Finite Group Databasebeta

← D3×C3⋊(#131⋊C3) #13166153

Subgroups — D3×C3⋊(#131⋊C3) #13166153

Order 1728.

376 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C3×D3×#41⋊C3C211yesmaximal, characteristic
2C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4D3×#948C311yesmaximal, characteristic, complemented
5C2×C3⋊(#131⋊C3)—31nomaximal
6D3×#131⋊C3—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×D3×SL(2,𝔽3)C411yescharacteristic
9C3×C3⋊(#41⋊C3)C411yescharacteristic
10C3×D3×C3⋊C8—41nomaximal
11C3×D3×#41C2×C311yescharacteristic
12C2×C3×#41⋊C3—31no—
13D3×#41⋊C3—21no—
14D3×#41⋊C3D311yescharacteristic
15#64157C2×C311yescharacteristic, complemented
16C3×#948C2×C311yescharacteristic, complemented
17C3×#131⋊C3—31no—
18#63815—31no—
19#63814—31no—
20D3×#208—31no—
21C2×D3×C3⋊C8—31no—
22C3⋊(#131⋊C3)—21no—
23C3⋊(#131⋊C3)—31no—
24C3⋊(#131⋊C3)—31no—
25C3⋊(#131⋊C3)—61no—
26C3⋊(#131⋊C3)D311yescharacteristic, direct product factor
27C32×C4×D3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29C3×C3⋊(C3⋊C8)—41no—
30C32×C3⋊C8—41no—
31C2×#948—31no—
32D3×#131—31no—
33C2×#131⋊C3—91no—
34C32×#41C22×C311yescharacteristic
35C3×#239C3×C411yescharacteristic
36C3×D3×Q8C3×C411yescharacteristic
37C3×#237—31no—
38C3×D3×D4—31no—
39C3×#235—31no—
40C2×C3×C4×D3—31no—
41C3×#41⋊C3—21no—
42C3×#41⋊C3—21no—
43C3×#41⋊C3—21no—
44C3×#41⋊C3—31no—
45C3×#41⋊C3—41no—
46C3×#41⋊C3C2×D311yescharacteristic
47C3×#41⋊C3C2×D311yescharacteristic
48C2×C3×SL(2,𝔽3)—31no—
49D3×SL(2,𝔽3)—21no—
50D3×SL(2,𝔽3)Dic311yescharacteristic
51C3⋊(#41⋊C3)—21no—
52C3⋊(#41⋊C3)Dic311yescharacteristic
53#3748—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×#208—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—121no—
58C3×D3×C8—121no—
59#3711—32no—
60D3×C3⋊C8—31no—
61D3×C3⋊C8—31no—
62D3×C3⋊C8—31no—
63D3×C3⋊C8A411yescharacteristic
64C2×C32×D3—41no—
65C33×C4—41no—
66C32×Dic3—41no—
67C2×C3×#41—31no—
68D3×#41C3×D311yescharacteristic
69C2×#41⋊C3—31no—
70C2×#41⋊C3—61no—
71C3×#131—31no—
72#948—31no—
73#948—31no—
74#948—61no—
75#948—91no—
76#948—91no—
77#948C3×D311yescharacteristic, complemented
78C2×#208—91no—
79C22×C3⋊C8—91no—
80#907—91no—
81D3×M16—91no—
82#901—91no—
83C2×D3×C8—91no—
84#131⋊C3—31no—
85#131⋊C3—61no—
86#131⋊C3—91no—
87C22×C3×D3—31no—
88C32×Q8C2×C3×C411yescharacteristic
89C32×D4—31no—
90C2×C32×C4—31no—
91C2×C32×C4—121no—
92C3×#67—32no—
93C2×C3×Dic3—31no—
94C3×D12—31no—
95C3×C4×D3—31no—
96C3×C4×D3—31no—
97C3×C4×D3—31no—
98C3×C4×D3—41no—
99C3×C4×D3—81no—
100C3×C4×D3C2×A411yescharacteristic
101C3×Dic6—31no—
102C3×SL(2,𝔽3)—21no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—41no—
