Finite Group Databasebeta

← #5964⋊C9 #13164628

Subgroups — #5964⋊C9 #13164628

Order 1728.

757 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 109 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×#41⋊C9C211yesmaximal, characteristic, complemented
2#1010⋊C9C211yesmaximal, characteristic, complemented
3D3×#41⋊C9C212yesmaximal, complemented
4C3⋊#63700C211yesmaximal, characteristic, complemented
5#1007⋊C9C212yesmaximal, complemented
6C3×#5964C311yesmaximal, characteristic
7#10593120—31nomaximal
8C9×#235—41nomaximal
9C3×#41⋊C9C2211yescharacteristic, complemented
10C3×#41⋊C9C2211yescharacteristic, complemented
11C2×C3×Q8⋊C9C2211yescharacteristic, complemented
12D3×Q8⋊C9C2212yescomplemented
13C3⋊(#41⋊C9)C2212yescomplemented
14C2×C32×#41C2×C311yescharacteristic
15C3×#1010—31no—
16C3×#1010C2×C311yescharacteristic
17C3×#1009—31no—
18C3×#1009C2×C311yescharacteristic
19C3×D3×#41—32no—
20C3×D3×#41C2×C312yes—
21C3×#1007C2×C312yes—
22C3×#1004—32no—
23C2×C3×#235—31no—
24C2×C3×#235—31no—
252−1+4⋊C9—31no—
262−1+4⋊C9—32no—
27#63700—31no—
28C2×#41⋊C9—21no—
29C2×#41⋊C9—32no—
30C2×#41⋊C9D311yescharacteristic, complemented
31C2×C3×C4×C9—41no—
32C9×#67—42no—
33C9×D12—41no—
34C4×D3×C9—42no—
35C9×Dic6—41no—
36C3×Q8⋊C9C2311yescharacteristic
37C3×#585—31no—
38#5964C911yescharacteristic, complemented
39C32×#41—31no—
40C32×#41—31no—
41C32×#41C22×C311yescharacteristic
42C32×#41C22×C311yescharacteristic
43C2×C32×Q8C22×C311yescharacteristic
44C2×C32×D4—31no—
45C22×C32×C4—31no—
46C2×C3×#67—32no—
47C2×C3×#67—32no—
48C3×#239—32no—
49C3×#239C22×C312yes—
50C3×D3×Q8—32no—
51C3×D3×Q8C22×C312yes—
52C3×#237—34no—
53C3×#237—32no—
54C3×#237—32no—
55C3×D3×D4—34no—
56C3×D3×D4—32no—
57C3×D3×D4—32no—
58C3×#235—32no—
59C3×#235—31no—
60C3×#235—32no—
61C3×#235—32no—
62C3×#235—31no—
63C3×#235—32no—
64C3×#235—32no—
65C3×#235—31no—
66C3×#235A411yescharacteristic
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C9×#41—121no—
74#41⋊C9—21no—
75#41⋊C9—21no—
76#41⋊C9—32no—
77#41⋊C9C2×D311yescharacteristic, complemented
78#41⋊C9C2×D311yescharacteristic, complemented
79C2×Q8⋊C9—21no—
80C2×Q8⋊C9—32no—
81C2×Q8⋊C9C2×D311yescharacteristic, complemented
82C22×C3×C9—41no—
83C2×D3×C9—42no—
84C3×C4×C9—41no—
85C3×C4×C9—41no—
86C9×Dic3—42no—
87C3×2−1+4—31no—
88C3×2−1+4—32no—
89C3×2−1+4—91no—
90C3×H(5,𝔽2)—31no—
91C3×H(5,𝔽2)—92no—
92C3×H(5,𝔽2)—91no—
93C2×C3×#41—21no—
94C2×C3×#41—32no—
95C2×C3×#41—91no—
96C2×C3×#41—91no—
97C2×C3×#41—92no—
98C2×C3×#41C2×C911yescharacteristic, complemented
99C2×C3×#41C3×D311yescharacteristic
100#1010—31no—
101#1010C2×C911yescharacteristic, complemented
102#1009—31no—
103#1009C2×C911yescharacteristic, complemented
