Finite Group Databasebeta

← #13141007

Subgroups — #13141007

Order 1728.

674 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#41×C3⋊D9C211yesmaximal, characteristic
2C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3#11720863C211yesmaximal, characteristic, complemented
4C9×#837C211yesmaximal, characteristic, complemented
5C3×#63396C211yesmaximal, characteristic, complemented
6#11720292C211yesmaximal, characteristic, complemented
7#11720261C211yesmaximal, characteristic, complemented
8#10591671—31nomaximal
9D9×#104—31nomaximal
10C3×C9×#41C2211yescharacteristic
11#88746C411yescharacteristic, complemented
12Q8×C3⋊D9C411yescharacteristic, complemented
13#88742—21no—
14#88742C411yescharacteristic, complemented
15D4×C3⋊D9—21no—
16D4×C3⋊D9C411yescharacteristic, complemented
17#88736—21no—
18#88736C2211yescharacteristic
19C2×C4×C3⋊D9—21no—
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×(C3⋊D9)⋊C4—21no—
22C4×C9×Dic3C2211yescharacteristic, complemented
23C3×C4×Dic9C2211yescharacteristic, complemented
24C9×#208C2211yescharacteristic, complemented
25C3×#3667C2211yescharacteristic, complemented
26Dic3×Dic9—21no—
27#88414—21no—
28(C3⋊D9)⋊C8—21no—
29#41×C3⋊D3—31no—
30C4×#624—31no—
31#63818—31no—
32#41×D9—31no—
33#41×D9—61no—
34C3×#837—31no—
35C3×#837D311yescharacteristic
36#63570—31no—
37#63570—31no—
38M16×D9—31no—
39C42×D9—31no—
40C9×#104—31no—
41#63396D311yescharacteristic, complemented
42#63384—31no—
43#63364—31no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9D411yescharacteristic
46C3×Q8×C9C2×C411yescharacteristic, complemented
47C3×D4×C9—21no—
48C3×D4×C9C2×C411yescharacteristic, complemented
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9C2311yescharacteristic
51#6233—21no—
52#6233—21no—
53#6233—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9D411yescharacteristic
56C4⋊(C3⋊D9)—21no—
57C4⋊(C3⋊D9)C2×C411yescharacteristic, complemented
58C4×C3⋊D9—21no—
59C4×C3⋊D9—21no—
60C4×C3⋊D9—21no—
61C4×C3⋊D9C2×C411yescharacteristic
62C4×C3⋊D9C2×C411yescharacteristic
63C4×C3⋊D9D411yescharacteristic
64C4×C3⋊D9D411yescharacteristic
65#6226—21no—
66#6226C2×C411yescharacteristic, complemented
67C2×C9×Dic3—21no—
68C2×C3×Dic9—21no—
69(C3⋊D9)⋊C4—24no—
70C9×C3⋊C8—21no—
71C3×C9⋊C8—21no—
72D3×#104—91no—
73D3×#104—91no—
74C32×#41C2×D311yescharacteristic
75#3832—31no—
76Q8×C3⋊D3—31no—
77#3830—31no—
78#3830—61no—
79D4×C3⋊D3—31no—
80D4×C3⋊D3—61no—
81#3828—31no—
82#3828—61no—
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—61no—
85C2×#624—61no—
86C3×C4×Dic3—31no—
87C3×C4×Dic3C2×D311yescharacteristic
88C3×#208—31no—
89C3×#208C2×D311yescharacteristic
90Dic32—61no—
91#3712—61no—
92#3710—61no—
93C9×#41—21no—
94C9×#41C2×D311yescharacteristic
95#3701—31no—
96#3701—61no—
97Q8×D9—31no—
98Q8×D9—61no—
99#3699—31no—
100#3699—61no—
101#3699—61no—
102#3699—62no—
103D4×D9—31no—
104D4×D9—61no—
105D4×D9—61no—
106D4×D9—62no—
107#3697—31no—
108#3697—61no—
109#3697—61no—
110#3697—62no—
111C2×C4×D9—31no—
112C2×C4×D9—61no—
113C2×C4×D9—61no—
114C2×C4×D9—61no—
115C2×C4×D9—62no—
116C9×M16—31no—
117C42×C9—31no—
118C4×Dic9—61no—
119C4×Dic9C2×D311yescharacteristic, complemented
120#3667C2×D311yescharacteristic, complemented
