—
31C2×D3×C3⋊C8—31no—
32C22×C3⋊(C3⋊D3)—41no—
33C22×C3×C3⋊D3—41no—
34C22×C32×D3—41no—
35C2×D3×C3⋊D3—44no—
36C2×D3×C3⋊D3—48no—
37C32×SL(2,𝔽3)C2311yescharacteristic, complemented
38C22×GL(2,𝔽3)—93no—
39C22×#215—91no—
40C2×D3×SD16—91no—
41C23×C3⋊D3—121no—
42C22×D32—61no—
43C22×D32—123no—
44C2×C32×Q8C2×D311yescharacteristic, complemented
45C2×C4⋊(C3⋊D3)—31no—
46C3×D3×Q8—32no—
47C3×D3×Q8C2×D312yescomplemented
48C2×C3×D12—31no—
49C2×C3×C4×D3—31no—
50C2×C3×Dic6—31no—
51C2×C3×SL(2,𝔽3)—21no—
52C2×C3×SL(2,𝔽3)—23no—
53C2×C3×SL(2,𝔽3)—32no—
54C2×C3×SL(2,𝔽3)C2×D313yescomplemented
55C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
56C2×#626—61no—
57D3×D12—34no—
58D3×D12—62no—
59D3×SL(2,𝔽3)C2×D316yescomplemented
60C3⋊GL(2,𝔽3)—32no—
61C3⋊GL(2,𝔽3)—36no—
62C3⋊GL(2,𝔽3)—62no—
63C3⋊GL(2,𝔽3)—66no—
64C3⋊GL(2,𝔽3)C2×D312yesdirect product factor
65C3×GL(2,𝔽3)—36no—
66#3755—32no—
67#3755—32no—
68C2×C3⋊(C3⋊C8)—31no—
69C3×#215—32no—
70C3×#215—32no—
71C2×C3×C3⋊C8—31no—
72#3717—34no—
73#3715—34no—
74D3×C3⋊C8—34no—
75C22×C33—41no—
76C2×C3⋊(C3⋊D3)—42no—
77C2×C3⋊(C3⋊D3)—44no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—44no—
80C2×C32×D3—44no—
81C2×C32×D3—42no—
82D3×C3⋊D3—416no—
83C22×C3×Q8—31no—
84C2×D3×Q8C3⋊D311yescharacteristic, complemented
85C2×D3×D4—91no—
86C22×D12—91no—
87C22×SL(2,𝔽3)—33no—
88C2×GL(2,𝔽3)—33no—
89C2×GL(2,𝔽3)—912no—
90C2×GL(2,𝔽3)—93no—
91C2×GL(2,𝔽3)—183no—
92C2×C3×SD16—91no—
93C2×#215—31no—
94C2×#215—91no—
95C2×#215—94no—
96C2×#215—94no—
97C2×#215—91no—
98C2×#215—91no—
99C2×#215—91no—
100C2×#215—181no—
101C2×#214—91no—
102C22×C3⋊C8—91no—
103D3×SD16—94no—
104D3×SD16—94no—
105C2×#204—91no—
106C2×D3×C8—91no—
107C23×C32—121no—
108C22×C3⋊D3—41no—
109C22×C3⋊D3—61no—
110C22×C3⋊D3—124no—
111C22×C3⋊D3—121no—
112C22×C3⋊D3—123no—
113C22×C3⋊D3—128no—
114C22×C3⋊D3—241no—
115C22×C3⋊D3—243no—
116C22×C3×D3—43no—
117C22×C3×D3—61no—
118C22×C3×D3—123no—
119C22×C3×D3S411yescharacteristic
120C2×D32—64no—
121C2×D32—1212no—
122C2×D32—1224no—
123C2×D32—124no—
124C32×Q8—31no—
125C32×Q8C22×D311yescharacteristic, complemented
126C2×C32×C4—31no—
127C4⋊(C3⋊D3)—32no—
128C4⋊(C3⋊D3)—61no—
129C2×C3×Dic3—31no—
130C3×D12—32no—
131C3×D12—61no—
132C3×C4×D3—32no—
133C3×C4×D3—32no—
134C3×Dic6—32no—
135C3×Dic6—61no—
136C3×SL(2,𝔽3)—21no—
137C3×SL(2,𝔽3)—23no—
138C3×SL(2,𝔽3)C22×D313yescomplemented
139C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
140#626—64no—
