Finite Group Databasebeta

← #13148836

Subgroups — #13148836

Order 1728.

1006 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88641C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3#11721743C211yesmaximal, characteristic, complemented
4C9×#895C211yesmaximal, characteristic, complemented
5C3×#63457C211yesmaximal, characteristic, complemented
6#11721043C212yesmaximal, complemented
7#11721043C211yesmaximal, characteristic, complemented
8#11721028C212yesmaximal, complemented
9C4×#6194C211yesmaximal, characteristic, complemented
10#11721016C212yesmaximal, complemented
11#11720982C212yesmaximal, complemented
12#10594527—31nomaximal
13#10589701—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C9×D12C2212yes—
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×C3×D36C2212yes—
20C2×C3×C4×D9C2211yescharacteristic
21C2×#6194C2212yes—
22C2×#6194C2212yes—
23C2×#6194C2211yescharacteristic
24C2×#6194C2212yes—
25C2×#6194C2212yes—
26C2×D3×Dic9C2212yes—
27C2×Dic3×D9C2212yes—
28#88641—24no—
29#88638—24no—
30C3×C9×C4⋊C4C2211yescharacteristic
31#88534C2212yes—
32C4×C3⋊Dic9C2211yescharacteristic
33C4×C3⋊Dic9C2212yes—
34C9×(C2×D3)⋊C4C2212yes—
35C3×(C2×D9)⋊C4C2212yes—
36(C2×D3)⋊Dic9C2211yescharacteristic
37(C2×D9)⋊Dic3C2211yescharacteristic
38#88442C2211yescharacteristic
39C2×#3795—31no—
40C2×#3795—31no—
41#64096—31no—
42C3×#895—31no—
43C3×#895D311yescharacteristic
44#63914—31no—
45#63914—32no—
46#63902—32no—
47C4×#625—31no—
48#63885—32no—
49#63885—32no—
50C2×#3699—31no—
51C2×#3697—31no—
52C9×#121—31no—
53#63502—31no—
54#63502—32no—
55#63501—32no—
56C4×#611—31no—
57#63457D311yescharacteristic, complemented
58#63449—31no—
59#63449—61no—
60#63446—32no—
61#611⋊C4—32no—
62C22×D3×C9C2312yes—
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2312yes—
70C2×C3⋊Dic9C2311yescharacteristic
71#6226D414yescomplemented
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C9×D12—22no—
75C9×D12—42no—
76C4×D3×C9—22no—
77C3×D36—22no—
78C3×D36—42no—
79C3×C4×D9—22no—
80#6194—24no—
81#6194—24no—
82#6194—22no—
83#6194—24no—
84#6194—42no—
85D3×Dic9—24no—
86Dic3×D9—24no—
87#5599—91no—
88#5599—91no—
89C2×#634—31no—
90C2×C3×D12—31no—
91C2×C3×D12—32no—
92C2×C3×D12C2×D311yescharacteristic
93C2×C3×D12C2×D312yes—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3C2×D311yescharacteristic
96C2×#625—32no—
97C2×#625—32no—
98C2×#625—31no—
99C2×#625—32no—
100C2×#625—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103#3795—64no—
104#3795—64no—
105C32×C4⋊C4C2×D311yescharacteristic
106#3750—32no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D312yes—
111#3724—31no—
112(C2×D3)⋊Dic3—31no—
113(C2×D3)⋊Dic3—31no—
114C2×D4×C9—31no—
115C2×D4×C9—32no—
116C22×C4×C9—31no—
117C2×#611—31no—
118C2×#611—32no—
119C2×#611—32no—
120C2×#611—32no—
121C2×#611—32no—
122C22×Dic9—32no—
123#3699—64no—
124#3697—64no—
125C2×D36C2×D311yescharacteristic
126C2×D36C2×D312yes—
127C2×C4×D9—32no—
128C2×C4×D9C2×D311yescharacteristic
129C2×Dic18—31no—
130C2×Dic18—61no—
131C9×C4⋊C4—21no—
