Finite Group Databasebeta

← #87287⋊C3 #11923401

Subgroups — #87287⋊C3 #11923401

Order 1152.

717 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 83 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
2#5964⋊C3C211yesmaximal, characteristic, complemented
3#5837⋊C3C212yesmaximal, complemented
4#5838⋊C3C212yesmaximal, complemented
5#5819⋊C3C211yesmaximal, characteristic, complemented
6#87287C311yesmaximal, characteristic, complemented
7#3586⋊C3—31nomaximal
8C3×2−1+4⋊C3—21no—
9C3×2−1+4⋊C3C2212yescomplemented
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)—21no—
12#1010⋊C3C2211yescharacteristic, complemented
13D3×#41⋊C3—21no—
14C3⋊#994C2211yescharacteristic, complemented
15#1007⋊C3—21no—
16C3×#932—41nomaximal
17C3⋊(#131⋊C3)C2212yescomplemented
18C2×C3×2−1+4C2×C311yescharacteristic, complemented
19#5964C2×C311yescharacteristic, complemented
20#5945—31no—
21C2×2−1+4⋊C3—21no—
22C2×2−1+4⋊C3D311yescharacteristic, complemented
23#5838—32no—
24#5838C2×C312yescomplemented
25#5837—32no—
26#5837C2×C312yescomplemented
27#5822—31no—
28#5822—31no—
29#5819C2×C311yescharacteristic, complemented
30C2×#942—31no—
31C2×#942—31no—
32#578⋊C3—32no—
33#577⋊C3—32no—
34#568⋊C3—31no—
35C2×C32×D4—41no—
36C3×#235—41no—
37C3×#41⋊C3D411yescharacteristic
38C3×#41⋊C3C2311yescharacteristic
39C2×C3×SL(2,𝔽3)—22no—
40C2×C3×SL(2,𝔽3)D411yescharacteristic
41D3×SL(2,𝔽3)—21no—
42C3⋊(#41⋊C3)—21no—
43C3×#214—42no—
44C3×#213—42no—
45C3×#208—41no—
46#3586—31noSylow
47C3×2−1+4—21no—
48C3×2−1+4—32no—
49C3×2−1+4—61no—
50C3×2−1+4C22×C312yes—
51C2×C3×#41—31no—
52C2×C3×#41—61no—
53C2×C3×#41C22×C311yescharacteristic
54C22×C3×Q8—21no—
55C22×C3×Q8—31no—
56#1010—31no—
57#1010C22×C311yescharacteristic, complemented
58#1009—31no—
59#1009C22×C311yescharacteristic, complemented
60D3×#41—21no—
61D3×#41—61no—
62#1007—21no—
63#1004—61no—
64C2×#235—31no—
65C2×#235—31no—
662−1+4⋊C3—22no—
672−1+4⋊C3—21no—
682−1+4⋊C3—22no—
692−1+4⋊C3—31no—
702−1+4⋊C3—61no—
712−1+4⋊C3C2×D312yescomplemented
72#994—31no—
73C2×#41⋊C3—21no—
74C2×#41⋊C3—61no—
75C2×#41⋊C3C2×D311yescharacteristic
76C22×SL(2,𝔽3)—22no—
77C22×SL(2,𝔽3)—21no—
78C3×AGL(1,ℤ/8)—121no—
79#951—32no—
80#951—32no—
81#950—34no—
82#950—32no—
83#950—32no—
84#949—32no—
85#949—32no—
86#948—32no—
87#948C22×C312yescomplemented
88C2×#216—32no—
89C2×#216—32no—
90#942—32no—
91#942—34no—
92#942—31no—
93#942—32no—
94#942—32no—
95#942—31no—
96C2×#215—32no—
97C2×#215—32no—
98#932—31no—
99#932A411yescharacteristic
100C2×#208—31no—
101C2×#208—31no—
102#131⋊C3—32no—
103C23×C32—81no—
104C32×D4—42no—
105C32×D4—81no—
106C2×C32×C4—41no—
107C3×#67—81no—
108C3×D12—41no—
109C3×C4×D3—81no—
110C3×Dic6—41no—
111C3×SL(2,𝔽3)C2×D411yescharacteristic
112C3×C3⋊C8—42no—
113#585—31no—
114C2×2−1+4C3×D311yescharacteristic, complemented
115#578—32no—
116#578—92no—
117#577—32no—
118#577—92no—
119#575—91no—
120#575—91no—
121C2×#137—91no—
122C2×#137—91no—
123#568—31no—
124C3×#41—31no—
125C3×#41—31no—
126C3×#41—32no—
127C3×#41—32no—
128C3×#41—61no—
129C3×#41—61no—
130C3×#41—64no—
131C3×#41—62no—
132C3×#41—62no—
133C3×#41—121no—
134C3×#41C3×D411yescharacteristic
135C3×#41C23×C311yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—32no—
138C2×C3×Q8—32no—
139C2×C3×Q8—31no—
140C2×C3×Q8—61no—
141C2×C3×Q8—61no—
142C2×C3×Q8—61no—
143C2×C3×Q8—62no—
144C2×C3×Q8C3×D411yescharacteristic
145C2×C3×D4—31no—
146C2×C3×D4—41no—
147C2×C3×D4—61no—
148C2×C3×D4—81no—
149C2×C3×D4C2×A411yescharacteristic
