Finite Group Databasebeta

← #86720⋊C3 #11923241

Subgroups — #86720⋊C3 #11923241

Order 1152.

555 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 77 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×#548⋊C3C211yesmaximal, characteristic, complemented
2#5831⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
5#5812⋊C3C211yesmaximal, characteristic, complemented
6#5572⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
8#86720C311yesmaximal, characteristic, complemented
9#3518⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C3×#885—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15Dic3×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#548C2×C311yescharacteristic, complemented
19#5831C2×C311yescharacteristic, complemented
20Dic3×#41C2×C311yescharacteristic, complemented
21Q8×#67C2×C311yescharacteristic
22#5812C2×C311yescharacteristic, complemented
23#5682—31no—
24#5671—31no—
25#5641—31no—
26#5622—31no—
27#5616—31no—
28#5603—31no—
29#5572C2×C311yescharacteristic, complemented
30Q8×Dic6C2×C311yescharacteristic, complemented
31#5542—31no—
32#5538—31no—
33Q8×SL(2,𝔽3)—31no—
34D4×SL(2,𝔽3)—31no—
35#548⋊C3—21no—
36#548⋊C3—31no—
37#548⋊C3D311yescharacteristic, complemented
38#519⋊C3—31no—
39#519⋊C3—31no—
40C4×#41⋊C3—31no—
41C2×C3×#67—41no—
42C2×C3×Dic6—41no—
43C3×#41⋊C3—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46D3×SL(2,𝔽3)—21no—
47C32×C22⋊C4—41no—
48C3×#217—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×(C2×D3)⋊C4—41no—
51C3×C4×Dic3—41no—
52#3518—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#41⋊C3—21no—
57C2×#41⋊C3C2×D311yescharacteristic
58C22×SL(2,𝔽3)—21no—
59C22×SL(2,𝔽3)—31no—
60C22×SL(2,𝔽3)C2×D311yescharacteristic
61C3×#124—31no—
62C3×#124—121no—
63C3×C22⋊Q8—31no—
64C3×#121—31no—
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×D4—31no—
67#946—31no—
68Q8×Dic3—31no—
69Q8×Dic3—31no—
70Q8×Dic3C22×C311yescharacteristic
71Q8×Dic3C22×C311yescharacteristic
72Q8×Dic3C22×C311yescharacteristic
73#944—31no—
74#939—31no—
75#938—31no—
76#936—32no—
77#936—31no—
78#935—31no—
79D4×Dic3—31no—
80D4×Dic3—31no—
81C4×#67—31no—
82C4×#67—31no—
83#926—31no—
84#926—31no—
85#926—31no—
86#924—31no—
87#924—31no—
88C2×C4×Dic3—31no—
89#898—31no—
90#892—31no—
91#889—31no—
92#888—31no—
93#888—31no—
94#887—32no—
95#887—31no—
96#886—31no—
97#885—32no—
98#885A411yescharacteristic
99#879—32no—
100#879—32no—
101#879—31no—
102#878—32no—
103#878—31no—
104#877—32no—
105#872—31no—
106#869—31no—
107C4×Dic6—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—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C3×#67—82no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C3×Dic6—82no—
123C3×SL(2,𝔽3)#4111yescharacteristic
124Q82—31no—
125#556—91no—
126#556—91no—
127#556—91no—
128#555—91no—
129D4×Q8—31no—
130#548—31no—
131#548C3×D311yescharacteristic, complemented
132#542—91no—
133#542—91no—
134#533—91no—
135#529—91no—
136#519—31no—
137#519—31no—
138C4×#41—31no—
139C3×#41—21no—
140C3×#41—61no—
141C2×C3×Q8—21no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8—121no—
145C2×C3×Q8C23×C311yescharacteristic
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—121no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C2×#67—31no—
