Finite Group Databasebeta

← #13148772

Subgroups — #13148772

Order 1728.

632 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721745C211yesmaximal, characteristic, complemented
2C9×#892C211yesmaximal, characteristic
3C3×#63451C211yesmaximal, characteristic, complemented
4#11720970C211yesmaximal, characteristic, complemented
5C4×#6189C211yesmaximal, characteristic, complemented
6#11720966C211yesmaximal, characteristic
7#11720965C211yesmaximal, characteristic
8#11720963C211yesmaximal, characteristic
9#11720962C211yesmaximal, characteristic
10#11720955C211yesmaximal, characteristic
11#11720955C211yesmaximal, characteristic
12#11720925C211yesmaximal, characteristic, complemented
13#11720918C211yesmaximal, characteristic, complemented
14#11720917C211yesmaximal, characteristic, complemented
15#11720900C211yesmaximal, characteristic, complemented
16#10594598—31nomaximal
17#10589620—31nomaximal
18C2×C4×D3×C9C2211yescharacteristic
19C2×#6189C2211yescharacteristic
20C2×#6189C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2211yescharacteristic
24#88534C2211yescharacteristic
25#88534C2211yescharacteristic
26#88534C2211yescharacteristic
27C4×C3⋊Dic9C2211yescharacteristic
28C9×(C2×D3)⋊C4C2211yescharacteristic
29C9×(C2×D3)⋊C4C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31C4×C9×Dic3C2211yescharacteristic
32C4×C9×Dic3C2211yescharacteristic
33C3×C4⋊Dic9C2211yescharacteristic
34C3×Dic9⋊C4C2211yescharacteristic
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×C4×Dic9C2211yescharacteristic
39(C2×D3)⋊Dic9C2211yescharacteristic
40(C2×D3)⋊Dic9C2211yescharacteristic
41(C2×D3)⋊Dic9C2211yescharacteristic
42(C2×D3)⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64098—31no—
54C3×#892D311yescharacteristic
55C3×#889—31no—
56#63918—31no—
57C4×#627—31no—
58#63867—31no—
59#63862—31no—
60#63861—31no—
61#63859—31no—
62#63858—31no—
63#63851—31no—
64#63851—31no—
65#63847—31no—
66#63846—31no—
67#63837—31no—
68C9×#117—31no—
69#63486—31no—
70#63486—31no—
71#63451—31no—
72#63451—31no—
73#63451—31no—
74#63451—61no—
75#63451D311yescharacteristic, complemented
76#63441—31no—
77#63441—31no—
78#63441—31no—
79#63441—31no—
80#63432—31no—
81#63431—31no—
82C4×Dic18—31no—
83C4×Dic18—31no—
84C22×D3×C9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C9×Dic3C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C3×Dic9C2311yescharacteristic
96C2×C3×Dic9C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C4×D3×C9—22no—
100#6189—22no—
101#6189—22no—
102#5696—91no—
103#5518—91no—
104C2×C3×C4×D3C2×D311yescharacteristic
105C2×#627—31no—
106C2×#627—31no—
107C32×C4⋊C4C2×D311yescharacteristic
108C4⋊(C3⋊Dic3)—31no—
109#3750—31no—
110#3750—31no—
111#3750—31no—
112#3750—31no—
113C4×C3⋊Dic3—31no—
114C3×(C2×D3)⋊C4C2×D311yescharacteristic
115C3×(C2×D3)⋊C4C2×D311yescharacteristic
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3C2×D311yescharacteristic
118C3×Dic3⋊C4—31no—
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—31no—
121C3×Dic3⋊C4—31no—
122C3×C4×Dic3—31no—
123C3×C4×Dic3C2×D311yescharacteristic
124C3×C4×Dic3C2×D311yescharacteristic
125#3724—31no—
126#3724—31no—
127Dic3⋊Dic3—31no—
128Dic3⋊Dic3—31no—
129Dic3⋊Dic3—31no—