106C3×SL(2,𝔽3)C4×D311yescharacteristic
107C3×SL(2,𝔽3)C2×Dic311yescharacteristic
108C32×C8—121no—
109C3⋊(C3⋊C8)—31no—
110C3⋊(C3⋊C8)C2×A411yescharacteristic
111C3×C3⋊C8—31no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—81no—
114C3×C3⋊C8—121no—
115C3×C3⋊C8—121no—
116C3×C3⋊C8—241no—
117C3×C3⋊C8C2×A411yescharacteristic
118C2×#131—91noSylow
119C2×C33—41no—
120C32×D3—42no—
121C3×#41—21no—
122C3×#41—31no—
123C3×#41—91no—
124C3×#41—91no—
125C3×#41C2×C3×D311yescharacteristic
126C3×#41C2×C3×D311yescharacteristic
127C2×C3×Q8—31no—
128C2×C3×D4—91no—
129C22×C3×C4—91no—
130#239C3×Dic311yescharacteristic
131D3×Q8C3×Dic311yescharacteristic
132#237—31no—
133D3×D4—31no—
134#235—31no—
135C2×C4×D3—31no—
136#41⋊C3—21no—
137#41⋊C3—21no—
138#41⋊C3—31no—
139#41⋊C3—41no—
140#41⋊C3—61no—
141#41⋊C3D3211yescharacteristic
142C2×SL(2,𝔽3)—31no—
143C2×SL(2,𝔽3)—61no—
144C3×M16—91no—
145C2×C3×C8—91no—
146C2×C3×C8—361no—
147#208—31no—
148#208—92no—
149#208—91no—
150#208—91no—
151#208—181no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—181no—
160ΓL(1,𝔽25)—92no—
161D3×C8—31no—
162D3×C8—91no—
163D3×C8—91no—
164D3×C8—91no—
165C22×C32—31no—
166C22×C32—121no—
167C2×C3×D3—31no—
168C2×C3×D3—41no—
169C2×C3×D3—62no—
170C2×C3×D3—81no—
171C2×C3×D3C4×A411yescharacteristic
172C32×C4—31no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4—81no—
176C32×C4—81no—
177C32×C4—81no—
178C32×C4—121no—
179C32×C4—161no—
180C32×C4C22×A411yescharacteristic
181C3×Dic3—31no—
182C3×Dic3—41no—
183C3×Dic3—81no—
184C3×Dic3C4×A411yescharacteristic
185C2×#41—31no—
186#131—31no—
187#131—91no—
188#131—271no—
189#131—271no—
190C2×M16—271no—
191C22×C8—271no—
192C33—41noSylow
193C23×C3—91no—
194C22×D3—31no—
195C3×Q8—21no—
196C3×Q8—91no—
197C3×Q8C3×C4×D311yescharacteristic
198C3×Q8C2×C3×Dic311yescharacteristic
199C3×D4—31no—
200C3×D4—31no—
201C3×D4—61no—
202C3×D4—91no—
203C3×D4—92no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—121no—
213C2×C3×C4—241no—
214#67—32no—
215C2×Dic3—31no—
216D12—31no—
217C4×D3—31no—
218C4×D3—31no—
219C4×D3—31no—
220C4×D3D3×A411yescharacteristic
221Dic6—31no—
222SL(2,𝔽3)—21no—
223SL(2,𝔽3)—21no—
224SL(2,𝔽3)—41no—
225SL(2,𝔽3)D3×Dic311yescharacteristic
226C3×C8—31no—
227C3×C8—91no—
228C3×C8—121no—
229C3×C8—241no—
230C3×C8—361no—
231C3⋊C8—31no—
232C3⋊C8—31no—
233C3⋊C8—31no—
234C3⋊C8—61no—
235C3⋊C8—91no—
236C3⋊C8—91no—
237C3⋊C8—181no—
238C3⋊C8D3×A411yescharacteristic
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—61no—
242C2×C32—81no—
243C2×C32—81no—
244C2×C32—81no—
245C2×C32—122no—
246C2×C32—161no—
