Finite Group Databasebeta

← #86749⋊C3 #11923257

Subgroups — #86749⋊C3 #11923257

Order 1152.

569 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 79 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5812⋊C3C211yesmaximal, characteristic, complemented
3#5817⋊C3C211yesmaximal, characteristic, complemented
4#5817⋊C3C211yesmaximal, characteristic, complemented
5C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5569⋊C3C211yesmaximal, characteristic, complemented
7#5574⋊C3C211yesmaximal, characteristic, complemented
8#86749C311yesmaximal, characteristic, complemented
9#3516⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#935—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15Dic3×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×D4×Q8C2×C311yescharacteristic
19#5817C2×C311yescharacteristic, complemented
20#5817C2×C311yescharacteristic, complemented
21#5812C2×C311yescharacteristic, complemented
22C2×Q8×Dic3C2×C311yescharacteristic, complemented
23#5732—31no—
24#5731—31no—
25#5643—31no—
26#5639—31no—
27#5635—31no—
28#5624—31no—
29#5574C2×C311yescharacteristic, complemented
30#5569C2×C311yescharacteristic, complemented
31#5565—31no—
32#5538—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—21no—
35D4×SL(2,𝔽3)D311yescharacteristic, complemented
36#548⋊C3—31no—
37#548⋊C3—31no—
38#520⋊C3—31no—
39#519⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C32×D4—41no—
42C22×C3×Dic3—41no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C3⋊(#41⋊C3)—21no—
47C3×#217—41no—
48C3×#217—41no—
49C3×#217—41no—
50C3×Dic3⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3516—31noSylow
53C22×C3×Q8C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×#239C22×C311yescharacteristic
56C2×#41⋊C3—31no—
57C22×SL(2,𝔽3)—21no—
58C22×SL(2,𝔽3)—21no—
59C22×SL(2,𝔽3)C2×D311yescharacteristic
60C22×SL(2,𝔽3)C2×D311yescharacteristic
61C3×C4⋊Q8—31no—
62C3×#123—121no—
63C3×C22⋊Q8—31no—
64C3×C22⋊Q8—31no—
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×D4—31no—
67#947—31no—
68#947—31no—
69#946—31no—
70#946—31no—
71Q8×Dic3—31no—
72Q8×Dic3—31no—
73Q8×Dic3—31no—
74Q8×Dic3—31no—
75Q8×Dic3C22×C311yescharacteristic
76Q8×Dic3C22×C311yescharacteristic
77Q8×Dic3C22×C311yescharacteristic
78#939—31no—
79#935A411yescharacteristic
80#930—31no—
81#930—31no—
82C4×#67—31no—
83C4×#67—31no—
84#926—31no—
85#926—31no—
86C2×C4⋊Dic3—31no—
87C2×C4×Dic3—31no—
88#898—32no—
89#898—32no—
90#898—31no—
91#897—32no—
92#893—31no—
93#892—32no—
94D3×C4⋊C4—31no—
95#890—31no—
96#889—32no—
97#889—31no—
98#888—31no—
99#886—31no—
100#885—31no—
101#883—32no—
102#67⋊C4—32no—
103#879—31no—
104#878—31no—
105C4×D12—31no—
106#872—31no—
107#870—31no—
108C4×Dic6—31no—
109C4×SL(2,𝔽3)—21no—
110C4×SL(2,𝔽3)—31no—
111C4×SL(2,𝔽3)—31no—
112C4×SL(2,𝔽3)—31no—
113C4×SL(2,𝔽3)C2×D311yescharacteristic
114C23×C32—41no—
115C23×C32—41no—
116C32×D4—82no—
117C2×C32×C4—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—82no—
123C3×SL(2,𝔽3)#4111yescharacteristic
124#557—31no—
125#556—91no—
126#554—91no—
127#553—91no—
128#552—91no—
129D4×Q8C3×D311yescharacteristic
130#548—31no—
131#548—31no—
132#542—91no—
133#541—91no—
134#529—91no—
135#529—91no—
136#520—31no—
137#519—31no—
138C2×C4×Q8—31no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8—61no—
