Finite Group Databasebeta

← #87299⋊C3 #11923406

Subgroups — #87299⋊C3 #11923406

Order 1152.

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

normal subgroups only

This group has 710 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
1C3×#5945C211yesmaximal, characteristic, complemented
2C2×#1010⋊C3C211yesmaximal, characteristic, complemented
3#5838⋊C3C212yesmaximal, complemented
4#5836⋊C3C212yesmaximal, complemented
5#5819⋊C3C211yesmaximal, characteristic, complemented
6#87299C311yesmaximal, characteristic, complemented
7#3586⋊C3—31nomaximal
8C3×2−1+4⋊C3—21no—
9C3×2−1+4⋊C3C2211yescharacteristic, complemented
10C3×#994C2211yescharacteristic, complemented
11C2×C3×#41⋊C3—21no—
12C2×C3×#41⋊C3C2211yescharacteristic
13#1010⋊C3—21no—
14#1010⋊C3C2212yescomplemented
15C2×D3×SL(2,𝔽3)—21no—
16C3×#949—41nomaximal
17C3⋊(#131⋊C3)C2212yescomplemented
18C3×#585C2×C311yescharacteristic, complemented
19C2×#1010C2×C311yescharacteristic, complemented
20#5945—21no—
21#5945D311yescharacteristic, complemented
22C2×2−1+4⋊C3—31no—
23#5838—32no—
24#5838C2×C312yescomplemented
25#5836—32no—
26#5836C2×C312yescomplemented
27C2×#951—31no—
28C2×#951—31no—
29#5822—31no—
30#5822—31no—
31#5819C2×C311yescharacteristic, complemented
32#578⋊C3—32no—
33#577⋊C3—32no—
34#568⋊C3—31no—
35C32×#41—41no—
36C2×C3×D12—41no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3D411yescharacteristic
39C3×#41⋊C3C2311yescharacteristic
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)D411yescharacteristic
42D3×SL(2,𝔽3)—22no—
43C3×#215—42no—
44C3×#213—42no—
45C3×#208—41no—
46#3586—31noSylow
47C3×2−1+4—21no—
48C3×2−1+4—31no—
49C3×2−1+4C22×C311yescharacteristic, complemented
50C3×H(5,𝔽2)—31no—
51C3×H(5,𝔽2)—61no—
52C3×H(5,𝔽2)C22×C311yescharacteristic, complemented
53C2×C3×#41—21no—
54C2×C3×#41—31no—
55C2×C3×#41—31no—
56C2×C3×#41—61no—
57C2×C3×#41C22×C311yescharacteristic
58#1010—21no—
59#1010—32no—
60#1010—61no—
61#1010C22×C312yes—
62C2×D3×Q8—21no—
63C2×#237—61no—
64C2×#235—31no—
65C22×Dic6—31no—
662−1+4⋊C3—22no—
672−1+4⋊C3—21no—
682−1+4⋊C3—21no—
692−1+4⋊C3—32no—
702−1+4⋊C3—61no—
712−1+4⋊C3C2×D311yescharacteristic, complemented
72#994—21no—
73#994C2×D311yescharacteristic, complemented
74C2×#41⋊C3—22no—
75C2×#41⋊C3—21no—
76C2×#41⋊C3—21no—
77C2×#41⋊C3C2×D311yescharacteristic
78C22×SL(2,𝔽3)—61no—
79C3×AGL(1,ℤ/8)—121no—
80#951—32no—
81#951—34no—
82#951—31no—
83#951—32no—
84#951—32no—
85#951—31no—
86#950—34no—
87#950—32no—
88#950—32no—
89#949—31no—
90#949A411yescharacteristic
91#948—32no—
92#948C22×C312yescomplemented
93C2×#216—32no—
94C2×#216—32no—
95#942—32no—
96#942—32no—
97C2×#214—32no—
98C2×#214—32no—
99#932—32no—
100#932—32no—
101C2×#208—31no—
102C2×#208—31no—
103#131⋊C3—32no—
