Finite Group Databasebeta

← #86752⋊C3 #11923256

Subgroups — #86752⋊C3 #11923256

Order 1152.

589 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 81 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
4#5812⋊C3C211yesmaximal, characteristic, complemented
5C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5572⋊C3C211yesmaximal, characteristic, complemented
7#5569⋊C3C211yesmaximal, characteristic, complemented
8#86752C311yesmaximal, characteristic, complemented
9#3523⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C3×#939—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15Dic3×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×D4×Q8C2×C311yescharacteristic, complemented
19Q8×#67C2×C311yescharacteristic, complemented
20Q8×#67C2×C311yescharacteristic, complemented
21#5812C2×C311yescharacteristic, complemented
22C2×Q8×Dic3C2×C311yescharacteristic, complemented
23#5731—31no—
24#5721—31no—
25#5642—31no—
26#5641—31no—
27#5625—31no—
28#5622—31no—
29#5572C2×C311yescharacteristic, complemented
30#5569C2×C311yescharacteristic, complemented
31#5541—31no—
32#5538—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—21no—
35D4×SL(2,𝔽3)—31no—
36D4×SL(2,𝔽3)—31no—
37D4×SL(2,𝔽3)D311yescharacteristic, complemented
38#519⋊C3—31no—
39#519⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C32×D4—41no—
42C2×C3×#67—41no—
43C2×C3×#67—41no—
44C22×C3×Dic3—41no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)—21no—
47C2×C3×SL(2,𝔽3)C2311yescharacteristic
48D3×SL(2,𝔽3)—21no—
49C3×#217—41no—
50C3×(C2×D3)⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3523—31noSylow
53C22×C3×Q8C22×C311yescharacteristic
54C22×C3×Q8C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C22×SL(2,𝔽3)—21no—
57C22×SL(2,𝔽3)—21no—
58C22×SL(2,𝔽3)—31no—
59C22×SL(2,𝔽3)C2×D311yescharacteristic
60C22×SL(2,𝔽3)C2×D311yescharacteristic
61C3×C4⋊Q8—31no—
62C3×C22⋊Q8—31no—
63C3×C22⋊Q8—31no—
64C3×#121—121no—
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×D4—31no—
67#946—31no—
68#946—31no—
69Q8×Dic3—31no—
70Q8×Dic3—31no—
71Q8×Dic3—31no—
72Q8×Dic3—31no—
73Q8×Dic3C22×C311yescharacteristic
74Q8×Dic3C22×C311yescharacteristic
75Q8×Dic3C22×C311yescharacteristic
76#944—31no—
77#944—31no—
78#939A411yescharacteristic
79#935—31no—
80C4×#67—31no—
81C4×#67—31no—
82#926—31no—
83#926—31no—
84C2×C4⋊Dic3—31no—
85#924—31no—
86#924—31no—
87C2×C4×Dic3—31no—
88#898—32no—
89#898—31no—
90#892—31no—
91#890—32no—
92#890—31no—
93#889—32no—
94#889—32no—
95#889—31no—
96#888—31no—
97#887—32no—
98#887—31no—
99#886—31no—
100#885—31no—
101#879—32no—
102#879—31no—
103#878—31no—
104#877—32no—
105#872—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—
115C23×C32—41no—
116C22×C3×D3—41no—
117C32×D4—82no—
118C2×C32×C4—41no—
119C3×#67—82no—
120C3×#67—82no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—82no—
125C3×SL(2,𝔽3)#4111yescharacteristic
126#557—31no—
127#557—91no—
128#556—91no—
129#552—91no—
130#552—91no—
131D4×Q8—31no—
132D4×Q8—31no—
133D4×Q8C3×D311yescharacteristic, complemented
134#541—91no—
135#541—91no—
136#533—91no—
137#529—91no—
138#519—31no—
139#519—31no—
140C2×C4×Q8—31no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—31no—
144C2×C3×Q8—31no—
145C2×C3×Q8—61no—
146C2×C3×Q8—61no—
147C2×C3×Q8C23×C311yescharacteristic