130Dic3⋊Dic3—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133(C2×D3)⋊Dic3—31no—
134(C2×D3)⋊Dic3—31no—
135(C2×D3)⋊Dic3—31no—
136(C2×D3)⋊Dic3—31no—
137Dic32—31no—
138Dic32—31no—
139C22×C4×C9—31no—
140C2×Dic18—31no—
141C2×Dic18—31no—
142C9×C4⋊C4—21no—
143C9×C4⋊C4—31no—
144C9×C4⋊C4C2×D311yescharacteristic
145C9×C22⋊C4—31no—
146C9×C22⋊C4—31no—
147C42×C9—31no—
148C42×C9—31no—
149#3676—31no—
150#3676—31no—
151#3676—31no—
152#3676—31no—
153C4⋊Dic9—31no—
154C4⋊Dic9—31no—
155C4⋊Dic9—31no—
156C4⋊Dic9—31no—
157C4⋊Dic9—31no—
158C4⋊Dic9—61no—
159C4⋊Dic9C2×D311yescharacteristic
160Dic9⋊C4—31no—
161Dic9⋊C4—31no—
162Dic9⋊C4—31no—
163Dic9⋊C4—31no—
164Dic9⋊C4—31no—
165Dic9⋊C4—31no—
166Dic9⋊C4—31no—
167Dic9⋊C4—31no—
168Dic9⋊C4—61no—
169Dic9⋊C4—61no—
170Dic9⋊C4—61no—
171Dic9⋊C4—61no—
172Dic9⋊C4C2×D311yescharacteristic
173Dic9⋊C4C2×D311yescharacteristic
174Dic9⋊C4C2×D311yescharacteristic
175Dic9⋊C4C2×D311yescharacteristic
176C4×Dic9—31no—
177C4×Dic9—31no—
178C4×Dic9—31no—
179C4×Dic9—61no—
180C4×Dic9C2×D311yescharacteristic
181C22×C3×C9C2411yescharacteristic
182C2×D3×C9—21no—
183C2×D3×C9—21no—
184C2×D3×C9—21no—
185C3×C4×C9—21no—
186C3×C4×C9—21no—
187C3×C4×C9—21no—
188C3⋊Dic9—21no—
189C3⋊Dic9—21no—
190C3⋊Dic9—21no—
191C3⋊Dic9—21no—
192C9×Dic3—21no—
193C9×Dic3—21no—
194C9×Dic3#4112yes—
195C3×Dic9—21no—
196C3×Dic9—21no—
197C3×Dic9—21no—
198C3×Dic9—21no—
199C3×#125—91no—
200C3×#117—31no—
201#924—91no—
202#924—91no—
203#898—91no—
204#898—91no—
205#898—91no—
206#896—91no—
207#896—91no—
208#892D911yescharacteristic
209#890—91no—
210#889—31no—
211#889—91no—
212#889—91no—
213#889—91no—
214#889—91no—
215#889—91no—
216#889—91no—
217#889—181no—
218#888—91no—
219#887—91no—
220#879—91no—
221#879—91no—
222#879—91no—
223#879—91no—
224#876—91no—
225#870—91no—
226#869—91no—
227C4×Dic6—91no—
228C4×Dic6—91no—
229C4×Dic6—91no—
230C22×C3×D3C22×D311yescharacteristic
231C2×C32×C4C22×D311yescharacteristic
232C2×C32×C4C22×D311yescharacteristic
233C2×C32×C4C22×D311yescharacteristic
234C2×C3⋊Dic3—31no—
235C2×C3⋊Dic3—31no—
236C2×C3⋊Dic3—31no—
237C2×C3⋊Dic3—31no—
238C2×C3×Dic3—31no—
239C2×C3×Dic3—31no—
240C2×C3×Dic3—31no—
241C2×C3×Dic3—31no—
242C2×C3×Dic3C22×D311yescharacteristic
243C2×C3×Dic3C22×D311yescharacteristic
244C2×C3×Dic3C22×D311yescharacteristic
245C3×C4×D3—22no—
246#627—62no—
247#627—62no—
248C23×C9—31no—
249C2×C4×C9—21no—
250C2×C4×C9—21no—
251C2×C4×C9—21no—
252C2×C4×C9—31no—
253C2×C4×C9—31no—
254C2×C4×C9—31no—
255C2×C4×C9—62no—
256C2×C4×C9C22×D311yescharacteristic
257C2×C4×C9C22×D311yescharacteristic
258C2×C4×C9C22×D311yescharacteristic
259C2×Dic9—31no—
260C2×Dic9—31no—
261C2×Dic9—31no—
262C2×Dic9—31no—
263C2×Dic9—61no—
264C2×Dic9—61no—
265C2×Dic9—61no—
266C2×Dic9—61no—
267C2×Dic9C22×D311yescharacteristic
268C2×Dic9C22×D311yescharacteristic
269C2×Dic9C22×D311yescharacteristic
270C2×Dic9C22×D311yescharacteristic
271Dic18—62no—
272Dic18—62no—
273#541—271noSylow
274C2×C3×C9C2×#4111yescharacteristic
275C2×C3×C92−1+411yescharacteristic
276C2×C3×C92−1+411yescharacteristic
277D3×C9—41no—
278C22×C3×C4—31no—
279C2×C4×D3C2×D911yescharacteristic
280C2×Dic6—91no—
281C2×Dic6—91no—
282C2×Dic6—91no—
283C2×Dic6—91no—
284C3×C4⋊C4—21no—