104C22×C3×D3—81no—
105C32×Q8—41no—
106C32×D4—41no—
107C32×D4—81no—
108C2×C32×C4—41no—
109C2×C32×C4—81no—
110C3×D12—42no—
111C3×D12—81no—
112C3×SL(2,𝔽3)C2×D411yescharacteristic
113C3×C3⋊C8—42no—
114#585C3×D311yescharacteristic, complemented
115C2×2−1+4—31no—
116#578—32no—
117#578—92no—
118#577—32no—
119#577—92no—
120#575—91no—
121#575—91no—
122C2×#137—91no—
123C2×#137—91no—
124#568—31no—
125C3×#41—21no—
126C3×#41—31no—
127C3×#41—31no—
128C3×#41—32no—
129C3×#41—31no—
130C3×#41—31no—
131C3×#41—31no—
132C3×#41—41no—
133C3×#41—61no—
134C3×#41—61no—
135C3×#41—61no—
136C3×#41—61no—
137C3×#41—62no—
138C3×#41—61no—
139C3×#41—61no—
140C3×#41—61no—
141C3×#41—81no—
142C3×#41C3×D411yescharacteristic
143C3×#41C2×A411yescharacteristic
144C3×#41C23×C311yescharacteristic
145C2×C3×Q8—21no—
146C2×C3×Q8—31no—
147C2×C3×Q8—31no—
148C2×C3×Q8—61no—
149C2×C3×Q8C3×D411yescharacteristic
150C2×C3×D4—31no—
151C2×C3×D4—31no—
152C2×C3×D4—31no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4—62no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—121no—
159C22×C3×C4—31no—
160C22×C3×C4—61no—
161C22×C3×C4—61no—
162C2×#67—61no—
163C22×Dic3—61no—
164D3×Q8—22no—
165D3×Q8—62no—
166#237—64no—
167#237—62no—
168#237—62no—
169#235—32no—
170#235—32no—
171#235—61no—
172#235—61no—
173C2×D12C2×A411yescharacteristic
174C2×C4×D3—61no—
175C2×Dic6—32no—
176C2×Dic6—31no—
177C2×Dic6—32no—
178C2×Dic6—61no—
179C2×Dic6—61no—
180C2×Dic6—61no—
181#41⋊C3—22no—
182#41⋊C3—21no—
183#41⋊C3—21no—
184#41⋊C3—21no—
185#41⋊C3#6711yescharacteristic
186#41⋊C3C22×D311yescharacteristic
187C2×SL(2,𝔽3)—22no—
188C2×SL(2,𝔽3)—21no—
189C2×SL(2,𝔽3)—21no—
190C2×SL(2,𝔽3)—62no—
191C2×SL(2,𝔽3)#6711yescharacteristic
192C3×SD16—122no—
193C3×D8—122no—
194C3×M16—121no—
195#216—34no—
196#216—32no—
197#216—32no—
198#215—32no—
199#215C2×A412yes—
200#214—34no—
201#214—32no—
202#214—32no—
203#213—32no—
204#213C2×A412yes—
205#208—31no—
206#208—32no—
207#208—32no—
208#208C2×A411yescharacteristic
209C2×C3⋊C8—32no—
210C2×C3⋊C8—32no—
211C22×C32—41no—
212C22×C32—81no—
213C2×C3×D3—82no—
214C2×C3×D3—162no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—81no—
2182−1+4—21no—
2192−1+4—31no—
2202−1+4—32no—
2212−1+4—61no—
2222−1+4—92no—
2232−1+4—181no—
2242−1+4C2×C3×D311yescharacteristic, complemented
225H(5,𝔽2)—31no—
226H(5,𝔽2)—61no—
227H(5,𝔽2)C2×C3×D311yescharacteristic, complemented
228C2×#41—21no—
229C2×#41—31no—
230C2×#41—31no—
231C2×#41—61no—
232C2×#41—91no—
233C2×#41—181no—
234C2×#41C2×C3×D311yescharacteristic
235C22×Q8—61no—
236C22×Q8—91no—
237#137—91no—
238#137—92no—
239#137—92no—
240#137—94no—
241#137—91no—
242#137—92no—
243#137—92no—
244#137—92no—
245AGL(1,ℤ/8)—31no—
246AGL(1,ℤ/8)—92no—
247AGL(1,ℤ/8)—91no—
