Finite Group Databasebeta

← #13148853

Subgroups — #13148853

Order 1728.

692 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721752C211yesmaximal, characteristic
2C9×#889C211yesmaximal, characteristic, complemented
3C3×#63450C211yesmaximal, characteristic, complemented
4#11720995C211yesmaximal, characteristic
5#11720985C211yesmaximal, characteristic
6#11720980C211yesmaximal, characteristic
7#11720936C211yesmaximal, characteristic
8#11720933C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#11720917C211yesmaximal, characteristic, complemented
11#11720916C211yesmaximal, characteristic, complemented
12#11720905C211yesmaximal, characteristic, complemented
13#11720904C211yesmaximal, characteristic, complemented
14#11720899C211yesmaximal, characteristic
15#11720898C211yesmaximal, characteristic
16#10594400—31nomaximal
17#10589791—31nomaximal
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2211yescharacteristic
23C2×(C3⋊D9)⋊C4C2211yescharacteristic
24C2×#6189C2211yescharacteristic
25C3×C9×C4⋊C4C2211yescharacteristic
26#88536C2211yescharacteristic
27#88536C2211yescharacteristic
28#88534C2211yescharacteristic
29#88534C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31C9×Dic3⋊C4C2211yescharacteristic
32C9×Dic3⋊C4C2211yescharacteristic
33C9×Dic3⋊C4C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C4×C9×Dic3C2211yescharacteristic
36C3×C4⋊Dic9C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×C4×Dic9C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64105—31no—
54C3×#889D311yescharacteristic
55C3×#888—31no—
56#63895—31no—
57#63888—31no—
58#63883—31no—
59#63871—31no—
60#63867—31no—
61#63866—31no—
62#63846—31no—
63#63846—31no—
64#63840—31no—
65#63839—31no—
66#63836—31no—
67#63836—31no—
68C9×#125—31no—
69#63459—31no—
70#63459—31no—
71#63458—31no—
72#63458—31no—
73#63458—61no—
74#63454—31no—
75C4⋊C4×D9—31no—
76#63452—31no—
77#63451—31no—
78#63451—31no—
79#63450—31no—
80#63450—31no—
81#63450D311yescharacteristic, complemented
82#63434—31no—
83#63431—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—41no—
92#6226—41no—
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C3×Dic18—41no—
102C3×Dic18—41no—
103(C3⋊D9)⋊C4—41no—
104(C3⋊D9)⋊C4Q814yescomplemented
105#6189—41no—
106#5729—91no—
107#5689—91no—
108C2×C4×C3⋊D3—31no—
109C2×#634—31no—
110C2×C3×Dic6—31no—
111C2×C3×Dic6—31no—
112C2×#627—31no—
113C2×#624—31no—
114C2×#624—31no—
115C32×C4⋊C4C2×D311yescharacteristic
116#3752—31no—
117#3752—31no—
118#3750—31no—
119#3750—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3C2×D311yescharacteristic
122C3×Dic3⋊C4—31no—
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4C2×D311yescharacteristic
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×C4×Dic3—31no—
129C3×C4×Dic3C2×D311yescharacteristic
130#3724—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139#3722—31no—
140#3722—31no—
141Dic32—31no—
142Dic32—31no—
143C2×C4×D9—31no—
144C2×C4×D9—31no—
145C2×C4×D9—31no—
146C2×C4×D9—61no—
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18—61no—
150C2×Dic18C2×D311yescharacteristic
151C2×Dic18C2×D311yescharacteristic