132C9×C4⋊C4C2×D311yescharacteristic
133C9×C22⋊C4—32no—
134#3676—31no—
135(C2×D9)⋊C4—31no—
136(C2×D9)⋊C4C2×D312yes—
137Dic9⋊C4—32no—
138Dic9⋊C4—31no—
139Dic9⋊C4—62no—
140C4×Dic9—31no—
141C4×Dic9—32no—
142C4×Dic9—62no—
143C4×Dic9—61no—
144C22×C3×C9C2411yescharacteristic
145C2×D3×C9—22no—
146C2×D3×C9—21no—
147C2×D3×C9—22no—
148C2×D3×C9—21no—
149C2×D3×C9—22no—
150C2×D3×C9—21no—
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156C2×C3×D9—21no—
157C3×C4×C9—22no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—21no—
160C3⋊Dic9C2×D414yes—
161C3⋊Dic9#4112yescomplemented
162C9×Dic3—21no—
163C3×Dic9—21no—
164C2×#237—91no—
165C2×#237—91no—
166C2×#235—91no—
167C2×#235—91no—
168C3×#121—31no—
169C3×#121—91no—
170#940—92no—
171#940—91no—
172#940—92no—
173#940—91no—
174#939—92no—
175#939—92no—
176C4×#67—91no—
177C4×#67—91no—
178#895—31no—
179#895D911yescharacteristic, complemented
180#887—91no—
181#887—91no—
182#887—181no—
183#884—92no—
184#884—92no—
185#67⋊C4—92no—
186#67⋊C4—92no—
187C22×C3×D3—31no—
188C22×C3×D3—32no—
189C22×C3×D3C22×D311yescharacteristic
190C22×C3×D3C22×D312yes—
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—31no—
196#634—34no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3C22×D311yescharacteristic
199C3×D12—22no—
200C3×D12—42no—
201C3×D12—62no—
202C3×D12—122no—
203C3×C4×D3—22no—
204C3×C4×D3—62no—
205#625—64no—
206#625—64no—
207#625—62no—
208#625—64no—
209#625—122no—
210D3×Dic3—64no—
211D3×Dic3—64no—
212C23×C9—31no—
213C23×C9—32no—
214C22×D9C22×D311yescharacteristic
215C22×D9C22×D312yes—
216D4×C9—62no—
217D4×C9—122no—
218C2×C4×C9—21no—
219C2×C4×C9—22no—
220C2×C4×C9—31no—
221C2×C4×C9—62no—
222C2×C4×C9C22×D312yes—
223C2×C4×C9C22×D311yescharacteristic
224#611—64no—
225#611—64no—
226#611—62no—
227#611—64no—
228#611—122no—
229C2×Dic9—31no—
230C2×Dic9—32no—
231C2×Dic9—31no—
232C2×Dic9—61no—
233C2×Dic9—62no—
234C2×Dic9—61no—
235C2×Dic9—64no—
236C2×Dic9C22×D311yescharacteristic
237D36—22no—
238D36—42no—
239C4×D9—22no—
240C4×D9—64no—
241Dic18—34no—
242Dic18—64no—
243#550—271noSylow
244C2×C3×C9C22×D411yescharacteristic
245C2×C3×C9C2×#4111yescharacteristic
246C2×C3×C9H(5,𝔽2)11yescharacteristic
247D3×C9—42no—
248D3×C9—41no—
249C3×D9—42no—
250C3×D9—41no—
251C2×C3×D4—31no—
252C2×C3×D4—32no—
253C2×C3×D4—91no—
254C2×C3×D4—92no—
255C22×C3×C4—31no—
256C22×C3×C4—91no—
257C2×#67—92no—
258C2×#67—92no—
259C2×#67—91no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—92no—
263C2×#67—91no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C22×Dic3—92no—
268C22×Dic3—92no—
269#237—184no—
270#237—184no—
271#235—184no—
272#235—184no—
273C2×D12—31no—
274C2×D12—32no—
275C2×D12C2×D911yescharacteristic
276C2×D12C2×D912yes—
277C2×C4×D3—31no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3C2×D911yescharacteristic
281C2×Dic6—91no—
282C2×Dic6—91no—
283C2×Dic6—181no—
284C3×C4⋊C4—21no—
285C3×C4⋊C4C2×D911yescharacteristic
286C3×C4⋊C4D3211yescharacteristic
287C3×C22⋊C4—32no—
288C3×C22⋊C4—92no—
289#217—91no—
