Finite Group Databasebeta

← #13148869

Subgroups — #13148869

Order 1728.

696 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721754C211yesmaximal, characteristic, complemented
2C3×#63460C211yesmaximal, characteristic, complemented
3C9×#887C211yesmaximal, characteristic, complemented
4#11720986C211yesmaximal, characteristic, complemented
5#11720984C211yesmaximal, characteristic, complemented
6#11720977C211yesmaximal, characteristic
7#11720975C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9C4×#6189C211yesmaximal, characteristic, complemented
10#11720949C211yesmaximal, characteristic, complemented
11#11720940C211yesmaximal, characteristic
12#11720940C211yesmaximal, characteristic
13#11720921C211yesmaximal, characteristic, complemented
14#11720900C211yesmaximal, characteristic, complemented
15#11720899C211yesmaximal, characteristic, complemented
16#10594421—31nomaximal
17#10589686—31nomaximal
18C2×C9×Dic6C2211yescharacteristic
19C2×#6192C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C4⋊C4C2211yescharacteristic
25#88536C2211yescharacteristic
26#88536C2211yescharacteristic
27#88536C2211yescharacteristic
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29#88534C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C4×C9×Dic3C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C4×C9×Dic3C2211yescharacteristic
39C3×C4⋊Dic9C2211yescharacteristic
40C3×Dic9⋊C4C2211yescharacteristic
41C3×C4×Dic9C2211yescharacteristic
42#88445C2211yescharacteristic
43#88445C2211yescharacteristic
44(C2×D9)⋊Dic3C2211yescharacteristic
45(C2×D9)⋊Dic3C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47Dic3⋊Dic9C2211yescharacteristic
48Dic3⋊Dic9C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64107—31no—
54C3×#898—31no—
55C3×#887D311yescharacteristic
56#63918—31no—
57C4×#627—31no—
58#63888—31no—
59#63887—31no—
60#63880—31no—
61#63878—31no—
62#63861—31no—
63#63852—31no—
64#63852—31no—
65#63848—31no—
66#63837—31no—
67#63836—31no—
68C4×Q8×C9—31no—
69#63508—31no—
70#63508—31no—
71#63460—31no—
72#63460—31no—
73#63460—31no—
74#63460—61no—
75#63460D311yescharacteristic, complemented
76#63454—31no—
77#63454—31no—
78#63437—31no—
79#63437—31no—
80C4×D36—31no—
81#63431—31no—
82C4×Dic18—31no—
83C4×Dic18—31no—
84C22×C3⋊D9C2311yescharacteristic
85C22×C3×D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C9×Dic3C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C3×Dic9C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C9×Dic6—22no—
100#6192—22no—
101(C3⋊D9)⋊C4—22no—
102Dic3×D9—22no—
103#6189—22no—
104#6189—22no—
105#5702—91no—
106#5584—91no—
107C2×C3×Dic6C2×D311yescharacteristic
108C2×#627—31no—
109C2×#627—31no—
110C2×#626—31no—
111C2×#624—31no—
112C2×D3×Dic3—31no—
113C32×C4⋊C4C2×D311yescharacteristic
114#3752—31no—
115#3752—31no—
116#3752—31no—
117C4⋊(C3⋊Dic3)—31no—
118#3750—31no—
119C4×C3⋊Dic3—31no—
120C3×(C2×D3)⋊C4—31no—
121C3×(C2×D3)⋊C4—31no—
122C3×(C2×D3)⋊C4—31no—
123C3×C4⋊Dic3—31no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×C4×Dic3—31no—
128C3×C4×Dic3C2×D311yescharacteristic
129C3×C4×Dic3C2×D311yescharacteristic
130C3×C4×Dic3C2×D311yescharacteristic
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136#3722—31no—
137#3722—31no—
138(C2×D3)⋊Dic3—31no—
