Finite Group Databasebeta

← #13148792

Subgroups — #13148792

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721752C211yesmaximal, characteristic
2C9×#889C211yesmaximal, characteristic, complemented
3C3×#63452C211yesmaximal, characteristic, complemented
4#11720934C211yesmaximal, characteristic
5#11720926C211yesmaximal, characteristic
6#11720923C211yesmaximal, characteristic
7#11720922C211yesmaximal, characteristic
8#11720921C211yesmaximal, characteristic
9#11720921C211yesmaximal, characteristic
10#11720919C211yesmaximal, characteristic, complemented
11#11720919C211yesmaximal, characteristic, complemented
12#11720917C211yesmaximal, characteristic, complemented
13#11720912C211yesmaximal, characteristic, complemented
14#11720905C211yesmaximal, characteristic, complemented
15#11720901C211yesmaximal, characteristic, complemented
16#10594323—31nomaximal
17#10589798—31nomaximal
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×#6189C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22#88536C2211yescharacteristic
23#88536C2211yescharacteristic
24#88534C2211yescharacteristic
25#88534C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2211yescharacteristic
28C9×Dic3⋊C4C2211yescharacteristic
29C9×Dic3⋊C4C2211yescharacteristic
30C9×Dic3⋊C4C2211yescharacteristic
31C4×C9×Dic3C2211yescharacteristic
32C3×C4⋊Dic9C2211yescharacteristic
33C3×C4⋊Dic9C2211yescharacteristic
34C3×C4⋊Dic9C2211yescharacteristic
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2211yescharacteristic
37C3×C4×Dic9C2211yescharacteristic
38#88445C2211yescharacteristic
39#88445C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43#88442C2211yescharacteristic
44#88442C2211yescharacteristic
45#88442C2211yescharacteristic
46Dic9⋊Dic3C2211yescharacteristic
47Dic9⋊Dic3C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic3×Dic9C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64105—31no—
54C3×#890—31no—
55C3×#889D311yescharacteristic
56#63868—31no—
57#63868—31no—
58#63865—31no—
59#63849—31no—
60#63848—31no—
61#63848—31no—
62#63847—31no—
63#63847—31no—
64#63846—31no—
65#63844—31no—
66#63840—31no—
67#63837—31no—
68C9×#125—31no—
69#63460—31no—
70#63460—31no—
71#63460—31no—
72#63458—31no—
73#63458—31no—
74#63458—61no—
75#63454—31no—
76#63452D311yescharacteristic, complemented
77#63451—31no—
78#63451—31no—
79#63451—31no—
80#63450—31no—
81#63449—31no—
82#63438—31no—
83C4×Dic18—31no—
84C22×C3⋊D9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C4×C3⋊D9—22no—
92#6226—22no—
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101#6189—22no—
102#5696—91no—
103#5696—91no—
104C2×C4×C3⋊D3—31no—
105C2×#634—31no—
106C2×#627—31no—
107C32×C4⋊C4C2×D311yescharacteristic
108#3752—31no—
109#3752—31no—
110#3750—31no—
111#3750—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3C2×D311yescharacteristic
116C3×Dic3⋊C4—31no—
117C3×Dic3⋊C4—31no—
118C3×Dic3⋊C4C2×D311yescharacteristic
119C3×Dic3⋊C4C2×D311yescharacteristic
120C3×Dic3⋊C4C2×D311yescharacteristic
121C3×Dic3⋊C4C2×D311yescharacteristic
122C3×C4×Dic3—31no—
123C3×C4×Dic3C2×D311yescharacteristic
124#3724—31no—
125#3724—31no—
126#3724—31no—
127Dic3⋊Dic3—31no—
128Dic3⋊Dic3—31no—
129Dic3⋊Dic3—31no—
130Dic3⋊Dic3—31no—
131#3722—31no—
132#3722—31no—
133#3722—31no—