121#3663—61no—
122C8×D9—61no—
123C22×C3×C9—21no—
124C22×C3×C9C2×D411yescharacteristic
125C2×C3⋊D9—22no—
126C2×C3⋊D9—21no—
127C2×C3⋊D9—42no—
128C2×C3⋊D9C22⋊C411yescharacteristic
129C2×C3⋊D9C22⋊C411yescharacteristic
130C3×C4×C9—21no—
131C3×C4×C9C22×C411yescharacteristic
132C3×C4×C9C2×D411yescharacteristic
133C3⋊Dic9—21no—
134C3⋊Dic9C22⋊C411yescharacteristic
135C3⋊Dic9C22⋊C411yescharacteristic
136C9×Dic3—22no—
137C3×Dic9—22no—
138D3×#41—91no—
139D3×#41—91no—
140D3×#41—181no—
141D3×M16—91no—
142D3×M16—91no—
143C42×D3—91no—
144C42×D3—91no—
145C3×#104—31no—
146C3×#104—91no—
147#837—31no—
148#837D911yescharacteristic, complemented
149#825—91no—
150#825—91no—
151#805—91no—
152#805—91no—
153C22×C3⋊D3—31no—
154C22×C3⋊D3—61no—
155C32×Q8C4×D311yescharacteristic
156C32×D4—21no—
157C32×D4C4×D311yescharacteristic
158C2×C32×C4—21no—
159C2×C32×C4C22×D311yescharacteristic
160#638—61no—
161#638—61no—
162#638—61no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—61no—
165C4⋊(C3⋊D3)—31no—
166C4⋊(C3⋊D3)—61no—
167C4×C3⋊D3—31no—
168C4×C3⋊D3—31no—
169C4×C3⋊D3—31no—
170C4×C3⋊D3—31no—
171C4×C3⋊D3—61no—
172C4×C3⋊D3—61no—
173C4×C3⋊D3—61no—
174#634—31no—
175#634—61no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—61no—
178#624—64no—
179C22×D9—31no—
180C22×D9—61no—
181C22×D9—61no—
182C22×D9—62no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8—61no—
185Q8×C9—21no—
186Q8×C9C4×D311yescharacteristic, complemented
187D4×C9—22no—
188D4×C9—21no—
189D4×C9—21no—
190D4×C9C4×D311yescharacteristic, complemented
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—61no—
195C2×C4×C9C22×D311yescharacteristic
196#611—61no—
197#611—61no—
198#611—62no—
199#611—62no—
200#611—62no—
201#611—61no—
202C2×Dic9—21no—
203C2×Dic9—31no—
204C2×Dic9—61no—
205C2×Dic9—61no—
206C2×Dic9—62no—
207D36—31no—
208D36—61no—
209D36—61no—
210D36—62no—
211C4×D9—31no—
212C4×D9—31no—
213C4×D9—31no—
214C4×D9—31no—
215C4×D9—61no—
216C4×D9—61no—
217C4×D9—61no—
218C4×D9—61no—
219C4×D9—64no—
220C4×D9—61no—
221C4×D9—61no—
222C4×D9—62no—
223C4×D9—62no—
224C4×D9—62no—
225C4×D9—61no—
226Dic18—31no—
227Dic18—61no—
228Dic18—61no—
229Dic18—62no—
230C8×C9—61no—
231C9⋊C8—21no—
232C2×#104—271noSylow
233C2×C3×C9—21no—
234C2×C3×C9—41no—
235C2×C3×C9C2×C22⋊C411yescharacteristic
236C3⋊D9—21no—
237C3⋊D9—41no—
238C3⋊D9#10412yescomplemented
239C3×#41—21no—
240C3×#41C2×D911yescharacteristic
241C3×#41D3211yescharacteristic
242#239—91no—
243#239—91no—
244#239—181no—
245D3×Q8—91no—
246D3×Q8—91no—
247D3×Q8—181no—
248#237—91no—
249#237—91no—
250#237—181no—
251#237—181no—
252#237—181no—
253#237—182no—
254D3×D4—91no—
255D3×D4—91no—
256D3×D4—181no—
257D3×D4—181no—
258D3×D4—181no—
259D3×D4—182no—
260#235—91no—
261#235—91no—
262#235—181no—
263#235—181no—
264#235—181no—
265#235—182no—
266C2×C4×D3—91no—
267C2×C4×D3—91no—
268C2×C4×D3—181no—
269C2×C4×D3—181no—
270C2×C4×D3—181no—
271C2×C4×D3—181no—
272C2×C4×D3—181no—
273C2×C4×D3—182no—
274C3×M16—31no—
275C3×M16—91no—
276C3×C42—31no—
277C3×C42—91no—
278C4×Dic3—31no—
279C4×Dic3—181no—
280C4×Dic3—181no—
281C4×Dic3C2×D911yescharacteristic, complemented
282#208—31no—
283#208C2×D911yescharacteristic, complemented
284ΓL(1,𝔽25)—181no—