152C9×C4⋊C4—21no—
153C9×C4⋊C4—31no—
154C9×C4⋊C4—31no—
155C9×C4⋊C4—31no—
156C9×C4⋊C4—31no—
157C9×C4⋊C4—31no—
158C9×C4⋊C4C2×D311yescharacteristic
159C42×C9—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—31no—
164(C2×D9)⋊C4—61no—
165(C2×D9)⋊C4—61no—
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171C4⋊Dic9C2×D311yescharacteristic
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—61no—
179Dic9⋊C4—61no—
180Dic9⋊C4C2×D311yescharacteristic
181Dic9⋊C4C2×D311yescharacteristic
182C4×Dic9—31no—
183C4×Dic9—31no—
184C4×Dic9C2×D311yescharacteristic
185C22×C3×C9C2411yescharacteristic
186C2×C3⋊D9—21no—
187C2×C3⋊D9—21no—
188C2×C3⋊D9C2×Q812yes—
189C3×C4×C9—21no—
190C3×C4×C9—21no—
191C3×C4×C9—21no—
192C3⋊Dic9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C9×Dic3—21no—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3C2×Q812yes—
199C3×Dic9—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9C2×Q812yes—
203C3×C4⋊Q8—91no—
204C3×#125—31no—
205#946—91no—
206#946—91no—
207#944—91no—
208#944—91no—
209#897—91no—
210#897—91no—
211#896—91no—
212#896—91no—
213#896—91no—
214#896—91no—
215#896—181no—
216#892—91no—
217#892—91no—
218D3×C4⋊C4—91no—
219D3×C4⋊C4—91no—
220#890—91no—
221#890—91no—
222#889—91no—
223#889—91no—
224#889—91no—
225#889D911yescharacteristic, complemented
226#888—31no—
227#888—91no—
228#888—91no—
229#888—91no—
230#872—91no—
231#872—91no—
232#870—91no—
233#869—91no—
234C22×C3⋊D3—31no—
235C2×C32×C4C22×D311yescharacteristic
236C2×C32×C4C22×D311yescharacteristic
237C2×C32×C4C22×D311yescharacteristic
238C2×C3⋊Dic3—31no—
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C4×C3⋊D3—121no—
242#634—121no—
243C2×C3×Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C3×Dic6—121no—
252C3×Dic6—121no—
253#627—121no—
254#624—34no—
255#624—121no—
256C22×D9—31no—
257C22×D9—61no—
258C2×C4×C9—21no—
259C2×C4×C9—21no—
260C2×C4×C9—21no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9C22×D311yescharacteristic
266C2×C4×C9C22×D311yescharacteristic
267C2×C4×C9C22×D311yescharacteristic
268C2×Dic9—31no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—61no—
272C2×Dic9—61no—
273C2×Dic9—61no—
274C2×Dic9C22×D311yescharacteristic
275C2×Dic9C22×D311yescharacteristic
276C2×Dic9C22×D311yescharacteristic
277C2×Dic9C22×D311yescharacteristic
278C4×D9—34no—
279C4×D9—121no—
280C4×D9—121no—
281C4×D9—122no—
282Dic18—41no—
283Dic18—41no—
284Dic18—121no—
285Dic18—121no—
286Dic18—122no—
287#544—271noSylow
288C2×C3×C9C22×Q811yescharacteristic
289C2×C3×C9H(5,𝔽2)11yescharacteristic
290C2×C3×C92−1+411yescharacteristic
291C3⋊D9—22no—
292C2×C3×Q8—91no—
293C2×C3×Q8—91no—
294C2×C4×D3—91no—
295C2×C4×D3—91no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×C4×D3—91no—
300C2×C4×D3—181no—
301C2×Dic6—31no—
302C2×Dic6—31no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—91no—
306C2×Dic6—91no—
307C2×Dic6—181no—
308C3×C4⋊C4—21no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—31no—
311C3×C4⋊C4—31no—
312C3×C4⋊C4—31no—
313C3×C4⋊C4—31no—
314C3×C4⋊C4—91no—
315C3×C4⋊C4—91no—
316C3×C4⋊C4—91no—
317C3×C4⋊C4C2×D911yescharacteristic
