Finite Group Databasebeta

← #13181147

Subgroups — #13181147

Order 1728.

1162 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 123 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#11724618C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4#11724354C214yesmaximal, complemented
5#11724344C212yesmaximal, complemented
6#11724343C214yesmaximal, complemented
7#11724334C212yesmaximal, complemented
8#10596512—31nomaximal
9C22×#1113C2212yes—
10C2×#157⋊D3C2218yescomplemented
11C3×#3828—21no—
12C2×C3×C4⋊(C3⋊D3)—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14#89049C2212yes—
15C2×#6295—21no—
16#89046—21no—
17C4⋊#1113—21no—
18C4⋊#1113—21no—
19C4⋊#1113C2212yes—
20C4⋊#1113C2212yes—
21C4⋊#1113C2211yescharacteristic, complemented
22C4×#1113C2212yescomplemented
23#89042—21no—
24#89041—21no—
25#89040C2211yescharacteristic, complemented
26C3×C22⋊#157C2212yes—
27C3×C4⋊#157C2212yescomplemented
28C3×C4×#157C2212yescomplemented
29#88939C2218yescomplemented
30#64306—31no—
31D3×#235—61no—
32C2×#3802—31no—
33C2×D3×D12—61no—
34#64286D311yescharacteristic, complemented
35#64241—34no—
36#64231—32no—
37#64230—34no—
38#64221—32no—
39C22×C3×C3⋊D3—22no—
40C22×C3×C3⋊D3C2311yescharacteristic
41C2×#1113—22no—
42C2×#1113—44no—
43C2×#1113C2312yes—
44C2×#1113C2314yes—
45C2×C3×#157C2314yes—
46#157⋊D3—48no—
47C2×C33×C4D411yescharacteristic, complemented
48C3×#638—22no—
49C2×C3×C3⋊Dic3D411yescharacteristic, complemented
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—22no—
52C3×C4⋊(C3⋊D3)—22no—
53C3×C4×C3⋊D3—22no—
54C3×C4×C3⋊D3D411yescharacteristic, complemented
55C3×C4×C3⋊D3D411yescharacteristic, complemented
56C3×C4×C3⋊D3C2311yescharacteristic
57C3×C4×C3⋊D3C2311yescharacteristic
58C3×#634—21no—
59#6296—21no—
60#6295—22no—
61#6295—21no—
62#6295—22no—
63#6295—22no—
64#6294—22no—
65#6294C2312yes—
66#5525—91nomaximal
67C22×D32—32no—
68C22×D32—62no—
69C2×GO+(4,𝔽2)—38no—
70#3828—21no—
71C2×C4⋊(C3⋊D3)—21no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C3×#235—61no—
74C3×#235—61no—
75C3×#235—121no—
76C2×C3×D12—61no—
77C2×C3×D12—61no—
78C2×C3×D12—121no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81#3811—32no—
82D3×#67—62no—
83C2×#626—62no—
84C2×#625—61no—
85#3805—61no—
86#3804—62no—
87#3802—32no—
88#3802—32no—
89#3802—31no—
90#3802—61no—
91#3802—61no—
92D3×D12—62no—
93D3×D12—62no—
94D3×D12—61no—
95D3×D12—62no—
96D3×D12—62no—
97C4×D32—32no—
98C4×D32—62no—
99#3799—61no—
100#3798—62no—
101#3798—61no—
102#3797—31no—
103#3796—61no—
104#3795—61no—
105D3×Dic6—61no—
106C22⋊#157C2×D312yes—
107C4⋊#157C2×D312yescomplemented
108C4×#157C2×D312yescomplemented
109D32⋊C4—38no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—24no—
114C2×C3×C3⋊D3—44no—
115C2×C3×C3⋊D3C2×D411yescharacteristic
116C2×C3×C3⋊D3C2411yescharacteristic
117#1113—24no—
118#1113—28no—
119#1113—44no—
120C3×#157—24no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3C2×D411yescharacteristic
