Finite Group Databasebeta

← #13150565

Subgroups — #13150565

Order 1728.

1166 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×D3×D36C211yesmaximal, characteristic, complemented
3C4×C9×#67C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721256C212yesmaximal, complemented
7#11721048C212yesmaximal, complemented
8#11720993C211yesmaximal, characteristic, complemented
9#11720961C211yesmaximal, characteristic, complemented
10#11720948C211yesmaximal, characteristic, complemented
11#11720941C212yesmaximal, complemented
12#10595078—31nomaximal
13#10589652—31nomaximal
14C22×D3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic, complemented
16C2×#6233C2212yescomplemented
17#88736—24no—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C2×C4×C3⋊D9C2212yescomplemented
20C2×#6226C2211yescharacteristic, complemented
21C2×C9×#67C2211yescharacteristic
22C2×C3×#611C2212yescomplemented
23C2×C4×D3×C9C2211yescharacteristic
24C2×C3×D36C2211yescharacteristic
25C2×#6192C2212yes—
26C2×(C3⋊D9)⋊C4C2212yescomplemented
27D3×D36—24no—
28C9×#217C2211yescharacteristic, complemented
29C9×(C2×D3)⋊C4C2211yescharacteristic
30C3×(C2×D9)⋊C4C2212yescomplemented
31C9×Dic3⋊C4C2211yescharacteristic, complemented
32C4×C9×Dic3C2211yescharacteristic, complemented
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34(C2×D3)⋊Dic9C2212yes—
35#88445C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37(C2×D9)⋊Dic3C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C2×#3828—31no—
40C2×D3×D12—31no—
41C3×#930—31no—
42C3×C4×#67D311yescharacteristic
43#63983—32no—
44#63958—32no—
45#63921—32no—
46#63894—31no—
47#63860—31no—
48#63857—31no—
49#63853—32no—
50C2×#3697—31no—
51C2×#3697—61no—
52C22×D36—31no—
53C4×D4×C9—31no—
54#63499—32no—
55#63492D311yescharacteristic, complemented
56#63486—31no—
57#63456—32no—
58#63446—32no—
59C22⋊C4×D9—32no—
60#63437—31no—
61C4×D36—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2312yes—
66C2×D3×D9—24no—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233—24no—
73C2×C3⋊Dic9C2312yes—
74C4⋊(C3⋊D9)D414yescomplemented
75C4×C3⋊D9—24no—
76#6226—22no—
77C9×#67—22no—
78C3×#611—42no—
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2311yescharacteristic
81C2×C9×Dic3C2311yescharacteristic
82C2×C3×Dic9C2312yes—
83C4×D3×C9—22no—
84C3×D36—22no—
85#6192—24no—
86(C3⋊D9)⋊C4—42no—
87#5612—91no—
88#5550—91no—
89C22×D32—32no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#638—32no—
92#3828—64no—
93C2×C4⋊(C3⋊D3)—31no—
94C2×C4×C3⋊D3—32no—
95C2×#634—31no—
96C2×C3×#67—32no—
97C2×C3×#67C2×D311yescharacteristic
98C2×C3×D12—31no—
99C2×C3×C4×D3C2×D311yescharacteristic
100C2×#626—32no—
101C2×#624—32no—
102D3×D12—64no—
103C23×D9—32no—
104C3×#217C2×D311yescharacteristic
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4C2×D311yescharacteristic
107C3×C4⋊Dic3—31no—
108C3×Dic3⋊C4C2×D311yescharacteristic
109C3×C4×Dic3C2×D311yescharacteristic
110Dic3⋊Dic3—32no—
111#3722—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9—31no—
118C22×C4×C9C2×D311yescharacteristic, complemented
119C2×#611—32no—
120C2×#611—62no—
121C2×#611C2×D312yescomplemented
122#3697—64no—
123#3697—68no—
124C2×D36—31no—
125C2×D36—32no—
126C2×D36—64no—
127C2×D36—61no—
128C2×D36C2×D311yescharacteristic
129C2×C4×D9—32no—
130C2×C4×D9—32no—
131C2×C4×D9—62no—
132C2×Dic18—31no—
133C2×Dic18—61no—
134C9×C4⋊C4—31no—
135C9×C22⋊C4—31no—
136C9×C22⋊C4—31no—