148C2×C3×D4—41no—
149C2×C3×D4—81no—
150C2×C3×D4—121no—
151C2×C3×D4—121no—
152C2×C3×D4C2×A411yescharacteristic
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—121no—
156C2×#67C2×A411yescharacteristic
157C2×#67C2×A411yescharacteristic
158C22×Dic3—31no—
159C22×Dic3C2×A411yescharacteristic
160D3×Q8—21no—
161D3×Q8—61no—
162C2×C4×D3—31no—
163C2×Dic6—31no—
164C2×Dic6—31no—
165C2×SL(2,𝔽3)—21no—
166C2×SL(2,𝔽3)—21no—
167C2×SL(2,𝔽3)—21no—
168C2×SL(2,𝔽3)—41no—
169C2×SL(2,𝔽3)—41no—
170C2×SL(2,𝔽3)—61no—
171C2×SL(2,𝔽3)C22×D311yescharacteristic
172C3×C4⋊C4—31no—
173C3×C4⋊C4—32no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—121no—
176C3×C22⋊C4—31no—
177C3×C22⋊C4—31no—
178C3×C22⋊C4—121no—
179C3×C22⋊C4—121no—
180C3×C42—31no—
181#217—31no—
182#217—31no—
183#217—31no—
184#217C2×A411yescharacteristic
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4C2×A411yescharacteristic
189C4⋊Dic3—32no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
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⋊C4—31no—
203Dic3⋊C4—31no—
204Dic3⋊C4C2×A411yescharacteristic
205C4×Dic3—32no—
206C4×Dic3—32no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C4×Dic3—31no—
211C4×Dic3—31no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—81no—
215C22×C32—81no—
216C22×C32—81no—
217C22×C32—81no—
218C22×C32—81no—
219C2×C3×D3—81no—
220C2×C3×D3—81no—
221C2×C3×D3—81no—
222C32×C4—81no—
223C3×Dic3—42no—
224C3×Dic3—81no—
225C3×Dic3—81no—
226C22×Q8—31no—
227C22×Q8C2×C3×D311yescharacteristic
228C22×Q8C2×C3×D311yescharacteristic
229C4⋊Q8—31no—
230C4⋊Q8—91no—
231C4⋊Q8—91no—
232C4⋊Q8—91no—
233#126—91no—
234#126—91no—
235#126—92no—
236#126—92no—
237#125—91no—
238#125—91no—
239#125—91no—
240#125—92no—
241#125—92no—
242#125—92no—
243#124—91no—
244#123—91no—
245#123—91no—
246C22⋊Q8—31no—
247C22⋊Q8—31no—
248C22⋊Q8—91no—
249C22⋊Q8—91no—
250C22⋊Q8—91no—
251C22⋊Q8—91no—
252C22⋊Q8—91no—
253#121—31no—
254C4×Q8—31no—
255C4×Q8—31no—
256C4×Q8—31no—
257C4×Q8—91no—
258C4×Q8—91no—
259C4×Q8—92no—
260C4×Q8—91no—
261C4×Q8—91no—
262C4×Q8—91no—
263C4×Q8—91no—
264C4×Q8—91no—
265C4×Q8C2×C3×D311yescharacteristic
266C4×D4—31no—
267C4×D4—91no—
268C4×D4—91no—
269#117—91no—
270#117—91no—
271#117—92no—
272#117—91no—
273#117—91no—
274C2×C4⋊C4—91no—
275C2×C42—91no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—81no—
279C23×C3—81no—
280C23×C3—121no—
281C23×C3C22×A411yescharacteristic
282C23×C3C22×A411yescharacteristic
283C22×D3C22×A411yescharacteristic
284C3×Q8—31no—
285C3×Q8—61no—
286C3×Q8—61no—
287C3×Q8—61no—
288C3×Q8—61no—
289C3×Q8C3×#4111yescharacteristic
290C3×D4—22no—
291C3×D4—82no—
292C3×D4—84no—
293C3×D4—242no—
294C3×D4—242no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—41no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—81no—
305C2×C3×C4—121no—
306C2×C3×C4—121no—
307C2×C3×C4—121no—
308C2×C3×C4—242no—
309C2×C3×C4C22×A411yescharacteristic
310#67—22no—
311#67—22no—
312C2×Dic3—22no—
313C2×Dic3—31no—
314C2×Dic3—31no—
315C2×Dic3—31no—
316C2×Dic3—31no—
317C2×Dic3—62no—
318C2×Dic3C22×A411yescharacteristic
319C2×Dic3C22×A411yescharacteristic
320C2×Dic3C22×A411yescharacteristic
321C4×D3—61no—
322C4×D3—61no—
323Dic6—62no—
324Dic6—61no—
325Dic6—61no—
326SL(2,𝔽3)—21no—
327SL(2,𝔽3)#23711yescharacteristic