141C3⋊(C3⋊C8)—32no—
142C3×C3⋊C8—32no—
143C22×SD16—271noSylow
144C2×C33—42no—
145C2×C33—41no—
146C3⋊(C3⋊D3)—44no—
147C3×C3⋊D3—44no—
148C32×D3—44no—
149C23×D3—181no—
150C23×D3—181no—
151C23×D3—363no—
152C2×C3×Q8—21no—
153C2×C3×Q8—32no—
154C2×C3×Q8—91no—
155C2×C3×Q8—92no—
156C2×C3×Q8D3211yescharacteristic, complemented
157C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
158C2×C3×D4—91no—
159C22×C3×C4—91no—
160C2×#67—181no—
161D3×Q8—32no—
162D3×Q8C2×C3⋊D312yescomplemented
163D3×D4—94no—
164D3×D4—182no—
165C2×D12—31no—
166C2×D12—94no—
167C2×D12—91no—
168C2×D12—91no—
169C2×D12—181no—
170C2×D12—182no—
171C2×D12—181no—
172C2×C4×D3—31no—
173C2×Dic6—31no—
174C2×SL(2,𝔽3)—23no—
175C2×SL(2,𝔽3)—36no—
176C2×SL(2,𝔽3)D3213yescomplemented
177GL(2,𝔽3)—36no—
178GL(2,𝔽3)—96no—
179GL(2,𝔽3)—186no—
180C3×SD16—92no—
181C3×SD16—92no—
182C2×C3×C8—91no—
183#215—32no—
184#215—32no—
185#215—94no—
186#215—94no—
187#215—92no—
188#215—92no—
189#215—92no—
190#215—92no—
191#215—182no—
192#215—182no—
193#214—94no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—94no—
196C2×C3⋊C8—91no—
197C2×C3⋊C8—91no—
198C2×C3⋊C8—181no—
199#204—94no—
200D3×C8—94no—
201C22×C32—41no—
202C22×C32—43no—
203C22×C32—83no—
204C22×C32—81no—
205C22×C32—124no—
206C22×C32—122no—
207C22×C32C2×S411yescharacteristic
208C2×C3⋊D3—42no—
209C2×C3⋊D3—44no—
210C2×C3⋊D3—62no—
211C2×C3⋊D3—122no—
212C2×C3⋊D3—126no—
213C2×C3⋊D3—1216no—
214C2×C3⋊D3—124no—
215C2×C3⋊D3—1212no—
216C2×C3⋊D3—122no—
217C2×C3⋊D3—242no—
218C2×C3⋊D3—246no—
219C2×C3⋊D3—2412no—
220C2×C3⋊D3—244no—
221C2×C3×D3—46no—
222C2×C3×D3—412no—
223C2×C3×D3—62no—
224C2×C3×D3—126no—
225C2×C3×D3—1212no—
226C2×C3×D3—122no—
227C2×C3×D3GL(2,𝔽3)14yescomplemented
228C2×C3×D3C2×S412yes—
229D32—1248no—
230D32—128no—
231C32×C4—31no—
232C32×C4—31no—
233C3×Dic3—32no—
234C22×Q8—31no—
235C22×D4—271no—
236C2×SD16—91no—
237C2×SD16—271no—
238C2×SD16—274no—
239C2×SD16—274no—
240C2×SD16—271no—
241C2×SD16—271no—
242C22×C8—271no—
243C33—41noSylow
244C23×C3—31no—
245C23×C3—123no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—123no—
249C22×D3—184no—
250C22×D3—181no—
251C22×D3—181no—
252C22×D3—184no—
253C22×D3—3612no—
254C22×D3—363no—
255C22×D3—361no—
256C22×D3—364no—
257C22×D3—3624no—
258C22×D3—364no—
259C22×D3—723no—
260C22×D3C3⋊S411yescharacteristic
261C3×Q8—21no—
262C3×Q8—31no—
263C3×Q8—31no—
264C3×Q8—61no—
265C3×Q8—92no—
266C3×Q8—181no—
