Finite Group Databasebeta

← #85817⋊C3 #11923187

Subgroups — #85817⋊C3 #11923187

Order 1152.

623 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 91 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
5C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5569⋊C3C211yesmaximal, characteristic, complemented
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#85817C311yesmaximal, characteristic, complemented
9#3482⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×C4×#67—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#5581—31no—
22#5580—31no—
23#5578—31no—
24#5576C2×C311yescharacteristic, complemented
25C4×#239C2×C311yescharacteristic, complemented
26C4×D3×Q8C2×C311yescharacteristic, complemented
27#5569C2×C311yescharacteristic, complemented
28#5568—31no—
29#5513—31no—
30#5507—31no—
31#5479—31no—
32C4×#235—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37C4×#41⋊C3—31no—
38C2×C4×SL(2,𝔽3)—21no—
39C2×C4×SL(2,𝔽3)—31no—
40C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
41C22×C32×C4—41no—
42C2×C3×#67—41no—
43C2×C3×C4×D3—41no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46D3×SL(2,𝔽3)—21no—
47C3⋊(#41⋊C3)—21no—
48C3×#217—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3482—31noSylow
53C22×C3×Q8C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#235—31no—
57C2×#41⋊C3—31no—
58C22×SL(2,𝔽3)—21no—
59C22×SL(2,𝔽3)—31no—
60C22×SL(2,𝔽3)C2×D311yescharacteristic
61C3×C4×Q8—31no—
62C3×C4×Q8—31no—
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×Q8C22×C311yescharacteristic
65C3×C4×D4—121no—
66C2×C3×C4⋊C4—31no—
67C2×C3×C42—31no—
68#947—31no—
69#946—31no—
70#946—31no—
71Q8×Dic3C22×C311yescharacteristic
72Q8×Dic3C22×C311yescharacteristic
73#944—31no—
74#930—31no—
75#929—32no—
76C4×#67—31no—
77C4×#67—31no—
78C4×#67—31no—
79C4×#67A411yescharacteristic
80#926—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#875—31no—
96C4×D12—31no—
97C4×D12—31no—
98C4×D12—31no—
99#872—32no—
100#872—31no—
101#872—31no—
102C42×D3—31no—
103C42×D3—31no—
104#870—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
114C23×C32—41no—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—82no—
119C3×#67—44no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
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#548—31no—
132#547—91no—
133#533—91no—
134#529—91no—
135#529—91no—
136C4×#41—31no—
137C4×#41—91no—
138C2×C4×Q8—31no—
139C2×C4×Q8C3×D311yescharacteristic, complemented
140C2×C3×Q8—21no—
141C2×C3×Q8—31no—
142C2×C3×Q8—61no—
143C2×C3×Q8C23×C311yescharacteristic
144C2×C3×D4—121no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—41no—
148C22×C3×C4—81no—
149C22×C3×C4—121no—
150C22×C3×C4C2×A411yescharacteristic
151C2×#67—31no—
152C2×#67C2×A411yescharacteristic
153#239—21no—
154#239—61no—
155D3×Q8—21no—
156D3×Q8—61no—
157#235—64no—
158C2×D12—31no—
159C2×D12—31no—
160C2×C4×D3—31no—
161C2×C4×D3—31no—
162C2×C4×D3—31no—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3C2×A411yescharacteristic
166C2×Dic6—31no—
167C2×Dic6—31no—
168#41⋊C3—61no—
169C2×SL(2,𝔽3)—22no—
170C2×SL(2,𝔽3)—21no—
171C2×SL(2,𝔽3)—21no—
172C2×SL(2,𝔽3)—61no—
173C2×SL(2,𝔽3)C22×D311yescharacteristic
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—32no—
177C3×C4⋊C4—121no—
178C3×C22⋊C4—121no—
179C3×C22⋊C4—121no—
180C3×C42—31no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—31no—
184C3×C42—121no—
185#217—31no—
186#217C2×A411yescharacteristic
187(C2×D3)⋊C4—32no—
188(C2×D3)⋊C4—32no—
189(C2×D3)⋊C4—32no—
190(C2×D3)⋊C4—31no—
191(C2×D3)⋊C4—31no—
192(C2×D3)⋊C4—31no—
193(C2×D3)⋊C4C2×A411yescharacteristic
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198Dic3⋊C4—32no—
199Dic3⋊C4—32no—
200Dic3⋊C4—32no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4—31no—
204Dic3⋊C4C2×A411yescharacteristic
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C4×Dic3C2×A411yescharacteristic
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—81no—
214C22×C32—81no—
215C2×C3×D3—42no—
216C2×C3×D3—81no—
217C2×C3×D3—81no—
218C32×C4—42no—
219C32×C4—81no—
220C3×Dic3—42no—
221C3×Dic3—81no—
222C3×Dic3—81no—
223C2×#41—31no—
224C2×#41—91no—
225C22×Q8—31no—
226C22×Q8C2×C3×D311yescharacteristic
227C4⋊Q8—91no—
228C4⋊Q8—91no—
229#126—92no—
230#126—92no—
231#125—91no—
232#125—91no—
233#125—92no—
234#124—91no—
235#124—91no—
236#123—92no—
237#123—92no—
238C22⋊Q8—92no—
239C22⋊Q8—91no—
240C22⋊Q8—91no—
241C22⋊Q8—91no—
242C22⋊Q8—91no—
243#121—91no—
244#121—91no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—31no—
248C4×Q8—31no—
249C4×Q8—91no—
250C4×Q8—91no—
251C4×Q8—91no—
252C4×Q8—91no—
253C4×Q8C2×C3×D311yescharacteristic
254C4×Q8C2×C3×D311yescharacteristic
255C4×D4—31no—
