Finite Group Databasebeta

← #86722⋊C3 #11923240

Subgroups — #86722⋊C3 #11923240

Order 1152.

583 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 81 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#5827⋊C3C211yesmaximal, characteristic, complemented
3#5834⋊C3C211yesmaximal, characteristic, complemented
4#5817⋊C3C211yesmaximal, characteristic, complemented
5C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5574⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
8#86722C311yesmaximal, characteristic, complemented
9#3522⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#878—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
19#5834C2×C311yescharacteristic, complemented
20#5827C2×C311yescharacteristic, complemented
21#5817C2×C311yescharacteristic
22C2×Q8×Dic3C2×C311yescharacteristic
23#5682—31no—
24#5681—31no—
25#5643—31no—
26#5635—31no—
27#5611—31no—
28#5596—31no—
29#5574C2×C311yescharacteristic, complemented
30Q8×Dic6C2×C311yescharacteristic, complemented
31#5555—31no—
32#5538—31no—
33Q8×SL(2,𝔽3)—31no—
34C4⋊(#41⋊C3)—31no—
35#548⋊C3—21no—
36#548⋊C3—31no—
37#548⋊C3D311yescharacteristic, complemented
38#520⋊C3—31no—
39#520⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C22×C3×Dic3—41no—
42C2×C3×Dic6—41no—
43C3×#41⋊C3—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C3⋊(#41⋊C3)—21no—
47C32×C22⋊C4—41no—
48C3×#217—41no—
49C3×C4⋊Dic3—41no—
50C3×Dic3⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3522—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×#239C22×C311yescharacteristic
56C2×#237—32no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3—31no—
59C2×#41⋊C3C2×D311yescharacteristic
60C22×SL(2,𝔽3)—21no—
61C22×SL(2,𝔽3)C2×D311yescharacteristic
62C3×#124—31no—
63C3×C22⋊Q8—31no—
64C3×C22⋊Q8—121no—
65C3×#121—31no—
66C3×C4×Q8C22×C311yescharacteristic
67C3×C4×D4—31no—
68#947—31no—
69#946—31no—
70Q8×Dic3—31no—
71Q8×Dic3—31no—
72Q8×Dic3C22×C311yescharacteristic
73Q8×Dic3C22×C311yescharacteristic
74Q8×Dic3C22×C311yescharacteristic
75#940—31no—
76#939—31no—
77#938—31no—
78#935—31no—
79D4×Dic3—31no—
80D4×Dic3—31no—
81#930—31no—
82#930—31no—
83C4×#67—31no—
84C4×#67—31no—
85#926—31no—
86C2×C4⋊Dic3—31no—
87C2×C4⋊Dic3—31no—
88C2×C4×Dic3—31no—
89#898—31no—
90#897—32no—
91#893—31no—
92D3×C4⋊C4—31no—
93#889—31no—
94#888—31no—
95#888—31no—
96#886—32no—
97#886—31no—
98#885—32no—
99#885—31no—
100#883—32no—
101#67⋊C4—32no—
102#879—31no—
103#878—32no—
104#878A411yescharacteristic
105C4×D12—31no—
106#872—31no—
107#869—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—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—82no—
122C3×Dic6—82no—
123C3×SL(2,𝔽3)#4111yescharacteristic, complemented
124#559—91no—
125Q82—31no—
126#554—91no—
127#550—31no—
128#548—31no—
129#548C3×D311yescharacteristic
130#547—91no—
131#547—91no—
132#542—91no—
133#542—91no—
134#532—91no—
135#529—91no—
136#520—31no—
137#520—31no—
138C2×C4×Q8—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—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—121no—
151C2×#67—31no—
152C2×#67—32no—
153C2×#67—31no—
154C22×Dic3—31no—
