Finite Group Databasebeta

← #86712⋊C3 #11923246

Subgroups — #86712⋊C3 #11923246

Order 1152.

613 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 89 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#5831⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
5C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#5569⋊C3C211yesmaximal, characteristic, complemented
8#86712C311yesmaximal, characteristic, complemented
9#3482⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C3×#67⋊C4—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, complemented
19#5831C2×C311yescharacteristic, complemented
20Dic3×#41C2×C311yescharacteristic, complemented
21Q8×#67C2×C311yescharacteristic
22C2×Q8×Dic3C2×C311yescharacteristic, complemented
23#5680—31no—
24#5671—31no—
25#5639—31no—
26#5622—31no—
27#5611—31no—
28#5600—31no—
29C4×D3×Q8C2×C311yescharacteristic, complemented
30#5569C2×C311yescharacteristic, complemented
31#5541—31no—
32C4×#237—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—31no—
35#548⋊C3—21no—
36#548⋊C3—31no—
37#548⋊C3D311yescharacteristic, complemented
38C4×#41⋊C3—31no—
39C2×C4×SL(2,𝔽3)—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×#67—41no—
42C22×C3×Dic3—41no—
43C2×C3×C4×D3—41no—
44C3×#41⋊C3—21no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)C2311yescharacteristic
47D3×SL(2,𝔽3)—21no—
48C32×C22⋊C4—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3482—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#237—31no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3C2×D311yescharacteristic
59C22×SL(2,𝔽3)—21no—
60C22×SL(2,𝔽3)—31no—
61C22×SL(2,𝔽3)C2×D311yescharacteristic
62C3×#124—31no—
63C3×C22⋊Q8—31no—
64C3×#121—31no—
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×D4—31no—
67C3×C4×D4—121no—
68#946—31no—
69Q8×Dic3—31no—
70Q8×Dic3—31no—
71Q8×Dic3C22×C311yescharacteristic
72Q8×Dic3C22×C311yescharacteristic
73Q8×Dic3C22×C311yescharacteristic
74#944—31no—
75#938—31no—
76#936—31no—
77#935—32no—
78D4×Dic3—31no—
79D4×Dic3—31no—
80D4×Dic3—31no—
81C4×#67—31no—
82C4×#67—31no—
83#926—31no—
84C2×C4⋊Dic3—31no—
85#924—31no—
86#924—31no—
87C2×C4×Dic3—31no—
88C2×C4×Dic3—31no—
89#892—31no—
90D3×C4⋊C4—31no—
91#890—31no—
92#889—32no—
93#888—31no—
94#887—31no—
95#887—31no—
96#883—32no—
97#67⋊C4—31no—
98#67⋊C4A411yescharacteristic
99#879—32no—
100#879—32no—
101#878—32no—
102#877—32no—
103#877—31no—
104#877—31no—
105C42×D3—31no—
106#870—31no—
107C4×Dic6—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—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C3×#67—44no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—82no—
123C3×C4×D3—82no—
124C3×SL(2,𝔽3)#4111yescharacteristic, complemented
125#557—31no—
126#556—91no—
127#552—91no—
128#552—91no—
129D4×Q8—31no—
130#548—31no—
131#548C3×D311yescharacteristic, complemented
132#547—91no—
133#533—91no—
134#529—91no—
135#529—91no—
136C4×#41—31no—
137C4×#41—91no—
138C2×C4×Q8—31no—
139C2×C4×Q8—31no—
140C3×#41—21no—
141C3×#41—61no—
142C2×C3×Q8—21no—
143C2×C3×Q8—31no—
144C2×C3×Q8—61no—
145C2×C3×Q8C23×C311yescharacteristic
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—121no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—121no—
152C22×C3×C4—121no—
