Finite Group Databasebeta

← C2×C3×D3×#226 #13180697

Subgroups — C2×C3×D3×#226 #13180697

Order 1728.

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

normal subgroups only

This group has 733 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
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4C3×D3×#226C214yesmaximal, direct product factor
5C22×C3×#226—31nomaximal
6C2×D3×#226C311yesmaximal, characteristic, direct product factor
7C2×C3×D3×Q16—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×D3×Dic3—41nomaximal
10C3×D3×SL(2,𝔽3)C2212yes—
11C3×C3⋊#226C2212yescomplemented
12C32×#226C2212yescomplemented
13C2×C3×D3×Q8—31no—
14C2×C3×D3×Q8D311yescharacteristic
15C22×C3×SL(2,𝔽3)—31no—
16C2×D3×SL(2,𝔽3)—21no—
17C2×D3×SL(2,𝔽3)C2×C311yescharacteristic
18C2×C3⋊#226C2×C311yescharacteristic, complemented
19C2×C3×#226—21no—
20C2×C3×#226—31no—
21C2×C3×#226—34no—
22C2×C3×#226—61no—
23C2×C3×#226D311yescharacteristic, direct product factor
24C2×C3×#226C2×C311yescharacteristic, complemented
25D3×#226C2×C314yesdirect product factor
26C2×C32×Q16—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C3×D3×Q16—34no—
30C3×D3×Q16—34no—
31C2×C3×Dic12—31no—
32C2×C3×D3×C8—31no—
33C22×C32×D3—41no—
34C2×C3×C3⋊Dic3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×D3×Dic3—44no—
38C22×#226—31no—
39C22×C3×Q16—91no—
40C2×D3×Q16—31no—
41C2×C32×Q8—31no—
42C2×C32×Q8C2×D311yescharacteristic
43C22×C3×Dic3—121no—
44C3×D3×Q8—32no—
45C3×D3×Q8—34no—
46C3×D3×Q8—62no—
47C3×D3×Q8C2×D312yes—
48C2×C3×C4×D3—31no—
49C2×C3×C4×D3—61no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—61no—
53C2×C3×SL(2,𝔽3)—21no—
54C2×C3×SL(2,𝔽3)—21no—
55C2×C3×SL(2,𝔽3)—21no—
56C2×C3×SL(2,𝔽3)—32no—
57C2×C3×SL(2,𝔽3)—41no—
58C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
60C2×D3×Dic3—41no—
61D3×SL(2,𝔽3)—22no—
62D3×SL(2,𝔽3)C22×C312yes—
63C3⋊#226C22×C312yescomplemented
64C3×#226—22no—
65C3×#226—32no—
66C3×#226—62no—
67C3×#226C2×D312yesdirect product factor
68C3×#226C22×C312yescomplemented
69C32×Q16—32no—
70C32×Q16—32no—
71C2×C32×C8—31no—
72C3×#216—34no—
73C3×#216—32no—
74C3×#216—32no—
75C2×C3×C3⋊C8—31no—
76C3×Dic12—34no—
77C3×D3×C8—34no—
78C22×C33—41no—
79C2×C32×D3—44no—
80C2×C32×D3—42no—
81C3×C3⋊Dic3—42no—
82C32×Dic3—42no—
83C22×C3×Q8—31no—
84C22×C3×Q8—91no—
85C2×D3×Q8—31no—
86C2×D3×Q8C3×D311yescharacteristic
87C22×SL(2,𝔽3)—31no—
88C22×SL(2,𝔽3)—61no—
89C2×#226—31no—
90C2×#226—34no—
91C2×#226—61no—
92C2×#226C3×D311yescharacteristic, direct product factor
93C2×C3×Q16—31no—
94C2×C3×Q16—31no—
95C2×C3×Q16—61no—
96C2×C3×Q16—91no—
97C2×C3×Q16—94no—
98C2×C3×Q16—94no—
99C2×C3×Q16—91no—
100C2×C3×Q16—91no—
101C22×C3×C8—91no—
102C2×#216—31no—
103C2×#216—31no—
104D3×Q16—34no—
105D3×Q16—34no—
106C2×Dic12—31no—
107C2×D3×C8—31no—
108C23×C32—121no—
109C22×C3×D3—41no—
110C22×C3×D3—81no—
111C22×C3×D3S411yescharacteristic
112C32×Q8—31no—
113C32×Q8—32no—
114C32×Q8—61no—
115C32×Q8C22×D311yescharacteristic
116C2×C32×C4—31no—
117C2×C32×C4—61no—
118C2×C3⋊Dic3—41no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—81no—
124C2×C3×Dic3—124no—
125C2×C3×Dic3—121no—
126C2×C3×Dic3—241no—
127C3×C4×D3—32no—
128C3×C4×D3—32no—
129C3×C4×D3—64no—
130C3×Dic6—32no—
131C3×Dic6—32no—
132C3×Dic6—61no—