285ΓL(1,𝔽25)—181no—
286D3×C8—181no—
287D3×C8—181no—
288C22×C32—21no—
289C22×C32D3×D411yescharacteristic
290C2×C3⋊D3—31no—
291C2×C3⋊D3—31no—
292C2×C3⋊D3—61no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—122no—
295C32×C4—21no—
296C32×C4C2×C4×D311yescharacteristic
297C32×C4D3×D411yescharacteristic
298C3⋊Dic3—31no—
299C3⋊Dic3—31no—
300C3⋊Dic3—61no—
301C3×Dic3—22no—
302C3×Dic3—62no—
303C22×C9—22no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9D3×D411yescharacteristic
307C2×D9—31no—
308C2×D9—31no—
309C2×D9—61no—
310C2×D9—61no—
311C2×D9—62no—
312C2×D9—62no—
313C2×D9—61no—
314C2×D9—122no—
315C2×D9—122no—
316C2×D9—124no—
317C4×C9—22no—
318C4×C9—21no—
319C4×C9—21no—
320C4×C9—21no—
321C4×C9—62no—
322C4×C9C2×C4×D311yescharacteristic
323C4×C9D3×D411yescharacteristic
324Dic9—22no—
325Dic9—31no—
326Dic9—31no—
327Dic9—61no—
328Dic9—61no—
329Dic9—62no—
330Dic9—61no—
331C2×#41—271no—
332C2×M16—271no—
333C2×C42—271no—
334#104—31no—
335#104—91no—
336#104—271no—
337#104—271no—
338C3×C9C2×#10411yescharacteristic, complemented, Sylow
339C22×D3—91no—
340C22×D3—91no—
341C22×D3—181no—
342C22×D3—181no—
343C22×D3—181no—
344C22×D3—182no—
345C3×Q8—21no—
346C3×Q8C4×D911yescharacteristic, complemented
347C3×Q8#62411yescharacteristic
348C3×D4—22no—
349C3×D4—21no—
350C3×D4—21no—
351C3×D4—21no—
352C3×D4C4×D911yescharacteristic, complemented
353C3×D4#62411yescharacteristic
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—61no—
359C2×C3×C4—181no—
360C2×C3×C4C22×D911yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362#67—181no—
363#67—181no—
364#67—181no—
365#67—181no—
366#67—182no—
367#67—182no—
368#67—182no—
369#67—181no—
370#67—181no—
371C2×Dic3—21no—
372C2×Dic3—61no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—181no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3—182no—
379D12—91no—
380D12—91no—
381D12—181no—
382D12—181no—
383D12—181no—
384D12—182no—
385C4×D3—91no—
386C4×D3—91no—
387C4×D3—91no—
388C4×D3—91no—
389C4×D3—91no—
390C4×D3—91no—
391C4×D3—91no—
392C4×D3—91no—
393C4×D3—181no—
394C4×D3—181no—
395C4×D3—181no—
396C4×D3—181no—
397C4×D3—184no—
398C4×D3—184no—
399C4×D3—181no—
400C4×D3—181no—
401C4×D3—181no—
402C4×D3—181no—
403C4×D3—182no—
404C4×D3—182no—
405C4×D3—182no—
406C4×D3—181no—
407C4×D3—181no—
408Dic6—91no—
409Dic6—91no—
410Dic6—181no—
411Dic6—181no—
412Dic6—181no—
413Dic6—182no—
414C3×C8—61no—
415C3×C8—181no—
416C3⋊C8—21no—
417C3⋊C8—61no—
418C2×C32—21no—
419C2×C32—41no—
420C2×C32D3×C22⋊C411yescharacteristic
421C3⋊D3—32no—
422C3⋊D3—61no—
423C3⋊D3—121no—
424C2×C9—21no—
425C2×C9—21no—
426C2×C9—41no—
427C2×C9—42no—
428C2×C9—41no—
429C2×C9D3×C22⋊C411yescharacteristic
430D9—32no—
431D9—62no—
432D9—61no—
433D9—121no—
434D9—122no—
435D9—121no—
436#41—271no—
437#41—271no—
438#41—271no—
439#41—541no—
440#41—541no—
441#41D3×D911yescharacteristic
442C2×Q8—271no—
443C2×D4—271no—
444C2×D4—541no—
445C22×C4—271no—
446C22×C4—541no—
447C22×C4—541no—
448M16—31no—
449M16—91no—
450M16—541no—
451C2×C8—541no—
452C42—31no—
453C42—91no—
454C42—541no—
455C22×C3—22no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3D4×D911yescharacteristic
460C22×C3#381111yescharacteristic
461C2×D3—91no—
462C2×D3—91no—