137C42×C9—31no—
138#3676—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4C2×D312yescomplemented
143C4⋊Dic9C2×D311yescharacteristic, complemented
144Dic9⋊C4—32no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9—21no—
148C22×C3×C9C2411yescharacteristic
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×C3⋊D9C2×D414yes—
152C2×D3×C9—21no—
153C2×D3×C9—21no—
154C2×D3×C9#4112yescomplemented
155C2×C3×D9—22no—
156C2×C3×D9—22no—
157C2×C3×D9—22no—
158D3×D9—48no—
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C9×Dic3—21no—
163C9×Dic3—21no—
164C9×Dic3—21no—
165C3×Dic9—22no—
166C2×D3×D4—91no—
167C2×#235—91no—
168C2×#235—91no—
169C2×#235—181no—
170C22×D12—91no—
171C3×#121—91no—
172C3×C4×D4—31no—
173#938—92no—
174#937—92no—
175#937—92no—
176#936—91no—
177#930—31no—
178C4×#67D911yescharacteristic, complemented
179#924—91no—
180#897—91no—
181#894—92no—
182#893—91no—
183#884—92no—
184#883—92no—
185D3×C22⋊C4—92no—
186D3×C22⋊C4—92no—
187#875—91no—
188C4×D12—91no—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—32no—
191C22×C3×D3—32no—
192C22×C3×D3C22×D311yescharacteristic
193C2×D32—64no—
194C2×D32—64no—
195C2×D32—122no—
196C2×C32×C4—22no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199#638—64no—
200C2×C3⋊Dic3—32no—
201C4⋊(C3⋊D3)—34no—
202C4×C3⋊D3—64no—
203#634—62no—
204C3×#67—22no—
205C3×#67—122no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3C22×D311yescharacteristic
208C2×C3×Dic3C22×D311yescharacteristic
209C2×C3×Dic3C22×D311yescharacteristic
210C3×D12—62no—
211C3×C4×D3—22no—
212#626—64no—
213#624—122no—
214C23×C9—21no—
215C23×C9—31no—
216C23×C9C22×D311yescharacteristic
217C22×D9—32no—
218C22×D9—62no—
219C22×D9—64no—
220C22×D9—64no—
221C22×D9—122no—
222C22×D9C22×D312yes—
223D4×C9—62no—
224C2×C4×C9—24no—
225C2×C4×C9—22no—
226C2×C4×C9—21no—
227C2×C4×C9—21no—
228C2×C4×C9—31no—
229C2×C4×C9—31no—
230C2×C4×C9—31no—
231C2×C4×C9—62no—
232C2×C4×C9C22×D311yescharacteristic
233C2×C4×C9C22×D311yescharacteristic
234#611—42no—
235#611—64no—
236#611—68no—
237C2×Dic9—32no—
238C2×Dic9—62no—
239C2×Dic9C22×D312yes—
240D36—22no—
241D36—34no—
242D36—64no—
243D36—64no—
244C4×D9—64no—
245C4×D9—68no—
246C4×D9—122no—
247Dic18—64no—
248Dic18—62no—
249#546—271noSylow
250C2×C3×C9—41no—
251C2×C3×C9C22×D411yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253C2×C3×C9H(5,𝔽2)11yescharacteristic
254C3⋊D9—24no—
255D3×C9—22no—
256C3×D9—42no—
257C23×D3—92no—
258C23×D3—92no—
259C2×C3×D4—31no—
260C2×C3×D4—92no—
261C2×C3×D4—91no—
262C22×C3×C4—21no—
263C22×C3×C4—31no—
264C22×C3×C4C2×D911yescharacteristic, complemented
265C22×C3×C4D3211yescharacteristic
266C2×#67—32no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—182no—
271C2×#67C2×D911yescharacteristic
272D3×D4—184no—
273#235—184no—
274#235—184no—
275#235—188no—
276C2×D12—31no—
277C2×D12—91no—
278C2×D12—91no—
279C2×D12—92no—
280C2×D12—181no—
281C2×D12—184no—
282C2×C4×D3—92no—
283C2×C4×D3—92no—
284C2×C4×D3—92no—
285C2×C4×D3—92no—
286C2×C4×D3—182no—
287C2×C4×D3C2×D911yescharacteristic
288C2×Dic6—91no—
289C2×Dic6—91no—
290C2×Dic6—181no—
291C3×C4⋊C4—31no—
292C3×C4⋊C4—91no—
293C3×C22⋊C4—31no—
294C3×C22⋊C4—31no—
295C3×C22⋊C4—92no—
296C3×C42—31no—
297#217—92no—
298#217—92no—
299#217—92no—
300#217C2×D911yescharacteristic, complemented
301(C2×D3)⋊C4—32no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4C2×D911yescharacteristic
308C4⋊Dic3—31no—