248AGL(1,ℤ/8)—92no—
249#135—92no—
250#135—92no—
251#135—94no—
252C2×Q16—92no—
253C2×Q16—92no—
254C2×SD16—92no—
255C2×SD16—92no—
256#131—32no—
257#131—92no—
258C2×M16—91no—
259C2×M16—91no—
260C23×C3—31no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—241no—
264C22×D3—21no—
265C3×Q8—31no—
266C3×Q8—31no—
267C3×Q8—41no—
268C3×Q8—61no—
269C3×Q8—61no—
270C3×Q8—81no—
271C3×Q8C2×C3×D411yescharacteristic
272C3×Q8C22×A411yescharacteristic
273C3×D4—21no—
274C3×D4—31no—
275C3×D4—31no—
276C3×D4—32no—
277C3×D4—31no—
278C3×D4—41no—
279C3×D4—61no—
280C3×D4—61no—
281C3×D4—61no—
282C3×D4—61no—
283C3×D4—62no—
284C3×D4—61no—
285C3×D4—81no—
286C3×D4—82no—
287C3×D4—81no—
288C3×D4—122no—
289C3×D4—241no—
290C3×D4C22×A411yescharacteristic
291C2×C3×C4—21no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—41no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—62no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—81no—
305C2×C3×C4—82no—
306C2×C3×C4—81no—
307C2×C3×C4C22×A411yescharacteristic
308#67—62no—
309#67—62no—
310C2×Dic3—62no—
311C2×Dic3—61no—
312C2×Dic3—62no—
313D12—21no—
314D12C22×A412yes—
315C4×D3—62no—
316C4×D3—62no—
317Dic6—32no—
318Dic6—62no—
319Dic6—61no—
320Dic6—62no—
321SL(2,𝔽3)—21no—
322SL(2,𝔽3)C2×#6711yescharacteristic
323C3×C8—122no—
324C3⋊C8—32no—
325C3⋊C8C22×A412yes—
326C2×C32—41no—
327C2×C32—81no—
328C2×C32—161no—
329C3×D3—162no—
330#41—21no—
331#41—31no—
332#41—31no—
333#41—32no—
334#41—31no—
335#41—31no—
336#41—31no—
337#41—61no—
338#41—61no—
339#41—61no—
340#41—61no—
341#41—62no—
342#41—61no—
343#41—61no—
344#41—61no—
345#41—92no—
346#41—92no—
347#41—181no—
348#41—181no—
349#41—184no—
350#41—182no—
351#41—182no—
352#41C3×#6711yescharacteristic
353#41D3×A411yescharacteristic
354#41C22×C3×D311yescharacteristic
355C2×Q8—21no—
356C2×Q8—31no—
357C2×Q8—31no—
358C2×Q8—61no—
359C2×Q8—62no—
360C2×Q8—92no—
361C2×Q8—92no—
362C2×Q8—91no—
363C2×Q8—181no—
364C2×Q8—181no—
365C2×Q8—181no—
366C2×Q8—182no—
367C2×Q8C3×#6711yescharacteristic
368C2×D4—31no—
369C2×D4—31no—
370C2×D4—31no—
371C2×D4—31no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—62no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4—181no—
378C22×C4—31no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—181no—
382C22×C4—181no—
383Q16—92no—
384Q16—94no—
385Q16—92no—
386SD16—32no—
387SD16—92no—
388SD16—92no—
389SD16—94no—
390SD16—92no—
391D8—32no—
392D8—92no—
393M16—31no—
394M16—92no—
395M16—91no—
396M16—92no—
397C2×C8—92no—
398C2×C8—92no—
399C22×C3—21no—
400C22×C3—31no—
401C22×C3—31no—
402C22×C3—41no—
403C22×C3—61no—
404C22×C3—81no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—121no—
408C22×C3—242no—
409C22×C3—241no—
410C22×C3—241no—
411C22×C3—482no—