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—61no—
153C2×#67—61no—
154C2×#67—61no—
155#239—21no—
156#239—61no—
157D3×Q8—21no—
158D3×Q8—61no—
159#237—62no—
160#237—61no—
161#237—61no—
162D3×D4—61no—
163D3×D4—62no—
164D3×D4—61no—
165#235—32no—
166#235—31no—
167#235—31no—
168#235—31no—
169#235—61no—
170#235—61no—
171#235—61no—
172#235—61no—
173#235C2×A411yescharacteristic
174C2×D12—31no—
175C2×D12—31no—
176C2×C4×D3—61no—
177C2×C4×D3—61no—
178C2×Dic6—31no—
179C2×Dic6—31no—
180#41⋊C3—21no—
181#41⋊C3—21no—
182#41⋊C3—61no—
183#41⋊C3#6711yescharacteristic
184#41⋊C3C22×D311yescharacteristic
185C2×SL(2,𝔽3)—24no—
186C2×SL(2,𝔽3)—22no—
187C2×SL(2,𝔽3)—21no—
188C2×SL(2,𝔽3)—61no—
189C2×SL(2,𝔽3)#6711yescharacteristic
190C3×SD16—122no—
191C3×D8—122no—
192C3×M16—121no—
193#216—34no—
194#216—32no—
195#216—32no—
196#215—34no—
197#215—32no—
198#215—32no—
199#214—32no—
200#214C2×A412yes—
201#213—32no—
202#213C2×A412yes—
203#208—31no—
204#208—32no—
205#208—32no—
206#208C2×A411yescharacteristic
207C2×C3⋊C8—32no—
208C2×C3⋊C8—32no—
209C22×C32—41no—
210C22×C32—82no—
211C22×C32—162no—
212C2×C3×D3—81no—
213C32×C4—41no—
214C32×C4—41no—
215C3×Dic3—81no—
2162−1+4—21no—
2172−1+4—32no—
2182−1+4—31no—
2192−1+4—61no—
2202−1+4—61no—
2212−1+4—91no—
2222−1+4C2×C3×D312yes—
223H(5,𝔽2)—31no—
224H(5,𝔽2)—91no—
225H(5,𝔽2)—181no—
226C2×#41—31no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—91no—
230C2×#41—91no—
231C2×#41—181no—
232C2×#41C2×C3×D311yescharacteristic
233C22×Q8—21no—
234C22×Q8—31no—
235#137—91no—
236#137—92no—
237#137—91no—
238#137—92no—
239#137—94no—
240#137—92no—
241#137—92no—
242#137—92no—
243AGL(1,ℤ/8)—31no—
244AGL(1,ℤ/8)—92no—
245AGL(1,ℤ/8)—91no—
246AGL(1,ℤ/8)—92no—
247#135—92no—
248#135—92no—
249#135—94no—
250C2×Q16—92no—
251C2×Q16—92no—
252C2×SD16—92no—
253C2×SD16—92no—
254#131—32no—
255#131—92no—
256C2×M16—91no—
257C2×M16—91no—
258C23×C3—21no—
259C23×C3—31no—
260C23×C3—81no—
261C23×C3—82no—
262C22×D3—61no—
263C22×D3—61no—
264C3×Q8—31no—
265C3×Q8—32no—
266C3×Q8—62no—
267C3×Q8—62no—
268C3×Q8—61no—
269C3×Q8—121no—
270C3×Q8C2×C3×D411yescharacteristic
271C3×D4—21no—
272C3×D4—31no—
273C3×D4—31no—
274C3×D4—32no—
275C3×D4—42no—
276C3×D4—62no—
277C3×D4—62no—
278C3×D4—81no—
279C3×D4—82no—
280C3×D4—82no—
281C3×D4—121no—
282C3×D4—241no—
283C3×D4C22×A412yes—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—41no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—62no—
295C2×C3×C4—81no—
296C2×C3×C4—241no—
297C2×C3×C4C22×A411yescharacteristic
298#67—21no—
299#67—61no—
300#67—61no—
301#67—62no—
302#67—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305D12—31no—
306D12—61no—
307D12—61no—
308D12C22×A411yescharacteristic
309C4×D3—21no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—62no—
313C4×D3—61no—
314Dic6—31no—
315Dic6—61no—
316Dic6—61no—
317Dic6C22×A411yescharacteristic
318SL(2,𝔽3)—21no—
319SL(2,𝔽3)C2×#6711yescharacteristic
320C3×C8—122no—
321C3⋊C8—32no—
322C3⋊C8C22×A412yes—
323C2×C32—41no—
324C2×C32—81no—
325C2×C32—162no—
326C3×D3—161no—
327#41—31no—
328#41—31no—
329#41—32no—
330#41—32no—
331#41—31no—
332#41—61no—
333#41—61no—
334#41—64no—
335#41—62no—
336#41—62no—
337#41—61no—
338#41—91no—
339#41—91no—
340#41—92no—
341#41—91no—
342#41—181no—
343#41—181no—
344#41—181no—
345#41—181no—
346#41—182no—
347#41—181no—
348#41—181no—
349#41—181no—
350#41C3×#6711yescharacteristic
351#41C22×C3×D311yescharacteristic
352C2×Q8—22no—
353C2×Q8—32no—
354C2×Q8—32no—
