Finite Group Databasebeta

← C2×D3×C3⋊#226 #13180808

Subgroups — C2×D3×C3⋊#226 #13180808

Order 1728.

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

normal subgroups only

This group has 901 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⋊#226)C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
4D3×C3⋊#226C214yesmaximal, direct product factor
5C22×C3⋊#226—31nomaximal
6C2×D3×#226—33nomaximal
7C2×D3×#216—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×D3×C3⋊Dic3—41nomaximal
10C3×D3×SL(2,𝔽3)C2212yes—
11C3⋊(C3⋊#226)C2212yescomplemented
12C3×C3⋊#226C2212yescomplemented
13C2×C3×D3×Q8D311yescharacteristic
14C22×C3×SL(2,𝔽3)—31no—
15C2×D3×Dic6—31no—
16C2×D3×SL(2,𝔽3)D313yes—
17C2×C3⋊#226—31no—
18C2×C3⋊#226—33no—
19C2×C3⋊#226—34no—
20C2×C3⋊#226—61no—
21C2×C3⋊#226—63no—
22C2×C3⋊#226D311yescharacteristic, direct product factor
23C2×C3×#226—33no—
24D3×#226—312no—
25C2×#3756—31no—
26C2×C3×#216—31no—
27D3×#216—34no—
28D3×#216—34no—
29C2×#3719—31no—
30C2×#3718—31no—
31C2×D3×C3⋊C8—31no—
32C22×C32×D3—41no—
33C2×C3⋊(C3⋊Dic3)—41no—
34C2×C3×C3⋊Dic3—41no—
35C32×SL(2,𝔽3)C2311yescharacteristic
36D3×C3⋊Dic3—44no—
37C22×#226—93no—
38C22×#216—91no—
39C2×D3×Q16—91no—
40C2×C32×Q8C2×D311yescharacteristic
41C22×C3⋊Dic3—121no—
42C2×#634—31no—
43C3×D3×Q8—32no—
44C3×D3×Q8C2×D312yes—
45C2×C3×C4×D3—31no—
46C2×C3×Dic6—31no—
47C2×C3×Dic6—31no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)—23no—
50C2×C3×SL(2,𝔽3)—32no—
51C2×C3×SL(2,𝔽3)C2×D313yes—
52C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
53C2×#627—61no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—123no—
56D3×Dic6—34no—
57D3×Dic6—62no—
58D3×SL(2,𝔽3)C2×D316yes—
59C3⋊#226—32no—
60C3⋊#226—36no—
61C3⋊#226—62no—
62C3⋊#226—66no—
63C3⋊#226C2×D312yesdirect product factor
64C3×#226—36no—
65#3756—32no—
66#3756—32no—
67C2×C3⋊(C3⋊C8)—31no—
68C3×#216—32no—
69C3×#216—32no—
70C2×C3×C3⋊C8—31no—
71#3719—34no—
72#3718—34no—
73D3×C3⋊C8—34no—
74C22×C33—41no—
75C2×C32×D3—44no—
76C2×C32×D3—42no—
77C3⋊(C3⋊Dic3)—42no—
78C3×C3⋊Dic3—42no—
79C22×C3×Q8—31no—
80C2×D3×Q8—91no—
81C2×D3×Q8C3⋊D311yescharacteristic
82C22×Dic6—91no—
83C22×SL(2,𝔽3)—33no—
84C2×#226—33no—
85C2×#226—912no—
86C2×#226—93no—
87C2×#226—183no—
88C2×C3×Q16—91no—
89C2×#216—31no—
90C2×#216—91no—
91C2×#216—94no—
92C2×#216—94no—
93C2×#216—91no—
94C2×#216—91no—
95C2×#216—91no—
96C2×#216—91no—
97C2×#216—181no—
98C22×C3⋊C8—91no—
99D3×Q16—94no—
100D3×Q16—94no—
101C2×Dic12—91no—
102C2×D3×C8—91no—
103C23×C32—121no—
104C22×C3×D3—43no—
105C22×C3×D3S411yescharacteristic
106C32×Q8—31no—
107C32×Q8C22×D311yescharacteristic
108C2×C32×C4—31no—
109C2×C3⋊Dic3—41no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—124no—
112C2×C3⋊Dic3—121no—
113C2×C3⋊Dic3—123no—
114C2×C3⋊Dic3—241no—
115C2×C3⋊Dic3—243no—
116#634—32no—