318C3×C4⋊C4D3211yescharacteristic
319C3×C42—31no—
320C3×C42—91no—
321(C2×D3)⋊C4—91no—
322(C2×D3)⋊C4—91no—
323(C2×D3)⋊C4—91no—
324(C2×D3)⋊C4—91no—
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—181no—
330(C2×D3)⋊C4—181no—
331C4⋊Dic3—31no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—91no—
340C4⋊Dic3C2×D911yescharacteristic
341Dic3⋊C4—31no—
342Dic3⋊C4—31no—
343Dic3⋊C4—91no—
344Dic3⋊C4—91no—
345Dic3⋊C4—91no—
346Dic3⋊C4—91no—
347Dic3⋊C4—91no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—181no—
358Dic3⋊C4—181no—
359Dic3⋊C4C2×D911yescharacteristic
360Dic3⋊C4C2×D911yescharacteristic
361Dic3⋊C4C2×D911yescharacteristic
362Dic3⋊C4C2×D911yescharacteristic
363C4×Dic3—31no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3C2×D911yescharacteristic
369C22×C32C23×D311yescharacteristic
370C2×C3⋊D3—32no—
371C2×C3⋊D3—61no—
372C2×C3⋊D3—61no—
373C32×C4—21no—
374C32×C4—21no—
375C32×C4—21no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—32no—
383C3×Dic3—61no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C3×Dic3D3×Q812yes—
387C22×C9—21no—
388C22×C9C23×D311yescharacteristic
389C2×D9—32no—
390C2×D9—62no—
391C2×D9—61no—
392C2×D9—61no—
393C2×D9—121no—
394C2×D9—121no—
395C4×C9—21no—
396C4×C9—21no—
397C4×C9—21no—
398C4×C9—32no—
399C4×C9—41no—
400C4×C9—41no—
401C4×C9—41no—
402C4×C9—61no—
403C4×C9—61no—
404C4×C9—61no—
405Dic9—21no—
406Dic9—21no—
407Dic9—21no—
408Dic9—62no—
409Dic9—61no—
410Dic9—61no—
411Dic9—61no—
412Dic9—121no—
413Dic9—121no—
414Dic9D3×Q812yes—
415C4⋊Q8—91no—
416C4⋊Q8—271no—
417C4⋊Q8—271no—
418C4⋊Q8—271no—
419#125—31no—
420#125—271no—
421#125—271no—
422#125—271no—
423C22⋊Q8—271no—
424C22⋊Q8—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427#117—271no—
428#117—271no—
429C2×C4⋊C4—271no—
430C3×C9#54411yescharacteristic, complemented, Sylow
431C22×D3—91no—
432C22×D3—91no—
433C22×D3—181no—
434C3×Q8—361no—
435C3×Q8—361no—
436C2×C3×C4—21no—
437C2×C3×C4—21no—
438C2×C3×C4—21no—
439C2×C3×C4—31no—
440C2×C3×C4—31no—
441C2×C3×C4—31no—
442C2×C3×C4—31no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4—91no—
447C2×C3×C4C22×D911yescharacteristic
448C2×C3×C4C22×D911yescharacteristic
449C2×C3×C4C22×D911yescharacteristic
450C2×C3×C4C2×D3211yescharacteristic
451C2×C3×C4C2×D3211yescharacteristic
452C2×C3×C4C2×D3211yescharacteristic
453C2×Dic3—31no—
454C2×Dic3—31no—
455C2×Dic3—31no—
456C2×Dic3—31no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—181no—
464C2×Dic3—181no—
465C2×Dic3—181no—
466C2×Dic3C22×D911yescharacteristic
467C2×Dic3C22×D911yescharacteristic
468C2×Dic3C22×D911yescharacteristic
469C2×Dic3C22×D911yescharacteristic
470C4×D3—94no—
471C4×D3—94no—
472C4×D3—361no—
473C4×D3—361no—
474C4×D3—361no—
475C4×D3—361no—
476C4×D3—362no—
477Dic6—121no—
478Dic6—121no—
479Dic6—361no—
480Dic6—361no—
481Dic6—361no—
482Dic6—361no—
483Dic6—362no—
484C2×C32C2×D3×Q811yescharacteristic
485C2×C32#100911yescharacteristic
486C2×C32#101011yescharacteristic
487C3⋊D3—62no—
488C2×C9—21no—
489C2×C9—21no—
490C2×C9—21no—
491C2×C9#100411yescharacteristic
492C2×C9C2×D3×Q811yescharacteristic
493C2×C9#100711yescharacteristic
