Finite Group Databasebeta

← #85820⋊C3 #11923188

Subgroups — #85820⋊C3 #11923188

Order 1152.

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

normal subgroups only

This group has 569 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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
3#5817⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C211yesmaximal, characteristic, complemented
5#5572⋊C3C211yesmaximal, characteristic, complemented
6#5569⋊C3C211yesmaximal, characteristic, complemented
7#5574⋊C3C211yesmaximal, characteristic, complemented
8#85820C311yesmaximal, characteristic, complemented
9#3516⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#929—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C2×C3×C4×Q8C2×C311yescharacteristic, complemented
19#5817C2×C311yescharacteristic, complemented
20Q8×#67C2×C311yescharacteristic
21#5583—31no—
22#5582—31no—
23#5581—31no—
24#5576C2×C311yescharacteristic, complemented
25#5574C2×C311yescharacteristic, complemented
26#5572C2×C311yescharacteristic, complemented
27#5569C2×C311yescharacteristic, complemented
28#5568—31no—
29#5514—31no—
30#5511—31no—
31#5479—31no—
32#5479—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37#520⋊C3—31no—
38#519⋊C3—31no—
39C2×C4×SL(2,𝔽3)—21no—
40C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
41C22×C32×C4—41no—
42C2×C3×#67—41no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45D3×SL(2,𝔽3)—21no—
46C3⋊(#41⋊C3)—21no—
47C3×#217—41no—
48C3×(C2×D3)⋊C4—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3516—31noSylow
53C22×C3×Q8C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#41⋊C3—31no—
57C22×SL(2,𝔽3)—21no—
58C22×SL(2,𝔽3)—31no—
59C22×SL(2,𝔽3)C2×D311yescharacteristic
60C3×#123—121no—
61C3×C4×Q8—31no—
62C3×C4×Q8—31no—
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×Q8C22×C311yescharacteristic
65C2×C3×C4⋊C4—31no—
66C2×C3×C42—31no—
67#947—31no—
68#946—31no—
69#946—31no—
70Q8×Dic3C22×C311yescharacteristic
71Q8×Dic3C22×C311yescharacteristic
72#944—31no—
73#930—31no—
74#930—31no—
75#929—31no—
76#929A411yescharacteristic
77C4×#67—32no—
78C4×#67—31no—
79C4×#67—31no—
80#924—31no—
81#924—31no—
82#898—32no—
83#897—31no—
84#896—32no—
85#895—31no—
86#894—32no—
87#893—31no—
88#892—31no—
89D3×C4⋊C4—31no—
90#890—31no—
91#889—32no—
92#888—31no—
93#887—31no—
94#876—32no—
95#876—31no—
96#876—31no—
97#875—31no—
98#875—31no—
99C4×D12—31no—
100C4×D12—31no—
101#872—32no—
102#872—31no—
103#872—31no—
104#870—31no—
105#870—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
114C23×C32—41no—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—82no—
119C3×#67—82no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
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#520—31no—
137#519—31no—
138C2×C4×Q8C3×D311yescharacteristic, complemented
139C2×C3×Q8—21no—
140C2×C3×Q8—31no—
141C2×C3×Q8—61no—
142C2×C3×Q8C23×C311yescharacteristic
143C2×C3×D4—121no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—41no—
147C22×C3×C4—81no—
148C22×C3×C4C2×A411yescharacteristic
149C2×#67—31no—
150C2×#67C2×A411yescharacteristic
151#239—21no—
152#239—61no—
153D3×Q8—21no—
154D3×Q8—61no—
155C2×D12—31no—
156C2×D12—31no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—32no—
160C2×Dic6—31no—
161C2×Dic6—31no—
162#41⋊C3—61no—
163C2×SL(2,𝔽3)—22no—
164C2×SL(2,𝔽3)—21no—
165C2×SL(2,𝔽3)—21no—
166C2×SL(2,𝔽3)—61no—
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—32no—
171C3×C4⋊C4—121no—
172C3×C4⋊C4—121no—
173C3×C22⋊C4—121no—
174C3×C22⋊C4—121no—
175C3×C22⋊C4—121no—
176C3×C42—31no—
177C3×C42—31no—
178C3×C42—31no—
179C3×C42—31no—
180#217—31no—
181#217C2×A411yescharacteristic
182(C2×D3)⋊C4—32no—
183(C2×D3)⋊C4—32no—
184(C2×D3)⋊C4—31no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4—31no—
189(C2×D3)⋊C4—31no—
190(C2×D3)⋊C4C2×A411yescharacteristic
191(C2×D3)⋊C4C2×A411yescharacteristic
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⋊C4—31no—
204Dic3⋊C4C2×A411yescharacteristic
205Dic3⋊C4C2×A411yescharacteristic
206C4×Dic3—32no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32—81no—
212C22×C32—81no—
213C2×C3×D3—81no—
214C2×C3×D3—81no—
215C2×C3×D3—81no—
216C32×C4—81no—
217C32×C4—81no—
218C3×Dic3—81no—
219C3×Dic3—81no—
220C3×Dic3—81no—
221C2×#41—31no—
222C22×Q8—31no—
223C22×Q8C2×C3×D311yescharacteristic
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—91no—
233#125—92no—
234#124—91no—
235#124—91no—
236#124—91no—
237#123—31no—
238#123—92no—
239#123—91no—
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—31no—
252C4×Q8—31no—
253C4×Q8—91no—
254C4×Q8—91no—
255C4×Q8—91no—
256C4×Q8C2×C3×D311yescharacteristic
