Finite Group Databasebeta

← #10595394⋊C3 #13166142

Subgroups — #10595394⋊C3 #13166142

Order 1728.

428 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 66 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
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10595394C311yesmaximal, characteristic, complemented
5#5827⋊C3—31nomaximal
6#5574⋊C3—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊(#41⋊C3)C412yescomplemented
9C3×Dic3⋊Dic3—41nomaximal
10C2×C3×#239C2×C311yescharacteristic
11C2×C3×#41⋊C3—31no—
12C2×C3⋊(#41⋊C3)—21no—
13C2×C3⋊(#41⋊C3)D311yescharacteristic
14Q8×C3⋊Dic3C2×C311yescharacteristic
15C3×Q8×Dic3C2×C311yescharacteristic
16C3×C4×SL(2,𝔽3)—31no—
17#63898—31no—
18Dic3×D12—31no—
19D3×C4⋊Dic3—31no—
20#63855—31no—
21Dic3×SL(2,𝔽3)—21no—
22Dic3×SL(2,𝔽3)—31no—
23Dic3×SL(2,𝔽3)—31no—
24Dic3×SL(2,𝔽3)—61no—
25Dic3×SL(2,𝔽3)D311yescharacteristic, complemented
26C2×C3×C3⋊Dic3—41no—
27C2×C32×Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30#5827—31no—
31#5574—31no—
32#520⋊C3—91no—
33C2×C32×Q8C22×C311yescharacteristic
34C3×#239—32no—
35C3×#239C3×C412yescomplemented
36C2×C3×D12—31no—
37C2×C3×C4×D3—31no—
38C3×#41⋊C3—32no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—41no—
43C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
44C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
45C2×D3×Dic3—32no—
46C3⋊(#41⋊C3)—22no—
47C3⋊(#41⋊C3)Dic312yescomplemented
48C4⋊(C3⋊Dic3)—31no—
49C4×C3⋊Dic3—31no—
50C3×C4⋊Dic3—31no—
51C3×C4⋊Dic3—121no—
52C3×Dic3⋊C4—121no—
53C3×C4×Dic3—31no—
54Dic3⋊Dic3—31no—
55Dic3⋊Dic3A411yescharacteristic
56(C2×D3)⋊Dic3—31no—
57(C2×D3)⋊Dic3—31no—
58C22×C33—41no—
59C3×C3⋊Dic3—81no—
60C32×Dic3—42no—
61C32×Dic3—81no—
62C2×C3×#41—31no—
63C2×#239C3×D311yescharacteristic
64C2×#41⋊C3—31no—
65C2×#41⋊C3—61no—
66C3×C4×Q8—31no—
67Q8×Dic3—31no—
68Q8×Dic3—31no—
69Q8×Dic3—61no—
70Q8×Dic3C3×D311yescharacteristic
71D4×Dic3—91no—
72D4×Dic3—91no—
73#926—91no—
74C2×C4⋊Dic3—91no—
75#893—91no—
76D3×C4⋊C4—91no—
77C4×D12—91no—
78#872—91no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)—61no—
81C4×SL(2,𝔽3)—91no—
82C22×C3×D3—31no—
83C32×Q8—31no—
84C32×Q8C2×C3×C411yescharacteristic
85C2×C32×C4—31no—
86C2×C32×C4—121no—
87C2×C32×C4—121no—
88C2×C3⋊Dic3—31no—
89C2×C3⋊Dic3C2×A411yescharacteristic
90C2×C3×Dic3—31no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—81no—
94C2×C3×Dic3—81no—
95C2×C3×Dic3—121no—
96C2×C3×Dic3—121no—
97C2×C3×Dic3—241no—
98C2×C3×Dic3C2×A411yescharacteristic
99C2×C3×Dic3C2×A411yescharacteristic
100C3×D12—32no—
101C3×D12—32no—
102C3×C4×D3—32no—
103C3×C4×D3—32no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—21no—
106C3×SL(2,𝔽3)—21no—
107C3×SL(2,𝔽3)—41no—
