Finite Group Databasebeta

← #13175903

Subgroups — #13175903

Order 1728.

1472 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723919C211yesmaximal, characteristic, complemented
2C3×#64088C211yesmaximal, characteristic, complemented
3C32×#883C211yesmaximal, characteristic
4#11723642C211yesmaximal, characteristic, complemented
5#11723603C211yesmaximal, characteristic
6#11723600C211yesmaximal, characteristic
7#11723597C211yesmaximal, characteristic, complemented
8#11723535C211yesmaximal, characteristic, complemented
9#11723430C211yesmaximal, characteristic, complemented
10#11723428C211yesmaximal, characteristic, complemented
11#11723426C211yesmaximal, characteristic, complemented
12#11723415C211yesmaximal, characteristic, complemented
13#11723412C211yesmaximal, characteristic, complemented
14#11723402C211yesmaximal, characteristic, complemented
15#11723395C211yesmaximal, characteristic, complemented
16#10596122—31nomaximal
17#10594044—34nomaximal
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C2×C32×#67C2211yescharacteristic
21C2×C32×C4×D3C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
27#88857C2211yescharacteristic
28#88857C2211yescharacteristic
29C3×#3757C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×(C2×D3)⋊C4C2211yescharacteristic
37C32×C4⋊Dic3C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
48(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
49(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
50(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#123—31no—
54#64145—31no—
55#64144—31no—
56#64133—31no—
57#64133—31no—
58#64098—31no—
59#64098—31no—
60#64097—31no—
61#64097—31no—
62#64088—31no—
63#64088—31no—
64#64088—31no—
65#64088D311yescharacteristic, complemented
66#64087—31no—
67#64087—31no—
68#64087—34no—
69#64087—64no—
70C3×#883D314yes—
71C3×#879—34no—
72#63982—34no—
73#63962—34no—
74#63959—34no—
75#63956—34no—
76#63918—34no—
77#63861—34no—
78#63859—34no—
79#63857—34no—
80#63851—34no—
81#63847—34no—
82#63846—34no—
83#63840—34no—
84C23×C33C2311yescharacteristic
85C22×C32×D3C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C3⋊(C3⋊Dic3)—41no—
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
93#6308—41no—
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C32×#67—41no—
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C32×C4×D3—41no—
103#6293—41no—
104#6293—41no—
105#5712—91no—
106#5651—94no—
107C2×C32×D4—31no—
108C22×C32×C4—31no—
109C22×C3⋊Dic3—31no—
110C22×C3⋊Dic3—34no—
111C22×C3⋊Dic3—64no—
112C2×#634—31no—
113C2×#634—34no—
114C2×#634—31no—
115C2×#634—31no—
116C2×#634—64no—
117C2×C3×#67C2×D314yes—
118C2×C3×C4×D3C2×D314yes—
119C2×#627—34no—
120C2×#627—34no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4—31no—
123C32×C22⋊C4—24no—
124C32×C22⋊C4—31no—
125C32×C22⋊C4—31no—
126C32×C22⋊C4C2×D314yes—
127C32×C22⋊C4C2×D311yescharacteristic
128#3757—31no—
129#3757—34no—
130#3757—31no—
131#3757—31no—
132#3757—31no—
133#3757—31no—
134#3757—64no—
135#3757C2×D311yescharacteristic
136#3757C2×D311yescharacteristic
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—34no—
139C4⋊(C3⋊Dic3)—31no—
140C4⋊(C3⋊Dic3)—31no—
141C4⋊(C3⋊Dic3)—64no—
142C4⋊(C3⋊Dic3)C2×D311yescharacteristic
143#3750—31no—
144#3750—31no—
145#3750—34no—
146#3750—34no—
147#3750—31no—
148#3750—31no—
149#3750—31no—
150#3750—31no—
151#3750—31no—
152#3750—31no—
153#3750—64no—
154#3750—64no—
155#3750C2×D311yescharacteristic
156#3750C2×D311yescharacteristic
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3C2×D311yescharacteristic
160C3×#217—34no—
161C3×#217—34no—
162C3×#217C2×D314yes—
163C3×(C2×D3)⋊C4C2×D314yes—
164C3×C4⋊Dic3—34no—
165C3×C4⋊Dic3C2×D314yes—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4—34no—
168C3×Dic3⋊C4C2×D314yes—
169C3×C4×Dic3—34no—
170#3724—34no—
171#3724—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178(C2×D3)⋊Dic3—34no—
179(C2×D3)⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181(C2×D3)⋊Dic3—34no—
182Dic32—34no—
183Dic32—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33—21no—