139(C2×D3)⋊Dic3—31no—
140Dic32—31no—
141Dic32—31no—
142C2×Q8×C9—31no—
143C2×D36—31no—
144C2×C4×D9—31no—
145C2×C4×D9—31no—
146C2×Dic18—31no—
147C2×Dic18—31no—
148C9×C4⋊C4—21no—
149C9×C4⋊C4—31no—
150C9×C4⋊C4—31no—
151C9×C4⋊C4C2×D311yescharacteristic
152C42×C9—31no—
153C42×C9—31no—
154C42×C9—31no—
155(C2×D9)⋊C4—31no—
156(C2×D9)⋊C4—31no—
157(C2×D9)⋊C4—31no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—61no—
163(C2×D9)⋊C4—61no—
164(C2×D9)⋊C4—61no—
165(C2×D9)⋊C4C2×D311yescharacteristic
166(C2×D9)⋊C4C2×D311yescharacteristic
167(C2×D9)⋊C4C2×D311yescharacteristic
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171C4⋊Dic9—31no—
172C4⋊Dic9—61no—
173C4⋊Dic9C2×D311yescharacteristic
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—61no—
178Dic9⋊C4C2×D311yescharacteristic
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9—61no—
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9C2411yescharacteristic
185C2×C3⋊D9—21no—
186C2×C3⋊D9—21no—
187C2×C3⋊D9—21no—
188C2×C3×D9—21no—
189C2×C3×D9—21no—
190C2×C3×D9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3×C4×C9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3#4112yes—
200C9×Dic3#4112yes—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#126—91no—
205C3×C4×Q8—31no—
206#946—91no—
207#946—91no—
208#898—31no—
209#898—91no—
210#898—91no—
211#898—91no—
212#898—91no—
213#898—91no—
214#898—181no—
215#896—91no—
216#892—91no—
217#892—91no—
218#888—91no—
219#887—91no—
220#887D911yescharacteristic, complemented
221#885—91no—
222#885—91no—
223#67⋊C4—91no—
224#879—91no—
225#878—91no—
226#877—91no—
227#875—91no—
228#875—91no—
229C4×D12—91no—
230#872—91no—
231#869—91no—
232C4×Dic6—91no—
233C4×Dic6—91no—
234C4×Dic6—91no—
235C22×C3⋊D3—31no—
236C22×C3×D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3×Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3C22×D311yescharacteristic
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C3×Dic6—22no—
251#627—62no—
252#627—62no—
253#626—62no—
254#624—62no—
255D3×Dic3—62no—
256C22×D9—31no—
257C22×D9—61no—
258C22×D9C22×D311yescharacteristic
259Q8×C9—62no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—21no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9—31no—
267C2×C4×C9C22×D311yescharacteristic
268C2×C4×C9C22×D311yescharacteristic
269C2×C4×C9C22×D311yescharacteristic
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9C22×D311yescharacteristic
277C2×Dic9C22×D311yescharacteristic
278C2×Dic9C22×D311yescharacteristic
279D36—62no—
280C4×D9—62no—
281C4×D9—62no—
282Dic18—62no—
283Dic18—62no—
284#556—271noSylow
285C2×C3×C9C2×#4111yescharacteristic
286C2×C3×C9C2×#4111yescharacteristic
287C2×C3×C92−1+411yescharacteristic
288C3⋊D9—41no—
289C3×D9—41no—
290C2×C3×Q8—31no—
291C2×#67—91no—
292C22×Dic3—91no—
293C2×D12—91no—
294C2×C4×D3—91no—
295C2×C4×D3—91no—
296C2×C4×D3—91no—
297C2×Dic6—91no—
298C2×Dic6—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6C2×D911yescharacteristic
302C3×C4⋊C4—21no—
303C3×C4⋊C4—31no—
304C3×C4⋊C4—31no—
305C3×C4⋊C4—91no—
306C3×C4⋊C4—91no—
307C3×C4⋊C4C2×D911yescharacteristic
308C3×C4⋊C4D3211yescharacteristic
309C3×C22⋊C4—91no—
310C3×C22⋊C4—91no—
311C3×C22⋊C4—91no—
312C3×C42—31no—
313C3×C42—31no—
314C3×C42—31no—
315C3×C42—91no—
316#217—91no—
317#217—91no—