152C2×#67C2×A411yescharacteristic
153C22×Dic3—31no—
154C22×Dic3—31no—
155D3×Q8—21no—
156D3×Q8—61no—
157C2×C4×D3—31no—
158C2×Dic6—31no—
159C2×Dic6—32no—
160C2×Dic6C2×A411yescharacteristic
161#41⋊C3—21no—
162#41⋊C3—41no—
163C2×SL(2,𝔽3)—21no—
164C2×SL(2,𝔽3)—21no—
165C2×SL(2,𝔽3)—41no—
166C2×SL(2,𝔽3)—61no—
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C22⋊C4—32no—
171C3×C22⋊C4—31no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—81no—
174C3×C22⋊C4—121no—
175C3×C22⋊C4—121no—
176C3×C22⋊C4—121no—
177C3×C22⋊C4C2×A411yescharacteristic
178C3×C42—31no—
179C3×C42—121no—
180#217—32no—
181#217—32no—
182#217—31no—
183#217—31no—
184#217—31no—
185#217C2×A411yescharacteristic
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4C2×A411yescharacteristic
189(C2×D3)⋊C4C2×A411yescharacteristic
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196Dic3⋊C4—32no—
197Dic3⋊C4—31no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203C4×Dic3—32no—
204C4×Dic3—32no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3C2×A411yescharacteristic
210C22×C32—41no—
211C22×C32—81no—
212C22×C32—81no—
213C22×C32—81no—
214C2×C3×D3—81no—
215C2×C3×D3—81no—
216C2×C3×D3—81no—
217C32×C4—81no—
218C32×C4—81no—
219C3×Dic3—42no—
220C3×Dic3—81no—
221C3×Dic3—81no—
222C2×#41C2×C3×D311yescharacteristic
223C22×Q8—31no—
224C22×Q8C2×C3×D311yescharacteristic
225C4⋊Q8—91no—
226C4⋊Q8—91no—
227C4⋊Q8—91no—
228C4⋊Q8—91no—
229#126—91no—
230#126—92no—
231#126—92no—
232#126—91no—
233#125—91no—
234#124—31no—
235#124—31no—
236#124—92no—
237#124—92no—
238#124—91no—
239#123—91no—
240#123—91no—
241C22⋊Q8—31no—
242C22⋊Q8—91no—
243C22⋊Q8—92no—
244C22⋊Q8—91no—
245C22⋊Q8—91no—
246C22⋊Q8—91no—
247#121—31no—
248#121—91no—
249#121—91no—
250C4×Q8—31no—
251C4×Q8—31no—
252C4×Q8—31no—
253C4×Q8—91no—
254C4×Q8—91no—
255C4×Q8—92no—
256C4×Q8—91no—
257C4×Q8—91no—
258C4×Q8—91no—
259C4×Q8C2×C3×D311yescharacteristic
260C4×D4—31no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—91no—
264C4×D4—91no—
265#117—91no—
266#117—91no—
267#117—92no—
268#117—91no—
269#117—91no—
270#117—91no—
271C2×C42—91no—
272C23×C3—31no—
273C23×C3—41no—
274C23×C3—81no—
275C23×C3—121no—
276C23×C3C22×A411yescharacteristic
277C22×D3C22×A411yescharacteristic
278C3×Q8—31no—
279C3×Q8—61no—
280C3×Q8—61no—
281C3×Q8—242no—
282C3×Q8C3×#4111yescharacteristic
283C3×D4—61no—
284C3×D4—61no—
285C3×D4—121no—
286C3×D4—242no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—81no—
297C2×C3×C4—81no—
298C2×C3×C4—121no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4C22×A411yescharacteristic
302C2×C3×C4C22×A411yescharacteristic
303#67—22no—
304#67—121no—
305C2×Dic3—31no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—62no—
310C2×Dic3—121no—
311C2×Dic3C22×A411yescharacteristic
312C2×Dic3C22×A411yescharacteristic
313C2×Dic3C22×A411yescharacteristic
314C4×D3—61no—
315C4×D3—61no—
316Dic6—22no—
317Dic6—61no—
318Dic6—61no—
319Dic6—122no—
320SL(2,𝔽3)—21no—
321SL(2,𝔽3)#23711yescharacteristic
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—161no—
326C3×D3—161no—
327#41—21no—
328#41—61no—
329C2×Q8—21no—
330C2×Q8—31no—
331C2×Q8—31no—
332C2×Q8—61no—
333C2×Q8—61no—
334C2×Q8—92no—
335C2×Q8—91no—
336C2×Q8—181no—
