Finite Group Databasebeta

← #85906⋊C3 #11923226

Subgroups — #85906⋊C3 #11923226

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×#520⋊C3C211yesmaximal, characteristic, complemented
2#5831⋊C3C211yesmaximal, characteristic, complemented
3#5831⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
6C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#5569⋊C3C211yesmaximal, characteristic, complemented
8#85906C311yesmaximal, characteristic, complemented
9#3516⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C3×#894—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#520C2×C311yescharacteristic, complemented
19#5831C2×C311yescharacteristic, complemented
20#5831C2×C311yescharacteristic, complemented
21Q8×D12C2×C311yescharacteristic
22#5572C2×C311yescharacteristic, complemented
23C4×D3×Q8C2×C311yescharacteristic, complemented
24#5569C2×C311yescharacteristic, complemented
25#5561—31no—
26#5558—31no—
27#5557—31no—
28#5542—31no—
29#5528—31no—
30#5518—31no—
31#5513—31no—
32#5511—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37#520⋊C3—21no—
38#520⋊C3D311yescharacteristic, complemented
39#519⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×D12—41no—
42C2×C3×C4×D3—41no—
43C3×#41⋊C3—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45D3×SL(2,𝔽3)—21no—
46D3×SL(2,𝔽3)—21no—
47C32×C4⋊C4—41no—
48C3×(C2×D3)⋊C4—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×(C2×D3)⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3516—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C2×D3×Q8C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#41⋊C3—21no—
57C2×#41⋊C3C2×D311yescharacteristic
58C22×SL(2,𝔽3)—31no—
59C22×SL(2,𝔽3)—31no—
60C3×#123—121no—
61C3×C4×Q8C22×C311yescharacteristic
62C3×C4×Q8C22×C311yescharacteristic
63C3×C4×D4—31no—
64C3×C4×D4—31no—
65C3×#117—31no—
66C2×C3×C4⋊C4—31no—
67Q8×Dic3C22×C311yescharacteristic
68Q8×Dic3C22×C311yescharacteristic
69#938—31no—
70#938—31no—
71#936—31no—
72#936—31no—
73#929—32no—
74C4×#67—32no—
75C4×#67—31no—
76C4×#67—31no—
77#926—32no—
78#924—32no—
79#924—31no—
80#924—31no—
81#898—31no—
82#897—31no—
83#897—31no—
84#896—31no—
85#894A411yescharacteristic
86#892—31no—
87D3×C4⋊C4—31no—
88#890—31no—
89#889—32no—
90#889—31no—
91#888—31no—
92#887—31no—
93#887—31no—
94#885—31no—
95#884—31no—
96#883—31no—
97#879—32no—
98#879—32no—
99#879—31no—
100C4×D12—31no—
101#872—31no—
102C42×D3—31no—
103#870—31no—
104#869—31no—
105C4×Dic6—31no—
106C4×Dic6—31no—
107C4×Dic6—31no—
108C4×SL(2,𝔽3)—21no—
109C4×SL(2,𝔽3)—21no—
110C4×SL(2,𝔽3)—31no—
111C4×SL(2,𝔽3)—31no—
112C4×SL(2,𝔽3)C2×D311yescharacteristic
113C4×SL(2,𝔽3)C2×D311yescharacteristic
114C22×C3×D3—41no—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C3×D12—82no—
122C3×C4×D3—82no—
123C3×SL(2,𝔽3)#4111yescharacteristic
124#557—31no—
125#556—91no—
126#554—91no—
127#553—91no—
128#552—91no—
129D4×Q8—31no—
130#548—31no—
131#548—31no—
132#542—91no—
133#541—91no—
134#529—91no—
135#529—91no—
136#520C3×D311yescharacteristic, complemented
137#519—31no—
138C2×C4×Q8—31no—
139C3×#41—21no—
140C3×#41—61no—
141C2×C3×Q8C23×C311yescharacteristic
142C2×C3×D4—31no—
143C2×C3×D4—121no—
144C22×C3×C4—32no—
145C22×C3×C4—31no—
146C22×C3×C4—121no—
147C2×#67—31no—
