Finite Group Databasebeta

← #10595409⋊C3 #13166138

Subgroups — #10595409⋊C3 #13166138

Order 1728.

472 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 58 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, complemented
2C2×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10595409C311yesmaximal, characteristic, complemented
5#5831⋊C3—31nomaximal
6#5817⋊C3—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×D3×SL(2,𝔽3)—21no—
9C3×C3⋊(#41⋊C3)—21no—
10C3×(C2×D3)⋊Dic3—41nomaximal
11C2×C3×#239C2×C311yescharacteristic, complemented
12C2×C3×D3×Q8C2×C311yescharacteristic
13C2×C3×#41⋊C3—31no—
14C22×C3×SL(2,𝔽3)—31no—
15C2×D3×SL(2,𝔽3)—21no—
16C2×D3×SL(2,𝔽3)D311yescharacteristic, complemented
17C2×C3⋊(#41⋊C3)—21no—
18C2×C3⋊(#41⋊C3)D311yescharacteristic, complemented
19Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
20#63908—31no—
21C4×#625—31no—
22#63863—31no—
23#63856—31no—
24Dic3×SL(2,𝔽3)—31no—
25Dic3×SL(2,𝔽3)—31no—
26Dic3×SL(2,𝔽3)—61no—
27C22×C32×D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31#5831—31no—
32#5817—31no—
33#548⋊C3—91no—
34C2×C32×Q8C22×C311yescharacteristic
35C3×#239—21no—
36C3×#239—61no—
37C3×D3×Q8—21no—
38C3×D3×Q8—61no—
39C2×C3×D12—31no—
40C2×C3×C4×D3—31no—
41C2×C3×C4×D3—31no—
42C2×C3×Dic6—31no—
43C3×#41⋊C3—61no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)—21no—
47C2×C3×SL(2,𝔽3)—41no—
48C2×C3×SL(2,𝔽3)—61no—
49C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
50C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
51C2×#625—31no—
52D3×SL(2,𝔽3)—22no—
53D3×SL(2,𝔽3)—21no—
54C3⋊(#41⋊C3)—22no—
55C3⋊(#41⋊C3)—21no—
56C4⋊(C3⋊Dic3)—31no—
57C4×C3⋊Dic3—31no—
58C3×#217—121no—
59C3×(C2×D3)⋊C4—121no—
60#3724—31no—
61(C2×D3)⋊Dic3—32no—
62(C2×D3)⋊Dic3—31no—
63(C2×D3)⋊Dic3A411yescharacteristic
64C22×C33—41no—
65C2×C32×D3—42no—
66C2×C32×D3—81no—
67C2×C32×D3—81no—
68C3×C3⋊Dic3—81no—
69C32×Dic3—81no—
70C2×C3×#41—31no—
71C22×C3×Q8—31no—
72C2×#239C3×D311yescharacteristic, complemented
73C2×D3×Q8C3×D311yescharacteristic
74C2×#41⋊C3—31no—
75C2×#41⋊C3—61no—
76C22×SL(2,𝔽3)—31no—
77C22×SL(2,𝔽3)—61no—
78#947—91no—
79#946—91no—
80Q8×Dic3—31no—
81Q8×Dic3—31no—
82Q8×Dic3—61no—
83#938—91no—
84#936—91no—
85#930—91no—
86C4×#67—91no—
87C4×#67—91no—
88#924—91no—
89C4×SL(2,𝔽3)—91no—
90C23×C32—121no—
91C22×C3×D3—31no—
92C22×C3×D3—41no—
93C22×C3×D3—81no—
94C22×C3×D3C2×A411yescharacteristic
95C32×Q8—31no—
96C32×Q8C3×D411yescharacteristic
97C2×C32×C4—31no—
98C2×C32×C4—121no—
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3C2×A411yescharacteristic
101C2×C3×Dic3—31no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3—81no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—121no—
106C2×C3×Dic3—241no—
107C2×C3×Dic3C2×A411yescharacteristic
108C3×D12—61no—
109C3×D12—61no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×C4×D3—61no—
114C3×Dic6—61no—
115C3×Dic6—61no—
116C3×SL(2,𝔽3)—21no—
117C3×SL(2,𝔽3)—21no—
118C3×SL(2,𝔽3)—21no—
119C3×SL(2,𝔽3)—41no—
120C3×SL(2,𝔽3)#6711yescharacteristic, complemented