104D3×#41—32no—
105D3×#41C2×C912yescomplemented
106#1007C2×C912yescomplemented
107#1004—32no—
108C2×#235—31no—
109C2×#235—31no—
110C23×C32—31no—
111C22×C3×D3—32no—
112C22×C3×D3—32no—
113C32×Q8—31no—
114C32×Q8C23×C311yescharacteristic
115C32×D4—31no—
116C32×D4—31no—
117C32×D4—32no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4—31no—
122C2×C32×C4—31no—
123C2×C32×C4C2×A411yescharacteristic
124C3×#67—34no—
125C3×#67—32no—
126C3×#67—32no—
127C3×#67—32no—
128C3×#67C2×A412yes—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—32no—
131C3×D12—32no—
132C3×D12—32no—
133C3×D12—31no—
134C3×D12C2×A411yescharacteristic
135C3×C4×D3—34no—
136C3×C4×D3—32no—
137C3×C4×D3—32no—
138C3×C4×D3—32no—
139C3×C4×D3C2×A412yes—
140C3×Dic6—31no—
141C3×Dic6—32no—
142C3×Dic6—32no—
143C3×Dic6C2×A411yescharacteristic
144Q8×C9—121no—
145D4×C9—121no—
146D4×C9—122no—
147C2×C4×C9—41no—
148C2×C4×C9—81no—
149C2×C4×C9—122no—
150Q8⋊C9—21no—
151Q8⋊C9C22×D311yescharacteristic
152#585—31noSylow
153C2×C3×C9—41no—
154C2×C3×C9—81no—
155D3×C9—82no—
156C3×#41—21no—
157C3×#41—21no—
158C3×#41—31no—
159C3×#41—31no—
160C3×#41—31no—
161C3×#41—31no—
162C3×#41—31no—
163C3×#41—32no—
164C3×#41—61no—
165C3×#41—61no—
166C3×#41—92no—
167C3×#41—92no—
168C3×#41—91no—
169C3×#41—91no—
170C3×#41—92no—
171C3×#41—92no—
172C3×#41—92no—
173C3×#41—94no—
174C3×#41—92no—
175C3×#41—92no—
176C3×#41—91no—
177C3×#41—92no—
178C3×#41C22×C911yescharacteristic
179C3×#41C22×C911yescharacteristic
180C3×#41C2×C3×D311yescharacteristic
181C3×#41C2×C3×D311yescharacteristic
182C2×C3×Q8—21no—
183C2×C3×Q8—32no—
184C2×C3×Q8—91no—
185C2×C3×Q8—91no—
186C2×C3×Q8—92no—
187C2×C3×Q8C22×C911yescharacteristic
188C2×C3×Q8C2×C3×D311yescharacteristic
189C2×C3×D4—31no—
190C2×C3×D4—31no—
191C2×C3×D4—61no—
192C2×C3×D4—91no—
193C2×C3×D4—91no—
194C2×C3×D4—92no—
195C2×C3×D4—92no—
196C2×C3×D4—94no—
197C2×C3×D4—92no—
198C2×C3×D4—92no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—61no—
202C22×C3×C4—92no—
203C22×C3×C4—92no—
204C2×#67—32no—
205C2×#67—32no—
206#239—32no—
207#239C22×C912yes—
208D3×Q8—32no—
209D3×Q8C22×C912yes—
210#237—34no—
211#237—32no—
212#237—32no—
213D3×D4—32no—
214D3×D4—34no—
215D3×D4—32no—
216#235—32no—
217#235—31no—
218#235—32no—
219#235—32no—
220#235—31no—
221#235—32no—
222#235—32no—
223#235—31no—
224#235C22⋊C911yescharacteristic
225C2×D12—31no—
226C2×D12—31no—
227C2×C4×D3—32no—
228C2×C4×D3—32no—
229C2×Dic6—31no—
230C2×Dic6—31no—
231C22×C32—31no—
232C22×C32—31no—
233C22×C32—121no—
234C22×C32C22×A411yescharacteristic
235C2×C3×D3—32no—
236C2×C3×D3—122no—
237C2×C3×D3—122no—
238C2×C3×D3C22×A412yes—
239C32×C4—31no—
240C32×C4—31no—
