Finite Group Databasebeta

← C2×C3×#64196 #13180692

Subgroups — C2×C3×#64196 #13180692

Order 1728.

669 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 82 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×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4C3×#64196C214yesmaximal, direct product factor
5C2×C3×#984—31nomaximal
6C2×#64196C311yesmaximal, characteristic, direct product factor
7C2×C3×#918—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×#627—41nomaximal
10C3×C3⋊(#41⋊C3)C2212yes—
11C3×C3⋊#226C2212yescomplemented
12C32×#226C2212yescomplemented
13C2×C3×#239D311yescharacteristic
14C2×C3×D3×Q8—31no—
15C2×C3×#41⋊C3—31no—
16C2×C3⋊(#41⋊C3)—21no—
17C2×C3⋊(#41⋊C3)C2×C311yescharacteristic
18C2×C3⋊#226C2×C311yescharacteristic, complemented
19C3×#984—34no—
20C2×C3×#226—21no—
21C2×C3×#226—31no—
22C2×C3×#226—61no—
23C2×C3×#226D311yescharacteristic, complemented
24C2×C3×#226C2×C311yescharacteristic, complemented
25#64196C2×C314yesdirect product factor
26C2×C32×Q16—31no—
27C2×C3×#216—31no—
28C2×C3×#215—31no—
29C3×#918—34no—
30C3×#918—34no—
31C2×C3×#204—31no—
32C2×C3×ΓL(1,𝔽25)—31no—
33C2×C3×C3⋊Dic3—41no—
34C2×C32×Dic3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×#627—44no—
38C2×#984—31no—
39C2×C3×#137—91no—
40C2×#918—31no—
41C2×C32×Q8—31no—
42C2×C32×Q8C2×D311yescharacteristic
43C3×#239—32no—
44C3×#239C2×D312yes—
45C3×D3×Q8—34no—
46C3×D3×Q8—62no—
47C2×C3×D12—31no—
48C2×C3×C4×D3—31no—
49C2×C3×C4×D3—61no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—61no—
52C2×C3×Dic6—121no—
53C3×#41⋊C3—32no—
54C2×C3×SL(2,𝔽3)—21no—
55C2×C3×SL(2,𝔽3)—21no—
56C2×C3×SL(2,𝔽3)—21no—
57C2×C3×SL(2,𝔽3)—41no—
58C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
60C2×#627—41no—
61C3⋊(#41⋊C3)—22no—
62C3⋊(#41⋊C3)C22×C312yes—
63C3⋊#226C22×C312yescomplemented
64C3×#226—22no—
65C3×#226—32no—
66C3×#226—62no—
67C3×#226C2×D312yescomplemented
68C3×#226C22×C312yescomplemented
69C32×Q16—32no—
70C32×Q16—32no—
71C2×C32×C8—31no—
72C3×#216—32no—
73C3×#216—32no—
74C3×#215—34no—
75C2×C3×C3⋊C8—31no—
76C3×#204—34no—
77C3×ΓL(1,𝔽25)—34no—
78C22×C33—41no—
79C3×C3⋊Dic3—42no—
80C32×Dic3—42no—
81C32×Dic3—42no—
82C2×C3×#41—31no—
83C22×C3×Q8—91no—
84C2×#239C3×D311yescharacteristic
85C2×D3×Q8—31no—
86C2×#41⋊C3—31no—
87C2×#41⋊C3—61no—
88#984—34no—
89C2×#226—31no—
90C2×#226—61no—
91C2×#226C3×D311yescharacteristic, complemented
92C3×#137—94no—
93C3×#137—94no—
94C2×C3×Q16—31no—
95C2×C3×Q16—31no—
96C2×C3×Q16—61no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100C2×C3×M16—91no—
101C2×#216—31no—
102C2×#215—31no—
103#918—34no—
104#918—34no—
105C2×#204—31no—
106C2×ΓL(1,𝔽25)—31no—
107C22×C3×D3—31no—
108C32×Q8—31no—
109C32×Q8—32no—
110C32×Q8—61no—
111C32×Q8C22×D311yescharacteristic
112C2×C32×C4—31no—
113C2×C32×C4—61no—
114C2×C32×C4—121no—
115C2×C3⋊Dic3—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—81no—
121C2×C3×Dic3—81no—
122C2×C3×Dic3—121no—
123C2×C3×Dic3—241no—
124C2×C3×Dic3S411yescharacteristic
