Finite Group Databasebeta

← #11922934

Subgroups — #11922934

Order 1152.

834 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 71 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×#64267C211yesmaximal, characteristic, complemented
2C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4#10593661C211yesmaximal, characteristic, complemented
5#10593655C211yesmaximal, characteristic, complemented
6#10593628C211yesmaximal, characteristic, complemented
7C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
8#85382—33nomaximal
9#83542—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12#64267—22no—
13C2×C3⋊GL(2,𝔽3)—22no—
14C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
15C2×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
16C2×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
17#64091—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19C3⋊#859C2211yescharacteristic, complemented
20C2×#986—33no—
21C22×GL(2,𝔽3)—33no—
22C3×#548D311yescharacteristic, complemented
23C2×#949—31no—
24C22×#215—31no—
25D4×D12—31no—
26#548⋊C3D313yescomplemented
27#5427—33no—
28#5421—33no—
29#5394—33no—
30C4⋊GL(2,𝔽3)—33no—
31#5238—31no—
32#5227—31no—
33#5058—31no—
34#5057—31no—
35#5049—31no—
36#5046—31no—
37#5038—31no—
38#5036—31no—
39#5027—31no—
40#5026—31no—
41#5011—31no—
42C23×C3⋊D3—41no—
43C2×#638—41no—
44C2×C4⋊(C3⋊D3)—41no—
45C2×C4⋊(C3⋊D3)—41no—
46C3×#41⋊C3—21no—
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)C2311yescharacteristic
49C3⋊GL(2,𝔽3)—21no—
50C3⋊GL(2,𝔽3)—21no—
51C3⋊GL(2,𝔽3)D412yescomplemented
52C32×C22⋊C4—41no—
53#3752—41no—
54#3752—41no—
55#3265—91noSylow
56C2×C3×#41C2×D311yescharacteristic, complemented
57C22×C3×Q8C2×D311yescharacteristic
58C2×D3×D4—61no—
59C22×D12—31no—
60C22×D12—31no—
61C2×#41⋊C3C2×D313yescomplemented
62C22×SL(2,𝔽3)C2×D313yescomplemented
63#986—66no—
64C2×GL(2,𝔽3)—33no—
65C2×GL(2,𝔽3)—33no—
66C2×GL(2,𝔽3)—33no—
67C2×GL(2,𝔽3)—66no—
68C3×#124—31no—
69C3×C22⋊Q8—31no—
70C3×#121—31no—
71C3×C4×Q8C2×D311yescharacteristic, complemented
72C3×C4×D4—31no—
73#949—62no—
74#949—62no—
75C2×#215—31no—
76C2×#215—31no—
77C2×#215—31no—
78C2×#215—31no—
79C2×#215—31no—
80C2×#215—31no—
81C2×#215—62no—
82C2×#215—62no—
83C2×#213—31no—
84C2×#213—31no—
85C2×#213—31no—
86#930—31no—
87#930—31no—
88C2×#208—31no—
89C22×C3⋊C8—31no—
90#895—61no—
91#882—61no—
92#882—61no—
93#882—123no—
94#874—31no—
95C4×D12—31no—
96C4×SL(2,𝔽3)C2×D313yescomplemented
97#859—33no—
98#835—31no—
99#835—31no—
100#832—31no—
101#830—31no—
102#830—31no—
103#809—31no—
104#809—31no—
105#809—31no—
106#807—31no—
107C4⋊(C3⋊C8)—31no—
108#803—31no—
109C23×C32—41no—
110C22×C3⋊D3—41no—
111C22×C3⋊D3—41no—
112C22×C3⋊D3—41no—
113C22×C3⋊D3—82no—
114C22×C3⋊D3—82no—
115C22×C3⋊D3—82no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118#638—82no—
119C2×C3⋊Dic3—41no—
120C4⋊(C3⋊D3)—82no—
121C4⋊(C3⋊D3)—82no—
122C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
123C2×AGL(1,ℤ/8)—91no—
124C22×SD16—91no—
125#548C3⋊D311yescharacteristic, complemented
126D42—91no—
127#496—91no—
128#468—91no—
129#466—91no—
130#463—91no—
131#460—91no—
132#450—91no—
133#449—91no—
134#448—91no—
135#447—91no—
136#440—91no—
137#435—91no—
138C23×D3—61no—
139C23×D3—61no—
140C23×D3—123no—
141C3×#41—21no—
142C3×#41—61no—
143C2×C3×Q8—21no—
144C2×C3×Q8—31no—
145C2×C3×Q8—61no—
146C2×C3×Q8C22×D311yescharacteristic
147C2×C3×D4—31no—
148C2×C3×D4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C2×#67—61no—
152C2×#67—61no—
153C2×#67—123no—
154D3×D4—124no—
155C2×D12—31no—
156C2×D12—31no—
157C2×D12—31no—
158C2×D12—62no—
159C2×D12—62no—
160C2×D12—61no—