134#3722—31no—
135Dic32—31no—
136Dic32—31no—
137Dic32—31no—
138Dic32—31no—
139C2×C4×D9—31no—
140C2×C4×D9—61no—
141C2×Dic18—31no—
142C2×Dic18—31no—
143C2×Dic18—61no—
144C9×C4⋊C4—21no—
145C9×C4⋊C4—31no—
146C9×C4⋊C4—31no—
147C9×C4⋊C4—31no—
148C9×C4⋊C4—31no—
149C9×C4⋊C4—31no—
150C9×C4⋊C4C2×D311yescharacteristic
151C42×C9—31no—
152(C2×D9)⋊C4—31no—
153(C2×D9)⋊C4—31no—
154(C2×D9)⋊C4—31no—
155(C2×D9)⋊C4—31no—
156(C2×D9)⋊C4—31no—
157(C2×D9)⋊C4—31no—
158(C2×D9)⋊C4—61no—
159(C2×D9)⋊C4—61no—
160C4⋊Dic9—31no—
161C4⋊Dic9C2×D311yescharacteristic
162C4⋊Dic9C2×D311yescharacteristic
163C4⋊Dic9C2×D311yescharacteristic
164Dic9⋊C4—31no—
165Dic9⋊C4—31no—
166Dic9⋊C4—31no—
167Dic9⋊C4—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—61no—
173Dic9⋊C4—61no—
174Dic9⋊C4C2×D311yescharacteristic
175Dic9⋊C4C2×D311yescharacteristic
176C4×Dic9—31no—
177C4×Dic9—31no—
178C4×Dic9—31no—
179C4×Dic9—31no—
180C4×Dic9C2×D311yescharacteristic
181C22×C3×C9C2411yescharacteristic
182C2×C3⋊D9—21no—
183C2×C3⋊D9—21no—
184C2×C3⋊D9—21no—
185C3×C4×C9—21no—
186C3×C4×C9—21no—
187C3×C4×C9—21no—
188C3⋊Dic9—21no—
189C3⋊Dic9—21no—
190C3⋊Dic9#4112yes—
191C9×Dic3—21no—
192C9×Dic3—21no—
193C9×Dic3—21no—
194C9×Dic3—21no—
195C3×Dic9—21no—
196C3×Dic9—21no—
197C3×Dic9—21no—
198C3×Dic9—21no—
199C3×#125—31no—
200C3×#125—91no—
201#898—91no—
202#898—91no—
203#898—91no—
204#898—91no—
205#898—91no—
206#898—91no—
207#896—91no—
208#896—91no—
209#896—91no—
210#896—91no—
211#896—181no—
212#892—91no—
213#892—91no—
214#890—31no—
215#890—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#889—91no—
220#889—91no—
221#889D911yescharacteristic, complemented
222#888—91no—
223#888—91no—
224#887—91no—
225#887—91no—
226#876—91no—
227#876—91no—
228C4×Dic6—91no—
229C4×Dic6—91no—
230C22×C3⋊D3—31no—
231C2×C32×C4C22×D311yescharacteristic
232C2×C32×C4C22×D311yescharacteristic
233C2×C32×C4C22×D311yescharacteristic
234C2×C3⋊Dic3—31no—
235C2×C3⋊Dic3—31no—
236C2×C3⋊Dic3—31no—
237C4×C3⋊D3—62no—
238#634—62no—
239C2×C3×Dic3—31no—
240C2×C3×Dic3—31no—
241C2×C3×Dic3—31no—
242C2×C3×Dic3—31no—
243C2×C3×Dic3C22×D311yescharacteristic
244C2×C3×Dic3C22×D311yescharacteristic
245C2×C3×Dic3C22×D311yescharacteristic
246C2×C3×Dic3C22×D311yescharacteristic
247#627—62no—
248C22×D9—31no—
249C22×D9—61no—
250C2×C4×C9—21no—
251C2×C4×C9—21no—
252C2×C4×C9—21no—
253C2×C4×C9—31no—
254C2×C4×C9—31no—
255C2×C4×C9—31no—
256C2×C4×C9—31no—
257C2×C4×C9C22×D311yescharacteristic
258C2×C4×C9C22×D311yescharacteristic
259C2×C4×C9C22×D311yescharacteristic
260C2×Dic9—31no—
261C2×Dic9—31no—
262C2×Dic9—31no—
263C2×Dic9—61no—
264C2×Dic9—61no—
265C2×Dic9—61no—
266C2×Dic9C22×D311yescharacteristic
267C2×Dic9C22×D311yescharacteristic
268C2×Dic9C22×D311yescharacteristic
269C2×Dic9C22×D311yescharacteristic
270C4×D9—62no—
271C4×D9—122no—
272Dic18—62no—
273Dic18—62no—
274Dic18—122no—
275#541—271noSylow
276C2×C3×C9C2×#4111yescharacteristic
277C2×C3×C92−1+411yescharacteristic
278C2×C3×C92−1+411yescharacteristic
279C3⋊D9—41no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×C4×D3—181no—
283C2×Dic6—91no—
284C2×Dic6—91no—
285C2×Dic6—91no—
286C2×Dic6—91no—
287C2×Dic6—181no—
288C3×C4⋊C4—21no—
289C3×C4⋊C4—31no—
290C3×C4⋊C4—31no—
291C3×C4⋊C4—31no—
292C3×C4⋊C4—31no—
293C3×C4⋊C4—31no—
294C3×C4⋊C4—91no—