309C4⋊Dic3—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4C2×D911yescharacteristic, complemented
312C4×Dic3C2×D911yescharacteristic, complemented
313C22×C32—21no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—34no—
318C2×C3⋊D3—62no—
319C2×C3⋊D3—62no—
320C2×C3×D3—21no—
321C2×C3×D3—21no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3#23512yes—
326D32—128no—
327C32×C4—21no—
328C32×C4D3×D412yes—
329C3⋊Dic3—62no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—62no—
334C22×C9—22no—
335C22×C9—21no—
336C22×C9—21no—
337C22×C9—21no—
338C22×C9—21no—
339C22×C9—32no—
340C22×C9—41no—
341C22×C9—41no—
342C22×C9—61no—
343C22×C9—61no—
344C22×C9C23×D311yescharacteristic
345C2×D9—22no—
346C2×D9—22no—
347C2×D9—22no—
348C2×D9—34no—
349C2×D9—64no—
350C2×D9—62no—
351C2×D9—62no—
352C2×D9—122no—
353C2×D9—122no—
354C2×D9—128no—
355C4×C9—22no—
356C4×C9—21no—
357C4×C9—22no—
358C4×C9—61no—
359C4×C9—61no—
360C4×C9—61no—
361C4×C9D3×D412yes—
362Dic9—22no—
363Dic9—64no—
364Dic9—62no—
365C2×#41—271no—
366C22×D4—271no—
367#124—271no—
368#123—272no—
369C22⋊Q8—271no—
370#121—91no—
371#121—272no—
372#120—272no—
373C4×D4—31no—
374C4×D4—271no—
375C2×C22⋊C4—272no—
376C3×C9#54611yescharacteristic, complemented, Sylow
377C23×C3—21no—
378C23×C3—31no—
379C23×C3—92no—
380C23×C3C22×D911yescharacteristic
381C23×C3C2×D3211yescharacteristic
382C22×D3—32no—
383C22×D3—92no—
384C22×D3—92no—
385C22×D3—182no—
386C22×D3—184no—
387C22×D3—184no—
388C22×D3—184no—
389C22×D3—184no—
390C22×D3—362no—
391C22×D3—362no—
392C22×D3C22×D911yescharacteristic
393C3×D4—62no—
394C3×D4—182no—
395C3×D4—362no—
396C2×C3×C4—24no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—31no—
402C2×C3×C4—31no—
403C2×C3×C4—31no—
404C2×C3×C4—62no—
405C2×C3×C4—92no—
406C2×C3×C4C22×D911yescharacteristic
407C2×C3×C4C22×D911yescharacteristic
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410#67—22no—
411#67—122no—
412#67—184no—
413#67—184no—
414#67—184no—
415#67—188no—
416C2×Dic3—32no—
417C2×Dic3—92no—
418C2×Dic3—92no—
419C2×Dic3—182no—
420C2×Dic3C22×D911yescharacteristic
421C2×Dic3C22×D911yescharacteristic
422C2×Dic3C22×D911yescharacteristic
423D12—62no—
424D12—94no—
425D12—94no—
426D12—184no—
427D12—184no—
428C4×D3—22no—
429C4×D3—184no—
430C4×D3—184no—
431C4×D3—188no—
432C4×D3—362no—
433C4×D3—362no—
434Dic6—184no—
435Dic6—182no—
436Dic6—182no—
437C2×C32—41no—
438C2×C32C2×#23511yescharacteristic
439C2×C32C2×D3×D411yescharacteristic
440C2×C32#100911yescharacteristic
441C3⋊D3—64no—
442C3×D3—22no—
443C3×D3—122no—
444C2×C9—21no—
445C2×C9—21no—
446C2×C9—21no—
447C2×C9—42no—
448C2×C9—41no—
449C2×C9—62no—
450C2×C9C2×D3×D411yescharacteristic
451C2×C9#100411yescharacteristic
452C2×C9D3×#4111yescharacteristic
453D9—42no—
454D9—64no—
455D9—124no—
456C24—272no—
457#41—544no—
458C2×Q8—271no—
459C2×D4—31no—
460C2×D4—92no—
461C2×D4—91no—
462C2×D4—272no—
463C2×D4—272no—
464C2×D4—271no—
465C2×D4—544no—
466C22×C4—31no—
467C22×C4—272no—
468C22×C4—272no—
469C22×C4D3×D911yescharacteristic, complemented
470C4⋊C4—31no—
471C4⋊C4—91no—
472C4⋊C4—272no—
473C22⋊C4—31no—
474C22⋊C4—31no—
475C22⋊C4—92no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C42—31no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—32no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—61no—
493C22×C3—61no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3C23×D911yescharacteristic
498C22×C3C22×D3211yescharacteristic