108C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
109C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
110D3×Dic3—122no—
111#520—91noSylow
112C2×C33—41no—
113C2×C33—41no—
114C2×C33—41no—
115C3×#41—32no—
116C3×#41—92no—
117C2×C3×Q8—21no—
118C2×C3×Q8C2×C3×D311yescharacteristic
119C2×C3×Q8C2×C3×D311yescharacteristic
120C2×C3×D4—91no—
121C22×C3×C4—91no—
122C22×Dic3—92no—
123#239—32no—
124#239C3×Dic312yescomplemented
125C2×D12—31no—
126C2×C4×D3—31no—
127C2×C4×D3—92no—
128#41⋊C3—32no—
129#41⋊C3—62no—
130C2×SL(2,𝔽3)—21no—
131C2×SL(2,𝔽3)—21no—
132C2×SL(2,𝔽3)—41no—
133C2×SL(2,𝔽3)D3211yescharacteristic
134C3×C4⋊C4—91no—
135C3×C4⋊C4—361no—
136C3×C42—91no—
137#217—91no—
138#217—91no—
139(C2×D3)⋊C4—91no—
140(C2×D3)⋊C4—91no—
141C4⋊Dic3—31no—
142C4⋊Dic3—31no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—91no—
146C4⋊Dic3—181no—
147Dic3⋊C4—31no—
148Dic3⋊C4—91no—
149C4×Dic3—31no—
150C4×Dic3—91no—
151C4×Dic3—91no—
152C4×Dic3—181no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32—81no—
156C22×C32—81no—
157C22×C32—81no—
158C22×C32—161no—
159C22×C32C22×A411yescharacteristic
160C2×C3×D3—32no—
161C2×C3×D3—61no—
162C2×C3×D3—61no—
163C32×C4—31no—
164C32×C4—31no—
165C32×C4—122no—
166C32×C4—241no—
167C3⋊Dic3—21no—
168C3⋊Dic3—61no—
169C3×Dic3—21no—
170C3×Dic3—42no—
171C3×Dic3—61no—
172C3×Dic3—81no—
173C3×Dic3—82no—
174C3×Dic3—82no—
175C3×Dic3—241no—
176C3×Dic3—241no—
177C3×Dic3—242no—
178C3×Dic3C4×A412yes—
179C2×#41—31no—
180C4×Q8—31no—
181C4×Q8—91no—
182C4×D4—271no—
183C4×D4—271no—
184#117—271no—
185C2×C4⋊C4—271no—
186C33—41noSylow
187C23×C3—91no—
188C22×D3—31no—
189C3×Q8—21no—
190C3×Q8—31no—
191C3×Q8—31no—
192C3×Q8—61no—
193C3×Q8C3×C4×D311yescharacteristic
194C3×Q8C2×C3×Dic311yescharacteristic
195C3×D4—92no—
196C3×D4—92no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—92no—
203C2×C3×C4—92no—
204C2×C3×C4—91no—
205C2×C3×C4—121no—
206C2×C3×C4—121no—
207C2×C3×C4—241no—
208C2×C3×C4—241no—
209C2×C3×C4—361no—
210C2×Dic3—31no—
211C2×Dic3—31no—
212C2×Dic3—31no—
213C2×Dic3—61no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—181no—
217C2×Dic3—362no—
218C2×Dic3D3×A411yescharacteristic
219C2×Dic3D3×A411yescharacteristic
220D12—32no—
221D12—32no—
222C4×D3—32no—
223C4×D3—32no—
224C4×D3—362no—
225SL(2,𝔽3)—21no—
226SL(2,𝔽3)—21no—
227SL(2,𝔽3)—41no—
228SL(2,𝔽3)D3×Dic311yescharacteristic, complemented
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—81no—
239C2×C32—81no—
240C2×C32—81no—
241C2×C32—81no—
242C2×C32—81no—
243C2×C32—81no—
244C2×C32—161no—
245C2×C32—161no—
246C2×C32—161no—
247C2×C32C2×C4×A411yescharacteristic
248C2×C32#99411yescharacteristic
249C2×C322−1+4⋊C311yescharacteristic
250C3×D3—62no—
251#41—32no—
252#41—92no—
