Finite Group Databasebeta

← #86723⋊C3 #11923243

Subgroups — #86723⋊C3 #11923243

Order 1152.

535 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 75 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#5834⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4#5812⋊C3C211yesmaximal, characteristic, complemented
5#5817⋊C3C211yesmaximal, characteristic, complemented
6#5570⋊C3C211yesmaximal, characteristic, complemented
7#5574⋊C3C211yesmaximal, characteristic, complemented
8#86723C311yesmaximal, characteristic, complemented
9#3524⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#879—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
20Dic3×#41C2×C311yescharacteristic, complemented
21#5817C2×C311yescharacteristic
22#5812C2×C311yescharacteristic
23#5682—31no—
24#5681—31no—
25#5639—31no—
26#5624—31no—
27#5619—31no—
28#5614—31no—
29#5574C2×C311yescharacteristic, complemented
30#5570C2×C311yescharacteristic, complemented
31#5563—31no—
32#5538—31no—
33#557⋊C3—31no—
34C4⋊(#41⋊C3)—31no—
35#548⋊C3—21no—
36#548⋊C3—31no—
37#548⋊C3D311yescharacteristic, complemented
38#520⋊C3—31no—
39#519⋊C3—31no—
40C4×#41⋊C3—31no—
41C3×#41⋊C3—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44C3⋊(#41⋊C3)—21no—
45C32×C22⋊C4—41no—
46C3×#217—41no—
47C3×#217—41no—
48C3×C4⋊Dic3—41no—
49C3×Dic3⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3524—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×#239C22×C311yescharacteristic
56C2×#41⋊C3—21no—
57C2×#41⋊C3—31no—
58C2×#41⋊C3C2×D311yescharacteristic
59C22×SL(2,𝔽3)—21no—
60C22×SL(2,𝔽3)C2×D311yescharacteristic
61C3×#126—121no—
62C3×#124—31no—
63C3×C22⋊Q8—31no—
64C3×#121—31no—
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×D4—31no—
67#947—31no—
68#946—31no—
69Q8×Dic3—31no—
70Q8×Dic3—31no—
71Q8×Dic3C22×C311yescharacteristic
72Q8×Dic3C22×C311yescharacteristic
73Q8×Dic3C22×C311yescharacteristic
74#940—31no—
75#939—31no—
76#938—32no—
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—
86#926—31no—
87#926—31no—
88C2×C4×Dic3—31no—
89#898—31no—
90#893—31no—
91#892—32no—
92D3×C4⋊C4—31no—
93#890—31no—
94#890—31no—
95#889—31no—
96#886—32no—
97#886—31no—
98#885—32no—
99#885—31no—
100#883—32no—
101#67⋊C4—32no—
102#879—32no—
103#879A411yescharacteristic
104#878—31no—
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—
121C3×SL(2,𝔽3)#4111yescharacteristic
122#559—91no—
123#557—31no—
124#555—91no—
125#553—91no—
126#552—91no—
127#550—31no—
128#548—31no—
129#548C3×D311yescharacteristic
130#542—91no—
131#540—91no—
132#529—91no—
133#529—91no—
134#520—31no—
135#519—31no—
136C4×#41—31no—
137C3×#41—21no—
138C3×#41—61no—
139C2×C3×Q8—21no—
140C2×C3×Q8—31no—
141C2×C3×Q8—61no—
142C2×C3×Q8C23×C311yescharacteristic
143C2×C3×D4—31no—
144C2×C3×D4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C2×#67—32no—
148C2×#67—31no—
149C2×#67—31no—
150C22×Dic3—31no—
151C22×Dic3—31no—
152#239—21no—
153#239—61no—
154C2×D12—31no—
155C2×C4×D3—31no—
156C2×C4×D3—32no—
157C2×Dic6—31no—
158#41⋊C3—21no—
159#41⋊C3—41no—
160#41⋊C3—61no—
161C2×SL(2,𝔽3)—21no—
162C2×SL(2,𝔽3)—21no—
163C2×SL(2,𝔽3)—41no—
164C2×SL(2,𝔽3)C22×D311yescharacteristic
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—121no—
168C3×C4⋊C4—121no—
169C3×C4⋊C4—121no—
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⋊C4C2×A411yescharacteristic
177C3×C42—31no—
178C3×C42—121no—
179#217—32no—
180#217—31no—
181#217—31no—
182#217C2×A411yescharacteristic
183#217C2×A411yescharacteristic
184(C2×D3)⋊C4—32no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4—31no—
189C4⋊Dic3—32no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3C2×A411yescharacteristic
196Dic3⋊C4—31no—
197Dic3⋊C4—31no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4C2×A411yescharacteristic
201Dic3⋊C4C2×A411yescharacteristic
202C4×Dic3—32no—
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3C2×A411yescharacteristic
208C22×C32—41no—
209C22×C32—81no—
210C22×C32—81no—
211C22×C32—81no—
212C32×C4—81no—
213C32×C4—81no—
214C3×Dic3—81no—
215C3×Dic3—81no—
216C3×Dic3—81no—
217C3×Dic3—81no—
218C2×#41—31no—
219C2×#41C2×C3×D311yescharacteristic
220C22×Q8C2×C3×D311yescharacteristic
221C4⋊Q8—91no—
222#127—91no—
223#126—31no—
224#126—91no—
225#126—92no—
226#125—91no—
227#125—91no—
228#125—91no—
229#124—31no—
230#124—92no—
231#124—91no—
232#124—91no—
233#123—91no—
234#123—91no—
235#123—92no—
236#123—92no—
237C22⋊Q8—31no—
238C22⋊Q8—91no—
239C22⋊Q8—91no—
