Finite Group Databasebeta

← #85908⋊C3 #11923219

Subgroups — #85908⋊C3 #11923219

Order 1152.

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

normal subgroups only

This group has 579 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×#520⋊C3C211yesmaximal, characteristic, complemented
2#5831⋊C3C211yesmaximal, characteristic, complemented
3#5834⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C211yesmaximal, characteristic, complemented
5C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5574⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
8#85908C311yesmaximal, characteristic, complemented
9#3522⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#896—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#5834C2×C311yescharacteristic, complemented
20#5831C2×C311yescharacteristic
21#5576C2×C311yescharacteristic
22#5574C2×C311yescharacteristic, complemented
23C4×D3×Q8C2×C311yescharacteristic
24Q8×Dic6C2×C311yescharacteristic, complemented
25#5561—31no—
26#5559—31no—
27#5551—31no—
28#5548—31no—
29#5534—31no—
30#5528—31no—
31#5511—31no—
32#5507—31no—
33Q8×SL(2,𝔽3)—31no—
34C4⋊(#41⋊C3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37#520⋊C3—21no—
38#520⋊C3—31no—
39#520⋊C3D311yescharacteristic, complemented
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×C4×D3—41no—
42C2×C3×Dic6—41no—
43C3×#41⋊C3—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45D3×SL(2,𝔽3)—21no—
46C3⋊(#41⋊C3)—21no—
47C32×C4⋊C4—41no—
48C3×(C2×D3)⋊C4—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3522—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#235—32no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3—31no—
59C2×#41⋊C3C2×D311yescharacteristic
60C22×SL(2,𝔽3)—31no—
61C3×C22⋊Q8—121no—
62C3×C4×Q8C22×C311yescharacteristic
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×D4—31no—
65C3×C4×D4—31no—
66C3×#117—31no—
67C2×C3×C4⋊C4—31no—
68Q8×Dic3C22×C311yescharacteristic
69Q8×Dic3C22×C311yescharacteristic
70#940—31no—
71#938—31no—
72#938—31no—
73#936—31no—
74#930—31no—
75#929—32no—
76C4×#67—32no—
77C4×#67—31no—
78C4×#67—31no—
79#924—32no—
80#924—31no—
81#898—31no—
82#897—31no—
83#896—32no—
84#896A411yescharacteristic
85#895—31no—
86#894—31no—
87#893—31no—
88D3×C4⋊C4—31no—
89D3×C4⋊C4—31no—
90#889—31no—
91#888—31no—
92#888—31no—
93#887—31no—
94#885—32no—
95#885—31no—
96#884—31no—
97#883—32no—
98#883—31no—
99#879—31no—
100#875—31no—
101C4×D12—31no—
102C4×D12—31no—
103#872—31no—
104C42×D3—31no—
105#869—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—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C3×C4×D3—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#548—31no—
130#547—91no—
131#547—91no—
132#542—91no—
133#542—91no—
134#532—91no—
135#529—91no—
136#520—31no—
137#520C3×D311yescharacteristic, complemented
138C2×C4×Q8—31no—
139C3×#41—21no—
140C3×#41—61no—
141C2×C3×Q8—121no—
142C2×C3×Q8C23×C311yescharacteristic
143C2×C3×D4—31no—
144C22×C3×C4—32no—
145C22×C3×C4—31no—
146C22×C3×C4—121no—
147C2×#67—31no—
148C2×#67—32no—
149C2×#67—31no—
150#239—21no—
151#239—61no—
152D3×Q8—21no—
153D3×Q8—61no—
154#235—124no—
155C2×D12—31no—
156C2×D12—31no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—31no—
160C2×C4×D3—32no—
161C2×C4×D3C2×A411yescharacteristic
162C2×Dic6—31no—
163C2×Dic6—32no—
164C2×Dic6C2×A411yescharacteristic
165#41⋊C3—22no—
166#41⋊C3—21no—
167#41⋊C3—61no—
168C2×SL(2,𝔽3)—21no—
169C2×SL(2,𝔽3)—61no—
170C2×SL(2,𝔽3)C22×D311yescharacteristic
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—81no—
176C3×C4⋊C4—121no—
177C3×C4⋊C4—121no—
178C3×C4⋊C4C2×A411yescharacteristic
179C3×C22⋊C4—31no—
180C3×C22⋊C4—31no—
181C3×C22⋊C4—121no—
182C3×C22⋊C4—121no—
183C3×C42—31no—
184C3×C42—31no—
185#217—32no—
186#217—31no—
187#217—31no—
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—
193(C2×D3)⋊C4C2×A411yescharacteristic
194(C2×D3)⋊C4C2×A411yescharacteristic
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199Dic3⋊C4—32no—
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—
210C22×C32—41no—
211C2×C3×D3—42no—
212C2×C3×D3—81no—
213C2×C3×D3—81no—
214C32×C4—81no—
215C32×C4—81no—
216C32×C4—81no—
217C3×Dic3—42no—
218C3×Dic3—81no—
219C3×Dic3—81no—
220C2×#41—31no—
221C2×#41—92no—
222C2×#41C2×C3×D311yescharacteristic
223C22×Q8—31no—
224C4⋊Q8—91no—
225C4⋊Q8—91no—
226C4⋊Q8—91no—
227#127—91no—
228#126—91no—
229#126—91no—
230#125—91no—
231#124—91no—
232#124—91no—
233#124—92no—
234#124—91no—
235#123—91no—
236#123—91no—
237#123—92no—
238#123—92no—
239C22⋊Q8—31no—
240C22⋊Q8—92no—
241C22⋊Q8—91no—
242C22⋊Q8—92no—