148C2×#67—31no—
149D3×Q8—21no—
150D3×Q8—21no—
151D3×Q8—61no—
152D3×Q8—61no—
153C2×D12C2×A411yescharacteristic
154C2×C4×D3—31no—
155C2×C4×D3—31no—
156C2×C4×D3—31no—
157C2×C4×D3C2×A411yescharacteristic
158C2×Dic6—31no—
159C2×Dic6—31no—
160C2×Dic6—31no—
161#41⋊C3—22no—
162#41⋊C3—21no—
163C2×SL(2,𝔽3)—21no—
164C2×SL(2,𝔽3)—61no—
165C2×SL(2,𝔽3)—61no—
166C2×SL(2,𝔽3)C22×D311yescharacteristic
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—81no—
172C3×C4⋊C4—121no—
173C3×C4⋊C4C2×A411yescharacteristic
174C3×C22⋊C4—31no—
175C3×C22⋊C4—31no—
176C3×C22⋊C4—121no—
177C3×C22⋊C4—121no—
178C3×C22⋊C4—121no—
179C3×C42—31no—
180C3×C42—31no—
181#217—32no—
182#217—32no—
183#217—31no—
184#217—31no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4C2×A411yescharacteristic
189(C2×D3)⋊C4C2×A411yescharacteristic
190(C2×D3)⋊C4C2×A411yescharacteristic
191C4⋊Dic3—32no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196Dic3⋊C4—32no—
197Dic3⋊C4—32no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4C2×A411yescharacteristic
204C4×Dic3—32no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C22×C32—41no—
210C2×C3×D3—42no—
211C2×C3×D3—81no—
212C2×C3×D3—81no—
213C2×C3×D3—81no—
214C2×C3×D3—81no—
215C2×C3×D3—81no—
216C32×C4—81no—
217C32×C4—81no—
218C32×C4—81no—
219C3×Dic3—81no—
220C3×Dic3—81no—
221C2×#41C2×C3×D311yescharacteristic
222C22×Q8—31no—
223C22×Q8—31no—
224C4⋊Q8—91no—
225C4⋊Q8—91no—
226#126—91no—
227#126—91no—
228#126—92no—
229#126—92no—
230#125—91no—
231#125—91no—
232#125—92no—
233#125—91no—
234#124—91no—
235#124—91no—
236#124—91no—
237#123—31no—
238#123—91no—
239#123—92no—
240C22⋊Q8—91no—
241C22⋊Q8—92no—
242C22⋊Q8—91no—
243C22⋊Q8—91no—
244C22⋊Q8—91no—
245C22⋊Q8—91no—
246#121—91no—
247#121—91no—
248#121—91no—
249C4×Q8—31no—
250C4×Q8—31no—
251C4×Q8—91no—
252C4×Q8—91no—
253C4×Q8—91no—
254C4×Q8—91no—
255C4×Q8—91no—
256C4×Q8C2×C3×D311yescharacteristic
257C4×Q8C2×C3×D311yescharacteristic
258C4×D4—31no—
259C4×D4—31no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—92no—
264#117—31no—
265#117—91no—
266#117—92no—
267#117—91no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—91no—
270C2×C42—91no—
271C23×C3—31no—
272C23×C3—121no—
273C23×C3—121no—
274C22×D3C22×A411yescharacteristic
275C22×D3C22×A411yescharacteristic
276C3×Q8—31no—
277C3×Q8C3×#4111yescharacteristic
278C3×D4—61no—
279C3×D4—61no—
280C3×D4—242no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4—81no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4—122no—
295C2×C3×C4—242no—
296C2×C3×C4C22×A411yescharacteristic
297C2×C3×C4C22×A411yescharacteristic
298C2×C3×C4C22×A411yescharacteristic
299#67—62no—
300#67—121no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3C22×A411yescharacteristic
306C2×Dic3C22×A411yescharacteristic
307D12—22no—
308C4×D3—22no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—61no—
313C4×D3—121no—
314Dic6—61no—
315Dic6—61no—
316Dic6—61no—
317Dic6—61no—
318Dic6—121no—
319SL(2,𝔽3)—21no—
320SL(2,𝔽3)#23511yescharacteristic
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—41no—
324C3×D3—82no—
325C3×D3—161no—
326#41—21no—
327#41—61no—
328C2×Q8—61no—
329C2×Q8—61no—
330C2×Q8—91no—
331C2×Q8—91no—
332C2×Q8—91no—
333C2×Q8—181no—
334C2×Q8—181no—