121C3×SL(2,𝔽3)#6711yescharacteristic, complemented
122#625—62no—
123#548—91noSylow
124C2×C33—41no—
125C2×C33—41no—
126C2×C33—41no—
127C32×D3—82no—
128C3×#41—61no—
129C3×#41—181no—
130C2×C3×Q8—21no—
131C2×C3×Q8—61no—
132C2×C3×Q8—91no—
133C2×C3×Q8—181no—
134C2×C3×Q8C2×C3×D311yescharacteristic
135C2×C3×Q8C2×C3×D311yescharacteristic
136C2×C3×D4—91no—
137C22×C3×C4—91no—
138C22×C3×C4—91no—
139C2×#67—91no—
140C2×#67—91no—
141#239—21no—
142#239—61no—
143D3×Q8—21no—
144D3×Q8—61no—
145C2×D12—31no—
146C2×C4×D3—31no—
147C2×C4×D3—31no—
148C2×Dic6—31no—
149#41⋊C3—62no—
150#41⋊C3—61no—
151C2×SL(2,𝔽3)—21no—
152C2×SL(2,𝔽3)—21no—
153C2×SL(2,𝔽3)—41no—
154C2×SL(2,𝔽3)—62no—
155C2×SL(2,𝔽3)—61no—
156C2×SL(2,𝔽3)D3211yescharacteristic
157C3×C22⋊C4—361no—
158#217—31no—
159#217—91no—
160#217—92no—
161(C2×D3)⋊C4—31no—
162(C2×D3)⋊C4—91no—
163(C2×D3)⋊C4—92no—
164C4⋊Dic3—91no—
165C4⋊Dic3—91no—
166C4⋊Dic3—181no—
167Dic3⋊C4—91no—
168Dic3⋊C4—91no—
169C4×Dic3—91no—
170C4×Dic3—91no—
171C4×Dic3—181no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—81no—
175C22×C32—81no—
176C22×C32—81no—
177C22×C32—122no—
178C22×C32—161no—
179C22×C32—241no—
180C22×C32—241no—
181C22×C32C22×A411yescharacteristic
182C2×C3×D3—21no—
183C2×C3×D3—21no—
184C2×C3×D3—42no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—81no—
189C2×C3×D3—81no—
190C2×C3×D3—82no—
191C2×C3×D3—82no—
192C2×C3×D3—82no—
193C2×C3×D3#41⋊C312yescomplemented
194C32×C4—31no—
195C32×C4—31no—
196C32×C4—241no—
197C3⋊Dic3—21no—
198C3⋊Dic3—61no—
199C3×Dic3—21no—
200C3×Dic3—61no—
201C3×Dic3—81no—
202C3×Dic3—82no—
203C3×Dic3—241no—
204C3×Dic3—241no—
205C3×Dic3—242no—
206C2×#41—31no—
207C22×Q8—31no—
208#124—271no—
209C22⋊Q8—271no—
210#121—271no—
211C4×Q8—91no—
212C4×D4—271no—
213C33—41noSylow
214C23×C3—31no—
215C23×C3—91no—
216C23×C3—121no—
217C23×C3—241no—
218C22×D3—31no—
219C22×D3D3×A411yescharacteristic
220C3×Q8—21no—
221C3×Q8—31no—
222C3×Q8—31no—
223C3×Q8—61no—
224C3×Q8—181no—
225C3×Q8—181no—
226C3×Q8C3×#6711yescharacteristic
227C3×Q8C3×#6711yescharacteristic
228C3×D4—181no—
229C3×D4—181no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—91no—
235C2×C3×C4—121no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—241no—
241C2×C3×C4—361no—
242#67—182no—
243#67—182no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—91no—
250C2×Dic3—181no—
251C2×Dic3D3×A411yescharacteristic
252D12—61no—
253D12—61no—
254C4×D3—61no—
255C4×D3—61no—
256C4×D3—61no—
257C4×D3—61no—
258Dic6—61no—
259Dic6—61no—
260SL(2,𝔽3)—21no—
261SL(2,𝔽3)—21no—
262SL(2,𝔽3)—41no—
263SL(2,𝔽3)#62511yescharacteristic, complemented
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—81no—
271C2×C32—81no—
272C2×C32—81no—
273C2×C32—81no—
274C2×C32—81no—
275C2×C32—81no—
276C2×C32—81no—
277C2×C32—81no—
278C2×C32—81no—
279C2×C32—161no—
280C2×C32—161no—
281C2×C32—161no—
282C2×C32—242no—
283C2×C32D4×A411yescharacteristic
284C2×C32C2×#41⋊C311yescharacteristic
285C2×C322−1+4⋊C311yescharacteristic
286C3×D3—22no—
287C3×D3—82no—