412C22×C3D4×A411yescharacteristic
413C2×D3—22no—
414C2×D3—42no—
415C3×C4—21no—
416C3×C4—31no—
417C3×C4—31no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—61no—
421C3×C4—81no—
422C3×C4—82no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4D4×A411yescharacteristic
426C3×C4C23×A411yescharacteristic
427Dic3—62no—
428C32—41noSylow
429C23—31no—
430C23—61no—
431C23—61no—
432C23—61no—
433Q8—31no—
434Q8—31no—
435Q8—61no—
436Q8—61no—
437Q8—92no—
438Q8—182no—
439Q8—182no—
440Q8—181no—
441Q8C2×C3×#6711yescharacteristic
442Q8C2×D3×A411yescharacteristic
443D4—21no—
444D4—31no—
445D4—31no—
446D4—32no—
447D4—32no—
448D4—31no—
449D4—61no—
450D4—61no—
451D4—61no—
452D4—61no—
453D4—62no—
454D4—61no—
455D4—61no—
456D4—182no—
457D4—182no—
458D4C2×D3×A411yescharacteristic
459C2×C4—21no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—62no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—182no—
472C2×C4—182no—
473C2×C4—182no—
474C2×C4—181no—
475C2×C4—182no—
476C2×C4C2×D3×A411yescharacteristic
477C8—32no—
478C8—92no—
479C2×C3—21no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—61no—
483C2×C3—61no—
484C2×C3—81no—
485C2×C3—81no—
486C2×C3—121no—
487C2×C3—161no—
488C2×C3—161no—
489C2×C3—162no—
490C2×C3—482no—
491C2×C3C2×D4×A411yescharacteristic
492D3—42no—
493C22—21no—
494C22—31no—
495C22—31no—
496C22—61no—
497C22—62no—
498C22—121no—
499C22—122no—
500C22—241no—
501C22—241no—
502C22A4×#6711yescharacteristic
503C4—21no—
504C4—31no—
505C4—31no—
506C4—61no—
507C4—182no—
508C4A4×#6711yescharacteristic
509C4C22×D3×A411yescharacteristic
510C3—41no—
511C3—81no—
512C3#3586⋊C311yescharacteristic, complemented
513C2—21no—
514C2—41no—
515C2—61no—
516C2—61no—
517C2—122no—
518C2—121no—
519C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC2C2×A4×#67
commutator, cosocleC3×#41C23×C3
FrattiniC4C22×D3×A4
FittingC3×#585C2×C3
socleC2×C3C2×D4×A4
hypercenter#41D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#5945C2semidirect
2. C2×#1010⋊C3C2semidirect
3. #5838⋊C3C2semidirect
4. #5836⋊C3C2semidirect
5. #5819⋊C3C2semidirect
6. #87299C3semidirect
9. C3×2−1+4⋊C3C22semidirect
10. C3×#994C22semidirect
14. #1010⋊C3C22semidirect
17. C3⋊(#131⋊C3)C22semidirect
18. C3×#585C2×C3semidirect
19. C2×#1010C2×C3semidirect
21. #5945D3semidirect
24. #5838C2×C3semidirect
26. #5836C2×C3semidirect
31. #5819C2×C3semidirect
49. C3×2−1+4C22×C3semidirect
52. C3×H(5,𝔽2)C22×C3semidirect
71. 2−1+4⋊C3C2×D3semidirect
73. #994C2×D3semidirect
92. #948C22×C3semidirect
114. #585C3×D3semidirect
224. 2−1+4C2×C3×D3semidirect
227. H(5,𝔽2)C2×C3×D3semidirect
512. C3#3586⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#59452
C2×#1010⋊C32
#5838⋊C32
#5836⋊C32
#5819⋊C32
#872993
#3586⋊C33
C3×#9494