256C4×D4—91no—
257C4×D4—91no—
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—91no—
263#117—91no—
264#117—91no—
265#117—92no—
266#117—91no—
267#117—91no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—91no—
270C2×C42—31no—
271C2×C42—91no—
272C2×C42—91no—
273C23×C3—41no—
274C23×C3—81no—
275C23×C3—121no—
276C23×C3C22×A411yescharacteristic
277C22×D3—31no—
278C22×D3C22×A411yescharacteristic
279C3×Q8—31no—
280C3×Q8—61no—
281C3×Q8—61no—
282C3×Q8C3×#4111yescharacteristic
283C3×D4—124no—
284C2×C3×C4—22no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—62no—
294C2×C3×C4—82no—
295C2×C3×C4—84no—
296C2×C3×C4—81no—
297C2×C3×C4—81no—
298C2×C3×C4—121no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—242no—
302C2×C3×C4C22×A411yescharacteristic
303C2×C3×C4C22×A411yescharacteristic
304#67—62no—
305#67#41⋊C314yescomplemented
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3C22×A411yescharacteristic
310C2×Dic3C22×A411yescharacteristic
311C2×Dic3C22×A411yescharacteristic
312D12—61no—
313D12—61no—
314D12—62no—
315C4×D3—22no—
316C4×D3—61no—
317C4×D3—61no—
318C4×D3—61no—
319C4×D3—61no—
320C4×D3—62no—
321C4×D3—62no—
322C4×D3—62no—
323Dic6—62no—
324Dic6—61no—
325Dic6—61no—
326SL(2,𝔽3)—21no—
327SL(2,𝔽3)#23511yescharacteristic, complemented
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—82no—
332C3×D3—82no—
333#41—61no—
334#41—184no—
335#41—181no—
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—91no—
346C2×D4—91no—
347C2×D4—91no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—91no—
352C22×C4—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4D3×A411yescharacteristic
357C4⋊C4—31no—
358C4⋊C4—31no—
359C4⋊C4—32no—
360C4⋊C4—31no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—91no—
367C4⋊C4—92no—
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—91no—
371C22⋊C4—31no—
372C22⋊C4—31no—
373C22⋊C4—91no—
374C22⋊C4—91no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C22⋊C4—92no—
378C22⋊C4—91no—
379C22⋊C4—91no—
380C42—31no—
381C42—31no—
382C42—31no—
383C42—31no—
384C42—31no—
385C42—91no—
386C42—91no—
387C42—91no—
388C42—91no—
389C42—91no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—42no—
393C22×C3—41no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—81no—
400C22×C3—122no—
401C22×C3—241no—
402C22×C3—241no—
403C22×C3C2×#41⋊C312yes—
404C22×C3C23×A411yescharacteristic
405C2×D3—21no—
406C2×D3—21no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3C2×#41⋊C312yes—
411C3×C4—21no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—42no—
415C3×C4—61no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—81no—
419C3×C4—82no—
420C3×C4—82no—
421C3×C4—122no—
422C3×C4—241no—
423C3×C4—241no—
424C3×C42−1+4⋊C312yes—
425Dic3—21no—
426Dic3—21no—
427Dic3—61no—
428Dic3—61no—
429Dic3—61no—
430Dic3C2×#41⋊C312yes—
431C32—41noSylow
432C23—31no—
433C23—91no—
434C23C2×D3×A411yescharacteristic
435Q8—31no—
436Q8—61no—
437Q8—61no—
438Q8—182no—
439Q8—181no—
440Q8—181no—
441Q8C3×#23511yescharacteristic
442D4—34no—
443D4—181no—
444D4—181no—
445D4—182no—
446D4—182no—
447C2×C4—22no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—181no—
463C2×C4—182no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
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—84no—
477C2×C3—81no—
478C2×C3—81no—
479C2×C3—81no—
480C2×C3—242no—
481C2×C3C22×#41⋊C311yescharacteristic
482C2×C3A4×#4111yescharacteristic
483C2×C3C2×2−1+4⋊C311yescharacteristic
484D3—22no—
485D3—121no—
486C22—21no—
487C22—21no—
488C22—32no—
489C22—61no—
490C22—61no—
491C22—181no—
492C22—181no—
493C22—181no—
494C22D3×#41⋊C312yes—
495C22C22×D3×A411yescharacteristic
496C4—21no—
497C4—31no—
498C4—31no—
499C4—32no—
500C4—61no—
501C4—61no—
502C4—61no—
503C4—61no—
504C4—61no—
505C4—181no—
506C4—181no—
507C4—181no—
508C4#1007⋊C312yes—
509C3—41no—
510C3—81no—
511C3#3482⋊C311yescharacteristic, complemented
512C2—22no—
513C2—62no—
514C2—361no—
515C2A4×#23511yescharacteristic
516C2C2×#1007⋊C311yescharacteristic
517C2C2×D3×#41⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ 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. C4×D3×SL(2,𝔽3)C2semidirect
6. #5569⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #85817C3semidirect
18. C2×C3×C4×Q8C2×C3semidirect
19. #5817C2×C3semidirect
24. #5576C2×C3semidirect
25. C4×#239C2×C3semidirect
26. C4×D3×Q8C2×C3semidirect
27. #5569C2×C3semidirect
40. C2×C4×SL(2,𝔽3)D3semidirect
124. C3×SL(2,𝔽3)#41semidirect
139. C2×C4×Q8C3×D3semidirect
305. #67#41⋊C3semidirect
327. SL(2,𝔽3)#235semidirect
511. C3#3482⋊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
C4×D3×SL(2,𝔽3)2
#5569⋊C32
C4×C3⋊(#41⋊C3)2
#858173
#3482⋊C33
C3×C4×#674