247C2×C32C2×C4×A411yescharacteristic
248C3×D3—42no—
249C3×D3—61no—
250C3×D3—82no—
251C3×D3#131⋊C312yescomplemented
252#41—31no—
253#41—91no—
254#41—91no—
255#41C3×D3211yescharacteristic
256C2×Q8—31no—
257C2×D4—91no—
258C22×C4—91no—
259M16—91no—
260M16—272no—
261M16—271no—
262C2×C8—91no—
263C2×C8—91no—
264C2×C8—271no—
265C2×C8—271no—
266C2×C8—271no—
267C2×C8—271no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—61no—
272C22×C3—91no—
273C22×C3—121no—
274C22×C3—182no—
275C22×C3—241no—
276C2×D3—31no—
277C2×D3—62no—
278C2×D3Dic3×A411yescharacteristic
279C3×C4—21no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—41no—
284C3×C4—61no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—91no—
288C3×C4—121no—
289C3×C4—161no—
290C3×C4—241no—
291C3×C4C2×D3×A411yescharacteristic
292C3×C4C2×D3×A411yescharacteristic
293Dic3—31no—
294Dic3Dic3×A411yescharacteristic
295C32—41no—
296C32—41no—
297C32—81no—
298C32—81no—
299C32—81no—
300C32—161no—
301C32C2×#131⋊C311yescharacteristic, complemented
302C23—91no—
303Q8—91no—
304Q8C3×D3×Dic311yescharacteristic
305D4—31no—
306D4—91no—
307D4—92no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—91no—
314C8—31no—
315C8—91no—
316C8—91no—
317C8—271no—
318C2×C3—21no—
319C2×C3—32no—
320C2×C3—41no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3—122no—
326C2×C3—121no—
327C2×C3—161no—
328C2×C3—181no—
329C2×C3—242no—
330C2×C3C4×D3×A411yescharacteristic
331C2×C3C2×Dic3×A411yescharacteristic
332D3—61no—
333D3C3⋊(#131⋊C3)12yesdirect product factor
334C22—31no—
335C22—31no—
336C22—91no—
337C22—182no—
338C4—31no—
339C4—31no—
340C4—91no—
341C4D32×A411yescharacteristic
342C3—21no—
343C3—41no—
344C3—81no—
345C3—81no—
346C3—161no—
347C3D3×#131⋊C311yescharacteristic, complemented
348C3C2×C3⋊(#131⋊C3)11yescharacteristic, complemented
349C2—32no—
350C2—61no—
351C2—181no—
352C2D3×Dic3×A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D32×A4
commutator, hypocenterC32×Q8C2×C3×C4
Fitting, cosocleC32×#41C22×C3
socleC2×C32C2×C4×A4
prime radicalC3×D3×#41⋊C3C2
LCSG ▷ C32×Q8
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊(#131⋊C3))C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
4. D3×#948C3semidirect
15. #64157C2×C3semidirect
16. C3×#948C2×C3semidirect
26. C3⋊(#131⋊C3)D3direct (N × H)
77. #948C3×D3semidirect
251. C3×D3#131⋊C3semidirect
301. C32C2×#131⋊C3semidirect
333. D3C3⋊(#131⋊C3)direct (N × H)
347. C3D3×#131⋊C3semidirect
348. C3C2×C3⋊(#131⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C3×D3×#41⋊C32
C3⋊(C3⋊(#131⋊C3))2
C3×C3⋊(#131⋊C3)2
D3×#9483
C2×C3⋊(#131⋊C3)3
D3×#131⋊C33
C3×D3×C3⋊C84