335C2×Q8C22×C3×D311yescharacteristic
336C2×D4—31no—
337C2×D4—31no—
338C2×D4—91no—
339C2×D4—91no—
340C22×C4—32no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4—91no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—92no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4D3×A411yescharacteristic
363C22⋊C4—31no—
364C22⋊C4—31no—
365C22⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—31no—
368C22⋊C4—91no—
369C22⋊C4—91no—
370C22⋊C4—91no—
371C22⋊C4—91no—
372C22⋊C4—91no—
373C22⋊C4—92no—
374C22⋊C4—92no—
375C42—31no—
376C42—31no—
377C42—91no—
378C42—91no—
379C42—91no—
380C42—92no—
381C42—91no—
382C22×C3—41no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—81no—
387C22×C3—122no—
388C22×C3—241no—
389C22×C3—241no—
390C22×C3—241no—
391C22×C3—241no—
392C22×C3—241no—
393C22×C3C23×A411yescharacteristic
394C2×D3—21no—
395C2×D3—21no—
396C2×D3—21no—
397C2×D3—21no—
398C2×D3—21no—
399C2×D32−1+4⋊C312yes—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—31no—
404C3×C4—31no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4—81no—
410C3×C4—82no—
411C3×C4—161no—
412C3×C4—161no—
413C3×C4—241no—
414C3×C4—241no—
415Dic3—21no—
416Dic3—21no—
417Dic3—61no—
418Dic3—61no—
419Dic3—61no—
420Dic3—61no—
421C32—41noSylow
422C23—31no—
423C23—31no—
424C23—31no—
425Q8—31no—
426Q8—181no—
427Q8—181no—
428Q8—181no—
429Q8—181no—
430Q8—361no—
431Q8C3×#23511yescharacteristic
432D4—62no—
433D4—61no—
434D4—61no—
435D4—182no—
436D4—361no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—62no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—122no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—361no—
455C2×C4C2×D3×A411yescharacteristic
456C2×C4C2×D3×A411yescharacteristic
457C2×C4C2×D3×A411yescharacteristic
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—81no—
462C2×C3—81no—
463C2×C3—81no—
464C2×C3—121no—
465C2×C3—242no—
466C2×C3—481no—
467C2×C3A4×#4111yescharacteristic
468C2×C3C2×2−1+4⋊C311yescharacteristic
469C2×C3#594511yescharacteristic
470D3—22no—
471D3—41no—
472C22—32no—
473C22—61no—
474C22—61no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—61no—
479C22—61no—
480C22—61no—
481C22C22×D3×A411yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—31no—
486C4—31no—
487C4—61no—
488C4—61no—
489C4—61no—
490C4—61no—
491C4—181no—
492C4—181no—
493C4—181no—
494C4—181no—
495C3—41no—
496C3—81no—
497C3#3516⋊C311yescharacteristic, complemented
498C2—62no—
499C2—121no—
500C2—121no—
501C2A4×#23511yescharacteristic
502C2#1059662311yescharacteristic
503C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#520C2×C3
socleC22×C3C23×A4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#520⋊C3C2semidirect
2. #5831⋊C3C2semidirect
3. #5831⋊C3C2semidirect
4. #5572⋊C3C2semidirect
5. SL(2,𝔽3)×D12C2semidirect
6. C4×D3×SL(2,𝔽3)C2semidirect
7. #5569⋊C3C2semidirect
8. #85906C3semidirect
18. C3×#520C2×C3semidirect
19. #5831C2×C3semidirect
20. #5831C2×C3semidirect
22. #5572C2×C3semidirect
23. C4×D3×Q8C2×C3semidirect
24. #5569C2×C3semidirect
38. #520⋊C3D3semidirect
136. #520C3×D3semidirect
497. C3#3516⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#5831⋊C32
#5572⋊C32
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
#5569⋊C32
#859063
#3516⋊C33
C3×#8944