153C2×#67—31no—
154C2×#67C2×A411yescharacteristic
155C22×Dic3—31no—
156C22×Dic3—31no—
157C22×Dic3—31no—
158C22×Dic3C2×A411yescharacteristic
159D3×Q8—21no—
160D3×Q8—61no—
161#237—64no—
162C2×C4×D3—31no—
163C2×C4×D3—31no—
164C2×C4×D3C2×A411yescharacteristic
165C2×Dic6—31no—
166C2×Dic6—31no—
167#41⋊C3—21no—
168#41⋊C3—41no—
169C2×SL(2,𝔽3)—21no—
170C2×SL(2,𝔽3)—21no—
171C2×SL(2,𝔽3)—41no—
172C2×SL(2,𝔽3)—61no—
173C2×SL(2,𝔽3)C22×D311yescharacteristic
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
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—
184C3×C42—121no—
185#217—32no—
186#217—32no—
187#217—31no—
188#217—31no—
189(C2×D3)⋊C4—31no—
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—32no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4C2×A411yescharacteristic
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C4×Dic3—31no—
211C4×Dic3C2×A411yescharacteristic
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—81no—
215C22×C32—81no—
216C2×C3×D3—42no—
217C2×C3×D3—81no—
218C2×C3×D3—81no—
219C32×C4—81no—
220C32×C4—81no—
221C3×Dic3—42no—
222C3×Dic3—42no—
223C3×Dic3—81no—
224C2×#41—91no—
225C2×#41C2×C3×D311yescharacteristic
226C22×Q8—31no—
227C22×Q8C2×C3×D311yescharacteristic
228C4⋊Q8—91no—
229C4⋊Q8—91no—
230#126—92no—
231#126—92no—
232#125—91no—
233#125—91no—
234#125—92no—
235#124—31no—
236#124—91no—
237#123—92no—
238#123—92no—
239C22⋊Q8—31no—
240C22⋊Q8—92no—
241C22⋊Q8—91no—
242C22⋊Q8—91no—
243C22⋊Q8—91no—
244#121—31no—
245#121—91no—
246C4×Q8—31no—
247C4×Q8—31no—
248C4×Q8—31no—
249C4×Q8—91no—
250C4×Q8—91no—
251C4×Q8—91no—
252C4×Q8—91no—
253C4×Q8—91no—
254C4×Q8—91no—
255C4×Q8C2×C3×D311yescharacteristic
256C4×D4—31no—
257C4×D4—31no—
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—91no—
264#117—91no—
265#117—91no—
266#117—92no—
267#117—91no—
268#117—91no—
269C2×C4⋊C4—91no—
270C2×C4⋊C4—91no—
271C2×C42—91no—
272C2×C42—91no—
273C2×C42—91no—
274C23×C3—31no—
275C23×C3—41no—
276C23×C3—81no—
277C23×C3—121no—
278C23×C3C22×A411yescharacteristic
279C22×D3C22×A411yescharacteristic
280C3×Q8—31no—
281C3×Q8—61no—
282C3×Q8—61no—
283C3×Q8C3×#4111yescharacteristic
284C3×D4—61no—
285C3×D4—61no—
286C3×D4—62no—
287C3×D4—124no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—81no—
298C2×C3×C4—81no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—242no—
303C2×C3×C4—242no—
304C2×C3×C4C22×A411yescharacteristic
305C2×C3×C4C22×A411yescharacteristic
306#67—62no—
307#67#41⋊C314yescomplemented
308C2×Dic3—22no—
309C2×Dic3—31no—
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—31no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3C22×A411yescharacteristic
317C2×Dic3C22×A411yescharacteristic
318C2×Dic3C22×A411yescharacteristic
319C4×D3—22no—
320C4×D3—61no—
321C4×D3—61no—
322C4×D3—62no—
323Dic6—62no—
324Dic6—61no—
325Dic6—61no—
326SL(2,𝔽3)—21no—
327SL(2,𝔽3)#23711yescharacteristic, complemented
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—82no—
332C3×D3—82no—
333#41—21no—
334#41—61no—
335#41—184no—
336C2×Q8—21no—
337C2×Q8—31no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—91no—
341C2×Q8—91no—
342C2×Q8—181no—
343C2×Q8C22×C3×D311yescharacteristic
344C2×D4—31no—
345C2×D4—31no—
346C2×D4—31no—