125C3×D12—32no—
126C3×D12—61no—
127C3×C4×D3—32no—
128C3×C4×D3—32no—
129C3×C4×D3—64no—
130C3×Dic6—32no—
131C3×Dic6—61no—
132C3×Dic6—64no—
133C3×Dic6—124no—
134C3×SL(2,𝔽3)—21no—
135C3×SL(2,𝔽3)—21no—
136C3×SL(2,𝔽3)—21no—
137C3×SL(2,𝔽3)—41no—
138C3×SL(2,𝔽3)C22×D311yescharacteristic
139C3×SL(2,𝔽3)C23×C311yescharacteristic
140#627—44no—
141C32×C8—32no—
142C3×C3⋊C8—32no—
143C2×#137—91noSylow
144C2×C33—42no—
145C2×C33—41no—
146C3×#41—32no—
147C3×#41—92no—
148C2×C3×Q8—21no—
149C2×C3×Q8—31no—
150C2×C3×Q8—31no—
151C2×C3×Q8—61no—
152C2×C3×Q8—91no—
153C2×C3×Q8—94no—
154C2×C3×Q8—181no—
155C2×C3×Q8—182no—
156C2×C3×Q8D3211yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—91no—
159C22×C3×C4—91no—
160C22×C3×C4—181no—
161#239—32no—
162#239C2×C3×D312yes—
163D3×Q8—34no—
164D3×Q8—62no—
165C2×D12—31no—
166C2×C4×D3—31no—
167C2×C4×D3—61no—
168C2×Dic6—31no—
169C2×Dic6—61no—
170C2×Dic6—121no—
171#41⋊C3—32no—
172#41⋊C3—62no—
173C2×SL(2,𝔽3)—21no—
174C2×SL(2,𝔽3)—21no—
175C2×SL(2,𝔽3)—41no—
176C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
177#226—32no—
178#226—62no—
179#226C2×C3×D312yescomplemented
180C3×Q16—32no—
181C3×Q16—32no—
182C3×Q16—32no—
183C3×Q16—32no—
184C3×Q16—62no—
185C3×Q16—62no—
186C3×Q16—92no—
187C3×Q16—92no—
188C3×SD16—94no—
189C3×SD16—94no—
190C3×M16—94no—
191C2×C3×C8—31no—
192C2×C3×C8—31no—
193C2×C3×C8—61no—
194C2×C3×C8—91no—
195#216—32no—
196#216—32no—
197#215—34no—
198C2×C3⋊C8—31no—
199#204—34no—
200ΓL(1,𝔽25)—34no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—81no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32—161no—
207C22×C32C2×S411yescharacteristic
208C2×C3×D3—32no—
209C2×C3×D3—62no—
210C32×C4—31no—
211C32×C4—31no—
212C32×C4—62no—
213C32×C4—122no—
214C3⋊Dic3—42no—
215C3×Dic3—42no—
216C3×Dic3—42no—
217C3×Dic3—42no—
218C3×Dic3—62no—
219C3×Dic3—82no—
220C3×Dic3—82no—
221C3×Dic3—122no—
222C3×Dic3—242no—
223C3×Dic3C2×S412yes—
224C2×#41—31no—
225C22×Q8—91no—
226#137—94no—
227#137—94no—
228C2×Q16—31no—
229C2×Q16—91no—
230C2×SD16—91no—
231C2×SD16—91no—
232C2×M16—91no—
233C33—41noSylow
234C23×C3—91no—
235C22×D3—31no—
236C3×Q8—21no—
237C3×Q8—31no—
238C3×Q8—31no—
239C3×Q8—32no—
240C3×Q8—32no—
241C3×Q8—61no—
242C3×Q8—62no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—92no—
246C3×Q8—121no—
247C3×Q8—184no—
248C3×Q8—181no—
249C3×Q8C2×D3211yescharacteristic
250C3×Q8C22×C3×D311yescharacteristic
251C3×D4—92no—
252C3×D4—181no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—121no—
262C2×C3×C4—121no—
263C2×C3×C4—184no—
264C2×C3×C4—181no—
265C2×C3×C4—241no—
266C2×Dic3—41no—
267C2×Dic3—61no—
268C2×Dic3—121no—
269C2×Dic3—241no—
270C2×Dic3C3×S411yescharacteristic
271D12—32no—
272D12—61no—
273C4×D3—32no—
274C4×D3—32no—
275C4×D3—64no—
276Dic6—32no—
277Dic6—61no—
278Dic6—64no—
279Dic6—124no—
280SL(2,𝔽3)—21no—
281SL(2,𝔽3)—21no—
282SL(2,𝔽3)—41no—
283SL(2,𝔽3)C22×C3×D311yescharacteristic
284C3×C8—32no—
285C3×C8—32no—
286C3×C8—62no—
287C3×C8—92no—
288C3⋊C8—32no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—42no—