125C3×C3⋊Dic3C2×D411yescharacteristic
126D3×#41—181no—
127C2×D3×D4—181no—
128C2×#235—91no—
129C22×D12—91no—
130C3×#117—91no—
131#895—94no—
132#882—94no—
133#875—92no—
134#874—92no—
135C22×C3⋊D3—22no—
136C22×C3⋊D3C22×D311yescharacteristic
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—122no—
142C2×D32—32no—
143C2×D32—34no—
144C2×D32—64no—
145C2×D32—62no—
146C2×D32—64no—
147C2×D32—62no—
148C2×D32—124no—
149C2×D32—128no—
150C2×#157C22×D314yes—
151GO+(4,𝔽2)—128no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4D1211yescharacteristic, complemented
157#638—22no—
158C2×C3⋊Dic3D1211yescharacteristic, complemented
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)—22no—
161C4⋊(C3⋊D3)—22no—
162C4×C3⋊D3—22no—
163C4×C3⋊D3D1211yescharacteristic, complemented
164C4×C3⋊D3D1211yescharacteristic, complemented
165C4×C3⋊D3C22×D311yescharacteristic
166C4×C3⋊D3C22×D311yescharacteristic
167#634—21no—
168C3×#67—62no—
169C3×#67—62no—
170C3×#67—122no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C3×D12—61no—
174C3×D12—62no—
175C3×D12—62no—
176C3×D12—61no—
177C3×D12—62no—
178C3×D12—62no—
179C3×D12—122no—
180C3×D12—122no—
181C3×D12—121no—
182C3×C4×D3—61no—
183C3×C4×D3—61no—
184C3×C4×D3—61no—
185C3×C4×D3—61no—
186C3×C4×D3—61no—
187C3×C4×D3—61no—
188C3×C4×D3—61no—
189C3×C4×D3—61no—
190C3×C4×D3—62no—
191C3×C4×D3—62no—
192C3×C4×D3—122no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—121no—
196#627—62no—
197#627—61no—
198#626—62no—
199#626—62no—
200#626—64no—
201#626—64no—
202#626—62no—
203#625—62no—
204#625—61no—
205#625—62no—
206#625—62no—
207#624—32no—
208#624—62no—
209D3×Dic3—62no—
210D3×Dic3—62no—
211D3×Dic3—62no—
212#535—271noSylow
213C2×C33—21no—
214C2×C33C22×D411yescharacteristic
215C3×C3⋊D3—21no—
216C3×C3⋊D3—44no—
217C3×C3⋊D3—42no—
218C3×C3⋊D3H(5,𝔽2)12yes—
219C23×D3—92no—
220C23×D3—182no—
221C3×#41—181no—
222C2×C3×D4—181no—
223C22×C3×C4—91no—
224C2×#67—92no—
225C2×#67—182no—
226#239—181no—
227D3×Q8—181no—
228#237—181no—
229#237—182no—
230D3×D4—182no—
231D3×D4—182no—
232D3×D4—181no—
233D3×D4—182no—
234D3×D4—182no—
235D3×D4—182no—
236#235—61no—
237#235—61no—
238#235—121no—
239#235—181no—
240#235—181no—
241#235—182no—
242#235—181no—
243C2×D12—61no—
244C2×D12—61no—
245C2×D12—91no—
246C2×D12—92no—
247C2×D12—92no—
248C2×D12—98no—
249C2×D12—121no—
250C2×D12—181no—
251C2×D12—181no—
252C2×D12—181no—
253C2×C4×D3—61no—
254C2×C4×D3—61no—
255C2×C4×D3—92no—
256C2×C4×D3—182no—
257C2×Dic6—91no—
258C3×C4⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C42—92no—
261(C2×D3)⋊C4—98no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—41no—
266C22×C32C2×D1211yescharacteristic
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—24no—
270C2×C3⋊D3—44no—
271C2×C3⋊D3C2×D1211yescharacteristic
272C2×C3⋊D3C23×D311yescharacteristic
273C2×C3×D3—61no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—64no—