494D9—62no—
495D9—122no—
496C2×Q8—91no—
497C2×Q8—91no—
498C2×Q8—271no—
499C2×Q8—271no—
500C22×C4—271no—
501C22×C4—271no—
502C22×C4—271no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—31no—
506C4⋊C4—31no—
507C4⋊C4—31no—
508C4⋊C4—91no—
509C4⋊C4—91no—
510C4⋊C4—91no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4D3×D911yescharacteristic
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C42—31no—
528C42—91no—
529C42—271no—
530C42—271no—
531C22×C3—21no—
532C22×C3C23×D911yescharacteristic
533C22×C3C22×D3211yescharacteristic
534C2×D3—92no—
535C2×D3—92no—
536C2×D3—182no—
537C2×D3—181no—
538C2×D3—181no—
539C2×D3—181no—
540C2×D3—181no—
541C2×D3—361no—
542C2×D3—361no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—32no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—61no—
554C3×C4—61no—
555C3×C4—61no—
556C3×C4—92no—
557C3×C4—181no—
558C3×C4—181no—
559C3×C4—181no—
560Dic3—21no—
561Dic3—21no—
562Dic3—21no—
563Dic3—32no—
564Dic3—61no—
565Dic3—61no—
566Dic3—61no—
567Dic3—182no—
568Dic3—181no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—181no—
574Dic3—361no—
575Dic3—361no—
576Dic3Q8×D912yes—
577C32#568911yescharacteristic
578C9—21no—
579C9#572911yescharacteristic, complemented
580C23—271no—
581Q8—361no—
582Q8—361no—
583Q8—1081no—
584Q8—1081no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—31no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—91no—
592C2×C4—91no—
593C2×C4—274no—
594C2×C4—271no—
595C2×C4—271no—
596C2×C4—271no—
597C2×C4—1081no—
598C2×C4—1081no—
599C2×C4C2×D3×D911yescharacteristic
600C2×C4C2×D3×D911yescharacteristic
601C2×C4C2×D3×D911yescharacteristic
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3C2×Q8×D911yescharacteristic
606C2×C3#6371511yescharacteristic
607C2×C3#6371611yescharacteristic
608C2×C3#6429711yescharacteristic
609C2×C3#6431411yescharacteristic
610C2×C3D32×Q811yescharacteristic
611D3—182no—
612D3—182no—
613D3—362no—
614C22—272no—
615C22—541no—
616C22—541no—
617C22C22×D3×D911yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—32no—
622C4—61no—
623C4—61no—
624C4—61no—
625C4—92no—
626C4—181no—
627C4—181no—
628C4—181no—
629C4—541no—
630C4—541no—
631C4—541no—
632C3—21no—
633C3#1058979111yescharacteristic, complemented
634C3#1059440011yescharacteristic
635C2—542no—
636C2#1172246411yescharacteristic
637C2#1172250111yescharacteristic
638C2D3×Q8×D911yescharacteristic
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#544
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×#63450C2semidirect
8. #11720933C2semidirect
9. #11720932C2semidirect
10. #11720917C2semidirect
11. #11720916C2semidirect
12. #11720905C2semidirect
13. #11720904C2semidirect
81. #63450D3semidirect
104. (C3⋊D9)⋊C4Q8semidirect
225. #889D9semidirect
430. C3×C9#544semidirect
579. C9#5729semidirect
633. C3#10589791semidirect

Maximal subgroups

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

subgroupindex
#117217522
C9×#8892
C3×#634502
#117209952
#117209852
#117209802
#117209362
#117209332
#117209322
#117209172
#117209162
#117209052
#117209042
#117208992
#117208982
#105944003
#105897913