Finite Group Databasebeta

← #11923070

Subgroups — #11923070

Order 1152.

792 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#64266C211yesmaximal, characteristic, complemented
3#10593661C211yesmaximal, characteristic, complemented
4#10593659C211yesmaximal, characteristic, complemented
5#10593658C211yesmaximal, characteristic, complemented
6#10593655C211yesmaximal, characteristic, complemented
7#10593653C211yesmaximal, characteristic, complemented
8#85521—33nomaximal
9#84080—31nomaximal
10C2×C3×#41⋊C3—22no—
11C2×C3×#41⋊C3C2211yescharacteristic
12C2×C3×#41⋊C3C2211yescharacteristic
13C22×C3×SL(2,𝔽3)C2211yescharacteristic
14#64266—22no—
15C2×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
16C2×C3⋊#226C2211yescharacteristic, complemented
17C4×#638—41nomaximal
18C3⋊#859C2211yescharacteristic, complemented
19C3⋊#859C2211yescharacteristic, complemented
20C22×C3×#41D311yescharacteristic, complemented
21C22×#41⋊C3D313yescomplemented
22C2×#985—33no—
23C2×#950—31no—
24C2×#950—31no—
25#5791—31no—
26#5427—33no—
27#5425—33no—
28#5424—33no—
29#5421—33no—
30#5419—33no—
31#5239—31no—
32#5239—31no—
33#5238—31no—
34#5238—31no—
35#5233—31no—
36#5233—31no—
37#5228—31no—
38#5227—31no—
39#5218—31no—
40#5217—31no—
41C2×#830—31no—
42C22×C32×C4—41no—
43C2×#638—41no—
44C2×C4×C3⋊D3—41no—
45C3×#41⋊C3—21no—
46C3×#41⋊C3D412yescomplemented
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)C2311yescharacteristic
49C3⋊GL(2,𝔽3)—21no—
50C3⋊#226—21no—
51#3757—41no—
52#3752—41no—
53#3750—41no—
54C4×C3⋊Dic3—41no—
55#3002—91noSylow
56C2×C3×#41—22no—
57C2×C3×#41—31no—
58C2×C3×#41—31no—
59C2×C3×#41—62no—
60C2×C3×#41C2×D311yescharacteristic
61C2×C3×#41C2×D311yescharacteristic
62C22×C3×Q8C2×D311yescharacteristic
63C22×C3×D4—31no—
64C23×C3×C4—31no—
65C2×#235—31no—
66C2×#41⋊C3—26no—
67C2×#41⋊C3C2×D313yes—
68C2×#41⋊C3C2×D313yes—
69C22×SL(2,𝔽3)C2×D313yes—
70#985—66no—
71C2×GL(2,𝔽3)—33no—
72C2×#226—33no—
73#953—61no—
74#950—62no—
75#950—64no—
76#950—62no—
77C2×#216—31no—
78C2×#216—31no—
79C2×#215—31no—
80C2×#215—31no—
81C2×#214—31no—
82C2×#214—31no—
83C2×#213—31no—
84C2×#213—31no—
85#930—31no—
86#930—31no—
87#929—61no—
88C4×#67—61no—
89C4×#67—61no—
90C4×#67—123no—
91#926—31no—
92#924—31no—
93#924—31no—
94C22×C3⋊C8—31no—
95C22×C3⋊C8—31no—
96#859—33no—
97#859—33no—
98#835—31no—
99#835—31no—
100#835—31no—
101#835—31no—
102#832—31no—
103#832—31no—
104#832—31no—
105#832—31no—
106#830—31no—
107#830—31no—
108#830—31no—
109#830—31no—
110C23×C32—41no—
111C22×C3⋊D3—41no—
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4—82no—
115#638—44no—
116C2×C3⋊Dic3—41no—
117C2×C3⋊Dic3—41no—
118C2×C3⋊Dic3—41no—
119C4×C3⋊D3—82no—
120C3×SL(2,𝔽3)C2×D411yescharacteristic
121C22×#41C3⋊D311yescharacteristic, complemented
122C2×#135—91no—
123C2×#135—91no—
124#525—91no—
125#452—91no—
126#452—91no—
127#451—91no—
128#450—91no—
129#449—91no—
130#449—91no—
131#448—91no—
132#447—91no—
133#416—91no—
134#416—91no—
135C2×C22⋊C8—91no—
136C24×C3—31no—
137C3×#41—21no—
138C3×#41—32no—
139C3×#41—34no—
140C3×#41—62no—
141C3×#41—62no—
142C3×#41—61no—
143C3×#41—62no—
144C3×#41#6712yescomplemented
145C2×C3×Q8—21no—
146C2×C3×Q8—31no—
147C2×C3×Q8—61no—
148C2×C3×Q8C22×D311yescharacteristic
149C2×C3×D4—31no—
150C2×C3×D4—31no—
151C2×C3×D4—61no—
152C2×C3×D4—62no—