290#217—91no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—91no—
293(C2×D3)⋊C4—91no—
294(C2×D3)⋊C4C2×D912yes—
295Dic3⋊C4—91no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—91no—
299Dic3⋊C4—182no—
300C4×Dic3—92no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C4×Dic3—91no—
304C4×Dic3—182no—
305C4×Dic3—181no—
306C22×C32C23×D311yescharacteristic
307C2×C3×D3—22no—
308C2×C3×D3—21no—
309C2×C3×D3—22no—
310C2×C3×D3—21no—
311C2×C3×D3—21no—
312C2×C3×D3—22no—
313C2×C3×D3—62no—
314C2×C3×D3—61no—
315C2×C3×D3—62no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—62no—
319C32×C4—22no—
320C32×C4D3×D412yes—
321C3⋊Dic3—34no—
322C3⋊Dic3—32no—
323C3⋊Dic3—61no—
324C3×Dic3—21no—
325C3×Dic3—61no—
326C22×C9—21no—
327C22×C9—62no—
328C22×C9—61no—
329C22×C9—62no—
330C22×C9—61no—
331C22×C9—61no—
332C22×C9—62no—
333C22×C9C23×D311yescharacteristic
334C2×D9—21no—
335C2×D9—22no—
336C2×D9—21no—
337C2×D9—22no—
338C2×D9—21no—
339C2×D9—22no—
340C4×C9—22no—
341C4×C9—22no—
342C4×C9—42no—
343C4×C9—61no—
344C4×C9D3×D412yes—
345Dic9—21no—
346Dic9—34no—
347Dic9—32no—
348Dic9—64no—
349Dic9—62no—
350Dic9—61no—
351Dic9—121no—
352C2×#41—271no—
353C2×#41—271no—
354#127—271no—
355#127—272no—
356#121—31no—
357#121—91no—
358#121—272no—
359#121—272no—
360C4×Q8—271no—
361C4×D4—272no—
362C4×D4—271no—
363C3×C9#55011yescharacteristic, complemented, Sylow
364C23×C3—31no—
365C23×C3—32no—
366C23×C3—91no—
367C23×C3—92no—
368C22×D3—32no—
369C22×D3—31no—
370C22×D3C22×D911yescharacteristic
371C22×D3C22×D912yes—
372C3×D4—62no—
373C3×D4—122no—
374C3×D4—182no—
375C3×D4—362no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—31no—
379C2×C3×C4—62no—
380C2×C3×C4—91no—
381C2×C3×C4—182no—
382C2×C3×C4C22×D912yes—
383C2×C3×C4C22×D911yescharacteristic
384C2×C3×C4C2×D3212yes—
385C2×C3×C4C2×D3211yescharacteristic
386#67—184no—
387#67—184no—
388#67—184no—
389#67—182no—
390#67—184no—
391#67—184no—
392#67—184no—
393#67—182no—
394#67—362no—
395#67—362no—
396C2×Dic3—31no—
397C2×Dic3—91no—
398C2×Dic3—92no—
399C2×Dic3—91no—
400C2×Dic3—92no—
401C2×Dic3—91no—
402C2×Dic3—91no—
403C2×Dic3—181no—
404C2×Dic3—182no—
405C2×Dic3—181no—
406C2×Dic3—184no—
407C2×Dic3—184no—
408C2×Dic3C22×D911yescharacteristic
409D12—22no—
410D12—42no—
411D12—62no—
412D12—122no—
413C4×D3—22no—
414C4×D3—62no—
415C4×D3—184no—
416C4×D3—184no—
417Dic6—94no—
418Dic6—94no—
419Dic6—184no—
420C2×C32C2×D3×D411yescharacteristic
421C2×C32#100411yescharacteristic
422C2×C32D3×#4111yescharacteristic
423C3×D3—42no—
424C3×D3—41no—
425C3×D3—122no—
426C3×D3—121no—
427C2×C9—21no—
428C2×C9—21no—
429C2×C9—21no—
430C2×C9—122no—
431C2×C9—121no—
432C2×C9C2×D3×D411yescharacteristic
433C2×C9#100411yescharacteristic
434C2×C9D3×#4111yescharacteristic
435D9—41no—
436D9—42no—
437#41—544no—
438#41—544no—
439C2×Q8—271no—
440C2×D4—31no—
441C2×D4—32no—
442C2×D4—91no—
443C2×D4—92no—
444C2×D4—272no—
445C2×D4—271no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4—272no—
449C22×C4—31no—
450C22×C4—91no—
451C22×C4—272no—
452C22×C4—272no—
453C4⋊C4—271no—
