Finite Group Databasebeta

← #11923084

Subgroups — #11923084

Order 1152.

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

normal subgroups only

This group has 772 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
1C2×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#64264C211yesmaximal, characteristic, complemented
3#10593661C211yesmaximal, characteristic, complemented
4#10593659C211yesmaximal, characteristic, complemented
5#10593655C211yesmaximal, characteristic, complemented
6#10593653C211yesmaximal, characteristic, complemented
7C2×C3⋊#859C211yesmaximal, characteristic, complemented
8#85539—33nomaximal
9#84114—31nomaximal
10C3×2−1+4⋊C3—22no—
11C2×C3×#41⋊C3C2211yescharacteristic
12C22×C3×SL(2,𝔽3)C2211yescharacteristic
13C22×C3×SL(2,𝔽3)C2211yescharacteristic
14#64264—22no—
15C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
16C2×C3⋊#226C2211yescharacteristic
17#64148—41nomaximal
18C3⋊#859C2211yescharacteristic
19C3⋊#859C2211yescharacteristic
20C2×C3×2−1+4D311yescharacteristic, complemented
21C2×2−1+4⋊C3D313yescomplemented
22C2×#983—33no—
23#5833—31no—
24C2×#950—31no—
25C2×#942—31no—
26#5427—33no—
27#5425—33no—
28#5421—33no—
29#5419—33no—
30C2×#859—33no—
31#5239—31no—
32#5239—31no—
33#5238—31no—
34#5238—31no—
35#5234—31no—
36#5232—31no—
37#5228—31no—
38#5228—31no—
39#5227—31no—
40#5227—31no—
41C2×#835—31no—
42C2×C32×D4—41no—
43C2×#638—41no—
44C2×#638—41no—
45C22×C3⋊Dic3—41no—
46C3×#41⋊C3—21no—
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)D412yescomplemented
49C2×C3×SL(2,𝔽3)C2311yescharacteristic
50C3⋊GL(2,𝔽3)—21no—
51C3⋊#226—21no—
52#3757—41no—
53#3752—41no—
54#3750—41no—
55#3013—91noSylow
56C3×2−1+4—22no—
57C3×2−1+4—62no—
58C2×C3×#41—31no—
59C2×C3×#41—31no—
60C2×C3×#41—31no—
61C2×C3×#41C2×D311yescharacteristic
62C22×C3×Q8—31no—
63C22×C3×Q8C2×D311yescharacteristic
64C22×C3×Q8C2×D311yescharacteristic
65C2×#235—31no—
662−1+4⋊C3—26no—
67C2×#41⋊C3C2×D313yes—
68C22×SL(2,𝔽3)C2×D313yes—
69C22×SL(2,𝔽3)C2×D313yes—
70#983—66no—
71C2×GL(2,𝔽3)—33no—
72C2×#226—33no—
73#950—64no—
74#946—61no—
75C2×#216—31no—
76C2×#216—31no—
77C2×#216—31no—
78#942—62no—
79#942—62no—
80C2×#215—31no—
81C2×#215—31no—
82C2×#215—31no—
83#939—61no—
84#939—61no—
85#939—123no—
86#938—61no—
87C2×#214—31no—
88C2×#213—31no—
89#930—31no—
90#930—31no—
91C2×C4⋊Dic3—31no—
92#924—31no—
93#924—31no—
94C2×#208—31no—
95C22×C3⋊C8—31no—
96#859—33no—
97#859—33no—
98#835—31no—
99#835—31no—
100#835—31no—
101#835—31no—
102#835—31no—
103#835—31no—
104#832—31no—
105#832—31no—
106#830—31no—
107#830—31no—
108#830—31no—
109#830—31no—
110C23×C32—41no—
111C23×C32—41no—
112C22×C3⋊D3—41no—
113C32×D4—82no—
114C2×C32×C4—41no—
115#638—82no—
116#638—82no—
117C2×C3⋊Dic3—41no—
118C2×C3⋊Dic3—41no—
119C2×C3⋊Dic3—41no—
120C2×C3⋊Dic3—82no—
121C3×SL(2,𝔽3)C2×D411yescharacteristic
122C2×2−1+4C3⋊D311yescharacteristic, complemented
123C2×#137—91no—
124C2×#135—91no—
125#537—91no—
126#452—91no—
127#452—91no—
128#451—91no—
129#451—91no—
130#449—91no—
131#449—91no—
132#448—91no—
133#448—91no—
134#417—91no—
135C2×#103—91no—
136#408—91no—
137C3×#41—21no—
138C3×#41—34no—
139C3×#41—62no—
140C3×#41—62no—
141C3×#41—62no—
142C3×#41—62no—
143C3×#41—62no—
144C3×#41—61no—
145C2×C3×Q8—21no—
146C2×C3×Q8—31no—
147C2×C3×Q8—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—32no—