295C3×C4⋊C4—91no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—91no—
298C3×C4⋊C4—91no—
299C3×C4⋊C4C2×D911yescharacteristic
300C3×C4⋊C4D3211yescharacteristic
301C3×C42—31no—
302C3×C42—91no—
303(C2×D3)⋊C4—91no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—91no—
306(C2×D3)⋊C4—91no—
307(C2×D3)⋊C4—91no—
308(C2×D3)⋊C4—91no—
309(C2×D3)⋊C4—91no—
310(C2×D3)⋊C4—91no—
311(C2×D3)⋊C4—91no—
312(C2×D3)⋊C4—91no—
313(C2×D3)⋊C4—91no—
314(C2×D3)⋊C4—91no—
315(C2×D3)⋊C4—181no—
316(C2×D3)⋊C4—181no—
317C4⋊Dic3—31no—
318C4⋊Dic3—31no—
319C4⋊Dic3—31no—
320C4⋊Dic3—91no—
321C4⋊Dic3—91no—
322C4⋊Dic3—91no—
323C4⋊Dic3—91no—
324C4⋊Dic3C2×D911yescharacteristic
325Dic3⋊C4—31no—
326Dic3⋊C4—31no—
327Dic3⋊C4—91no—
328Dic3⋊C4—91no—
329Dic3⋊C4—91no—
330Dic3⋊C4—91no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340Dic3⋊C4—91no—
341Dic3⋊C4—181no—
342Dic3⋊C4—181no—
343Dic3⋊C4C2×D911yescharacteristic
344Dic3⋊C4C2×D911yescharacteristic
345Dic3⋊C4C2×D911yescharacteristic
346Dic3⋊C4C2×D911yescharacteristic
347C4×Dic3—31no—
348C4×Dic3—91no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—91no—
353C4×Dic3—91no—
354C4×Dic3—91no—
355C4×Dic3—91no—
356C4×Dic3C2×D911yescharacteristic
357C22×C32C23×D311yescharacteristic
358C2×C3⋊D3—61no—
359C2×C3⋊D3—61no—
360C2×C3⋊D3—61no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C3⋊Dic3—32no—
365C3⋊Dic3—61no—
366C3⋊Dic3—61no—
367C3×Dic3—21no—
368C3×Dic3—21no—
369C3×Dic3—21no—
370C3×Dic3—21no—
371C3×Dic3—61no—
372C3×Dic3—61no—
373C3×Dic3—61no—
374C3×Dic3—61no—
375C22×C9—21no—
376C22×C9C23×D311yescharacteristic
377C2×D9—62no—
378C2×D9—61no—
379C2×D9—61no—
380C2×D9—61no—
381C2×D9—121no—
382C2×D9—121no—
383C4×C9—21no—
384C4×C9—21no—
385C4×C9—21no—
386C4×C9—41no—
387C4×C9—41no—
388C4×C9—41no—
389C4×C9—61no—
390C4×C9—61no—
391C4×C9—61no—
392C4×C9—61no—
393Dic9—21no—
394Dic9—21no—
395Dic9—21no—
396Dic9—21no—
397Dic9—32no—
398Dic9—62no—
399Dic9—61no—
400Dic9—61no—
401Dic9—121no—
402Dic9—121no—
403C4⋊Q8—271no—
404#126—271no—
405#126—271no—
406#126—271no—
407#126—271no—
408#125—31no—
409#125—91no—
410#125—271no—
411#125—271no—
412#125—271no—
413C22⋊Q8—271no—
414C22⋊Q8—271no—
415C4×Q8—271no—
416C4×Q8—271no—
417#117—271no—
418C3×C9#54111yescharacteristic, complemented, Sylow
419C22×D3—91no—
420C22×D3—91no—
421C22×D3—181no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—31no—
426C2×C3×C4—31no—
427C2×C3×C4—31no—
428C2×C3×C4—31no—
429C2×C3×C4—91no—
430C2×C3×C4—91no—
431C2×C3×C4—91no—
432C2×C3×C4—91no—
433C2×C3×C4C22×D911yescharacteristic
434C2×C3×C4C22×D911yescharacteristic
435C2×C3×C4C22×D911yescharacteristic
436C2×C3×C4C2×D3211yescharacteristic
437C2×C3×C4C2×D3211yescharacteristic
438C2×C3×C4C2×D3211yescharacteristic
439C2×Dic3—31no—
440C2×Dic3—31no—
441C2×Dic3—31no—
442C2×Dic3—31no—
443C2×Dic3—91no—
444C2×Dic3—91no—
445C2×Dic3—91no—
446C2×Dic3—91no—
447C2×Dic3—91no—
448C2×Dic3—91no—
449C2×Dic3—181no—
450C2×Dic3—181no—
451C2×Dic3—181no—
452C2×Dic3C22×D911yescharacteristic
453C2×Dic3C22×D911yescharacteristic
454C2×Dic3C22×D911yescharacteristic
455C2×Dic3C22×D911yescharacteristic
456C4×D3—182no—
457C4×D3—182no—
458C4×D3—362no—
459Dic6—182no—
460Dic6—182no—
461Dic6—182no—
462Dic6—182no—
463Dic6—362no—
464C2×C32#100711yescharacteristic
465C2×C32D3×#4111yescharacteristic