240#121—31no—
241#121—91no—
242#121—91no—
243#121—92no—
244#121—91no—
245#121—91no—
246C4×Q8—31no—
247C4×Q8—31no—
248C4×Q8—31no—
249C4×Q8—91no—
250C4×Q8—91no—
251C4×Q8—91no—
252C4×Q8C2×C3×D311yescharacteristic
253C4×D4—31no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—92no—
257C4×D4—91no—
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—91no—
261#117—91no—
262#117—91no—
263#117—92no—
264#117—91no—
265#117—91no—
266C2×C4⋊C4—91no—
267C2×C42—91no—
268C23×C3—31no—
269C23×C3—41no—
270C23×C3—81no—
271C23×C3C22×A411yescharacteristic
272C22×D3—31no—
273C3×Q8—31no—
274C3×Q8—61no—
275C3×Q8—61no—
276C3×Q8C3×#4111yescharacteristic
277C3×D4—61no—
278C3×D4—61no—
279C3×D4—121no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4—121no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4—121no—
295C2×C3×C4C22×A411yescharacteristic
296C2×C3×C4C22×A411yescharacteristic
297#67—121no—
298#67—121no—
299#67—122no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3—121no—
304C2×Dic3—121no—
305C2×Dic3C22×A411yescharacteristic
306C2×Dic3C22×A411yescharacteristic
307C2×Dic3C22×A411yescharacteristic
308C2×Dic3C22×A411yescharacteristic
309D12—61no—
310D12—61no—
311C4×D3—61no—
312C4×D3—61no—
313C4×D3—122no—
314Dic6—121no—
315SL(2,𝔽3)—21no—
316SL(2,𝔽3)#23711yescharacteristic
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—161no—
321#41—21no—
322#41—61no—
323#41—61no—
324#41—181no—
325C2×Q8—21no—
326C2×Q8—31no—
327C2×Q8—61no—
328C2×Q8—91no—
329C2×Q8C22×C3×D311yescharacteristic
330C2×D4—31no—
331C2×D4—31no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—92no—
335C2×D4—91no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—91no—
339C22×C4—91no—
340C22×C4—91no—
341C22×C4—92no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—91no—
348C4⋊C4—91no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—91no—
356C4⋊C4—92no—
357C22⋊C4—32no—
358C22⋊C4—31no—
359C22⋊C4—31no—
360C22⋊C4—31no—
361C22⋊C4—91no—
362C22⋊C4—91no—
363C22⋊C4—91no—
364C22⋊C4—92no—
365C22⋊C4—92no—
366C22⋊C4—91no—
367C22⋊C4—91no—
368C22⋊C4—91no—
369C22⋊C4D3×A411yescharacteristic
370C42—31no—
371C42—31no—
372C42—91no—
373C42—91no—
374C42—91no—
375C42—92no—
376C42—91no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—61no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—82no—
388C22×C3—81no—
389C22×C3—161no—
390C22×C3—161no—
391C22×C3C23×A411yescharacteristic
392C2×D3—61no—
393C2×D3—61no—
394C2×D3—61no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—31no—
398C3×C4—31no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4—161no—
404C3×C4—161no—
405C3×C4—241no—
406C3×C4—241no—
407C3×C4—241no—
408C3×C4—241no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—61no—
414Dic3—61no—
415Dic3—61no—
416C32—41noSylow
417C23—31no—
418C23—91no—
419C23C2×D3×A411yescharacteristic
420Q8—31no—
421Q8—61no—
422Q8—61no—
423Q8—361no—
424Q8C3×#23711yescharacteristic
425D4—61no—
426D4—61no—
427D4—121no—
428D4—181no—
429D4—181no—
430D4—361no—
431D4—361no—
432D4—362no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—361no—
450C2×C4—361no—
451C2×C4—362no—
452C2×C4C2×D3×A411yescharacteristic
453C2×C4C2×D3×A411yescharacteristic
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—81no—
459C2×C3—81no—
460C2×C3—81no—
461C2×C3—121no—
462C2×C3—161no—
463C2×C3—162no—
464C2×C3A4×#4111yescharacteristic
465C2×C3#594511yescharacteristic
466C2×C3#594511yescharacteristic
467D3—121no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—181no—
475C22—181no—
476C22—181no—
477C22C22×D3×A411yescharacteristic
478C4—21no—
479C4—21no—
480C4—31no—
481C4—31no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—61no—
486C4—61no—
487C4—61no—
488C4—181no—
489C4—181no—
490C4—181no—
491C3—41no—
492C3—81no—
493C3#3524⋊C311yescharacteristic, complemented
494C2—41no—
495C2—121no—
496C2—361no—
497C2A4×#23711yescharacteristic
498C2#5964⋊C311yescharacteristic
499C2#5968⋊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. #5834⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. #5812⋊C3C2semidirect
5. #5817⋊C3C2semidirect
6. #5570⋊C3C2semidirect
7. #5574⋊C3C2semidirect
8. #86723C3semidirect
19. #5834C2×C3semidirect
20. Dic3×#41C2×C3semidirect
29. #5574C2×C3semidirect
30. #5570C2×C3semidirect
37. #548⋊C3D3semidirect
493. C3#3524⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#5834⋊C32
Dic3×#41⋊C32
#5812⋊C32
#5817⋊C32
#5570⋊C32
#5574⋊C32
#867233
#3524⋊C33
C3×#8794