161C2×D12—61no—
162C2×D12—61no—
163C2×D12—61no—
164C2×D12—123no—
165C2×D12—123no—
166C2×D12—121no—
167C2×D12—121no—
168C2×C4×D3—61no—
169#41⋊C3—23no—
170C2×SL(2,𝔽3)—23no—
171C2×SL(2,𝔽3)C22×D313yes—
172GL(2,𝔽3)—36no—
173GL(2,𝔽3)—63no—
174GL(2,𝔽3)—63no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—31no—
177C3×C22⋊C4—31no—
178C3×C22⋊C4—43no—
179C3×C22⋊C4—61no—
180C3×C22⋊C4S411yescharacteristic
181C3×C42—31no—
182#215—32no—
183#215—32no—
184#215—62no—
185#215—62no—
186#215—61no—
187#215—61no—
188#215—61no—
189#215—61no—
190#215—121no—
191#213—62no—
192#213—62no—
193#213—121no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—61no—
197(C2×D3)⋊C4—123no—
198(C2×D3)⋊C4—123no—
199C4⋊Dic3—31no—
200#208—62no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—62no—
206C22×C32—42no—
207C22×C32—41no—
208C22×C32—81no—
209C22×C32—81no—
210C2×C3⋊D3—42no—
211C2×C3⋊D3—42no—
212C2×C3⋊D3—81no—
213C2×C3⋊D3—81no—
214C2×C3⋊D3—81no—
215C2×C3⋊D3—81no—
216C2×C3⋊D3—81no—
217C2×C3⋊D3—81no—
218C2×C3⋊D3—81no—
219C2×C3⋊D3—88no—
220C32×C4—81no—
221C32×C4—81no—
222C3⋊Dic3—81no—
223C2×#41C2×C3⋊D311yescharacteristic, complemented
224C22×Q8C2×C3⋊D311yescharacteristic
225C22×D4—91no—
226C22×D4—91no—
227C22×D4—181no—
228AGL(1,ℤ/8)—182no—
229AGL(1,ℤ/8)—182no—
230C2×SD16—91no—
231C2×SD16—91no—
232C2×SD16—91no—
233C2×SD16—91no—
234C2×SD16—91no—
235C2×SD16—91no—
236C2×SD16—182no—
237C2×SD16—182no—
238C2×D8—91no—
239C2×D8—91no—
240C2×D8—91no—
241C2×M16—91no—
242C22×C8—91no—
243#127—91no—
244#124—31no—
245C22⋊Q8—31no—
246#121—31no—
247#121—91no—
248#121—91no—
249#121—181no—
250#120—181no—
251#120—181no—
252C4×Q8C2×C3⋊D311yescharacteristic, complemented
253C4×D4—31no—
254C4×D4—91no—
255#107—91no—
256C4⋊C8—91no—
257#103—91no—
258#103—91no—
259#102—91no—
260#102—91no—
261#102—91no—
262#102—91no—
263C22⋊C8—91no—
264C22⋊C8—91no—
265#97—91no—
266C23×C3—31no—
267C23×C3—43no—
268C23×C3C2×S411yescharacteristic
269C22×D3—61no—
270C22×D3—61no—
271C22×D3—61no—
272C22×D3—123no—
273C22×D3—123no—
274C22×D3—123no—
275C22×D3—122no—
276C22×D3—122no—
277C22×D3—122no—
278C22×D3—122no—
279C22×D3—246no—
280C22×D3—246no—
281C22×D3—241no—
282C22×D3—241no—
283C22×D3—246no—
284C3×Q8—31no—
285C3×Q8—61no—
286C3×Q8—61no—
287C3×Q8D3×D411yescharacteristic
288C3×D4—61no—
289C3×D4—61no—
290C3×D4—62no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—43no—
295C2×C3×C4—43no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4C2×S411yescharacteristic
301C2×C3×C4C2×S411yescharacteristic
302#67—122no—
303#67—122no—
304#67—246no—
305C2×Dic3—61no—
306C2×Dic3—123no—
307D12—32no—
308D12—61no—
309D12—61no—
310D12—61no—
311D12—61no—
312D12—61no—
313D12—122no—
314D12—122no—
315D12—122no—
316D12—246no—
317D12—241no—
318D12—246no—
319C4×D3—122no—
320SL(2,𝔽3)D3×D413yescomplemented
321C3⋊C8—32no—
322C3⋊C8—61no—
323C3⋊C8—61no—
324C3⋊C8—61no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—82no—
329C3⋊D3—82no—
330C3⋊D3—82no—
331C3⋊D3—161no—
332C24—181no—
333C24—181no—
334#41—21no—
335#41—61no—
336C2×Q8—21no—
337C2×Q8—31no—
338C2×Q8—61no—
339C2×Q8C22×C3⋊D311yescharacteristic
340C2×D4—31no—
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—91no—
344C2×D4—91no—
345C2×D4—181no—
346C2×D4—181no—
347C2×D4—182no—
348C2×D4—182no—
349C2×D4—181no—
350C2×D4—181no—
351C2×D4—181no—
352C2×D4—181no—
353C2×D4—361no—
354C2×D4—361no—
355C2×D4—364no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—181no—
359SD16—92no—
360SD16—92no—
361SD16—182no—
362SD16—182no—
363SD16—181no—
364SD16—181no—
365SD16—181no—
366SD16—181no—
367SD16—361no—
368D8—182no—
369D8—182no—