278C2×C3×D3—62no—
279C2×C3×D3—64no—
280C2×C3×D3—62no—
281C2×C3×D3—64no—
282C2×C3×D3—122no—
283C2×C3×D3—122no—
284C2×C3×D3—124no—
285C2×C3×D3—124no—
286C2×C3×D3—124no—
287C2×C3×D3—244no—
288D32—64no—
289D32—68no—
290D32—124no—
291D32—128no—
292D32—128no—
293D32—124no—
294#157—24no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4C2×D1211yescharacteristic
304C32×C4C2×D1211yescharacteristic
305C3⋊Dic3—22no—
306C3⋊Dic3C2×D1211yescharacteristic
307C3⋊Dic3C2×D1211yescharacteristic
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—122no—
315C2×#41—271no—
316C22×D4—271no—
317#127—272no—
318#124—272no—
319#121—274no—
320#120—274no—
321#117—91no—
322C33#53511yescharacteristic, complemented, Sylow
323C23×C3—91no—
324C23×C3—182no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—62no—
328C22×D3—62no—
329C22×D3—94no—
330C22×D3—92no—
331C22×D3—122no—
332C22×D3—182no—
333C22×D3—184no—
334C22×D3—182no—
335C22×D3—184no—
336C22×D3—368no—
337C22×D3—364no—
338C3×Q8—181no—
339C3×D4—182no—
340C3×D4—182no—
341C3×D4—181no—
342C3×D4—182no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—94no—
353C2×C3×C4—182no—
354C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
355#67—62no—
356#67—62no—
357#67—122no—
358#67—182no—
359#67—182no—
360#67—182no—
361#67—184no—
362#67—184no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—92no—
366C2×Dic3—182no—
367D12—62no—
368D12—62no—
369D12—61no—
370D12—62no—
371D12—62no—
372D12—61no—
373D12—122no—
374D12—122no—
375D12—121no—
376D12—182no—
377D12—182no—
378D12—181no—
379D12—182no—
380D12—368no—
381C4×D3—61no—
382C4×D3—61no—
383C4×D3—61no—
384C4×D3—61no—
385C4×D3—61no—
386C4×D3—61no—
387C4×D3—61no—
388C4×D3—61no—
389C4×D3—62no—
390C4×D3—62no—
391C4×D3—122no—
392C4×D3—182no—
393C4×D3—182no—
394C4×D3—182no—
395Dic6—61no—
396Dic6—61no—
397Dic6—121no—
398Dic6—181no—
399Dic6—182no—
400C2×C32—22no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—41no—
405C2×C32—41no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—81no—
409C2×C32C22×D1211yescharacteristic
410C3⋊D3—21no—
411C3⋊D3—44no—
412C3⋊D3—42no—
413C3⋊D3#100912yes—
414C3×D3—62no—
415C3×D3—62no—
416C3×D3—62no—
417C3×D3—121no—
418C3×D3—124no—
419C3×D3—122no—
420C3×D3—124no—
421C3×D3—122no—
422C3×D3—242no—
423C3×D3—244no—
424C24—272no—
425#41—181no—
426#41—541no—
427#41—541no—
428#41—541no—
429C2×Q8—271no—
430C2×D4—181no—
431C2×D4—272no—
432C2×D4—272no—
433C2×D4—271no—
434C2×D4—278no—
435C2×D4—272no—
436C2×D4—541no—
437C2×D4—541no—
438C2×D4—541no—
439C22×C4—91no—
440C22×C4—272no—
441C4⋊C4—92no—
442C22⋊C4—92no—
443C22⋊C4—278no—
444C42—92no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—91no—
450C22×C3—91no—
451C22×C3—182no—
452C22×C3—182no—
453C22×C3—184no—
454C22×C3—364no—
455C22×C3C2×GO+(4,𝔽2)11yescharacteristic
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—64no—
461C2×D3—62no—
462C2×D3—64no—
463C2×D3—62no—