253C2×Q8C3×D3211yescharacteristic
254C2×D4—91no—
255C22×C4—91no—
256C22×C4—272no—
257C4⋊C4—91no—
258C4⋊C4—91no—
259C4⋊C4—271no—
260C4⋊C4—271no—
261C22⋊C4—271no—
262C22⋊C4—271no—
263C42—91no—
264C42—271no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3—92no—
270C22×C3—161no—
271C22×C3—181no—
272C22×C3—181no—
273C22×C3C2×D3×A411yescharacteristic
274C22×C3C2×D3×A411yescharacteristic
275C2×D3—32no—
276C2×D3—61no—
277C2×D3—61no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—32no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—122no—
287C3×C4—181no—
288C3×C4—242no—
289C3×C4—241no—
290C3×C4—242no—
291C3×C4—721no—
292Dic3—21no—
293Dic3—62no—
294Dic3—61no—
295Dic3—61no—
296Dic3—61no—
297Dic3—181no—
298Dic3—181no—
299Dic3—361no—
300Dic3Dic3×A412yes—
301C32—41no—
302C32—41no—
303C32—81no—
304C32—81no—
305C32—81no—
306C32—161no—
307C32#520⋊C311yescharacteristic, complemented
308C23—91no—
309Q8—31no—
310Q8C3×D3×Dic311yescharacteristic
311D4—92no—
312D4—92no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—92no—
317C2×C4—92no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—271no—
321C2×C4—1082no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3—81no—
331C2×C3—81no—
332C2×C3—81no—
333C2×C3—81no—
334C2×C3—161no—
335C2×C3—161no—
336C2×C3—161no—
337C2×C3—182no—
338C2×C3C4×D3×A411yescharacteristic
339C2×C3#1007⋊C311yescharacteristic
340C2×C3C3⋊#99411yescharacteristic
341C2×C3C3⋊#99411yescharacteristic
342C2×C3#1010⋊C311yescharacteristic
343C2×C3C2×Dic3×A411yescharacteristic
344D3—62no—
345C22—92no—
346C22—181no—
347C22—181no—
348C22D32×A411yescharacteristic
349C4—31no—
350C4—31no—
351C4—32no—
352C4—61no—
353C4—181no—
354C4—181no—
355C4—541no—
356C3—21no—
357C3—41no—
358C3—81no—
359C3—81no—
360C3—161no—
361C3#5574⋊C311yescharacteristic, complemented
362C3#5827⋊C311yescharacteristic, complemented
363C2—182no—
364C2#64315⋊C311yescharacteristic
365C2#64320⋊C311yescharacteristic
366C2D3×Dic3×A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, hypocenterC32×Q8C2×C3×C4
Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C3×C3⋊(#41⋊C3)C2
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
4. #10595394C3semidirect
8. C3×C3⋊(#41⋊C3)C4semidirect
25. Dic3×SL(2,𝔽3)D3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
35. C3×#239C3×C4semidirect
47. C3⋊(#41⋊C3)Dic3semidirect
108. C3×SL(2,𝔽3)C4×D3semidirect
109. C3×SL(2,𝔽3)C2×Dic3semidirect
124. #239C3×Dic3semidirect
228. SL(2,𝔽3)D3×Dic3semidirect
307. C32#520⋊C3semidirect
361. C3#5574⋊C3semidirect
362. C3#5827⋊C3semidirect

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
SL(2,𝔽3)×C3⋊Dic32
C3×Dic3×SL(2,𝔽3)2
#105953943
#5827⋊C33
#5574⋊C33
C3×Dic3⋊Dic34