318(C2×D3)⋊C4—31no—
319(C2×D3)⋊C4—31no—
320(C2×D3)⋊C4—31no—
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—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—181no—
334(C2×D3)⋊C4—181no—
335(C2×D3)⋊C4—181no—
336C4⋊Dic3—31no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—91no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—91no—
343C4⋊Dic3—181no—
344Dic3⋊C4—31no—
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—181no—
354Dic3⋊C4C2×D911yescharacteristic
355Dic3⋊C4C2×D911yescharacteristic
356C4×Dic3—31no—
357C4×Dic3—91no—
358C4×Dic3—91no—
359C4×Dic3—91no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—181no—
364C4×Dic3C2×D911yescharacteristic
365C4×Dic3C2×D911yescharacteristic
366C4×Dic3C2×D911yescharacteristic
367C22×C32C23×D311yescharacteristic
368C2×C3⋊D3—61no—
369C2×C3⋊D3—61no—
370C2×C3⋊D3—61no—
371C2×C3×D3—61no—
372C2×C3×D3—61no—
373C2×C3×D3—61no—
374C32×C4—21no—
375C32×C4—21no—
376C32×C4—21no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—61no—
383C3×Dic3—61no—
384C3×Dic3—61no—
385C3×Dic3#23512yes—
386C3×Dic3#23912yes—
387C22×C9—21no—
388C22×C9C23×D311yescharacteristic
389C2×D9—21no—
390C2×D9—21no—
391C2×D9—21no—
392C2×D9—61no—
393C2×D9—61no—
394C2×D9—61no—
395C2×D9—121no—
396C2×D9—121no—
397C2×D9—121no—
398C4×C9—21no—
399C4×C9—21no—
400C4×C9—21no—
401C4×C9—32no—
402C4×C9—32no—
403C4×C9—41no—
404C4×C9—41no—
405C4×C9—41no—
406C4×C9—61no—
407C4×C9—61no—
408Dic9—21no—
409Dic9—21no—
410Dic9—21no—
411Dic9—61no—
412Dic9—61no—
413Dic9—61no—
414Dic9—121no—
415Dic9—121no—
416Dic9—121no—
417C4⋊Q8—271no—
418#126—91no—
419#126—271no—
420#126—271no—
421#126—271no—
422#124—271no—
423#124—271no—
424C22⋊Q8—271no—
425C22⋊Q8—271no—
426C4×Q8—31no—
427C4×Q8—271no—
428C4×Q8—271no—
429C4×D4—271no—
430#117—271no—
431#117—271no—
432C3×C9#55611yescharacteristic, complemented, Sylow
433C23×C3—91no—
434C22×D3—31no—
435C22×D3—91no—
436C22×D3—91no—
437C22×D3—181no—
438C3×Q8—62no—
439C2×C3×C4—21no—
440C2×C3×C4—21no—
441C2×C3×C4—21no—
442C2×C3×C4—31no—
443C2×C3×C4—31no—
444C2×C3×C4—31no—
445C2×C3×C4—31no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4C22×D911yescharacteristic
450C2×C3×C4C22×D911yescharacteristic
451C2×C3×C4C22×D911yescharacteristic
452C2×C3×C4C2×D3211yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454C2×C3×C4C2×D3211yescharacteristic
455#67—182no—
456C2×Dic3—31no—
457C2×Dic3—31no—
458C2×Dic3—31no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—181no—
466C2×Dic3—181no—
467C2×Dic3—181no—
468C2×Dic3—182no—
469C2×Dic3C22×D911yescharacteristic
470C2×Dic3C22×D911yescharacteristic
471C2×Dic3C22×D911yescharacteristic
472C2×Dic3C22×D911yescharacteristic
473D12—182no—
474C4×D3—182no—
475C4×D3—182no—
476C4×D3—182no—
477Dic6—22no—
478Dic6—182no—
479Dic6—182no—
480Dic6—182no—
481Dic6—182no—
482C2×C32C2×#23511yescharacteristic
483C2×C32C2×#23911yescharacteristic
484C2×C32#101011yescharacteristic
485C3⋊D3—121no—
486C3×D3—121no—
487C2×C9—21no—
488C2×C9—21no—
489C2×C9—21no—
490C2×C9D3×#4111yescharacteristic
491C2×C9D3×#4111yescharacteristic
492C2×C9#101011yescharacteristic
493D9—41no—
494D9—121no—
495D9—241no—
496C2×Q8—31no—
497C2×Q8—271no—
498C2×Q8—271no—
499C2×D4—271no—
500C22×C4—271no—
501C22×C4—271no—
502C4⋊C4—31no—
503C4⋊C4—31no—
504C4⋊C4—91no—