257C4×Q8C2×C3×D311yescharacteristic
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—92no—
263C4×D4—91no—
264#117—91no—
265#117—91no—
266#117—92no—
267#117—91no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—91no—
270C2×C42—31no—
271C23×C3—41no—
272C23×C3—81no—
273C23×C3—121no—
274C23×C3C22×A411yescharacteristic
275C22×D3—31no—
276C22×D3C22×A411yescharacteristic
277C3×Q8—31no—
278C3×Q8—61no—
279C3×Q8—61no—
280C3×Q8C3×#4111yescharacteristic
281C3×D4—242no—
282C2×C3×C4—22no—
283C2×C3×C4—31no—
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—82no—
292C2×C3×C4—84no—
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×C4C22×A411yescharacteristic
300C2×C3×C4C22×A411yescharacteristic
301#67—22no—
302#67—62no—
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—31no—
306C2×Dic3C22×A411yescharacteristic
307C2×Dic3C22×A411yescharacteristic
308C2×Dic3C22×A411yescharacteristic
309D12—61no—
310D12—61no—
311D12—121no—
312C4×D3—61no—
313C4×D3—61no—
314C4×D3—61no—
315C4×D3—61no—
316C4×D3—122no—
317Dic6—61no—
318Dic6—61no—
319Dic6—121no—
320SL(2,𝔽3)—21no—
321SL(2,𝔽3)#23511yescharacteristic
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—82no—
326C3×D3—161no—
327#41—61no—
328#41—181no—
329C2×Q8—21no—
330C2×Q8—31no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—91no—
334C2×Q8—91no—
335C2×Q8—181no—
336C2×Q8C22×C3×D311yescharacteristic
337C2×D4—31no—
338C2×D4—91no—
339C2×D4—91no—
340C2×D4—91no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4—92no—
346C22×C4D3×A411yescharacteristic
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—32no—
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—92no—
361C4⋊C4—92no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C22⋊C4—31no—
365C22⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—91no—
368C22⋊C4—91no—
369C22⋊C4—91no—
370C22⋊C4—91no—
371C22⋊C4—92no—
372C22⋊C4—92no—
373C22⋊C4—91no—
374C22⋊C4—91no—
375C22⋊C4—91no—
376C42—31no—
377C42—31no—
378C42—31no—
379C42—31no—
380C42—91no—
381C42—91no—
382C42—92no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—81no—
393C22×C3—241no—
394C22×C3—241no—
395C22×C3—241no—
396C22×C32−1+4⋊C312yes—
397C22×C3C23×A411yescharacteristic
398C2×D3—21no—
399C2×D3—21no—
400C2×D3—21no—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—61no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—31no—
407C3×C4—31no—
408C3×C4—61no—
409C3×C4—61no—
410C3×C4—61no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4—241no—
416C3×C4—241no—
417C3×C4—241no—
418Dic3—21no—
419Dic3—21no—
420Dic3—21no—
421Dic3—61no—
422Dic3—61no—
423Dic3—61no—
424C32—41noSylow
425C23—31no—
426C23—91no—
427C23C2×D3×A411yescharacteristic
428Q8—31no—
429Q8—61no—
430Q8—61no—
431Q8—181no—
432Q8—181no—
433Q8—361no—
434Q8C3×#23511yescharacteristic
435D4—62no—
436D4—181no—
437D4—181no—
438D4—182no—
439D4—361no—
440C2×C4—22no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—91no—
451C2×C4—91no—
452C2×C4—91no—
453C2×C4—121no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4—362no—
459C2×C4C2×D3×A411yescharacteristic
460C2×C4C2×D3×A411yescharacteristic
461C2×C3—22no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—82no—
466C2×C3—84no—
467C2×C3—81no—
468C2×C3—81no—
469C2×C3—81no—
470C2×C3—481no—
471C2×C3A4×#4111yescharacteristic
472C2×C3C2×2−1+4⋊C311yescharacteristic
473C2×C3#594511yescharacteristic
474D3—41no—
475D3—121no—
476C22—21no—
477C22—21no—
478C22—61no—
479C22—61no—
480C22—61no—
481C22—181no—
482C22—181no—
483C22—181no—
484C22#1007⋊C312yes—
485C22C22×D3×A411yescharacteristic
486C4—21no—
487C4—21no—
488C4—31no—
489C4—31no—
490C4—61no—
491C4—61no—
492C4—61no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—181no—
497C4—181no—
498C4—181no—
499C3—41no—
500C3—81no—
501C3#3516⋊C311yescharacteristic, complemented
502C2—22no—
503C2—121no—
504C2—361no—
505C2A4×#23511yescharacteristic
506C2C2×#1007⋊C311yescharacteristic
507C2#5964⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×C4×Q8C2×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. C2×C3×C4×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×#67C2semidirect
3. #5817⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. #5572⋊C3C2semidirect
6. #5569⋊C3C2semidirect
7. #5574⋊C3C2semidirect
8. #85820C3semidirect
18. C2×C3×C4×Q8C2×C3semidirect
19. #5817C2×C3semidirect
24. #5576C2×C3semidirect
25. #5574C2×C3semidirect
26. #5572C2×C3semidirect
27. #5569C2×C3semidirect
40. C2×C4×SL(2,𝔽3)D3semidirect
138. C2×C4×Q8C3×D3semidirect
501. C3#3516⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
SL(2,𝔽3)×#672
#5817⋊C32
#5576⋊C32
#5572⋊C32
#5569⋊C32
#5574⋊C32
#858203
#3516⋊C33
C3×#9294