288C3×D3—84no—
289C3×D3—121no—
290#41—61no—
291#41—181no—
292C2×Q8—61no—
293C2×Q8—91no—
294C2×Q8—181no—
295C2×Q8C3×D3211yescharacteristic
296C2×D4—91no—
297C2×D4—271no—
298C22×C4—91no—
299C22×C4—91no—
300C4⋊C4—271no—
301C4⋊C4—271no—
302C22⋊C4—91no—
303C22⋊C4—271no—
304C22⋊C4—272no—
305C42—271no—
306C22×C3—21no—
307C22×C3—32no—
308C22×C3—41no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—122no—
314C22×C3—161no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3—181no—
318C22×C3—242no—
319C22×C3—241no—
320C22×C3—241no—
321C22×C3—242no—
322C22×C3—242no—
323C22×C3C2×D3×A411yescharacteristic
324C22×C3C2×D3×A411yescharacteristic
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—61no—
328C2×D3—61no—
329C2×D3—61no—
330C2×D3C3⋊(#41⋊C3)12yescomplemented
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—181no—
339C3×C4—241no—
340C3×C4—242no—
341C3×C4—721no—
342Dic3—21no—
343Dic3—62no—
344Dic3—61no—
345Dic3—61no—
346Dic3—61no—
347Dic3—181no—
348Dic3—181no—
349Dic3—361no—
350C32—41no—
351C32—41no—
352C32—81no—
353C32—81no—
354C32—81no—
355C32—161no—
356C32#548⋊C311yescharacteristic, complemented
357C23—31no—
358C23—91no—
359Q8—31no—
360Q8—181no—
361Q8—181no—
362Q8C3×#62511yescharacteristic
363D4—181no—
364D4—181no—
365D4—542no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—271no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—62no—
382C2×C3—81no—
383C2×C3—81no—
384C2×C3—81no—
385C2×C3—81no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—161no—
389C2×C3—161no—
390C2×C3—161no—
391C2×C3—242no—
392C2×C3—244no—
393C2×C3—361no—
394C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
395C2×C3#1007⋊C311yescharacteristic
396C2×C3D3×#41⋊C311yescharacteristic
397C2×C3#1010⋊C311yescharacteristic
398C2×C3A4×#6711yescharacteristic
399C2×C3A4×#6711yescharacteristic
400D3—22no—
401D3—121no—
402C22—32no—
403C22—61no—
404C22—61no—
405C22—181no—
406C22—181no—
407C22—181no—
408C22D32×A411yescharacteristic
409C4—31no—
410C4—31no—
411C4—61no—
412C4—181no—
413C4—181no—
414C4—541no—
415C3—21no—
416C3—41no—
417C3—81no—
418C3—81no—
419C3—161no—
420C3#5817⋊C311yescharacteristic, complemented
421C3#5831⋊C311yescharacteristic, complemented
422C2—62no—
423C2—361no—
424C2D3×C3⋊(#41⋊C3)11yescharacteristic
425C2#64315⋊C311yescharacteristic
426C2A4×#62511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊(#41⋊C3)C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
4. #10595409C3semidirect
11. C2×C3×#239C2×C3semidirect
16. C2×D3×SL(2,𝔽3)D3semidirect
18. C2×C3⋊(#41⋊C3)D3semidirect
19. Q8×C3⋊Dic3C2×C3semidirect
30. C32×SL(2,𝔽3)D4semidirect
72. C2×#239C3×D3semidirect
120. C3×SL(2,𝔽3)#67semidirect
121. C3×SL(2,𝔽3)#67semidirect
193. C2×C3×D3#41⋊C3semidirect
263. SL(2,𝔽3)#625semidirect
330. C2×D3C3⋊(#41⋊C3)semidirect
356. C32#548⋊C3semidirect
420. C3#5817⋊C3semidirect
421. C3#5831⋊C3semidirect

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⋊(#41⋊C3)2
SL(2,𝔽3)×C3⋊Dic32
#105954093
#5831⋊C33
#5817⋊C33
C3×(C2×D3)⋊Dic34