241C32×C4C22×A411yescharacteristic
242C32×C4C22×A411yescharacteristic
243C3×Dic3—32no—
244C3×Dic3C22×A412yes—
245C22×C9—41no—
246C22×C9—81no—
247C22×C9—122no—
248C4×C9—41no—
249C4×C9—41no—
250C4×C9—81no—
251C4×C9—81no—
252C4×C9—122no—
2532−1+4—31no—
2542−1+4—32no—
2552−1+4—91no—
256H(5,𝔽2)—31no—
257H(5,𝔽2)—92no—
258H(5,𝔽2)—91no—
259C2×#41—32no—
260C2×#41—91no—
261C2×#41—91no—
262C2×#41—92no—
263C2×#41D3×C911yescharacteristic, complemented
264C3×C9—41noSylow
265C23×C3—31no—
266C23×C3—31no—
267C23×C3—61no—
268C23×C3—92no—
269C23×C3—92no—
270C22×D3—32no—
271C22×D3—32no—
272C3×Q8—21no—
273C3×Q8—31no—
274C3×Q8—31no—
275C3×Q8—31no—
276C3×Q8—61no—
277C3×Q8—92no—
278C3×Q8—92no—
279C3×Q8—91no—
280C3×Q8C23×C911yescharacteristic
281C3×Q8C22×C3×D311yescharacteristic
282C3×D4—31no—
283C3×D4—31no—
284C3×D4—31no—
285C3×D4—31no—
286C3×D4—32no—
287C3×D4—32no—
288C3×D4—32no—
289C3×D4—31no—
290C3×D4—61no—
291C3×D4—61no—
292C3×D4—62no—
293C3×D4—92no—
294C3×D4—92no—
295C3×D4—92no—
296C3×D4—92no—
297C3×D4—94no—
298C3×D4—92no—
299C3×D4—91no—
300C2×C3×C4—21no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—32no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—92no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—94no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4C2×C22⋊C911yescharacteristic
324C2×C3×C4D3×A411yescharacteristic
325#67—34no—
326#67—32no—
327#67—32no—
328#67—32no—
329#67C2×C22⋊C912yes—
330C2×Dic3—32no—
331C2×Dic3—32no—
332D12—32no—
333D12—32no—
334D12—31no—
335D12C2×C22⋊C911yescharacteristic
336C4×D3—34no—
337C4×D3—32no—
338C4×D3—32no—
339C4×D3—32no—
340C4×D3C2×C22⋊C912yes—
341Dic6—31no—
342Dic6—32no—
343Dic6—32no—
344Dic6C2×C22⋊C911yescharacteristic
345C2×C32—21no—
346C2×C32—61no—
347C2×C32—61no—
348C2×C32C23×A411yescharacteristic
349C3×D3—22no—
350C3×D3—62no—
351C2×C9—41no—
352C2×C9—81no—
353C2×C9—82no—
354C2×C9—81no—
355C2×C9—242no—
356#41—31no—
357#41—31no—
358#41—31no—
359#41—32no—
360#41—92no—
361#41—92no—
362#41—91no—
363#41—91no—
364#41—92no—
365#41—92no—
366#41—92no—
367#41—94no—
368#41—92no—
369#41—92no—
370#41—91no—
371#41—92no—
372#41C2×D3×C911yescharacteristic
373#41C2×D3×C911yescharacteristic
374C2×Q8—32no—
375C2×Q8—91no—
376C2×Q8—91no—
377C2×Q8—92no—
378C2×Q8C2×D3×C911yescharacteristic
379C2×D4—31no—
380C2×D4—91no—
381C2×D4—91no—
382C2×D4—92no—
383C2×D4—92no—
384C2×D4—94no—
385C2×D4—92no—
386C2×D4—92no—
387C22×C4—31no—
388C22×C4—92no—
389C22×C4—92no—
390C22×C3—21no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—32no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—92no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—122no—
402C22×C3—362no—