155C22×Dic3—32no—
156C22×Dic3C2×A411yescharacteristic
157#239—21no—
158#239—61no—
159#237—124no—
160C2×D12—31no—
161C2×C4×D3—31no—
162C2×C4×D3—32no—
163C2×Dic6—32no—
164C2×Dic6C2×A411yescharacteristic
165#41⋊C3—21no—
166#41⋊C3—41no—
167#41⋊C3—61no—
168C2×SL(2,𝔽3)—21no—
169C2×SL(2,𝔽3)—21no—
170C2×SL(2,𝔽3)—41no—
171C2×SL(2,𝔽3)C22×D311yescharacteristic
172C3×C4⋊C4—31no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—121no—
175C3×C4⋊C4—121no—
176C3×C4⋊C4—121no—
177C3×C22⋊C4—32no—
178C3×C22⋊C4—31no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—81no—
181C3×C22⋊C4—121no—
182C3×C22⋊C4C2×A411yescharacteristic
183C3×C42—31no—
184#217—31no—
185#217—31no—
186#217—31no—
187#217C2×A411yescharacteristic
188(C2×D3)⋊C4—32no—
189(C2×D3)⋊C4—31no—
190(C2×D3)⋊C4—31no—
191(C2×D3)⋊C4—31no—
192(C2×D3)⋊C4—31no—
193C4⋊Dic3—32no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199C4⋊Dic3C2×A411yescharacteristic
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4—31no—
204Dic3⋊C4C2×A411yescharacteristic
205Dic3⋊C4C2×A411yescharacteristic
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C4×Dic3—31no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—81no—
214C22×C32—81no—
215C32×C4—81no—
216C32×C4—81no—
217C3×Dic3—42no—
218C3×Dic3—81no—
219C3×Dic3—81no—
220C3×Dic3—81no—
221C2×#41—31no—
222C2×#41—92no—
223C2×#41C2×C3×D311yescharacteristic
224C22×Q8C2×C3×D311yescharacteristic
225C4⋊Q8—91no—
226C4⋊Q8—91no—
227C4⋊Q8—91no—
228#127—91no—
229#126—91no—
230#126—91no—
231#125—91no—
232#124—31no—
233#124—91no—
234#124—92no—
235#124—91no—
236#123—91no—
237#123—92no—
238#123—92no—
239#123—91no—
240C22⋊Q8—31no—
241C22⋊Q8—31no—
242C22⋊Q8—92no—
243C22⋊Q8—92no—
244C22⋊Q8—91no—
245#121—31no—
246#121—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×Q8C2×C3×D311yescharacteristic
257C4×D4—31no—
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—92no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—91no—
264C4×D4—91no—
265#117—91no—
266#117—91no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C4⋊C4—91no—
270C2×C42—91no—
271C23×C3—31no—
272C23×C3—41no—
273C23×C3—81no—
274C23×C3C22×A411yescharacteristic
275C22×D3—31no—
276C3×Q8—31no—
277C3×Q8—61no—
278C3×Q8—61no—
279C3×Q8—242no—
280C3×Q8C3×#4111yescharacteristic
281C3×D4—61no—
282C3×D4—61no—
283C3×D4—62no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—81no—
294C2×C3×C4—81no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4—121no—
298C2×C3×C4—121no—
299C2×C3×C4—242no—
300C2×C3×C4C22×A411yescharacteristic
301C2×C3×C4C22×A411yescharacteristic
302#67—62no—
303#67—62no—
304#67—122no—
305C2×Dic3—22no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—62no—
310C2×Dic3—122no—
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—22no—
321Dic6—122no—
322SL(2,𝔽3)—21no—
323SL(2,𝔽3)#23711yescharacteristic, complemented
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—82no—
328#41—21no—
329#41—61no—
330#41—61no—
331#41—181no—
332#41—364no—
333C2×Q8—21no—
334C2×Q8—31no—
335C2×Q8—31no—
336C2×Q8—61no—
337C2×Q8—92no—
338C2×Q8C22×C3×D311yescharacteristic
339C2×D4—31no—
340C2×D4—31no—