150C2×C3×Q8—62no—
151C2×C3×Q8—62no—
152C2×C3×Q8—61no—
153C2×C3×Q8#6712yescomplemented
154C2×C3×Q8C22×D311yescharacteristic
155C2×C3×D4—31no—
156C2×C3×D4—31no—
157C2×C3×D4—31no—
158C2×C3×D4—43no—
159C2×C3×D4S411yescharacteristic
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—61no—
164C2×#67—61no—
165C2×#67—61no—
166C2×#67—61no—
167C2×#67—123no—
168C2×#67—123no—
169C22×Dic3—61no—
170C22×Dic3—123no—
171#235—62no—
172#235—62no—
173C2×D12—31no—
174C2×C4×D3—61no—
175C2×Dic6—31no—
176#41⋊C3—23no—
177C2×SL(2,𝔽3)—23no—
178C2×SL(2,𝔽3)#6716yescomplemented
179C2×SL(2,𝔽3)C22×D313yes—
180GL(2,𝔽3)—63no—
181#226—63no—
182#217—61no—
183#217—61no—
184#217—123no—
185#216—62no—
186#216—61no—
187#216—62no—
188#216—61no—
189#215—62no—
190#215—61no—
191#215—62no—
192#215—61no—
193#214—62no—
194#213—62no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—61no—
197(C2×D3)⋊C4—123no—
198C4⋊Dic3—31no—
199C4⋊Dic3—31no—
200C4⋊Dic3—61no—
201Dic3⋊C4—61no—
202Dic3⋊C4—61no—
203Dic3⋊C4—123no—
204#208—62no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—62no—
210C22×C32—42no—
211C22×C32—41no—
212C22×C32—81no—
213C22×C32—81no—
214C22×C32—81no—
215C22×C32—81no—
216C22×C32—81no—
217C2×C3⋊D3—81no—
218C2×C3⋊D3—81no—
219C2×C3⋊D3—81no—
220C32×C4—81no—
221C3⋊Dic3—42no—
222C3⋊Dic3—81no—
223C3⋊Dic3—81no—
2242−1+4—22no—
2252−1+4—62no—
226C2×#41—31no—
227C2×#41—31no—
228C2×#41—31no—
229C2×#41—91no—
230C2×#41C2×C3⋊D311yescharacteristic
231C22×Q8—31no—
232C22×Q8C2×C3⋊D311yescharacteristic
233C22×Q8C2×C3⋊D311yescharacteristic
234#137—182no—
235#137—182no—
236#135—184no—
237C2×Q16—91no—
238C2×Q16—91no—
239C2×Q16—91no—
240C2×SD16—91no—
241C2×SD16—91no—
242C2×SD16—91no—
243C2×SD16—91no—
244C2×D8—91no—
245C2×M16—91no—
246C22×C8—91no—
247C22⋊Q8—91no—
248C22⋊Q8—91no—
249C22⋊Q8—181no—
250#121—91no—
251#121—91no—
252#121—181no—
253#121—181no—
254#121—181no—
255C2×C4⋊C4—91no—
256#103—91no—
257#103—91no—
258#103—91no—
259#103—91no—
260#103—91no—
261#103—91no—
262#102—91no—
263#102—91no—
264C22⋊C8—91no—
265C22⋊C8—91no—
266C22⋊C8—91no—
267C22⋊C8—91no—
268C23×C3—31no—
269C23×C3—43no—
270C23×C3—43no—
271C23×C3C2×S411yescharacteristic
272C23×C3C2×S411yescharacteristic
273C22×D3—61no—
274C22×D3—123no—
275C3×Q8—32no—
276C3×Q8—31no—
277C3×Q8—32no—
278C3×Q8—62no—
279C3×Q8—61no—
280C3×Q8—61no—
281C3×Q8C2×#6711yescharacteristic
282C3×D4—22no—
283C3×D4—32no—
284C3×D4—62no—
285C3×D4—61no—
286C3×D4—62no—
287C3×D4—86no—
288C2×C3×C4—32no—
289C2×C3×C4—32no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—43no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—64no—
299C2×C3×C4C2×S411yescharacteristic
300#67—122no—
301#67—246no—
302#67—241no—
303#67—241no—
304#67—246no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—123no—
309C2×Dic3—123no—
310C2×Dic3—123no—
311C2×Dic3—241no—
312C2×Dic3—246no—
313D12—61no—
314D12—61no—
315C4×D3—122no—
316Dic6—61no—
317Dic6—61no—
318SL(2,𝔽3)C2×#6713yes—
319C3⋊C8—61no—
320C3⋊C8—61no—
321C3⋊C8—61no—
322C3⋊C8—61no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—82no—
327C2×C32—161no—
328C3⋊D3—161no—
329#41—21no—
330#41—34no—
331#41—62no—
332#41—62no—
333#41—62no—
334#41—62no—
335#41—62no—
336#41—61no—
337#41—182no—
338#41—182no—
339C2×Q8—21no—
340C2×Q8—31no—
341C2×Q8—31no—
342C2×Q8—31no—