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—82no—
332C2×C32—161no—
333C3×D3—161no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—31no—
337C2×Q8—31no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—91no—
342C2×Q8—91no—
343C2×Q8—181no—
344C2×Q8C22×C3×D311yescharacteristic
345C2×D4—31no—
346C2×D4—31no—
347C2×D4D3×A411yescharacteristic
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—91no—
352C22×C4—91no—
353C4⋊C4—31no—
354C4⋊C4—32no—
355C4⋊C4—31no—
356C4⋊C4—31no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—91no—
371C4⋊C4—91no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C22⋊C4—31no—
375C22⋊C4—31no—
376C22⋊C4—91no—
377C22⋊C4—91no—
378C22⋊C4—91no—
379C22⋊C4—91no—
380C22⋊C4—91no—
381C22⋊C4—91no—
382C42—31no—
383C42—91no—
384C42—91no—
385C42—91no—
386C42—91no—
387C42—91no—
388C42—92no—
389C42—92no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—81no—
405C22×C3—161no—
406C22×C3—161no—
407C22×C3—161no—
408C22×C3—161no—
409C22×C3—241no—
410C22×C3—241no—
411C22×C3—241no—
412C22×C32−1+4⋊C312yes—
413C22×C3C23×A411yescharacteristic
414C2×D3—21no—
415C2×D3—21no—
416C2×D3—21no—
417C3×C4—21no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—61no—
421C3×C4—61no—
422C3×C4—61no—
423C3×C4—81no—
424C3×C4—82no—
425C3×C4—122no—
426C3×C4—241no—
427C3×C4—241no—
428Dic3—21no—
429Dic3—21no—
430Dic3—61no—
431Dic3—61no—
432Dic3—61no—
433Dic3—61no—
434Dic32−1+4⋊C312yes—
435C32—41noSylow
436C23—31no—
437C23C2×D3×A411yescharacteristic
438C23C2×D3×A411yescharacteristic
439Q8—31no—
440Q8—61no—
441Q8—61no—
442Q8—61no—
443Q8—61no—
444Q8—182no—
445Q8—181no—
446Q8—181no—
447Q8C3×#23711yescharacteristic
448D4—22no—
449D4—62no—
450D4—62no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—62no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—182no—
470C2×C4C2×D3×A411yescharacteristic
471C2×C3—22no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—82no—
477C2×C3—81no—
478C2×C3—81no—
479C2×C3—81no—
480C2×C3—162no—
481C2×C3—161no—
482C2×C3—162no—
483C2×C3—481no—
484C2×C3A4×#4111yescharacteristic
485C2×C3C2×2−1+4⋊C311yescharacteristic
486C2×C3C2×2−1+4⋊C311yescharacteristic
487D3—41no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—61no—
494C22—61no—
495C22—61no—
496C22#1007⋊C312yes—
497C22C22×D3×A411yescharacteristic
498C4—21no—
499C4—31no—
500C4—31no—
501C4—32no—
502C4—61no—
503C4—61no—
504C4—61no—
505C4—61no—
506C4—61no—
507C4—181no—
508C4—181no—
509C4—181no—
510C4—181no—
511C3—41no—
512C3—81no—
513C3#3523⋊C311yescharacteristic, complemented
514C2—22no—
515C2—41no—
516C2—121no—
517C2A4×#23711yescharacteristic
518C2C2×#1007⋊C311yescharacteristic
519C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×#67C2semidirect
3. SL(2,𝔽3)×#67C2semidirect
4. #5812⋊C3C2semidirect
5. C2×Dic3×SL(2,𝔽3)C2semidirect
6. #5572⋊C3C2semidirect
7. #5569⋊C3C2semidirect
8. #86752C3semidirect
18. C3×D4×Q8C2×C3semidirect
19. Q8×#67C2×C3semidirect
20. Q8×#67C2×C3semidirect
21. #5812C2×C3semidirect
22. C2×Q8×Dic3C2×C3semidirect
29. #5572C2×C3semidirect
30. #5569C2×C3semidirect
37. D4×SL(2,𝔽3)D3semidirect
133. D4×Q8C3×D3semidirect
513. C3#3523⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
SL(2,𝔽3)×#672
#5812⋊C32
C2×Dic3×SL(2,𝔽3)2
#5572⋊C32
#5569⋊C32
#867523
#3523⋊C33
C3×#9394