292C2×C32—42no—
293C2×C32—81no—
294C2×C32—82no—
295C2×C32—81no—
296C2×C32—81no—
297C2×C32—82no—
298C2×C32—82no—
299C2×C32—161no—
300C2×C32—162no—
301C2×C32#98412yescomplemented
302C2×C32C22×S411yescharacteristic
303C3×D3—62no—
304#41—32no—
305#41—92no—
306C2×Q8—31no—
307C2×Q8—91no—
308C2×Q8—94no—
309C2×Q8—181no—
310C2×Q8—182no—
311C2×Q8C3×D3211yescharacteristic
312C2×D4—91no—
313C22×C4—91no—
314C22×C4—181no—
315Q16—32no—
316Q16—32no—
317Q16—92no—
318Q16—92no—
319SD16—94no—
320SD16—94no—
321M16—94no—
322C2×C8—31no—
323C2×C8—91no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—92no—
329C22×C3—161no—
330C22×C3—182no—
331C22×C3D3×S411yescharacteristic
332C22×C3C2×C3×S411yescharacteristic
333C2×D3—32no—
334C2×D3—62no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—32no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—62no—
343C3×C4—62no—
344C3×C4—122no—
345C3×C4—122no—
346C3×C4—182no—
347C3×C4—242no—
348Dic3—42no—
349Dic3—62no—
350Dic3—122no—
351Dic3—242no—
352Dic3C2×C3×S412yes—
353C32—41no—
354C32—41no—
355C32—81no—
356C32—81no—
357C32—81no—
358C32—161no—
359C32C2×#98411yescharacteristic, complemented
360C23—91no—
361Q8—31no—
362Q8—32no—
363Q8—61no—
364Q8—92no—
365Q8—184no—
366Q8—181no—
367Q8C2×C3×D3211yescharacteristic
368D4—92no—
369D4—181no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—61no—
373C2×C4—92no—
374C2×C4—92no—
375C2×C4—184no—
376C2×C4—181no—
377C8—32no—
378C8—92no—
379C2×C3—22no—
380C2×C3—21no—
381C2×C3—42no—
382C2×C3—41no—
383C2×C3—82no—
384C2×C3—82no—
385C2×C3—81no—
386C2×C3—81no—
387C2×C3—162no—
388C2×C3—161no—
389C2×C3—182no—
390C2×C3#6419612yesdirect product factor
391C2×C3C3×#98412yescomplemented
392C2×C3C2×D3×S411yescharacteristic
393C2×C3C22×C3×S411yescharacteristic
394D3—62no—
395C22—92no—
396C22—182no—
397C22C3×D3×S411yescharacteristic
398C4—31no—
399C4—31no—
400C4—32no—
401C4—62no—
402C4—182no—
403C3—21no—
404C3—41no—
405C3—81no—
406C3—81no—
407C3—161no—
408C3C2×#6419611yescharacteristic, direct product factor
409C3C2×C3×#98411yescharacteristic, complemented
410C2—182no—
411C2C3×#6419612yesdirect product factor
412C2C2×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×C3⋊(#41⋊C3)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×#64196C2direct (N × H)
6. C2×#64196C3direct (N × H)
11. C3×C3⋊#226C22semidirect
12. C32×#226C22semidirect
18. C2×C3⋊#226C2×C3semidirect
23. C2×C3×#226D3semidirect
24. C2×C3×#226C2×C3semidirect
25. #64196C2×C3direct (N × H)
63. C3⋊#226C22×C3semidirect
67. C3×#226C2×D3semidirect
68. C3×#226C22×C3semidirect
91. C2×#226C3×D3semidirect
179. #226C2×C3×D3semidirect
301. C2×C32#984semidirect
359. C32C2×#984semidirect
390. C2×C3#64196direct (N × H)
391. C2×C3C3×#984semidirect
408. C3C2×#64196direct (N × H)
409. C3C2×C3×#984semidirect
411. C2C3×#64196direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3×C3⋊#2262
C2×C32×#2262
C3×#641962
C2×C3×#9843
C2×#641963
C2×C3×#9183
C2×C3×#6274