505C4⋊C4—91no—
506C4⋊C4—271no—
507C4⋊C4—271no—
508C4⋊C4—271no—
509C4⋊C4—271no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4D3×D911yescharacteristic
514C22⋊C4—91no—
515C22⋊C4—91no—
516C22⋊C4—91no—
517C22⋊C4—271no—
518C22⋊C4—271no—
519C22⋊C4—271no—
520C22⋊C4—271no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C42—31no—
525C42—31no—
526C42—31no—
527C42—91no—
528C42—271no—
529C42—271no—
530C42—271no—
531C22×C3—21no—
532C22×C3—181no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3C23×D911yescharacteristic
536C22×C3C22×D3211yescharacteristic
537C2×D3—61no—
538C2×D3—61no—
539C2×D3—61no—
540C2×D3—181no—
541C2×D3—181no—
542C2×D3—181no—
543C2×D3—181no—
544C2×D3—181no—
545C2×D3—181no—
546C2×D3—361no—
547C2×D3—361no—
548C2×D3—361no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—32no—
556C3×C4—32no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—61no—
561C3×C4—61no—
562C3×C4—181no—
563C3×C4—181no—
564C3×C4—181no—
565Dic3—21no—
566Dic3—21no—
567Dic3—61no—
568Dic3—61no—
569Dic3—61no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—181no—
574Dic3—181no—
575Dic3—181no—
576Dic3—361no—
577Dic3—361no—
578Dic3—361no—
579Dic3#369712yes—
580Dic3#370112yes—
581C32#558411yescharacteristic
582C9—21no—
583C9#570211yescharacteristic, complemented
584C23—91no—
585C23—271no—
586Q8—62no—
587Q8—542no—
588Q8—542no—
589D4—542no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—91no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—271no—
598C2×C4—271no—
599C2×C4—271no—
600C2×C4—542no—
601C2×C4—542no—
602C2×C4C2×D3×D911yescharacteristic
603C2×C4C2×D3×D911yescharacteristic
604C2×C4C2×D3×D911yescharacteristic
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—361no—
609C2×C3C2×#369711yescharacteristic
610C2×C3C2×#370111yescharacteristic
611C2×C3#6371611yescharacteristic
612C2×C3D3×#23511yescharacteristic
613C2×C3#6430911yescharacteristic
614C2×C3D3×#23911yescharacteristic
615D3—121no—
616D3—361no—
617D3—361no—
618D3—721no—
619C22—181no—
620C22—181no—
621C22—181no—
622C22—541no—
623C22—541no—
624C22—541no—
625C22C22×D3×D911yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—32no—
630C4—32no—
631C4—61no—
632C4—61no—
633C4—181no—
634C4—181no—
635C4—181no—
636C4—541no—
637C4—541no—
638C4—541no—
639C3—21no—
640C3#1058968611yescharacteristic, complemented
641C3#1059442111yescharacteristic
642C2—361no—
643C2—1081no—
644C2D3×#369711yescharacteristic
645C2#1172249411yescharacteristic
646C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#556
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. #11721754C2semidirect
2. C3×#63460C2semidirect
3. C9×#887C2semidirect
4. #11720986C2semidirect
5. #11720984C2semidirect
7. #11720975C2semidirect
8. #11720970C2semidirect
9. C4×#6189C2semidirect
10. #11720949C2semidirect
13. #11720921C2semidirect
14. #11720900C2semidirect
15. #11720899C2semidirect
75. #63460D3semidirect
220. #887D9semidirect
432. C3×C9#556semidirect
583. C9#5702semidirect
640. C3#10589686semidirect

Maximal subgroups

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

subgroupindex
#117217542
C3×#634602
C9×#8872
#117209862
#117209842
#117209772
#117209752
#117209702
C4×#61892
#117209492
#117209402
#117209212
#117209002
#117208992
#105944213
#105896863