355C2×Q8—31no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8—62no—
360C2×Q8—61no—
361C2×Q8—91no—
362C2×Q8—91no—
363C2×Q8—181no—
364C2×Q8C3×#6711yescharacteristic
365C2×D4—31no—
366C2×D4—61no—
367C2×D4—91no—
368C2×D4—91no—
369C2×D4—181no—
370C2×D4—182no—
371C2×D4—181no—
372C2×D4—181no—
373C2×D4—181no—
374C2×D4D3×A411yescharacteristic
375C22×C4—31no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—181no—
379C22×C4—181no—
380Q16—92no—
381Q16—92no—
382Q16—94no—
383SD16—32no—
384SD16—92no—
385SD16—92no—
386SD16—94no—
387SD16—92no—
388D8—32no—
389D8—92no—
390M16—31no—
391M16—92no—
392M16—91no—
393M16—92no—
394C2×C8—92no—
395C2×C8—92no—
396C22×C3—22no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—42no—
400C22×C3—41no—
401C22×C3—82no—
402C22×C3—84no—
403C22×C3—81no—
404C22×C3—121no—
405C22×C3—162no—
406C22×C3—164no—
407C22×C3—241no—
408C22×C3D4×A411yescharacteristic
409C2×D3—21no—
410C2×D3—61no—
411C2×D3—241no—
412C2×D3—241no—
413C3×C4—31no—
414C3×C4—31no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—62no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4—241no—
421C3×C4D4×A411yescharacteristic
422C3×C4C23×A411yescharacteristic
423Dic3—21no—
424Dic3—61no—
425C32—41noSylow
426C23—21no—
427C23—31no—
428C23—181no—
429C23—181no—
430Q8—31no—
431Q8—31no—
432Q8—32no—
433Q8—62no—
434Q8—62no—
435Q8—61no—
436Q8—91no—
437Q8—181no—
438Q8—181no—
439Q8C2×C3×#6711yescharacteristic
440D4—21no—
441D4—31no—
442D4—31no—
443D4—32no—
444D4—31no—
445D4—62no—
446D4—62no—
447D4—61no—
448D4—91no—
449D4—181no—
450D4—181no—
451D4—181no—
452D4—181no—
453D4—182no—
454D4—181no—
455D4C2×D3×A412yes—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—62no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—182no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—181no—
473C2×C4C2×D3×A411yescharacteristic
474C8—32no—
475C8—92no—
476C2×C3—21no—
477C2×C3—42no—
478C2×C3—41no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—81no—
482C2×C3—82no—
483C2×C3—81no—
484C2×C3—162no—
485C2×C3—164no—
486C2×C3—481no—
487C2×C3C2×D4×A411yescharacteristic
488D3—41no—
489D3—121no—
490C22—22no—
491C22—31no—
492C22—31no—
493C22—42no—
494C22—61no—
495C22—121no—
496C22—181no—
497C22—721no—
498C22—721no—
499C22A4×#6711yescharacteristic
500C4—31no—
501C4—31no—
502C4—62no—
503C4—61no—
504C4—181no—
505C4A4×#6711yescharacteristic
506C4C22×D3×A411yescharacteristic
507C3—41no—
508C3—81no—
509C3#3586⋊C311yescharacteristic, complemented
510C2—21no—
511C2—42no—
512C2—61no—
513C2—61no—
514C2—121no—
515C2—361no—
516C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC2C2×A4×#67
commutator, cosocleC3×#41C23×C3
FrattiniC4C22×D3×A4
FittingC2×C3×2−1+4C2×C3
socleC2×C3C2×D4×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. #5964⋊C3C2semidirect
3. #5837⋊C3C2semidirect
4. #5838⋊C3C2semidirect
5. #5819⋊C3C2semidirect
6. #87287C3semidirect
9. C3×2−1+4⋊C3C22semidirect
12. #1010⋊C3C22semidirect
14. C3⋊#994C22semidirect
17. C3⋊(#131⋊C3)C22semidirect
18. C2×C3×2−1+4C2×C3semidirect
19. #5964C2×C3semidirect
22. C2×2−1+4⋊C3D3semidirect
24. #5838C2×C3semidirect
26. #5837C2×C3semidirect
29. #5819C2×C3semidirect
57. #1010C22×C3semidirect
59. #1009C22×C3semidirect
71. 2−1+4⋊C3C2×D3semidirect
87. #948C22×C3semidirect
114. C2×2−1+4C3×D3semidirect
509. C3#3586⋊C3semidirect

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
#5964⋊C32
#5837⋊C32
#5838⋊C32
#5819⋊C32
#872873
#3586⋊C33
C3×#9324