285C3×C4⋊C4—31no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—91no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4C2×D911yescharacteristic
292C3×C4⋊C4D3211yescharacteristic
293C3×C22⋊C4—31no—
294C3×C22⋊C4—31no—
295C3×C42—31no—
296C3×C42—31no—
297C3×C42—91no—
298#217—91no—
299#217—91no—
300#217—91no—
301#217—91no—
302(C2×D3)⋊C4—91no—
303(C2×D3)⋊C4—91no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—91no—
306(C2×D3)⋊C4C2×D911yescharacteristic
307(C2×D3)⋊C4C2×D911yescharacteristic
308C4⋊Dic3—31no—
309C4⋊Dic3—91no—
310C4⋊Dic3—91no—
311C4⋊Dic3—91no—
312C4⋊Dic3—91no—
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—181no—
318C4⋊Dic3C2×D911yescharacteristic
319Dic3⋊C4—31no—
320Dic3⋊C4—31no—
321Dic3⋊C4—31no—
322Dic3⋊C4—31no—
323Dic3⋊C4—91no—
324Dic3⋊C4—91no—
325Dic3⋊C4—91no—
326Dic3⋊C4—91no—
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⋊C4—181no—
344Dic3⋊C4—181no—
345C4×Dic3—31no—
346C4×Dic3—91no—
347C4×Dic3—91no—
348C4×Dic3—91no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—181no—
353C4×Dic3C2×D911yescharacteristic
354C4×Dic3C2×D911yescharacteristic
355C22×C32C23×D311yescharacteristic
356C2×C3×D3—21no—
357C2×C3×D3—21no—
358C2×C3×D3—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C3⋊Dic3—61no—
363C3⋊Dic3—61no—
364C3⋊Dic3—61no—
365C3⋊Dic3—61no—
366C3×Dic3—21no—
367C3×Dic3—21no—
368C3×Dic3—61no—
369C3×Dic3—61no—
370C3×Dic3—61no—
371C3×Dic3—61no—
372C3×Dic3#23512yes—
373C22×C9—21no—
374C22×C9—61no—
375C22×C9—61no—
376C22×C9—61no—
377C22×C9C23×D311yescharacteristic
378C4×C9—21no—
379C4×C9—21no—
380C4×C9—21no—
381C4×C9—32no—
382C4×C9—41no—
383C4×C9—41no—
384C4×C9—41no—
385C4×C9—61no—
386C4×C9—61no—
387Dic9—21no—
388Dic9—21no—
389Dic9—21no—
390Dic9—21no—
391Dic9—62no—
392Dic9—61no—
393Dic9—61no—
394Dic9—61no—
395Dic9—61no—
396Dic9—121no—
397Dic9—121no—
398Dic9—121no—
399C4⋊Q8—271no—
400#126—271no—
401#126—271no—
402#126—271no—
403#126—271no—
404#125—91no—
405#125—271no—
406#125—271no—
407#125—271no—
408#125—271no—
409C22⋊Q8—271no—
410C22⋊Q8—271no—
411C4×Q8—271no—
412C4×Q8—271no—
413#117—31no—
414C3×C9#54111yescharacteristic, complemented, Sylow
415C23×C3—31no—
416C22×D3C22×D911yescharacteristic
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—31no—
421C2×C3×C4—31no—
422C2×C3×C4—31no—
423C2×C3×C4—62no—
424C2×C3×C4—91no—
425C2×C3×C4—91no—
426C2×C3×C4—91no—
427C2×C3×C4—91no—
428C2×C3×C4C22×D911yescharacteristic
429C2×C3×C4C22×D911yescharacteristic
430C2×C3×C4C22×D911yescharacteristic
431C2×C3×C4C2×D3211yescharacteristic
432C2×C3×C4C2×D3211yescharacteristic
433C2×C3×C4C2×D3211yescharacteristic
434C2×Dic3—31no—
435C2×Dic3—31no—
436C2×Dic3—31no—
437C2×Dic3—31no—
438C2×Dic3—91no—
439C2×Dic3—91no—
440C2×Dic3—91no—
441C2×Dic3—91no—
442C2×Dic3—91no—
443C2×Dic3—91no—
444C2×Dic3—91no—
445C2×Dic3—91no—
446C2×Dic3—181no—
447C2×Dic3—181no—
448C2×Dic3—181no—
449C2×Dic3—181no—
450C2×Dic3C22×D911yescharacteristic
451C2×Dic3C22×D911yescharacteristic
452C2×Dic3C22×D911yescharacteristic
453C4×D3—22no—
454Dic6—182no—
455Dic6—182no—
456Dic6—182no—
457Dic6—182no—
458C2×C32C2×#23511yescharacteristic
459C2×C32#101011yescharacteristic
460C2×C32#101011yescharacteristic
461C3×D3—41no—
462C2×C9—21no—
463C2×C9—21no—