464C2×D3—64no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—124no—
468C2×D3—124no—
469C2×D3—124no—
470C2×D3—184no—
471C2×D3—188no—
472C2×D3—244no—
473C2×D3—368no—
474C2×D3—364no—
475C2×D3—368no—
476C2×D3—364no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—91no—
486C3×C4—91no—
487C3×C4—182no—
488C3×C4—184no—
489C3×C4C2×GO+(4,𝔽2)11yescharacteristic
490C3×C4C2×GO+(4,𝔽2)11yescharacteristic
491Dic3—61no—
492Dic3—61no—
493Dic3—61no—
494Dic3—61no—
495Dic3—62no—
496Dic3—62no—
497Dic3—122no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32—41no—
502C32#552511yescharacteristic, complemented
503C23—91no—
504C23—182no—
505C23—274no—
506C23—272no—
507C23—542no—
508C23—1084no—
509Q8—181no—
510Q8—541no—
511D4—182no—
512D4—182no—
513D4—182no—
514D4—181no—
515D4—542no—
516D4—542no—
517D4—541no—
518D4—542no—
519D4—1088no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—91no—
523C2×C4—91no—
524C2×C4—91no—
525C2×C4—94no—
526C2×C4—182no—
527C2×C4—272no—
528C2×C4—542no—
529C2×C4#157⋊D311yescharacteristic, complemented
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—42no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—81no—
539C2×C3—92no—
540C2×C3—181no—
541C2×C3—364no—
542C2×C3—362no—
543C2×C3C22×GO+(4,𝔽2)11yescharacteristic
544D3—62no—
545D3—62no—
546D3—62no—
547D3—121no—
548D3—124no—
549D3—122no—
550D3—124no—
551D3—122no—
552D3—244no—
553D3—242no—
554C22—91no—
555C22—91no—
556C22—182no—
557C22—182no—
558C22—184no—
559C22—364no—
560C22—544no—
561C22—548no—
562C22—1084no—
563C22C2×#157⋊D311yescharacteristic
564C4—91no—
565C4—91no—
566C4—184no—
567C4—182no—
568C4C2×#157⋊D311yescharacteristic
569C4C2×#157⋊D311yescharacteristic
570C3—21no—
571C3—21no—
572C3—41no—
573C3—41no—
574C3#1059651211yescharacteristic, complemented
575C2—21no—
576C2—92no—
577C2—181no—
578C2—364no—
579C2—362no—
580C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#535
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11724618C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C3×#64286C2semidirect
4. #11724354C2semidirect
5. #11724344C2semidirect
6. #11724343C2semidirect
7. #11724334C2semidirect
10. C2×#157⋊D3C22semidirect
21. C4⋊#1113C22semidirect
22. C4×#1113C22semidirect
25. #89040C22semidirect
27. C3×C4⋊#157C22semidirect
28. C3×C4×#157C22semidirect
29. #88939C22semidirect
34. #64286D3semidirect
47. C2×C33×C4D4semidirect
49. C2×C3×C3⋊Dic3D4semidirect
54. C3×C4×C3⋊D3D4semidirect
55. C3×C4×C3⋊D3D4semidirect
107. C4⋊#157C2×D3semidirect
108. C4×#157C2×D3semidirect
156. C2×C32×C4D12semidirect
158. C2×C3⋊Dic3D12semidirect
163. C4×C3⋊D3D12semidirect
164. C4×C3⋊D3D12semidirect
322. C33#535semidirect
354. C2×C3×C4GO+(4,𝔽2)semidirect
502. C32#5525semidirect
529. C2×C4#157⋊D3semidirect
574. C3#10596512semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C4⋊#11132
C3×#642862
#117243542
#117243442
#117243432
#117243342
#105965123
#55259