347C2×D4—91no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C4⋊C4—31no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4—92no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4—92no—
369C4⋊C4—91no—
370C4⋊C4—91no—
371C22⋊C4—32no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C22⋊C4—91no—
375C22⋊C4—91no—
376C22⋊C4—92no—
377C22⋊C4—92no—
378C22⋊C4—91no—
379C22⋊C4D3×A411yescharacteristic
380C42—31no—
381C42—31no—
382C42—91no—
383C42—91no—
384C42—91no—
385C42—91no—
386C42—91no—
387C42—91no—
388C42—91no—
389C42—91no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—42no—
393C22×C3—41no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—82no—
400C22×C3—81no—
401C22×C3—122no—
402C22×C3—161no—
403C22×C3—161no—
404C22×C3—241no—
405C22×C3—241no—
406C22×C3C2×#41⋊C312yes—
407C22×C3C23×A411yescharacteristic
408C2×D3—21no—
409C2×D3—21no—
410C2×D3C2×#41⋊C312yes—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—31no—
414C3×C4—31no—
415C3×C4—61no—
416C3×C4—61no—
417C3×C4—81no—
418C3×C4—81no—
419C3×C4—122no—
420C3×C4—122no—
421C3×C4—161no—
422C3×C4—161no—
423C3×C4—241no—
424Dic3—21no—
425Dic3—61no—
426Dic3—61no—
427Dic3—61no—
428Dic3—61no—
429Dic3C2×#41⋊C312yes—
430Dic32−1+4⋊C312yes—
431C32—41noSylow
432C23—31no—
433C23—31no—
434C23C2×D3×A411yescharacteristic
435Q8—31no—
436Q8—61no—
437Q8—61no—
438Q8—182no—
439Q8—181no—
440Q8—181no—
441Q8C3×#23711yescharacteristic
442D4—34no—
443D4—61no—
444D4—61no—
445D4—62no—
446D4—182no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—62no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4—182no—
468C2×C4—182no—
469C2×C4C2×D3×A411yescharacteristic
470C2×C4C2×D3×A411yescharacteristic
471C2×C3—22no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—82no—
476C2×C3—81no—
477C2×C3—81no—
478C2×C3—81no—
479C2×C3—121no—
480C2×C3—162no—
481C2×C3—242no—
482C2×C3C22×#41⋊C311yescharacteristic
483C2×C3A4×#4111yescharacteristic
484C2×C3C2×2−1+4⋊C311yescharacteristic
485D3—22no—
486C22—21no—
487C22—21no—
488C22—32no—
489C22—61no—
490C22—61no—
491C22—61no—
492C22—61no—
493C22—61no—
494C22D3×#41⋊C312yes—
495C22C22×D3×A411yescharacteristic
496C4—21no—
497C4—21no—
498C4—31no—
499C4—31no—
500C4—32no—
501C4—32no—
502C4—61no—
503C4—61no—
504C4—61no—
505C4—181no—
506C4—181no—
507C4—181no—
508C4—181no—
509C3—41no—
510C3—81no—
511C3#3482⋊C311yescharacteristic, complemented
512C2—22no—
513C2—62no—
514C2—121no—
515C2A4×#23711yescharacteristic
516C2C2×D3×#41⋊C311yescharacteristic
517C2D3×2−1+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. #5831⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. SL(2,𝔽3)×#67C2semidirect
5. C2×Dic3×SL(2,𝔽3)C2semidirect
6. C4×D3×SL(2,𝔽3)C2semidirect
7. #5569⋊C3C2semidirect
8. #86712C3semidirect
18. C3×#548C2×C3semidirect
19. #5831C2×C3semidirect
20. Dic3×#41C2×C3semidirect
22. C2×Q8×Dic3C2×C3semidirect
29. C4×D3×Q8C2×C3semidirect
30. #5569C2×C3semidirect
37. #548⋊C3D3semidirect
124. C3×SL(2,𝔽3)#41semidirect
131. #548C3×D3semidirect
307. #67#41⋊C3semidirect
327. SL(2,𝔽3)#237semidirect
511. C3#3482⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
#5831⋊C32
Dic3×#41⋊C32
SL(2,𝔽3)×#672
C2×Dic3×SL(2,𝔽3)2
C4×D3×SL(2,𝔽3)2
#5569⋊C32
#867123
#3482⋊C33
C3×#67⋊C44