145C2×C3×Q8C23×C311yescharacteristic
146C2×C3×D4—41no—
147C2×C3×D4—81no—
148C2×C3×D4C2×A411yescharacteristic
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—121no—
152C2×#67—31no—
153C2×#67—31no—
154C22×Dic3—31no—
155C22×Dic3C2×A411yescharacteristic
156#239—21no—
157#239—61no—
158C2×D12—31no—
159C2×C4×D3—31no—
160C2×C4×D3—32no—
161C2×Dic6—31no—
162#41⋊C3—61no—
163C2×SL(2,𝔽3)—21no—
164C2×SL(2,𝔽3)—21no—
165C2×SL(2,𝔽3)—21no—
166C2×SL(2,𝔽3)—41no—
167C2×SL(2,𝔽3)—41no—
168C2×SL(2,𝔽3)C22×D311yescharacteristic
169C3×C4⋊C4—31no—
170C3×C4⋊C4—32no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—121no—
173C3×C4⋊C4—121no—
174C3×C22⋊C4—31no—
175C3×C22⋊C4—31no—
176C3×C22⋊C4—121no—
177C3×C22⋊C4—121no—
178C3×C22⋊C4—121no—
179C3×C42—31no—
180#217—31no—
181#217C2×A411yescharacteristic
182#217C2×A411yescharacteristic
183#217C2×A411yescharacteristic
184(C2×D3)⋊C4—32no—
185(C2×D3)⋊C4—32no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4—31no—
189(C2×D3)⋊C4—31no—
190C4⋊Dic3—32no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197Dic3⋊C4—32no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4C2×A411yescharacteristic
203Dic3⋊C4C2×A411yescharacteristic
204C4×Dic3—32no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C22×C32—42no—
211C22×C32—41no—
212C22×C32—81no—
213C22×C32—81no—
214C22×C32—81no—
215C22×C32—81no—
216C22×C32—81no—
217C32×C4—81no—
218C3×Dic3—81no—
219C3×Dic3—81no—
220C3×Dic3—81no—
221C3×Dic3—81no—
222C2×#41—31no—
223C22×Q8C2×C3×D311yescharacteristic
224C22×Q8C2×C3×D311yescharacteristic
225C4⋊Q8—31no—
226C4⋊Q8—91no—
227#126—91no—
228#126—91no—
229#126—92no—
230#126—92no—
231#125—91no—
232#125—91no—
233#125—91no—
234#125—92no—
235#124—91no—
236#124—91no—
237#124—91no—
238#123—31no—
239#123—91no—
240#123—92no—
241C22⋊Q8—31no—
242C22⋊Q8—31no—
243C22⋊Q8—91no—
244C22⋊Q8—92no—
245C22⋊Q8—91no—
246C22⋊Q8—91no—
247#121—91no—
248#121—91no—
249#121—91no—
250C4×Q8—31no—
251C4×Q8—31no—
252C4×Q8—31no—
253C4×Q8—91no—
254C4×Q8—91no—
255C4×Q8—91no—
256C4×Q8—91no—
257C4×Q8—91no—
258C4×Q8C2×C3×D311yescharacteristic
259C4×D4—31no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—92no—
264C4×D4—91no—
265#117—91no—
266#117—91no—
267#117—92no—
268#117—91no—
269C2×C4⋊C4—91no—
270C2×C4⋊C4—91no—
271C2×C42—91no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—81no—
275C23×C3—81no—
276C23×C3C22×A411yescharacteristic
277C23×C3C22×A411yescharacteristic
278C22×D3—31no—
279C3×Q8—31no—
280C3×Q8—61no—
281C3×Q8—61no—
282C3×Q8—61no—
283C3×Q8—61no—
284C3×Q8C3×#4111yescharacteristic
285C3×D4—22no—
286C3×D4—82no—
287C3×D4—84no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—41no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—81no—
298C2×C3×C4—121no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—242no—
303C2×C3×C4C22×A411yescharacteristic
304#67—62no—
305#67—121no—
306C2×Dic3—22no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—31no—
310C2×Dic3—121no—
311C2×Dic3C22×A411yescharacteristic
312C2×Dic3C22×A411yescharacteristic
313C2×Dic3C22×A411yescharacteristic
314C2×Dic3C22×A411yescharacteristic
315D12—61no—
316D12—61no—
317C4×D3—61no—
318C4×D3—61no—
319C4×D3—122no—
320Dic6—121no—
321SL(2,𝔽3)—21no—
322SL(2,𝔽3)#23711yescharacteristic