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—91no—
344C2×D4—92no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—91no—
349C22×C4—92no—
350C22×C4—91no—
351C22×C4—92no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—31no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C4⋊C4—92no—
365C4⋊C4—91no—
366C4⋊C4—91no—
367C22⋊C4—32no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C22⋊C4—91no—
371C22⋊C4—91no—
372C22⋊C4—91no—
373C22⋊C4—91no—
374C22⋊C4—91no—
375C22⋊C4—92no—
376C22⋊C4—91no—
377C22⋊C4—91no—
378C22⋊C4D3×A411yescharacteristic
379C42—31no—
380C42—91no—
381C42—91no—
382C42—91no—
383C42—91no—
384C42—91no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—42no—
388C22×C3—41no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—82no—
395C22×C3—81no—
396C22×C3—161no—
397C22×C3—161no—
398C22×C3#99412yes—
399C22×C3C23×A411yescharacteristic
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—61no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—122no—
412C3×C4—161no—
413C3×C4—161no—
414C3×C4—241no—
415C3×C4—241no—
416C3×C4—241no—
417Dic3—21no—
418Dic3—21no—
419Dic3—21no—
420Dic3—61no—
421Dic3—61no—
422Dic3—61no—
423Dic32−1+4⋊C312yes—
424C32—41noSylow
425C23—31no—
426C23—91no—
427C23C2×D3×A411yescharacteristic
428Q8—31no—
429Q8—62no—
430Q8—61no—
431Q8—61no—
432Q8—362no—
433Q8C3×#23711yescharacteristic
434D4—61no—
435D4—61no—
436D4—62no—
437D4—182no—
438D4—181no—
439D4—181no—
440D4—182no—
441D4—362no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—62no—
454C2×C4—91no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—181no—
458C2×C4—181no—
459C2×C4—182no—
460C2×C4—362no—
461C2×C4—362no—
462C2×C4C2×D3×A411yescharacteristic
463C2×C4C2×D3×A411yescharacteristic
464C2×C3—22no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—82no—
469C2×C3—81no—
470C2×C3—81no—
471C2×C3—81no—
472C2×C3—121no—
473C2×C3—162no—
474C2×C3A4×#4111yescharacteristic
475C2×C3C2×#99411yescharacteristic
476C2×C3C2×2−1+4⋊C311yescharacteristic
477D3—121no—
478C22—21no—
479C22—21no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—181no—
484C22—181no—
485C22—181no—
486C22C3⋊#99412yes—
487C22C22×D3×A411yescharacteristic
488C4—21no—
489C4—21no—
490C4—31no—
491C4—31no—
492C4—32no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4—61no—
498C4—181no—
499C4—181no—
500C4—181no—
501C3—41no—
502C3—81no—
503C3#3522⋊C311yescharacteristic, complemented
504C2—22no—
505C2—121no—
506C2—361no—
507C2A4×#23711yescharacteristic
508C2C2×C3⋊#99411yescharacteristic
509C2D3×2−1+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. #5827⋊C3C2semidirect
3. #5834⋊C3C2semidirect
4. #5817⋊C3C2semidirect
5. C2×Dic3×SL(2,𝔽3)C2semidirect
6. #5574⋊C3C2semidirect
7. SL(2,𝔽3)×Dic6C2semidirect
8. #86722C3semidirect
19. #5834C2×C3semidirect
20. #5827C2×C3semidirect
29. #5574C2×C3semidirect
30. Q8×Dic6C2×C3semidirect
37. #548⋊C3D3semidirect
123. C3×SL(2,𝔽3)#41semidirect
323. SL(2,𝔽3)#237semidirect
503. C3#3522⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#5827⋊C32
#5834⋊C32
#5817⋊C32
C2×Dic3×SL(2,𝔽3)2
#5574⋊C32
SL(2,𝔽3)×Dic62
#867223
#3522⋊C33
C3×#8784