343C2×Q8—32no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—61no—
347C2×Q8—91no—
348C2×Q8#63812yescomplemented
349C2×Q8C22×C3⋊D311yescharacteristic
350C2×D4—31no—
351C2×D4—31no—
352C2×D4—31no—
353C2×D4—91no—
354C2×D4—181no—
355C2×D4—181no—
356C2×D4—181no—
357C2×D4C3⋊S411yescharacteristic
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—61no—
362C22×C4—181no—
363C22×C4—181no—
364Q16—182no—
365Q16—182no—
366Q16—181no—
367Q16—181no—
368SD16—182no—
369SD16—182no—
370SD16—182no—
371SD16—181no—
372SD16—181no—
373D8—182no—
374M16—182no—
375C2×C8—91no—
376C2×C8—91no—
377C2×C8—91no—
378C2×C8—91no—
379C2×C8—182no—
380C4⋊C4—91no—
381C4⋊C4—91no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C4⋊C4—181no—
385C22⋊C4—181no—
386C22⋊C4—181no—
387C22⋊C4—181no—
388C22⋊C4—181no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—46no—
395C22×C3—43no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—83no—
400C22×C3—83no—
401C22×C3—83no—
402C22×C3—83no—
403C22×C3—83no—
404C22×C3GL(2,ℤ/4)12yes—
405C22×C3C22×S411yescharacteristic
406C2×D3—121no—
407C2×D3—121no—
408C2×D3—121no—
409C2×D3—243no—
410C2×D3—243no—
411C2×D3—243no—
412C3×C4—21no—
413C3×C4—31no—
414C3×C4—31no—
415C3×C4—32no—
416C3×C4—61no—
417C3×C4—83no—
418Dic3—126no—
419Dic3—121no—
420Dic3—121no—
421Dic3—121no—
422Dic3—243no—
423Dic3—243no—
424C32—41noSylow
425C23—31no—
426C23—181no—
427C23C2×C3⋊S411yescharacteristic
428C23C2×C3⋊S411yescharacteristic
429Q8—32no—
430Q8—31no—
431Q8—32no—
432Q8—62no—
433Q8—61no—
434Q8—61no—
435Q8—181no—
436Q8—181no—
437Q8C2×#63811yescharacteristic
438D4—22no—
439D4—32no—
440D4—62no—
441D4—61no—
442D4—62no—
443D4—181no—
444D4—181no—
445D4—362no—
446D4—721no—
447D4—721no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—64no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—362no—
462C2×C4—721no—
463C2×C4C2×C3⋊S411yescharacteristic
464C8—181no—
465C8—181no—
466C8—181no—
467C8—181no—
468C2×C3—22no—
469C2×C3—41no—
470C2×C3—43no—
471C2×C3—43no—
472C2×C3—43no—
473C2×C3—86no—
474C2×C3—121no—
475C2×C3—163no—
476C2×C3#592611yescharacteristic
477C2×C3#592711yescharacteristic
478C2×C3C2×GL(2,ℤ/4)11yescharacteristic
479D3—241no—
480D3—483no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—61no—
487C22—61no—
488C22—61no—
489C22—361no—
490C22—361no—
491C22—361no—
492C22#6426912yes—
493C22C22×C3⋊S411yescharacteristic
494C4—21no—
495C4—31no—
496C4—31no—
497C4—32no—
498C4—61no—
499C4—361no—
500C4—361no—
501C4—361no—
502C3—43no—
503C3#8553911yescharacteristic, complemented
504C2—22no—
505C2—41no—
506C2—121no—
507C2—721no—
508C2#1059658911yescharacteristic
509C2#1059659011yescharacteristic
510C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×2−1+4D3
socleC22×C3C22×S4
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×#64264C2semidirect
3. #10593661C2semidirect
4. #10593659C2semidirect
5. #10593655C2semidirect
6. #10593653C2semidirect
7. C2×C3⋊#859C2semidirect
20. C2×C3×2−1+4D3semidirect
21. C2×2−1+4⋊C3D3semidirect
48. C2×C3×SL(2,𝔽3)D4semidirect
122. C2×2−1+4C3⋊D3semidirect
153. C2×C3×Q8#67semidirect
178. C2×SL(2,𝔽3)#67semidirect
348. C2×Q8#638semidirect
503. C3#85539semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×#642642
#105936612
#105936592
#105936552
#105936532
C2×C3⋊#8592
#855393
#841143
#641484