243C22⋊Q8—91no—
244#121—91no—
245#121—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×Q8C2×C3×D311yescharacteristic
255C4×Q8C2×C3×D311yescharacteristic
256C4×D4—31no—
257C4×D4—31no—
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—92no—
263C4×D4—91no—
264#117—31no—
265#117—91no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C42—91no—
270C23×C3—31no—
271C23×C3—121no—
272C22×D3—31no—
273C22×D3C22×A411yescharacteristic
274C3×Q8—31no—
275C3×Q8—242no—
276C3×Q8C3×#4111yescharacteristic
277C3×D4—61no—
278C3×D4—61no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×C3×C4—81no—
290C2×C3×C4—121no—
291C2×C3×C4—121no—
292C2×C3×C4—121no—
293C2×C3×C4—122no—
294C2×C3×C4—242no—
295C2×C3×C4C22×A411yescharacteristic
296C2×C3×C4C22×A411yescharacteristic
297C2×C3×C4C22×A411yescharacteristic
298#67—62no—
299#67—62no—
300#67—122no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3—31no—
304C2×Dic3C22×A411yescharacteristic
305C2×Dic3C22×A411yescharacteristic
306C2×Dic3C22×A411yescharacteristic
307D12—61no—
308D12—61no—
309D12—62no—
310C4×D3—22no—
311C4×D3—61no—
312C4×D3—61no—
313C4×D3—61no—
314C4×D3—61no—
315C4×D3—62no—
316C4×D3—122no—
317Dic6—22no—
318Dic6—61no—
319Dic6—61no—
320Dic6—122no—
321SL(2,𝔽3)—21no—
322SL(2,𝔽3)#23511yescharacteristic, complemented
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—41no—
326C3×D3—82no—
327#41—21no—
328#41—61no—
329#41—61no—
330#41—181no—
331#41—364no—
332C2×Q8—31no—
333C2×Q8—61no—
334C2×Q8—92no—
335C2×Q8—91no—
336C2×Q8—181no—
337C2×Q8C22×C3×D311yescharacteristic
338C2×D4—31no—
339C2×D4—91no—
340C2×D4—91no—
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—92no—
344C22×C4—32no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—91no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—92no—
351C4⋊C4—31no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—92no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4D3×A411yescharacteristic
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—
377C22⋊C4—91no—
378C42—31no—
379C42—31no—
380C42—91no—
381C42—91no—
382C42—91no—
383C42—91no—
384C22×C3—41no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—61no—
388C22×C3—81no—
389C22×C3—122no—
390C22×C3—241no—
391C22×C3—241no—
392C22×C3C23×A411yescharacteristic
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—61no—
396C2×D3—61no—
397C2×D3—61no—
398C2×D3#99412yes—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—31no—
403C3×C4—31no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4—82no—
410C3×C4—122no—
411C3×C4—161no—
412C3×C4—161no—
413C3×C4—241no—
414C3×C4—241no—
415Dic3—21no—
416Dic3—21no—
417Dic3—61no—
418Dic3—61no—
419Dic3—61no—
420Dic32−1+4⋊C312yes—
421C32—41noSylow
422C23—31no—
423C23—31no—
424C23—91no—
425Q8—31no—
426Q8—62no—
427Q8—181no—
428Q8—181no—
429Q8—362no—
430Q8C3×#23511yescharacteristic
431D4—61no—
432D4—61no—
433D4—182no—
434D4—181no—
435D4—181no—
436D4—182no—
437D4—182no—
438D4—362no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—62no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—122no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—182no—
457C2×C4—362no—
458C2×C4C2×D3×A411yescharacteristic
459C2×C4C2×D3×A411yescharacteristic
460C2×C4C2×D3×A411yescharacteristic
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—81no—
465C2×C3—81no—
466C2×C3—81no—
467C2×C3—121no—
468C2×C3—242no—
469C2×C3A4×#4111yescharacteristic
470C2×C3C2×#99411yescharacteristic
471C2×C3C2×2−1+4⋊C311yescharacteristic
472D3—22no—
473D3—121no—
474C22—32no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—61no—
479C22—61no—
480C22—181no—
481C22—181no—
482C22—181no—
483C22C22×D3×A411yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—31no—
488C4—31no—
489C4—32no—
490C4—61no—
491C4—61no—
492C4—61no—
493C4—61no—
494C4—181no—
495C4—181no—
496C4—181no—
497C3—41no—
498C3—81no—
499C3#3522⋊C311yescharacteristic, complemented
500C2—62no—
501C2—121no—
502C2—361no—
503C2A4×#23511yescharacteristic
504C2D3×#99411yescharacteristic
505C2D3×2−1+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. #5834⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. #5574⋊C3C2semidirect
7. SL(2,𝔽3)×Dic6C2semidirect
8. #85908C3semidirect
18. C3×#520C2×C3semidirect
19. #5834C2×C3semidirect
22. #5574C2×C3semidirect
24. Q8×Dic6C2×C3semidirect
39. #520⋊C3D3semidirect
123. C3×SL(2,𝔽3)#41semidirect
137. #520C3×D3semidirect
322. SL(2,𝔽3)#235semidirect
499. C3#3522⋊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
#5834⋊C32
#5576⋊C32
C4×D3×SL(2,𝔽3)2
#5574⋊C32
SL(2,𝔽3)×Dic62
#859083
#3522⋊C33
C3×#8964