403C22×C3—362no—
404C22×C3C22×C22⋊C911yescharacteristic
405C22×C3C2×D3×A411yescharacteristic
406C2×D3—32no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3C22×C22⋊C912yes—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—32no—
415C3×C4—31no—
416C3×C4—31no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—92no—
420C3×C4C22×C22⋊C911yescharacteristic
421C3×C4C22×C22⋊C911yescharacteristic
422C3×C4C2×D3×A411yescharacteristic
423C3×C4C2×D3×A411yescharacteristic
424Dic3—32no—
425Dic3C22×C22⋊C912yes—
426C32#594511yescharacteristic
427C9—41no—
428C9—81no—
429C23—31no—
430C23—92no—
431C23—92no—
432Q8—31no—
433Q8—31no—
434Q8—92no—
435Q8—92no—
436Q8—91no—
437Q8C22×D3×C911yescharacteristic
438D4—31no—
439D4—31no—
440D4—32no—
441D4—32no—
442D4—31no—
443D4—92no—
444D4—92no—
445D4—92no—
446D4—92no—
447D4—94no—
448D4—92no—
449D4—91no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—32no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—94no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4D3×C22⋊C911yescharacteristic
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—62no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3—121no—
473C2×C3—121no—
474C2×C3—182no—
475C2×C3C23×C22⋊C911yescharacteristic
476C2×C3C22×D3×A411yescharacteristic
477D3—22no—
478D3—62no—
479C22—31no—
480C22—31no—
481C22—32no—
482C22—92no—
483C22—121no—
484C22—362no—
485C22—362no—
486C22C2×D3×C22⋊C911yescharacteristic
487C4—31no—
488C4—31no—
489C4—32no—
490C4—92no—
491C4C2×D3×C22⋊C911yescharacteristic
492C4C2×D3×C22⋊C911yescharacteristic
493C3—21no—
494C3#1059312011yescharacteristic, complemented
495C3#5964⋊C311yescharacteristic
496C2—21no—
497C2—61no—
498C2—61no—
499C2—62no—
500C2—182no—
501C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC3×C4C2×D3×A4
commutator, hypocenterC3×Q8C23×C9
FrattiniC2×C3C22×D3×A4
FittingC2×C32×#41C2×C3
socleC2×C32C23×A4
prime radicalC3×#5964C3
cosocleC32×Q8C23×C3
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. #1010⋊C9C2semidirect
3. D3×#41⋊C9C2semidirect
4. C3⋊#63700C2semidirect
5. #1007⋊C9C2semidirect
9. C3×#41⋊C9C22semidirect
10. C3×#41⋊C9C22semidirect
11. C2×C3×Q8⋊C9C22semidirect
12. D3×Q8⋊C9C22semidirect
13. C3⋊(#41⋊C9)C22semidirect
30. C2×#41⋊C9D3semidirect
38. #5964C9semidirect
77. #41⋊C9C2×D3semidirect
78. #41⋊C9C2×D3semidirect
81. C2×Q8⋊C9C2×D3semidirect
98. C2×C3×#41C2×C9semidirect
101. #1010C2×C9semidirect
103. #1009C2×C9semidirect
105. D3×#41C2×C9semidirect
106. #1007C2×C9semidirect
263. C2×#41D3×C9semidirect
494. C3#10593120semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
#1010⋊C92
D3×#41⋊C92
C3⋊#637002
#1007⋊C92
C3×#59643
#105931203
C9×#2354