117#634—61no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—123no—
121C3×C4×D3—32no—
122C3×C4×D3—32no—
123C3×Dic6—32no—
124C3×Dic6—32no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C3×SL(2,𝔽3)—21no—
128C3×SL(2,𝔽3)—23no—
129C3×SL(2,𝔽3)C22×D313yes—
130C3×SL(2,𝔽3)C22×D311yescharacteristic
131#627—64no—
132D3×Dic3—64no—
133D3×Dic3—1212no—
134C3⋊(C3⋊C8)—32no—
135C3×C3⋊C8—32no—
136C22×Q16—271noSylow
137C2×C33—42no—
138C2×C33—41no—
139C32×D3—44no—
140C2×C3×Q8—21no—
141C2×C3×Q8—32no—
142C2×C3×Q8—91no—
143C2×C3×Q8—91no—
144C2×C3×Q8—92no—
145C2×C3×Q8D3211yescharacteristic
146C2×C3×Q8C2×C3⋊D311yescharacteristic
147C22×C3×C4—91no—
148C22×Dic3—181no—
149C22×Dic3—363no—
150D3×Q8—32no—
151D3×Q8—94no—
152D3×Q8—182no—
153D3×Q8C2×C3⋊D312yes—
154C2×C4×D3—31no—
155C2×C4×D3—181no—
156C2×Dic6—31no—
157C2×Dic6—31no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—91no—
161C2×Dic6—181no—
162C2×Dic6—182no—
163C2×Dic6—181no—
164C2×Dic6—181no—
165C2×SL(2,𝔽3)—23no—
166C2×SL(2,𝔽3)—36no—
167C2×SL(2,𝔽3)D3213yes—
168#226—36no—
169#226—96no—
170#226—186no—
171C3×Q16—92no—
172C3×Q16—92no—
173C2×C3×C8—91no—
174#216—32no—
175#216—32no—
176#216—94no—
177#216—94no—
178#216—94no—
179#216—92no—
180#216—92no—
181#216—92no—
182#216—92no—
183#216—182no—
184#216—182no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—181no—
190Dic12—94no—
191D3×C8—94no—
192C22×C32—41no—
193C22×C32—43no—
194C22×C32—83no—
195C22×C32—81no—
196C22×C32—122no—
197C22×C32—124no—
198C22×C32C2×S411yescharacteristic
199C2×C3×D3—46no—
200C2×C3×D3—412no—
201C2×C3×D3#22614yescomplemented
202C2×C3×D3C2×S412yes—
203C32×C4—31no—
204C32×C4—31no—
205C3⋊Dic3—42no—
206C3⋊Dic3—62no—
207C3⋊Dic3—122no—
208C3⋊Dic3—126no—
209C3⋊Dic3—242no—
210C3⋊Dic3—246no—
211C3×Dic3—32no—
212C3×Dic3—62no—
213C3×Dic3—126no—
214C22×Q8—31no—
215C22×Q8—271no—
216C2×Q16—91no—
217C2×Q16—271no—
218C2×Q16—274no—
219C2×Q16—274no—
220C2×Q16—271no—
221C2×Q16—271no—
222C22×C8—271no—
223C33—41noSylow
224C23×C3—31no—
225C23×C3—123no—
226C22×D3C3⋊S411yescharacteristic
227C3×Q8—21no—
228C3×Q8—31no—
229C3×Q8—31no—
230C3×Q8—61no—
231C3×Q8—92no—
232C3×Q8—92no—
233C3×Q8—181no—
234C3×Q8—181no—
235C3×Q8C2×D3211yescharacteristic
236C3×Q8C22×C3⋊D311yescharacteristic
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—91no—
243C2×C3×C4—181no—
244C2×Dic3—31no—
245C2×Dic3—61no—
246C2×Dic3—123no—
247C2×Dic3—184no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250C2×Dic3—3612no—
251C2×Dic3—361no—
252C2×Dic3—363no—
253C2×Dic3—723no—
254C4×D3—32no—
255C4×D3—32no—
256C4×D3—184no—
257Dic6—32no—
258Dic6—32no—
259Dic6—61no—
260Dic6—61no—
261Dic6—92no—
262Dic6—92no—
263Dic6—182no—
264Dic6—184no—
265Dic6—184no—
266Dic6—181no—
267Dic6—181no—
268Dic6—361no—
269SL(2,𝔽3)—23no—
270SL(2,𝔽3)C2×D3213yes—
271C3×C8—92no—
272C3⋊C8—32no—
273C3⋊C8—92no—
274C3⋊C8—92no—
275C3⋊C8—182no—
276C2×C32—41no—