370D8—361no—
371M16—182no—
372C2×C8—91no—
373C2×C8—91no—
374C2×C8—91no—
375C2×C8—91no—
376C2×C8—182no—
377C4⋊C4—31no—
378C4⋊C4—31no—
379C4⋊C4—91no—
380C22⋊C4—31no—
381C22⋊C4—61no—
382C22⋊C4—181no—
383C22⋊C4—181no—
384C22⋊C4—181no—
385C22⋊C4C3⋊S411yescharacteristic
386C42—31no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—46no—
390C22×C3—43no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—83no—
395C22×C3—83no—
396C22×C3#98612yescomplemented
397C22×C3C22×S411yescharacteristic
398C2×D3—62no—
399C2×D3—126no—
400C2×D3—121no—
401C2×D3—121no—
402C2×D3—126no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—121no—
407C2×D3—121no—
408C2×D3—121no—
409C2×D3—243no—
410C2×D3—243no—
411C2×D3—243no—
412C2×D3—243no—
413C2×D3—243no—
414C2×D3—243no—
415C2×D3—243no—
416C2×D3—244no—
417C2×D3—244no—
418C2×D3—2424no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—83no—
426C3×C4—83no—
427Dic3—121no—
428Dic3—243no—
429C32—41noSylow
430C23—31no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23—362no—
435C23—362no—
436C23—362no—
437C23—362no—
438C23—721no—
439C23—721no—
440C23C2×C3⋊S411yescharacteristic
441Q8—31no—
442Q8—61no—
443Q8—61no—
444Q8D4×C3⋊D311yescharacteristic
445D4—61no—
446D4—61no—
447D4—62no—
448D4—92no—
449D4—181no—
450D4—181no—
451D4—181no—
452D4—181no—
453D4—181no—
454D4—362no—
455D4—362no—
456D4—362no—
457D4—362no—
458D4—362no—
459D4—721no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—181no—
468C2×C4—362no—
469C2×C4C2×C3⋊S411yescharacteristic
470C2×C4C2×C3⋊S411yescharacteristic
471C8—92no—
472C8—181no—
473C8—181no—
474C8—181no—
475C2×C3—22no—
476C2×C3—43no—
477C2×C3—43no—
478C2×C3—43no—
479C2×C3—86no—
480C2×C3—121no—
481C2×C3D4×S411yescharacteristic
482C2×C3C2×#98611yescharacteristic
483C2×C3#592611yescharacteristic
484D3—122no—
485D3—246no—
486D3—246no—
487D3—241no—
488D3—241no—
489D3—483no—
490C22—21no—
491C22—21no—
492C22—61no—
493C22—61no—
494C22—61no—
495C22—182no—
496C22—361no—
497C22—361no—
498C22—361no—
499C22—361no—
500C22—361no—
501C22—361no—
502C22—361no—
503C22—361no—
504C22—724no—
505C22—724no—
506C22#6426712yescomplemented
507C22C22×C3⋊S411yescharacteristic
508C4—21no—
509C4—21no—
510C4—31no—
511C4—31no—
512C4—61no—
513C4—61no—
514C4—361no—
515C3—43no—
516C3#8538211yescharacteristic, complemented
517C2—22no—
518C2—121no—
519C2—362no—
520C2—721no—
521C2—721no—
522C2D4×C3⋊S411yescharacteristic
523C2C2×#6426711yescharacteristic
524C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#548D3
socleC22×C3C22×S4
hypercenterC22⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64267C2semidirect
2. C22×C3⋊GL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. #10593661C2semidirect
5. #10593655C2semidirect
6. #10593628C2semidirect
7. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
10. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
15. C2×C3⋊GL(2,𝔽3)C22semidirect
16. C2×C3⋊GL(2,𝔽3)C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. C3⋊#859C22semidirect
22. C3×#548D3semidirect
26. #548⋊C3D3semidirect
51. C3⋊GL(2,𝔽3)D4semidirect
56. C2×C3×#41C2×D3semidirect
61. C2×#41⋊C3C2×D3semidirect
62. C22×SL(2,𝔽3)C2×D3semidirect
71. C3×C4×Q8C2×D3semidirect
96. C4×SL(2,𝔽3)C2×D3semidirect
122. C3×SL(2,𝔽3)C2×D4semidirect
125. #548C3⋊D3semidirect
223. C2×#41C2×C3⋊D3semidirect
252. C4×Q8C2×C3⋊D3semidirect
320. SL(2,𝔽3)D3×D4semidirect
396. C22×C3#986semidirect
506. C22#64267semidirect
516. C3#85382semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C22×C3⋊GL(2,𝔽3)2
C3×#548⋊C32
#105936612
#105936552
#105936282
C4⋊(C3⋊GL(2,𝔽3))2
#853823
#835423
#640914