Finite Group Databasebeta

← #87294⋊C3 #11923403

Subgroups — #87294⋊C3 #11923403

Order 1152.

847 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 91 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#5831⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×#67C212yesmaximal, complemented
5#5812⋊C3C211yesmaximal, characteristic, complemented
6#87294C311yesmaximal, characteristic, complemented
7#3486⋊C3—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic, complemented
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×D3×SL(2,𝔽3)—22no—
13C2×D3×SL(2,𝔽3)C2212yes—
14C3×#937—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4C2×C311yescharacteristic, complemented
17C22×D3×Q8C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3—21no—
19C2×2−1+4⋊C3D311yescharacteristic, complemented
20C23×SL(2,𝔽3)—31no—
21#5831—32no—
22#5831C2×C312yescomplemented
23#5828—31no—
24#5828—31no—
25#5825—31no—
26#5825—31no—
27Q8×#67—32no—
28Q8×#67C2×C312yescomplemented
29#5812C2×C311yescharacteristic, complemented
30D4×SL(2,𝔽3)—32no—
31#548⋊C3—32no—
32#519⋊C3—31no—
33C23×C3×D3—41no—
34C2×C32×D4—41no—
35C2×C3×#67—42no—
36C3×#41⋊C3—21no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)D412yescomplemented
39C2×C3×SL(2,𝔽3)C2311yescharacteristic
40D3×SL(2,𝔽3)—22no—
41C3×#217—41no—
42C3×(C2×D3)⋊C4—42no—
43#3486—31noSylow
44C3×2−1+4—22no—
45C3×2−1+4—62no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41C22×C311yescharacteristic, complemented
50C22×C3×Q8—31no—
51C22×C3×Q8C22×C311yescharacteristic
52C22×C3×Q8C22×C311yescharacteristic, complemented
53C2×D3×Q8—22no—
54C2×D3×Q8—32no—
55C2×D3×Q8—62no—
56C2×D3×Q8C22×C312yes—
57C22×C4×D3—31no—
58C22×Dic6—31no—
592−1+4⋊C3—24no—
602−1+4⋊C3—22no—
61C2×#41⋊C3—21no—
62C2×#41⋊C3C2×D311yescharacteristic, complemented
63C22×SL(2,𝔽3)—21no—
64C22×SL(2,𝔽3)—21no—
65C22×SL(2,𝔽3)—32no—
66C22×SL(2,𝔽3)—62no—
67C22×SL(2,𝔽3)C2×D311yescharacteristic
68C22×SL(2,𝔽3)C2×D311yescharacteristic, complemented
69C3×#120—121no—
70#946—32no—
71#946—32no—
72Q8×Dic3—32no—
73Q8×Dic3C22×C312yescomplemented
74#944—32no—
75#944—32no—
76#938—32no—
77#938—32no—
78#937—31no—
79#937A411yescharacteristic
80#936—32no—
81#936—32no—
82#935—32no—
83#935—31no—
84#935—31no—
85C4×#67—34no—
86C4×#67—32no—
87C4×#67—32no—
88#926—31no—
89#926—31no—
90#924—34no—
91#924—32no—
92#924—32no—
93C4×SL(2,𝔽3)—32no—
94C23×C32—41no—
95C23×C32—41no—
96C22×C3×D3—42no—
97C22×C3×D3—84no—
98C22×C3×D3—82no—
99C32×D4—82no—
100C2×C32×C4—41no—
101C3×#67—84no—
102C2×C3×Dic3—42no—
103C3×SL(2,𝔽3)C2×D411yescharacteristic
104C2×2−1+4C3×D311yescharacteristic, complemented
105C23×Q8—31no—
106D4×Q8—32no—
107D4×Q8—92no—
108#548—32no—
109#548—92no—
110#536—91no—
111#536—91no—
112#525—91no—
113#525—91no—
114#519—31no—
115C24×C3—121no—
116C23×D3C2×A411yescharacteristic
117C3×#41—21no—
118C3×#41—62no—
119C3×#41—62no—
120C3×#41—62no—
121C3×#41—62no—
122C3×#41—62no—
123C3×#41—62no—
124C3×#41—61no—
125C2×C3×Q8—21no—
126C2×C3×Q8—31no—
127C2×C3×Q8—31no—
128C2×C3×Q8—32no—
129C2×C3×Q8—31no—
130C2×C3×Q8—62no—
131C2×C3×Q8—62no—
132C2×C3×Q8—61no—
133C2×C3×Q8C3×D412yescomplemented
134C2×C3×Q8C23×C311yescharacteristic
135C2×C3×D4—31no—
136C2×C3×D4—31no—
137C2×C3×D4—31no—
138C2×C3×D4—41no—
139C2×C3×D4—81no—
140C2×C3×D4—122no—
141C2×C3×D4C2×A411yescharacteristic