267C3×Q8C2×D3211yescharacteristic, complemented
268C3×Q8C22×C3⋊D311yescharacteristic, complemented
269C3×D4—92no—
270C3×D4—181no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—61no—
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—91no—
277#67—184no—
278C2×Dic3—31no—
279D12—32no—
280D12—61no—
281D12—92no—
282D12—92no—
283D12—182no—
284D12—181no—
285D12—181no—
286D12—184no—
287D12—361no—
288C4×D3—32no—
289C4×D3—32no—
290Dic6—32no—
291Dic6—61no—
292SL(2,𝔽3)—23no—
293SL(2,𝔽3)C2×D3213yescomplemented
294C3×C8—92no—
295C3⋊C8—32no—
296C3⋊C8—92no—
297C3⋊C8—92no—
298C3⋊C8—182no—
299C2×C32—41no—
300C2×C32—42no—
301C2×C32—46no—
302C2×C32—43no—
303C2×C32—83no—
304C2×C32—86no—
305C2×C32—81no—
306C2×C32—82no—
307C2×C32—124no—
308C2×C32C2×GL(2,𝔽3)12yescomplemented
309C2×C32C22×S411yescharacteristic
310C3⋊D3—44no—
311C3⋊D3—124no—
312C3⋊D3—1212no—
313C3⋊D3—122no—
314C3⋊D3—2412no—
315C3⋊D3—244no—
316C3×D3—412no—
317C3×D3—1212no—
318C3×D3—122no—
319C3×D3C2×GL(2,𝔽3)14yescomplemented
320C24—541no—
321C2×Q8—32no—
322C2×Q8—91no—
323C2×Q8—92no—
324C2×Q8D3×C3⋊D311yescharacteristic, complemented
325C2×D4—91no—
326C2×D4—271no—
327C2×D4—274no—
328C2×D4—542no—
329C2×D4—541no—
330C22×C4—91no—
331SD16—92no—
332SD16—92no—
333SD16—274no—
334SD16—274no—
335SD16—272no—
336SD16—272no—
337C2×C8—91no—
338C2×C8—274no—
339C2×C8—271no—
340C22×C3—21no—
341C22×C3—32no—
342C22×C3—34no—
343C22×C3—43no—
344C22×C3—83no—
345C22×C3—1212no—
346C22×C3—126no—
347C22×C3—182no—
348C22×C3—362no—
349C22×C3D3×S411yescharacteristic
350C22×C3C2×C3⋊S411yescharacteristic
351C2×D3—62no—
352C2×D3—126no—
353C2×D3—1212no—
354C2×D3—122no—
355C2×D3—182no—
356C2×D3—182no—
357C2×D3—366no—
358C2×D3—362no—
359C2×D3—368no—
360C2×D3—3648no—
361C2×D3—3612no—
362C2×D3—362no—
363C2×D3—362no—
364C2×D3—368no—
365C2×D3—726no—
366C2×D3—722no—
367C2×D3—7212no—
368C2×D3C3⋊GL(2,𝔽3)14yesdirect product factor
369C2×D3C2×C3⋊S412yes—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—92no—
377Dic3—32no—
378C32—41no—
379C32—43no—
380C32—83no—
381C32—81no—
382C32C22×GL(2,𝔽3)11yescharacteristic, complemented
383C23—31no—
384C23—181no—
385C23—544no—
386C23—541no—
387C23—1084no—
388Q8—31no—
389Q8—92no—
390Q8—181no—
391Q8C2×D3×C3⋊D311yescharacteristic, complemented
392D4—92no—
393D4—181no—
394D4—272no—
395D4—541no—
396D4—544no—
397C2×C4—31no—
398C2×C4—92no—
399C2×C4—92no—
400C2×C4—91no—
401C8—92no—