463C2×D3—91no—
464C2×D3—91no—
465C2×D3—181no—
466C2×D3—181no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—362no—
473C2×D3—362no—
474C2×D3—362no—
475C2×D3—364no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—62no—
482C3×C4—182no—
483C3×C4C2×C4×D911yescharacteristic
484C3×C4D4×D911yescharacteristic
485C3×C4C2×#62411yescharacteristic
486C3×C4#381111yescharacteristic
487Dic3—22no—
488Dic3—62no—
489Dic3—91no—
490Dic3—91no—
491Dic3—91no—
492Dic3—91no—
493Dic3—181no—
494Dic3—181no—
495Dic3—182no—
496Dic3—181no—
497Dic3—181no—
498C32D3×#10411yescharacteristic
499C9—21no—
500C9D3×#10411yescharacteristic, complemented
501C23—271no—
502C23—541no—
503Q8—271no—
504Q8—541no—
505Q8(C3⋊D9)⋊C411yescharacteristic, complemented
506D4—21no—
507D4—271no—
508D4—541no—
509D4—541no—
510D4—541no—
511D4—541no—
512D4(C3⋊D9)⋊C411yescharacteristic, complemented
513C2×C4—21no—
514C2×C4—61no—
515C2×C4—181no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—271no—
519C2×C4—271no—
520C2×C4—271no—
521C2×C4—541no—
522C2×C4—541no—
523C2×C4—544no—
524C2×C4—541no—
525C2×C4—541no—
526C2×C4C2×D3×D911yescharacteristic
527C8—61no—
528C8—181no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—41no—
533C2×C3—42no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3C22⋊C4×D911yescharacteristic
537C2×C3#6397411yescharacteristic
538D3—92no—
539D3—92no—
540D3—182no—
541D3—181no—
542D3—181no—
543D3—361no—
544D3—362no—
545D3—361no—
546D3—361no—
547C22—21no—
548C22—271no—
549C22—271no—
550C22—542no—
551C22—541no—
552C22—1082no—
553C22#8866811yescharacteristic
554C4—21no—
555C4—62no—
556C4—182no—
557C4—271no—
558C4—271no—
559C4—541no—
560C4C2×(C3⋊D9)⋊C411yescharacteristic
561C4#8866811yescharacteristic
562C3—21no—
563C3D9×#10411yescharacteristic, complemented
564C3#1059167111yescharacteristic
565C2—21no—
566C2—41no—
567C2—272no—
568C2—541no—
569C2—1081no—
570C2#1172127411yescharacteristic
subgroupG/H
centerC4#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×#41C22
socleC2×C32D3×C22⋊C4
prime radical#41×C3⋊D9C2
cosocleC2×C3×C4×C9C23
hypercenter#41D3×D9
hypocenterC3×C9C2×#104
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C4×(C3⋊D9)⋊C4C2semidirect
3. #11720863C2semidirect
4. C9×#837C2semidirect
5. C3×#63396C2semidirect
6. #11720292C2semidirect
7. #11720261C2semidirect
11. #88746C4semidirect
12. Q8×C3⋊D9C4semidirect
14. #88742C4semidirect
16. D4×C3⋊D9C4semidirect
22. C4×C9×Dic3C22semidirect
23. C3×C4×Dic9C22semidirect
24. C9×#208C22semidirect
25. C3×#3667C22semidirect
41. #63396D3semidirect
46. C3×Q8×C9C2×C4semidirect
48. C3×D4×C9C2×C4semidirect
57. C4⋊(C3⋊D9)C2×C4semidirect
66. #6226C2×C4semidirect
119. C4×Dic9C2×D3semidirect
120. #3667C2×D3semidirect
148. #837D9semidirect
186. Q8×C9C4×D3semidirect
190. D4×C9C4×D3semidirect
238. C3⋊D9#104semidirect
281. C4×Dic3C2×D9semidirect
283. #208C2×D9semidirect
338. C3×C9C2×#104semidirect
346. C3×Q8C4×D9semidirect
352. C3×D4C4×D9semidirect
500. C9D3×#104semidirect
505. Q8(C3⋊D9)⋊C4semidirect
512. D4(C3⋊D9)⋊C4semidirect
563. C3D9×#104semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊D92
C4×(C3⋊D9)⋊C42
#117208632
C9×#8372
C3×#633962
#117202922
#117202612
#105916713
D9×#1043