142C22×C3×C4—31no—
143C22×C3×C4—32no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C2×#67—32no—
147C2×#67C2×A412yes—
148C22×Dic3—31no—
149D3×Q8—22no—
150D3×Q8—64no—
151D3×Q8—64no—
152D3×Q8—62no—
153C2×C4×D3—32no—
154C2×C4×D3—32no—
155C2×C4×D3—62no—
156C2×C4×D3—62no—
157C2×Dic6—32no—
158C2×Dic6—32no—
159C2×Dic6—62no—
160C2×Dic6—62no—
161#41⋊C3—22no—
162#41⋊C3—21no—
163C2×SL(2,𝔽3)—22no—
164C2×SL(2,𝔽3)—21no—
165C2×SL(2,𝔽3)—22no—
166C2×SL(2,𝔽3)—21no—
167C2×SL(2,𝔽3)—62no—
168C2×SL(2,𝔽3)#6712yescomplemented
169C2×SL(2,𝔽3)C22×D311yescharacteristic
170C3×C22⋊C4—122no—
171C3×C22⋊C4—121no—
172#217—34no—
173#217—32no—
174#217—31no—
175#217—32no—
176#217C2×A411yescharacteristic
177(C2×D3)⋊C4—32no—
178(C2×D3)⋊C4C2×A412yes—
179C4⋊Dic3—32no—
180C4⋊Dic3—32no—
181Dic3⋊C4—34no—
182Dic3⋊C4—32no—
183Dic3⋊C4—32no—
184C4×Dic3—32no—
185C4×Dic3—32no—
186C22×C32—42no—
187C22×C32—41no—
188C22×C32—81no—
189C22×C32—82no—
190C22×C32—82no—
191C2×C3×D3—44no—
192C2×C3×D3—82no—
193C2×C3×D3—82no—
194C2×C3×D3—88no—
195C32×C4—81no—
196C3×Dic3—82no—
1972−1+4—22no—
1982−1+4—62no—
199C2×#41—31no—
200C2×#41—31no—
201C2×#41—31no—
202C2×#41C2×C3×D311yescharacteristic, complemented
203C22×Q8—31no—
204C22×Q8—32no—
205C22×Q8—62no—
206C22×Q8—91no—
207C22×Q8—92no—
208C22×Q8—182no—
209C22×Q8C2×C3×D311yescharacteristic
210C22×Q8C2×C3×D311yescharacteristic, complemented
211C23×C4—91no—
212C4⋊Q8—92no—
213C4⋊Q8—92no—
214#124—92no—
215#124—92no—
216#123—91no—
217#123—92no—
218#123—91no—
219C22⋊Q8—92no—
220C22⋊Q8—94no—
221C22⋊Q8—92no—
222C22⋊Q8—92no—
223C22⋊Q8—92no—
224#121—92no—
225#121—92no—
226#120—31no—
227#120—91no—
228C4×Q8—32no—
229C4×Q8—92no—
230C4×D4—92no—
231C4×D4—92no—
232C4×D4—94no—
233#117—91no—
234#117—91no—
235C23×C3—31no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—81no—
239C23×C3—81no—
240C23×C3—122no—
241C23×C3—242no—
242C23×C3—244no—
243C23×C3C22×A411yescharacteristic
244C23×C3C22×A411yescharacteristic
245C22×D3—22no—
246C22×D3—24no—
247C22×D3C22×A412yes—
248C3×Q8—32no—
249C3×Q8—31no—
250C3×Q8—32no—
251C3×Q8—62no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8C2×C3×D411yescharacteristic
255C3×D4—22no—
256C3×D4—62no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—62no—
260C3×D4—82no—
261C3×D4—84no—
262C3×D4—244no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—41no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—64no—
274C2×C3×C4—81no—
275C2×C3×C4—122no—
276C2×C3×C4C22×A411yescharacteristic
277#67—24no—
278#67—64no—
279C2×Dic3—32no—
280C2×Dic3—32no—
281C2×Dic3—62no—
282C2×Dic3C22×A412yes—
283C4×D3—62no—
284C4×D3—64no—
285C4×D3—62no—
286Dic6—64no—
287Dic6—62no—
288Dic6—62no—
289SL(2,𝔽3)—21no—
290SL(2,𝔽3)C2×#6711yescharacteristic
291C2×C32—41no—
292C2×C32—42no—
293C2×C32—82no—
294C2×C32—161no—
295C3×D3—84no—
296C24—31no—
297#41—21no—
298#41—62no—
299#41—62no—
300#41—62no—
301#41—62no—
302#41—62no—
303#41—62no—
304#41—61no—
305C2×Q8—21no—
306C2×Q8—31no—
307C2×Q8—31no—
308C2×Q8—32no—
309C2×Q8—31no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—61no—
313C2×Q8—62no—
314C2×Q8—92no—
315C2×Q8—92no—
316C2×Q8—182no—
317C2×Q8—182no—
318C2×Q8—184no—
319C2×Q8—182no—
320C2×Q8—184no—
321C2×Q8C3×#6712yescomplemented
322C2×Q8C22×C3×D311yescharacteristic
323C2×D4—31no—
324C2×D4—32no—
325C2×D4—31no—
326C2×D4—31no—