153C2×C3×D4—62no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—43no—
160C22×C3×C4—62no—
161C22×C3×C4—62no—
162C22×C3×C4S411yescharacteristic
163C2×#67—61no—
164C2×#67—61no—
165C2×#67—61no—
166C2×#67—123no—
167#235—62no—
168#235—62no—
169C2×D12—31no—
170C2×C4×D3—61no—
171C2×C4×D3—123no—
172C2×Dic6—31no—
173#41⋊C3—23no—
174#41⋊C3#6716yescomplemented
175C2×SL(2,𝔽3)—23no—
176C2×SL(2,𝔽3)C22×D313yes—
177GL(2,𝔽3)—63no—
178#226—63no—
179#217—61no—
180#217—61no—
181#217—61no—
182#217—123no—
183#216—61no—
184#216—62no—
185#216—61no—
186#215—61no—
187#215—62no—
188#215—61no—
189#214—62no—
190#214—62no—
191#213—62no—
192#213—62no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—123no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198Dic3⋊C4—61no—
199Dic3⋊C4—61no—
200Dic3⋊C4—123no—
201C4×Dic3—61no—
202C4×Dic3—123no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—62no—
208C2×C3⋊C8—62no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32—81no—
212C22×C32—81no—
213C2×C3⋊D3—42no—
214C2×C3⋊D3—81no—
215C2×C3⋊D3—81no—
216C32×C4—42no—
217C32×C4—81no—
218C3⋊Dic3—42no—
219C3⋊Dic3—81no—
220C3⋊Dic3—81no—
221C2×#41—22no—
222C2×#41—31no—
223C2×#41—31no—
224C2×#41—62no—
225C2×#41—91no—
226C2×#41C2×C3⋊D311yescharacteristic
227C2×#41C2×C3⋊D311yescharacteristic
228C22×Q8C2×C3⋊D311yescharacteristic
229C22×D4—31no—
230C23×C4—31no—
231#135—182no—
232#135—184no—
233#135—182no—
234C2×Q16—91no—
235C2×Q16—91no—
236C2×SD16—91no—
237C2×SD16—91no—
238C2×SD16—91no—
239C2×SD16—91no—
240C2×D8—91no—
241C2×D8—91no—
242C22×C8—91no—
243C22×C8—91no—
244#123—181no—
245C22⋊Q8—91no—
246C22⋊Q8—91no—
247#121—91no—
248#121—91no—
249#120—181no—
250C4×D4—181no—
251C4×D4—181no—
252#117—91no—
253#103—91no—
254#103—91no—
255#103—91no—
256#103—91no—
257#102—91no—
258#102—91no—
259#102—91no—
260#102—91no—
261C22⋊C8—91no—
262C22⋊C8—91no—
263C22⋊C8—91no—
264C22⋊C8—91no—
265C23×C3—31no—
266C23×C3—31no—
267C23×C3—43no—
268C23×C3—62no—
269C23×C3—62no—
270C23×C3—121no—
271C23×C3C2×S411yescharacteristic
272C22×D3—61no—
273C22×D3—123no—
274C3×Q8—31no—
275C3×Q8—32no—
276C3×Q8—61no—
277C3×Q8C2×#6711yescharacteristic
278C3×D4—32no—
279C3×D4—32no—
280C3×D4—61no—
281C3×D4—61no—
282C3×D4—64no—
283C2×C3×C4—22no—
284C2×C3×C4—31no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—31no—
288C2×C3×C4—43no—
289C2×C3×C4—43no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—62no—
296C2×C3×C4—86no—
297C2×C3×C4C2×S411yescharacteristic
298C2×C3×C4C2×S411yescharacteristic
299#67—122no—
300#67—1212no—
301#67—122no—
302#67—122no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—123no—
307C2×Dic3—123no—
308C2×Dic3—123no—
309D12—61no—
310D12—61no—
311C4×D3—122no—
312C4×D3—246no—
313Dic6—61no—
314Dic6—61no—
315SL(2,𝔽3)C2×#6713yes—
316C3⋊C8—61no—
317C3⋊C8—61no—
318C3⋊C8—61no—
319C3⋊C8—61no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—82no—
324C3⋊D3—82no—
325C24—31no—
326#41—21no—
327#41—32no—
328#41—34no—
329#41—62no—
330#41—62no—
331#41—61no—
332#41—62no—
333#41—182no—
334#41—182no—
335#41#63812yescomplemented
336C2×Q8—21no—
337C2×Q8—31no—
338C2×Q8—61no—
339C2×Q8—91no—
340C2×Q8C22×C3⋊D311yescharacteristic
341C2×D4—31no—
342C2×D4—31no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—91no—
347C2×D4—181no—
348C2×D4—181no—
349C2×D4—181no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—181no—