454C4⋊C4—272no—
455C4⋊C4D3×D911yescharacteristic
456C22⋊C4—32no—
457C22⋊C4—92no—
458C22⋊C4—271no—
459C22⋊C4—271no—
460C42—272no—
461C42—271no—
462C22×C3—21no—
463C22×C3—62no—
464C22×C3—61no—
465C22×C3—62no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—182no—
470C22×C3—181no—
471C22×C3—182no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3—182no—
475C22×C3C23×D911yescharacteristic
476C22×C3C22×D3211yescharacteristic
477C2×D3—21no—
478C2×D3—22no—
479C2×D3—21no—
480C2×D3—22no—
481C2×D3—21no—
482C2×D3—22no—
483C2×D3—61no—
484C2×D3—62no—
485C2×D3—61no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—61no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—42no—
493C3×C4—61no—
494C3×C4—181no—
495C3×C4D4×D912yes—
496C3×C4#380212yes—
497Dic3—21no—
498Dic3—61no—
499Dic3—94no—
500Dic3—92no—
501Dic3—94no—
502Dic3—92no—
503Dic3—184no—
504Dic3—182no—
505Dic3—181no—
506Dic3—181no—
507Dic3—361no—
508C32#559911yescharacteristic
509C9—21no—
510C9#559911yescharacteristic, complemented
511C23—32no—
512C23—31no—
513C23—92no—
514C23—91no—
515Q8—274no—
516D4—62no—
517D4—122no—
518D4—182no—
519D4—362no—
520D4—544no—
521D4—544no—
522D4—544no—
523D4—542no—
524D4—1082no—
525C2×C4—31no—
526C2×C4—62no—
527C2×C4—91no—
528C2×C4—182no—
529C2×C4—271no—
530C2×C4—272no—
531C2×C4—271no—
532C2×C4—544no—
533C2×C4—544no—
534C2×C4C2×D3×D912yes—
535C2×C4C2×D3×D911yescharacteristic
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—122no—
540C2×C3—121no—
541C2×C3—362no—
542C2×C3—361no—
543C2×C3C2×D4×D911yescharacteristic
544C2×C3#6371011yescharacteristic
545C2×C3#41×D911yescharacteristic
546C2×C3C2×#380211yescharacteristic
547C2×C3#6431311yescharacteristic
548C2×C3#6431911yescharacteristic
549D3—41no—
550D3—42no—
551D3—121no—
552D3—122no—
553C22—61no—
554C22—62no—
555C22—62no—
556C22—61no—
557C22—61no—
558C22—62no—
559C22—181no—
560C22—182no—
561C22—182no—
562C22—181no—
563C22—181no—
564C22—182no—
565C22C22×D3×D911yescharacteristic
566C4—22no—
567C4—61no—
568C4—181no—
569C4—272no—
570C4—274no—
571C4—541no—
572C4#8864612yes—
573C3—21no—
574C3#1058970111yescharacteristic, complemented
575C3#1059452711yescharacteristic
576C2—121no—
577C2—122no—
578C2—361no—
579C2—362no—
580C2C2×#8864611yescharacteristic
581C2#1172249911yescharacteristic
582C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#550
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88641C2semidirect
2. C2×#88638C2semidirect
3. #11721743C2semidirect
4. C9×#895C2semidirect
5. C3×#63457C2semidirect
6. #11721043C2semidirect
7. #11721043C2semidirect
8. #11721028C2semidirect
9. C4×#6194C2semidirect
10. #11721016C2semidirect
11. #11720982C2semidirect
57. #63457D3semidirect
71. #6226D4semidirect
161. C3⋊Dic9#41semidirect
179. #895D9semidirect
363. C3×C9#550semidirect
510. C9#5599semidirect
574. C3#10589701semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
C2×#886382
#117217432
C9×#8952
C3×#634572
#117210432
#117210282
C4×#61942
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