327C2×D4—92no—
328C2×D4D3×A411yescharacteristic
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—92no—
334C22×C4—92no—
335C22×C4—91no—
336C22×C4—182no—
337C22×C4—182no—
338C4⋊C4—92no—
339C4⋊C4—92no—
340C4⋊C4—92no—
341C4⋊C4—94no—
342C4⋊C4—92no—
343C22⋊C4—31no—
344C22⋊C4—32no—
345C22⋊C4—92no—
346C22⋊C4—92no—
347C22⋊C4—94no—
348C22⋊C4—91no—
349C22⋊C4—92no—
350C42—92no—
351C42—92no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—22no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—61no—
358C22×C3—62no—
359C22×C3—82no—
360C22×C3—81no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—81no—
365C22×C3—124no—
366C22×C3—162no—
367C22×C3—162no—
368C22×C3—242no—
369C22×C3—242no—
370C22×C3—248no—
371C22×C3D4×A412yes—
372C22×C3C23×A411yescharacteristic
373C2×D3—28no—
374C2×D3—22no—
375C2×D3—22no—
376C2×D32−1+4⋊C314yescomplemented
377C3×C4—21no—
378C3×C4—32no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—61no—
382C3×C4—81no—
383C3×C4—82no—
384C3×C4—242no—
385Dic3—22no—
386Dic3—62no—
387Dic3—62no—
388C32—41noSylow
389C23—32no—
390C23—31no—
391C23—64no—
392C23—62no—
393C23C2×D3×A411yescharacteristic
394C23C2×D3×A411yescharacteristic
395Q8—32no—
396Q8—31no—
397Q8—32no—
398Q8—62no—
399Q8—61no—
400Q8—61no—
401Q8—182no—
402Q8—182no—
403Q8—184no—
404Q8C2×C3×#6711yescharacteristic
405D4—22no—
406D4—62no—
407D4—61no—
408D4—61no—
409D4—64no—
410D4—62no—
411D4—184no—
412C2×C4—32no—
413C2×C4—32no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—64no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C4—184no—
428C2×C4—182no—
429C2×C4C2×D3×A411yescharacteristic
430C2×C3—22no—
431C2×C3—41no—
432C2×C3—42no—
433C2×C3—41no—
434C2×C3—82no—
435C2×C3—82no—
436C2×C3—81no—
437C2×C3—121no—
438C2×C3—162no—
439C2×C3—161no—
440C2×C3—162no—
441C2×C3—244no—
442C2×C3C2×D4×A411yescharacteristic
443C2×C3C2×2−1+4⋊C312yes—
444D3—24no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—34no—
449C22—68no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—61no—
454C22A4×#6712yes—
455C22C22×D3×A411yescharacteristic
456C4—21no—
457C4—31no—
458C4—32no—
459C4—31no—
460C4—62no—
461C4—61no—
462C4—182no—
463C4—182no—
464C3—41no—
465C3—81no—
466C3#3486⋊C311yescharacteristic, complemented
467C2—22no—
468C2—41no—
469C2—64no—
470C2—121no—
471C2D3×2−1+4⋊C312yes—
472C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×2−1+4C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C22×D3×SL(2,𝔽3)C2semidirect
3. #5831⋊C3C2semidirect
4. SL(2,𝔽3)×#67C2semidirect
5. #5812⋊C3C2semidirect
6. #87294C3semidirect
9. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
16. C2×C3×2−1+4C2×C3semidirect
17. C22×D3×Q8C2×C3semidirect
19. C2×2−1+4⋊C3D3semidirect
22. #5831C2×C3semidirect
28. Q8×#67C2×C3semidirect
29. #5812C2×C3semidirect
38. C2×C3×SL(2,𝔽3)D4semidirect
49. C2×C3×#41C22×C3semidirect
52. C22×C3×Q8C22×C3semidirect
62. C2×#41⋊C3C2×D3semidirect
68. C22×SL(2,𝔽3)C2×D3semidirect
73. Q8×Dic3C22×C3semidirect
104. C2×2−1+4C3×D3semidirect
133. C2×C3×Q8C3×D4semidirect
168. C2×SL(2,𝔽3)#67semidirect
202. C2×#41C2×C3×D3semidirect
210. C22×Q8C2×C3×D3semidirect
321. C2×Q8C3×#67semidirect
376. C2×D32−1+4⋊C3semidirect
466. C3#3486⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×D3×SL(2,𝔽3)2
#5831⋊C32
SL(2,𝔽3)×#672
#5812⋊C32
#872943
#3486⋊C33
C3×#9374