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—82no—
327C2×C32—161no—
328#41—61no—
329#41—181no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—31no—
333C2×Q8—31no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—91no—
337C2×Q8C22×C3×D311yescharacteristic
338C2×D4—91no—
339C2×D4—91no—
340C2×D4—91no—
341C2×D4D3×A411yescharacteristic
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—91no—
346C22×C4—92no—
347C22×C4—91no—
348C4⋊C4—31no—
349C4⋊C4—32no—
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—31no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—92no—
363C4⋊C4—92no—
364C4⋊C4—91no—
365C22⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C22⋊C4—91no—
371C22⋊C4—91no—
372C22⋊C4—91no—
373C22⋊C4—92no—
374C22⋊C4—92no—
375C22⋊C4—91no—
376C22⋊C4—91no—
377C42—31no—
378C42—91no—
379C42—91no—
380C42—91no—
381C42—91no—
382C42—91no—
383C42—92no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—81no—
399C22×C3—161no—
400C22×C3—161no—
401C22×C3—161no—
402C22×C3—161no—
403C22×C32−1+4⋊C312yes—
404C22×C3C23×A411yescharacteristic
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—61no—
408C3×C4—21no—
409C3×C4—31no—
410C3×C4—31no—
411C3×C4—61no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—81no—
415C3×C4—82no—
416C3×C4—241no—
417C3×C4—241no—
418C3×C4—241no—
419C3×C4—241no—
420Dic3—21no—
421Dic3—21no—
422Dic3—21no—
423Dic3—21no—
424Dic3—61no—
425Dic3—61no—
426Dic3—61no—
427C32—41noSylow
428C23—91no—
429C23C2×D3×A411yescharacteristic
430C23C2×D3×A411yescharacteristic
431Q8—31no—
432Q8—61no—
433Q8—61no—
434Q8—61no—
435Q8—61no—
436Q8—361no—
437Q8C3×#23711yescharacteristic
438D4—22no—
439D4—181no—
440D4—181no—
441D4—182no—
442D4—361no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—62no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—362no—
462C2×C4—361no—
463C2×C4C2×D3×A411yescharacteristic
464C2×C3—22no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—82no—
470C2×C3—81no—
471C2×C3—81no—
472C2×C3—81no—
473C2×C3—162no—
474C2×C3—161no—
475C2×C3—162no—
476C2×C3A4×#4111yescharacteristic
477C2×C3C2×2−1+4⋊C311yescharacteristic
478C2×C3#594511yescharacteristic
479D3—121no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22#1007⋊C312yes—
489C22C22×D3×A411yescharacteristic
490C4—21no—
491C4—31no—
492C4—31no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4—61no—
498C4—61no—
499C4—61no—
500C4—181no—
501C4—181no—
502C4—181no—
503C3—41no—
504C3—81no—
505C3#3516⋊C311yescharacteristic, complemented
506C2—22no—
507C2—41no—
508C2—361no—
509C2A4×#23711yescharacteristic
510C2C2×#1007⋊C311yescharacteristic
511C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×#41
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. C3×D4×SL(2,𝔽3)C2semidirect
2. #5812⋊C3C2semidirect
3. #5817⋊C3C2semidirect
4. #5817⋊C3C2semidirect
5. C2×Dic3×SL(2,𝔽3)C2semidirect
6. #5569⋊C3C2semidirect
7. #5574⋊C3C2semidirect
8. #86749C3semidirect
19. #5817C2×C3semidirect
20. #5817C2×C3semidirect
21. #5812C2×C3semidirect
22. C2×Q8×Dic3C2×C3semidirect
29. #5574C2×C3semidirect
30. #5569C2×C3semidirect
35. D4×SL(2,𝔽3)D3semidirect
505. C3#3516⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#5812⋊C32
#5817⋊C32
C2×Dic3×SL(2,𝔽3)2
#5569⋊C32
#5574⋊C32
#867493
#3516⋊C33
C3×#9354