277C2×C32—42no—
278C2×C32—46no—
279C2×C32—43no—
280C2×C32—83no—
281C2×C32—86no—
282C2×C32—81no—
283C2×C32—82no—
284C2×C32—124no—
285C2×C32C2×#22612yescomplemented
286C2×C32C22×S411yescharacteristic
287C3×D3—412no—
288C3×D3C2×#22614yescomplemented
289C2×Q8—32no—
290C2×Q8—91no—
291C2×Q8—91no—
292C2×Q8—92no—
293C2×Q8—271no—
294C2×Q8—274no—
295C2×Q8—541no—
296C2×Q8—542no—
297C2×Q8D3×C3⋊D311yescharacteristic
298C22×C4—91no—
299C22×C4—541no—
300Q16—92no—
301Q16—92no—
302Q16—274no—
303Q16—272no—
304Q16—274no—
305Q16—272no—
306C2×C8—91no—
307C2×C8—274no—
308C2×C8—271no—
309C22×C3—21no—
310C22×C3—32no—
311C22×C3—34no—
312C22×C3—43no—
313C22×C3—83no—
314C22×C3—126no—
315C22×C3—1212no—
316C22×C3D3×S411yescharacteristic
317C22×C3C2×C3⋊S411yescharacteristic
318C2×D3C3⋊#22614yesdirect product factor
319C2×D3C2×C3⋊S412yes—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—92no—
327C3×C4—182no—
328Dic3—32no—
329Dic3—62no—
330Dic3—126no—
331Dic3—182no—
332Dic3—182no—
333Dic3—362no—
334Dic3—366no—
335Dic3—726no—
336C32—41no—
337C32—43no—
338C32—83no—
339C32—81no—
340C32C22×#22611yescharacteristic, complemented
341C23—31no—
342Q8—31no—
343Q8—92no—
344Q8—92no—
345Q8—181no—
346Q8—181no—
347Q8—272no—
348Q8—541no—
349Q8—544no—
350Q8C2×D3×C3⋊D311yescharacteristic
351C2×C4—31no—
352C2×C4—92no—
353C2×C4—92no—
354C2×C4—91no—
355C2×C4—181no—
356C2×C4—544no—
357C2×C4—541no—
358C8—92no—
359C8—272no—
360C2×C3—22no—
361C2×C3—21no—
362C2×C3—34no—
363C2×C3—46no—
364C2×C3—43no—
365C2×C3—86no—
366C2×C3—83no—
367C2×C3—1212no—
368C2×C3D3×#22612yescomplemented
369C2×C3C2×C3⋊#22612yescomplemented
370C2×C3C2×D3×S411yescharacteristic
371C2×C3C22×C3⋊S411yescharacteristic
372D3C2×C3⋊#22614yesdirect product factor
373C22—34no—
374C22—32no—
375C22D3×C3⋊S411yescharacteristic
376C4—31no—
377C4—31no—
378C4—92no—
379C4—182no—
380C4—542no—
381C3—21no—
382C3—43no—
383C3—83no—
384C3C2×D3×#22611yescharacteristic, complemented
385C3C22×C3⋊#22611yescharacteristic, complemented
386C2—34no—
387C2D3×C3⋊#22612yesdirect product factor
388C2C2×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
prime radicalC2×C3×D3×SL(2,𝔽3)C2
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C2×C3×C3⋊#226C2semidirect
4. D3×C3⋊#226C2direct (N × H)
11. C3⋊(C3⋊#226)C22semidirect
12. C3×C3⋊#226C22semidirect
22. C2×C3⋊#226D3direct (N × H)
63. C3⋊#226C2×D3direct (N × H)
201. C2×C3×D3#226semidirect
285. C2×C32C2×#226semidirect
288. C3×D3C2×#226semidirect
318. C2×D3C3⋊#226direct (N × H)
340. C32C22×#226semidirect
368. C2×C3D3×#226semidirect
369. C2×C3C2×C3⋊#226semidirect
372. D3C2×C3⋊#226direct (N × H)
384. C3C2×D3×#226semidirect
385. C3C22×C3⋊#226semidirect
387. C2D3×C3⋊#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⋊#226)2
C2×C3×C3⋊#2262
D3×C3⋊#2262
C22×C3⋊#2263
C2×D3×#2263
C2×D3×#2163
C2×D3×C3⋊Dic34