464C2×C9—21no—
465C2×C9—121no—
466C2×C9#100711yescharacteristic
467C2×C9D3×#4111yescharacteristic
468C2×C9#101011yescharacteristic
469C2×Q8—271no—
470C2×Q8—271no—
471C22×C4—31no—
472C4⋊C4—31no—
473C4⋊C4—91no—
474C4⋊C4—91no—
475C4⋊C4—91no—
476C4⋊C4—91no—
477C4⋊C4—91no—
478C4⋊C4—271no—
479C4⋊C4—271no—
480C4⋊C4—271no—
481C4⋊C4—271no—
482C4⋊C4—271no—
483C4⋊C4—271no—
484C4⋊C4—271no—
485C4⋊C4—271no—
486C4⋊C4—271no—
487C4⋊C4—271no—
488C4⋊C4—271no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4D3×D911yescharacteristic
492C22⋊C4—31no—
493C22⋊C4—31no—
494C22⋊C4—271no—
495C22⋊C4—271no—
496C22⋊C4—271no—
497C22⋊C4—271no—
498C42—31no—
499C42—31no—
500C42—91no—
501C42—271no—
502C42—271no—
503C42—271no—
504C22×C3—21no—
505C22×C3—61no—
506C22×C3—61no—
507C22×C3—61no—
508C22×C3C23×D911yescharacteristic
509C22×C3C22×D3211yescharacteristic
510C2×D3—21no—
511C2×D3—21no—
512C2×D3—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—32no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—61no—
524C3×C4—61no—
525C3×C4—181no—
526C3×C4—181no—
527C3×C4—181no—
528C3×C4—181no—
529Dic3—21no—
530Dic3—21no—
531Dic3—61no—
532Dic3—61no—
533Dic3—61no—
534Dic3—61no—
535Dic3—182no—
536Dic3—181no—
537Dic3—181no—
538Dic3—181no—
539Dic3—181no—
540Dic3—181no—
541Dic3—181no—
542Dic3—181no—
543Dic3—181no—
544Dic3—361no—
545Dic3—361no—
546Dic3—361no—
547Dic3#369712yes—
548C32#551811yescharacteristic
549C9—21no—
550C9#569611yescharacteristic, complemented
551C23—31no—
552Q8—542no—
553Q8—542no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—62no—
558C2×C4—91no—
559C2×C4—91no—
560C2×C4—91no—
561C2×C4—91no—
562C2×C4—271no—
563C2×C4—271no—
564C2×C4—271no—
565C2×C4—271no—
566C2×C4C2×D3×D911yescharacteristic
567C2×C4C2×D3×D911yescharacteristic
568C2×C4C2×D3×D911yescharacteristic
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—121no—
573C2×C3C2×#369711yescharacteristic
574C2×C3#6371611yescharacteristic
575C2×C3#6371611yescharacteristic
576C2×C3D3×#23511yescharacteristic
577C2×C3#6430911yescharacteristic
578C2×C3#6431511yescharacteristic
579D3—41no—
580C22—61no—
581C22—61no—
582C22—61no—
583C22C22×D3×D911yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—32no—
588C4—61no—
589C4—61no—
590C4—181no—
591C4—181no—
592C4—181no—
593C4—181no—
594C4—541no—
595C4—541no—
596C4—541no—
597C4—541no—
598C3—21no—
599C3#1058962011yescharacteristic, complemented
600C3#1059459811yescharacteristic
601C2—121no—
602C2D3×#369711yescharacteristic
603C2#1172249411yescharacteristic
604C2#1172250811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D3C22×D3
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
1. #11721745C2semidirect
3. C3×#63451C2semidirect
4. #11720970C2semidirect
5. C4×#6189C2semidirect
12. #11720925C2semidirect
13. #11720918C2semidirect
14. #11720917C2semidirect
15. #11720900C2semidirect
75. #63451D3semidirect
414. C3×C9#541semidirect
550. C9#5696semidirect
599. C3#10589620semidirect

Maximal subgroups

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

subgroupindex
#117217452
C9×#8922
C3×#634512
#117209702
C4×#61892
#117209662
#117209652
#117209632
#117209622
#117209552
#117209252
#117209182
#117209172
#117209002
#105945983
#105896203