402C8—272no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—34no—
406C2×C3—46no—
407C2×C3—43no—
408C2×C3—86no—
409C2×C3—83no—
410C2×C3—1212no—
411C2×C3—362no—
412C2×C3D3×GL(2,𝔽3)12yescomplemented
413C2×C3C2×C3⋊GL(2,𝔽3)12yescomplemented
414C2×C3C2×D3×S411yescharacteristic
415C2×C3C22×C3⋊S411yescharacteristic
416D3—1212no—
417D3—122no—
418D3—3612no—
419D3—362no—
420D3—362no—
421D3—722no—
422D3—7212no—
423D3C2×C3⋊GL(2,𝔽3)14yesdirect product factor
424C22—34no—
425C22—32no—
426C22—182no—
427C22—362no—
428C22—542no—
429C22—1088no—
430C22—1082no—
431C22D3×C3⋊S411yescharacteristic
432C4—31no—
433C4—31no—
434C4—92no—
435C3—21no—
436C3—43no—
437C3—83no—
438C3C2×D3×GL(2,𝔽3)11yescharacteristic, complemented
439C3C22×C3⋊GL(2,𝔽3)11yescharacteristic, complemented
440C2—34no—
441C2—362no—
442C2—1082no—
443C2D3×C3⋊GL(2,𝔽3)12yesdirect product factor
444C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×D3×C3⋊S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
4. D3×C3⋊GL(2,𝔽3)C2direct (N × H)
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C3×D3×SL(2,𝔽3)C22semidirect
11. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
12. C3×C3⋊GL(2,𝔽3)C22semidirect
13. C2×C3×D3×Q8D3semidirect
16. C2×D3×SL(2,𝔽3)D3semidirect
22. C2×C3⋊GL(2,𝔽3)D3direct (N × H)
37. C32×SL(2,𝔽3)C23semidirect
44. C2×C32×Q8C2×D3semidirect
47. C3×D3×Q8C2×D3semidirect
54. C2×C3×SL(2,𝔽3)C2×D3semidirect
55. C2×C3×SL(2,𝔽3)C2×D3semidirect
59. D3×SL(2,𝔽3)C2×D3semidirect
64. C3⋊GL(2,𝔽3)C2×D3direct (N × H)
84. C2×D3×Q8C3⋊D3semidirect
125. C32×Q8C22×D3semidirect
138. C3×SL(2,𝔽3)C22×D3semidirect
139. C3×SL(2,𝔽3)C22×D3semidirect
156. C2×C3×Q8D32semidirect
157. C2×C3×Q8C2×C3⋊D3semidirect
162. D3×Q8C2×C3⋊D3semidirect
176. C2×SL(2,𝔽3)D32semidirect
227. C2×C3×D3GL(2,𝔽3)semidirect
267. C3×Q8C2×D32semidirect
268. C3×Q8C22×C3⋊D3semidirect
293. SL(2,𝔽3)C2×D32semidirect
308. C2×C32C2×GL(2,𝔽3)semidirect
319. C3×D3C2×GL(2,𝔽3)semidirect
324. C2×Q8D3×C3⋊D3semidirect
368. C2×D3C3⋊GL(2,𝔽3)direct (N × H)
382. C32C22×GL(2,𝔽3)semidirect
391. Q8C2×D3×C3⋊D3semidirect
412. C2×C3D3×GL(2,𝔽3)semidirect
413. C2×C3C2×C3⋊GL(2,𝔽3)semidirect
423. D3C2×C3⋊GL(2,𝔽3)direct (N × H)
438. C3C2×D3×GL(2,𝔽3)semidirect
439. C3C22×C3⋊GL(2,𝔽3)semidirect
443. C2D3×C3⋊GL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C3×C3⋊GL(2,𝔽3)2
D3×C3⋊GL(2,𝔽3)2
C22×C3⋊GL(2,𝔽3)3
C2×D3×GL(2,𝔽3)3
C2×D3×#2153
C22×D3×C3⋊D34