499C2×D3—21no—
500C2×D3—21no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—94no—
505C2×D3—94no—
506C2×D3—184no—
507C2×D3—182no—
508C2×D3—182no—
509C2×D3—182no—
510C2×D3—182no—
511C2×D3—362no—
512C2×D3—362no—
513C2×D3—368no—
514C2×D3—368no—
515C2×D3#369712yescomplemented
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—22no—
520C3×C4—61no—
521C3×C4—61no—
522C3×C4—61no—
523C3×C4—182no—
524C3×C4D4×D912yes—
525C3×C4#381112yes—
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—62no—
530Dic3—184no—
531Dic3—182no—
532Dic3—182no—
533C32#555011yescharacteristic
534C9—21no—
535C9#561211yescharacteristic, complemented
536C23—31no—
537C23—92no—
538C23—272no—
539C23—544no—
540C23—544no—
541C23—1082no—
542C23C2×D3×D911yescharacteristic
543Q8—542no—
544D4—62no—
545D4—182no—
546D4—274no—
547D4—362no—
548D4—544no—
549D4—544no—
550C2×C4—22no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—62no—
555C2×C4—92no—
556C2×C4—272no—
557C2×C4—544no—
558C2×C4—1082no—
559C2×C4C2×D3×D911yescharacteristic
560C2×C4C2×D3×D911yescharacteristic
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—42no—
565C2×C3—41no—
566C2×C3—41no—
567C2×C3—62no—
568C2×C3—362no—
569C2×C3C2×#369711yescharacteristic
570C2×C3C2×D4×D911yescharacteristic
571C2×C3#6371511yescharacteristic
572C2×C3D3×#23511yescharacteristic
573C2×C3#6430711yescharacteristic
574C2×C3C2×#381111yescharacteristic
575D3—22no—
576D3—122no—
577D3—184no—
578D3—184no—
579D3—364no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—32no—
584C22—61no—
585C22—61no—
586C22—182no—
587C22—182no—
588C22—182no—
589C22—274no—
590C22—542no—
591C22—542no—
592C22—1088no—
593C22C22×D3×D911yescharacteristic
594C4—21no—
595C4—61no—
596C4—61no—
597C4—61no—
598C4—182no—
599C4—542no—
600C4#8866812yes—
601C3—21no—
602C3#1058965211yescharacteristic, complemented
603C3#1059507811yescharacteristic
604C2—41no—
605C2—62no—
606C2—362no—
607C2—544no—
608C2D3×#369711yescharacteristic
609C2#1172247711yescharacteristic
610C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#546
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C2×D3×D36C2semidirect
3. C4×C9×#67C2semidirect
4. C3×#63492C2semidirect
5. #11721292C2semidirect
6. #11721256C2semidirect
7. #11721048C2semidirect
8. #11720993C2semidirect
9. #11720961C2semidirect
10. #11720948C2semidirect
11. #11720941C2semidirect
15. C22×C3×C4×C9C22semidirect
16. C2×#6233C22semidirect
19. C2×C4×C3⋊D9C22semidirect
20. C2×#6226C22semidirect
22. C2×C3×#611C22semidirect
26. C2×(C3⋊D9)⋊C4C22semidirect
28. C9×#217C22semidirect
30. C3×(C2×D9)⋊C4C22semidirect
31. C9×Dic3⋊C4C22semidirect
32. C4×C9×Dic3C22semidirect
33. C3×C4⋊Dic9C22semidirect
35. #88445C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. Dic9⋊Dic3C22semidirect
55. #63492D3semidirect
74. C4⋊(C3⋊D9)D4semidirect
118. C22×C4×C9C2×D3semidirect
121. C2×#611C2×D3semidirect
142. (C2×D9)⋊C4C2×D3semidirect
143. C4⋊Dic9C2×D3semidirect
154. C2×D3×C9#41semidirect
178. C4×#67D9semidirect
264. C22×C3×C4C2×D9semidirect
300. #217C2×D9semidirect
311. Dic3⋊C4C2×D9semidirect
312. C4×Dic3C2×D9semidirect
376. C3×C9#546semidirect
469. C22×C4D3×D9semidirect
515. C2×D3#3697semidirect
535. C9#5612semidirect
602. C3#10589652semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×D3×D362
C4×C9×#672
C3×#634922
#117212922
#117212562
#117210482
#117209932
#117209612
#117209482
#117209412
#105950783
#105896523