133C3×Dic6—64no—
134C3×Dic6—61no—
135C3×SL(2,𝔽3)—21no—
136C3×SL(2,𝔽3)—21no—
137C3×SL(2,𝔽3)—21no—
138C3×SL(2,𝔽3)—41no—
139C3×SL(2,𝔽3)C22×D311yescharacteristic
140C3×SL(2,𝔽3)C23×C311yescharacteristic
141D3×Dic3—44no—
142C32×C8—32no—
143C3×C3⋊C8—32no—
144C22×Q16—91noSylow
145C2×C33—42no—
146C2×C33—41no—
147C32×D3—44no—
148C2×C3×Q8—21no—
149C2×C3×Q8—31no—
150C2×C3×Q8—31no—
151C2×C3×Q8—32no—
152C2×C3×Q8—61no—
153C2×C3×Q8—91no—
154C2×C3×Q8—91no—
155C2×C3×Q8—92no—
156C2×C3×Q8—94no—
157C2×C3×Q8—181no—
158C2×C3×Q8—182no—
159C2×C3×Q8D3211yescharacteristic
160C2×C3×Q8C2×C3×D311yescharacteristic
161C22×C3×C4—91no—
162C22×C3×C4—181no—
163C22×Dic3—121no—
164D3×Q8—32no—
165D3×Q8—34no—
166D3×Q8—62no—
167D3×Q8C2×C3×D312yes—
168C2×C4×D3—31no—
169C2×C4×D3—61no—
170C2×Dic6—31no—
171C2×Dic6—31no—
172C2×Dic6—61no—
173C2×SL(2,𝔽3)—21no—
174C2×SL(2,𝔽3)—21no—
175C2×SL(2,𝔽3)—32no—
176C2×SL(2,𝔽3)—41no—
177C2×SL(2,𝔽3)—62no—
178C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
179#226—32no—
180#226—62no—
181#226C2×C3×D312yesdirect product factor
182C3×Q16—32no—
183C3×Q16—32no—
184C3×Q16—32no—
185C3×Q16—32no—
186C3×Q16—62no—
187C3×Q16—62no—
188C3×Q16—94no—
189C3×Q16—92no—
190C3×Q16—94no—
191C3×Q16—92no—
192C2×C3×C8—31no—
193C2×C3×C8—31no—
194C2×C3×C8—61no—
195C2×C3×C8—94no—
196C2×C3×C8—91no—
197#216—34no—
198#216—32no—
199#216—32no—
200C2×C3⋊C8—31no—
201Dic12—34no—
202D3×C8—34no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32—122no—
209C22×C32—124no—
210C22×C32—161no—
211C22×C32C2×S411yescharacteristic
212C2×C3×D3—42no—
213C2×C3×D3—44no—
214C2×C3×D3—82no—
215C2×C3×D3—84no—
216C2×C3×D3#22614yesdirect product factor
217C2×C3×D3C2×S412yes—
218C32×C4—31no—
219C32×C4—31no—
220C32×C4—62no—
221C3⋊Dic3—42no—
222C3×Dic3—32no—
223C3×Dic3—42no—
224C3×Dic3—42no—
225C3×Dic3—62no—
226C3×Dic3—82no—
227C3×Dic3—122no—
228C3×Dic3—242no—
229C22×Q8—31no—
230C22×Q8—91no—
231C2×Q16—31no—
232C2×Q16—91no—
233C2×Q16—94no—
234C2×Q16—94no—
235C2×Q16—91no—
236C2×Q16—91no—
237C22×C8—91no—
238C33—41noSylow
239C23×C3—31no—
240C23×C3—121no—
241C23×C3—241no—
242C22×D3C3×S411yescharacteristic
243C3×Q8—21no—
244C3×Q8—31no—
245C3×Q8—31no—
246C3×Q8—32no—
247C3×Q8—32no—
248C3×Q8—61no—
249C3×Q8—62no—
250C3×Q8—61no—
251C3×Q8—61no—
252C3×Q8—92no—
253C3×Q8—92no—
254C3×Q8—121no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×Q8—184no—
258C3×Q8C2×D3211yescharacteristic
259C3×Q8C22×C3×D311yescharacteristic
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—91no—
268C2×C3×C4—121no—
269C2×C3×C4—184no—
270C2×C3×C4—181no—
271C2×Dic3—31no—
272C2×Dic3—41no—
273C2×Dic3—61no—
274C2×Dic3—124no—
275C2×Dic3—121no—
276C2×Dic3—241no—
277C4×D3—32no—
278C4×D3—32no—
279C4×D3—64no—
280Dic6—32no—
281Dic6—32no—
282Dic6—61no—
283Dic6—64no—
284Dic6—61no—
285SL(2,𝔽3)—21no—
286SL(2,𝔽3)—21no—
287SL(2,𝔽3)—41no—
288SL(2,𝔽3)C22×C3×D311yescharacteristic
289C3×C8—32no—
290C3×C8—32no—
291C3×C8—62no—
292C3×C8—92no—
293C3⋊C8—32no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32—42no—
297C2×C32—42no—
298C2×C32—81no—
299C2×C32—82no—
300C2×C32—81no—
301C2×C32—81no—
302C2×C32—82no—
303C2×C32—82no—
304C2×C32—124no—
305C2×C32—161no—
306C2×C32—162no—
307C2×C32C2×#22612yescomplemented