358C22×C4C3⋊S411yescharacteristic
359Q16—181no—
360Q16—182no—
361Q16—181no—
362SD16—182no—
363SD16—182no—
364SD16—181no—
365SD16—182no—
366SD16—181no—
367D8—182no—
368D8—182no—
369C2×C8—91no—
370C2×C8—91no—
371C2×C8—91no—
372C2×C8—91no—
373C2×C8—182no—
374C2×C8—182no—
375C4⋊C4—91no—
376C4⋊C4—91no—
377C4⋊C4—181no—
378C4⋊C4—181no—
379C22⋊C4—181no—
380C22⋊C4—181no—
381C22⋊C4—181no—
382C22⋊C4—181no—
383C22⋊C4—181no—
384C42—181no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—32no—
388C22×C3—46no—
389C22×C3—43no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—83no—
396C22×C3—83no—
397C22×C3—124no—
398C22×C3#98512yescomplemented
399C22×C3C22×S411yescharacteristic
400C2×D3—126no—
401C2×D3—121no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—243no—
405C2×D3—243no—
406C3×C4—21no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—46no—
410C3×C4—61no—
411C3×C4—83no—
412C3×C4GL(2,ℤ/4)12yes—
413Dic3—126no—
414Dic3—121no—
415Dic3—121no—
416Dic3—121no—
417Dic3—243no—
418Dic3—243no—
419C32—41noSylow
420C23—31no—
421C23—31no—
422C23—62no—
423C23—62no—
424C23—121no—
425C23—181no—
426C23C2×C3⋊S411yescharacteristic
427Q8—31no—
428Q8—32no—
429Q8—61no—
430Q8—181no—
431Q8—181no—
432Q8C2×#63811yescharacteristic
433D4—32no—
434D4—32no—
435D4—61no—
436D4—61no—
437D4—64no—
438D4—181no—
439D4—181no—
440D4—362no—
441D4—362no—
442D4—362no—
443C2×C4—22no—
444C2×C4—31no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—31no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—62no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—62no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—362no—
458C2×C4C2×C3⋊S411yescharacteristic
459C2×C4C2×C3⋊S411yescharacteristic
460C8—181no—
461C8—181no—
462C8—181no—
463C8—181no—
464C2×C3—22no—
465C2×C3—43no—
466C2×C3—43no—
467C2×C3—43no—
468C2×C3—62no—
469C2×C3—86no—
470C2×C3—121no—
471C2×C3C2×#98511yescharacteristic
472C2×C3#592311yescharacteristic
473C2×C3C2×GL(2,ℤ/4)11yescharacteristic
474D3—246no—
475D3—241no—
476C22—21no—
477C22—21no—
478C22—32no—
479C22—61no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—61no—
484C22—124no—
485C22—361no—
486C22—361no—
487C22—361no—
488C22#6426612yescomplemented
489C22C22×C3⋊S411yescharacteristic
490C4—21no—
491C4—31no—
492C4—31no—
493C4—61no—
494C4—361no—
495C4—361no—
496C4—361no—
497C4#6426912yes—
498C3—43no—
499C3#8552111yescharacteristic, complemented
500C2—22no—
501C2—62no—
502C2—121no—
503C2—721no—
504C2C2×#6426611yescharacteristic
505C2#1059658611yescharacteristic
506C2C2×#6426911yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC22×C3×#41D3
socleC22×C3C22×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×#64266C2semidirect
3. #10593661C2semidirect
4. #10593659C2semidirect
5. #10593658C2semidirect
6. #10593655C2semidirect
7. #10593653C2semidirect
15. C2×C3⋊GL(2,𝔽3)C22semidirect
16. C2×C3⋊#226C22semidirect
18. C3⋊#859C22semidirect
19. C3⋊#859C22semidirect
20. C22×C3×#41D3semidirect
21. C22×#41⋊C3D3semidirect
46. C3×#41⋊C3D4semidirect
121. C22×#41C3⋊D3semidirect
144. C3×#41#67semidirect
174. #41⋊C3#67semidirect
335. #41#638semidirect
398. C22×C3#985semidirect
488. C22#64266semidirect
499. C3#85521semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642662
#105936612
#105936592
#105936582
#105936552
#105936532
#855213
#840803
C4×#6384