337C2×Q8C22×C3×D311yescharacteristic
338C2×D4—31no—
339C2×D4—31no—
340C2×D4—31no—
341C2×D4—91no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—91no—
345C22×C4—91no—
346C22×C4—91no—
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—92no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C22⋊C4—32no—
363C22⋊C4—31no—
364C22⋊C4—31no—
365C22⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—91no—
368C22⋊C4—91no—
369C22⋊C4—91no—
370C22⋊C4—92no—
371C22⋊C4—92no—
372C22⋊C4—91no—
373C22⋊C4—91no—
374C22⋊C4D3×A411yescharacteristic
375C42—31no—
376C42—31no—
377C42—91no—
378C42—91no—
379C42—91no—
380C42—92no—
381C42—92no—
382C42—91no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—41no—
387C22×C3—61no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—82no—
394C22×C3—81no—
395C22×C3—161no—
396C22×C3—161no—
397C22×C3—241no—
398C22×C3—241no—
399C22×C3—241no—
400C22×C3C23×A411yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—31no—
407C3×C4—31no—
408C3×C4—61no—
409C3×C4—61no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4—122no—
413C3×C4—161no—
414C3×C4—161no—
415C3×C4—241no—
416C3×C4—241no—
417Dic3—21no—
418Dic3—21no—
419Dic3—61no—
420Dic3—61no—
421Dic3—61no—
422Dic3—61no—
423Dic32−1+4⋊C312yes—
424C32—41noSylow
425C23—31no—
426C23—31no—
427C23C2×D3×A411yescharacteristic
428Q8—31no—
429Q8—62no—
430Q8—61no—
431Q8—61no—
432Q8—181no—
433Q8—181no—
434Q8—362no—
435Q8C3×#23711yescharacteristic
436D4—62no—
437D4—61no—
438D4—61no—
439D4—121no—
440D4—361no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—91no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—182no—
458C2×C4—361no—
459C2×C4C2×D3×A411yescharacteristic
460C2×C4C2×D3×A411yescharacteristic
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—81no—
466C2×C3—81no—
467C2×C3—81no—
468C2×C3—121no—
469C2×C3—161no—
470C2×C3—162no—
471C2×C3—481no—
472C2×C3A4×#4111yescharacteristic
473C2×C3C2×2−1+4⋊C311yescharacteristic
474C2×C3#594511yescharacteristic
475D3—41no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—61no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—61no—
484C22—61no—
485C22C22×D3×A411yescharacteristic
486C4—21no—
487C4—21no—
488C4—31no—
489C4—31no—
490C4—32no—
491C4—61no—
492C4—61no—
493C4—61no—
494C4—61no—
495C4—181no—
496C4—181no—
497C4—181no—
498C4—181no—
499C3—41no—
500C3—81no—
501C3#3518⋊C311yescharacteristic, complemented
502C2—41no—
503C2—121no—
504C2—121no—
505C2A4×#23711yescharacteristic
506C2#5964⋊C311yescharacteristic
507C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#548C2×C3
socleC22×C3C23×A4
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#548⋊C3C2semidirect
2. #5831⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. SL(2,𝔽3)×#67C2semidirect
5. #5812⋊C3C2semidirect
6. #5572⋊C3C2semidirect
7. SL(2,𝔽3)×Dic6C2semidirect
8. #86720C3semidirect
18. C3×#548C2×C3semidirect
19. #5831C2×C3semidirect
20. Dic3×#41C2×C3semidirect
22. #5812C2×C3semidirect
29. #5572C2×C3semidirect
30. Q8×Dic6C2×C3semidirect
37. #548⋊C3D3semidirect
131. #548C3×D3semidirect
501. C3#3518⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#5831⋊C32
Dic3×#41⋊C32
SL(2,𝔽3)×#672
#5812⋊C32
#5572⋊C32
SL(2,𝔽3)×Dic62
#867203
#3518⋊C33
C3×#8854