308C2×C32C22×S411yescharacteristic
309C3×D3—44no—
310C3×D3—84no—
311C3×D3C2×#22614yesdirect product factor
312C2×Q8—31no—
313C2×Q8—32no—
314C2×Q8—91no—
315C2×Q8—91no—
316C2×Q8—92no—
317C2×Q8—94no—
318C2×Q8—181no—
319C2×Q8—182no—
320C2×Q8C3×D3211yescharacteristic
321C22×C4—91no—
322C22×C4—181no—
323Q16—32no—
324Q16—32no—
325Q16—94no—
326Q16—92no—
327Q16—94no—
328Q16—92no—
329C2×C8—31no—
330C2×C8—94no—
331C2×C8—91no—
332C22×C3—21no—
333C22×C3—32no—
334C22×C3—34no—
335C22×C3—41no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—122no—
339C22×C3—124no—
340C22×C3—161no—
341C22×C3—244no—
342C22×C3—242no—
343C22×C3D3×S411yescharacteristic
344C22×C3C2×C3×S411yescharacteristic
345C2×D3C3×#22614yesdirect product factor
346C2×D3C2×C3×S412yes—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—62no—
354C3×C4—62no—
355C3×C4—92no—
356C3×C4—122no—
357C3×C4—182no—
358Dic3—32no—
359Dic3—42no—
360Dic3—62no—
361Dic3—122no—
362Dic3—242no—
363C32—41no—
364C32—41no—
365C32—81no—
366C32—81no—
367C32—81no—
368C32—161no—
369C32C22×#22611yescharacteristic, complemented
370C23—31no—
371Q8—31no—
372Q8—32no—
373Q8—61no—
374Q8—92no—
375Q8—92no—
376Q8—181no—
377Q8—181no—
378Q8—184no—
379Q8C2×C3×D3211yescharacteristic
380C2×C4—31no—
381C2×C4—61no—
382C2×C4—92no—
383C2×C4—92no—
384C2×C4—91no—
385C2×C4—184no—
386C2×C4—181no—
387C8—32no—
388C8—92no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—34no—
392C2×C3—42no—
393C2×C3—41no—
394C2×C3—82no—
395C2×C3—82no—
396C2×C3—81no—
397C2×C3—81no—
398C2×C3—124no—
399C2×C3—162no—
400C2×C3—161no—
401C2×C3—244no—
402C2×C3D3×#22612yesdirect product factor
403C2×C3C2×C3×#22612yescomplemented
404C2×C3C2×D3×S411yescharacteristic
405C2×C3C22×C3×S411yescharacteristic
406D3C2×C3×#22614yesdirect product factor
407C22—34no—
408C22—32no—
409C22C3×D3×S411yescharacteristic
410C4—31no—
411C4—31no—
412C4—62no—
413C4—92no—
414C4—182no—
415C3—21no—
416C3—41no—
417C3—81no—
418C3—81no—
419C3—161no—
420C3C2×D3×#22611yescharacteristic, direct product factor
421C3C22×C3×#22611yescharacteristic, complemented
422C2—34no—
423C2C3×D3×#22612yesdirect product factor
424C2C2×C3×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C2×C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×D3×SL(2,𝔽3)C2
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2semidirect
3. C2×C32×#226C2semidirect
4. C3×D3×#226C2direct (N × H)
6. C2×D3×#226C3direct (N × H)
11. C3×C3⋊#226C22semidirect
12. C32×#226C22semidirect
18. C2×C3⋊#226C2×C3semidirect
23. C2×C3×#226D3direct (N × H)
24. C2×C3×#226C2×C3semidirect
25. D3×#226C2×C3direct (N × H)
63. C3⋊#226C22×C3semidirect
67. C3×#226C2×D3direct (N × H)
68. C3×#226C22×C3semidirect
92. C2×#226C3×D3direct (N × H)
181. #226C2×C3×D3direct (N × H)
216. C2×C3×D3#226direct (N × H)
307. C2×C32C2×#226semidirect
311. C3×D3C2×#226direct (N × H)
345. C2×D3C3×#226direct (N × H)
369. C32C22×#226semidirect
402. C2×C3D3×#226direct (N × H)
403. C2×C3C2×C3×#226semidirect
406. D3C2×C3×#226direct (N × H)
420. C3C2×D3×#226direct (N × H)
421. C3C22×C3×#226semidirect
423. C2C3×D3×#226direct (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⋊#2262
C2×C32×#2262
C3×D3×#2262
C22×C3×#2263
C2×D3×#2263
C2×C3×D3×Q163
C2×C3×D3×Dic34