466C2×C32#101011yescharacteristic
467C3⋊D3—121no—
468C2×C9—21no—
469C2×C9—21no—
470C2×C9—21no—
471C2×C9#100711yescharacteristic
472C2×C9D3×#4111yescharacteristic
473C2×C9#101011yescharacteristic
474D9—122no—
475D9—121no—
476C2×Q8—271no—
477C2×Q8—271no—
478C22×C4—271no—
479C4⋊C4—31no—
480C4⋊C4—31no—
481C4⋊C4—31no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—91no—
485C4⋊C4—91no—
486C4⋊C4—91no—
487C4⋊C4—91no—
488C4⋊C4—91no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4—271no—
492C4⋊C4—271no—
493C4⋊C4—271no—
494C4⋊C4—271no—
495C4⋊C4—271no—
496C4⋊C4—271no—
497C4⋊C4—271no—
498C4⋊C4D3×D911yescharacteristic
499C22⋊C4—271no—
500C22⋊C4—271no—
501C22⋊C4—271no—
502C22⋊C4—271no—
503C22⋊C4—271no—
504C22⋊C4—271no—
505C42—31no—
506C42—91no—
507C42—271no—
508C42—271no—
509C42—271no—
510C42—271no—
511C22×C3—21no—
512C22×C3C23×D911yescharacteristic
513C22×C3C22×D3211yescharacteristic
514C2×D3—182no—
515C2×D3—181no—
516C2×D3—181no—
517C2×D3—181no—
518C2×D3—181no—
519C2×D3—181no—
520C2×D3—181no—
521C2×D3—361no—
522C2×D3—361no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—61no—
533C3×C4—61no—
534C3×C4—61no—
535C3×C4—61no—
536C3×C4—181no—
537C3×C4—181no—
538C3×C4—181no—
539C3×C4—181no—
540Dic3—21no—
541Dic3—21no—
542Dic3—21no—
543Dic3—21no—
544Dic3—61no—
545Dic3—61no—
546Dic3—61no—
547Dic3—61no—
548Dic3—92no—
549Dic3—92no—
550Dic3—182no—
551Dic3—181no—
552Dic3—181no—
553Dic3—181no—
554Dic3—181no—
555Dic3—361no—
556Dic3—361no—
557C32#569611yescharacteristic
558C9—21no—
559C9#569611yescharacteristic, complemented
560C23—271no—
561Q8—542no—
562Q8—542no—
563C2×C4—31no—
564C2×C4—31no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—91no—
568C2×C4—91no—
569C2×C4—91no—
570C2×C4—91no—
571C2×C4—271no—
572C2×C4—271no—
573C2×C4—271no—
574C2×C4—542no—
575C2×C4C2×D3×D911yescharacteristic
576C2×C4C2×D3×D911yescharacteristic
577C2×C4C2×D3×D911yescharacteristic
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3#6371311yescharacteristic
582C2×C3#41×D911yescharacteristic
583C2×C3#6371611yescharacteristic
584C2×C3#6429711yescharacteristic
585C2×C3#6431211yescharacteristic
586C2×C3#6431511yescharacteristic
587D3—362no—
588D3—361no—
589D3—361no—
590C22—541no—
591C22—541no—
592C22—541no—
593C22C22×D3×D911yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—61no—
598C4—61no—
599C4—61no—
600C4—61no—
601C4—181no—
602C4—181no—
603C4—181no—
604C4—181no—
605C4—272no—
606C4—541no—
607C4—541no—
608C3—21no—
609C3#1058979811yescharacteristic, complemented
610C3#1059432311yescharacteristic
611C2—1081no—
612C2#1172246411yescharacteristic
613C2#1172249811yescharacteristic
614C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#889C2semidirect
3. C3×#63452C2semidirect
10. #11720919C2semidirect
11. #11720919C2semidirect
12. #11720917C2semidirect
13. #11720912C2semidirect
14. #11720905C2semidirect
15. #11720901C2semidirect
76. #63452D3semidirect
221. #889D9semidirect
418. C3×C9#541semidirect
559. C9#5696semidirect
609. C3#10589798semidirect

Maximal subgroups

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

subgroupindex
#117217522
C9×#8892
C3×#634522
#117209342
#117209262
#117209232
#117209222
#117209212
#117209192
#117209172
#117209122
#117209052
#117209012
#105943233
#105897983