187C22×C33C2411yescharacteristic
188C2×C32×D3—21no—
189C2×C32×D3—21no—
190C2×C32×D3—21no—
191C33×C4—21no—
192C33×C4—21no—
193C3⋊(C3⋊Dic3)—21no—
194C3⋊(C3⋊Dic3)—21no—
195C3⋊(C3⋊Dic3)—21no—
196C3⋊(C3⋊Dic3)—21no—
197C3×C3⋊Dic3—21no—
198C3×C3⋊Dic3—21no—
199C3×C3⋊Dic3—21no—
200C3×C3⋊Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×#126—91no—
205C3×#123—34no—
206#936—94no—
207#935—94no—
208#924—94no—
209#924—94no—
210#898—91no—
211#897—91no—
212#896—91no—
213#890—91no—
214#889—94no—
215#889—94no—
216#889—91no—
217#888—94no—
218#888—94no—
219#888—91no—
220#886—91no—
221#885—91no—
222#883C3⋊D311yescharacteristic
223#879—34no—
224#879—94no—
225#879—94no—
226#879—94no—
227#879—91no—
228#878—94no—
229#878—94no—
230#878—91no—
231#878—91no—
232#878—184no—
233#876—91no—
234#869—91no—
235C23×C32—24no—
236C23×C32—31no—
237C23×C32C22×D314yes—
238C23×C32C22×D311yescharacteristic
239C22×C3×D3C22×D314yes—
240C32×D4—121no—
241C2×C32×C4—24no—
242C2×C32×C4—24no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—121no—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D311yescharacteristic
250C2×C32×C4C22×D311yescharacteristic
251C2×C3⋊Dic3—31no—
252C2×C3⋊Dic3—31no—
253C2×C3⋊Dic3—31no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—34no—
256C2×C3⋊Dic3—34no—
257C2×C3⋊Dic3—34no—
258C2×C3⋊Dic3—34no—
259C2×C3⋊Dic3—64no—
260C2×C3⋊Dic3—64no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—121no—
264C2×C3⋊Dic3—124no—
265C2×C3⋊Dic3—128no—
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270#634—121no—
271#634—121no—
272#634—121no—
273#634—124no—
274#634—128no—
275C3×#67—44no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3—34no—
280C2×C3×Dic3C22×D314yes—
281C2×C3×Dic3C22×D314yes—
282C2×C3×Dic3C22×D314yes—
283C3×C4×D3—44no—
284#627—124no—
285#627—124no—
286#563—271noSylow
287C2×C33—41no—
288C2×C33H(5,𝔽2)11yescharacteristic
289C2×C332−1+411yescharacteristic
290C2×C332−1+411yescharacteristic
291C32×D3—41no—
292C2×C3×D4—34no—
293C22×C3×C4—34no—
294C2×#67C2×C3⋊D311yescharacteristic
295C22×Dic3—94no—
296C22×Dic3—91no—
297C22×Dic3—184no—
298C2×C4×D3C2×C3⋊D311yescharacteristic
299C2×Dic6—94no—
300C2×Dic6—94no—
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—184no—
306C3×C4⋊C4—34no—
307C3×C4⋊C4—34no—
308C3×C4⋊C4—91no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—91no—
311C3×C22⋊C4—24no—
312C3×C22⋊C4—34no—
313C3×C22⋊C4—34no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4D3214yes—
317C3×C22⋊C4C2×C3⋊D311yescharacteristic
318C3×C42—91no—
319#217—34no—
320#217—34no—
321#217—94no—
322#217—94no—
323#217—94no—
324#217—94no—
325#217—94no—
326#217—91no—
327#217—184no—
328#217C2×C3⋊D311yescharacteristic
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
334C4⋊Dic3—34no—
335C4⋊Dic3—94no—
336C4⋊Dic3—94no—
337C4⋊Dic3—94no—
338C4⋊Dic3—91no—
339C4⋊Dic3—91no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—91no—
343C4⋊Dic3—184no—
344C4⋊Dic3C2×C3⋊D311yescharacteristic
345Dic3⋊C4—34no—
346Dic3⋊C4—34no—
347Dic3⋊C4—94no—
348Dic3⋊C4—94no—
349Dic3⋊C4—94no—
350Dic3⋊C4—94no—
351Dic3⋊C4—94no—
352Dic3⋊C4—94no—
353Dic3⋊C4—94no—
354Dic3⋊C4—94no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—91no—
361Dic3⋊C4—184no—
362Dic3⋊C4—184no—
363Dic3⋊C4C2×C3⋊D311yescharacteristic
364C4×Dic3—34no—
365C4×Dic3—94no—
366C4×Dic3—94no—
367C4×Dic3—91no—
368C4×Dic3—91no—
369C22×C32—28no—
370C22×C32—21no—
371C22×C32—24no—
372C22×C32—21no—
373C22×C32—21no—
374C22×C32—24no—
375C22×C32—24no—
376C22×C32—24no—
377C22×C32—44no—
378C22×C32—44no—
379C22×C32—61no—
380C22×C32—61no—
381C22×C32—61no—
382C22×C32C23×D314yes—
383C22×C32C23×D311yescharacteristic
384C2×C3×D3—24no—
385C2×C3×D3—24no—
386C2×C3×D3—24no—
387C32×C4—21no—
388C32×C4—21no—
389C32×C4—24no—
390C32×C4—24no—
391C32×C4—44no—
392C32×C4—44no—
393C32×C4—61no—
394C32×C4—61no—
395C32×C4—61no—
396C3⋊Dic3—21no—
397C3⋊Dic3—21no—
398C3⋊Dic3—21no—
399C3⋊Dic3—21no—
400C3⋊Dic3—68no—
401C3⋊Dic3—64no—
402C3⋊Dic3—61no—
403C3⋊Dic3—61no—
404C3⋊Dic3—61no—
405C3⋊Dic3—61no—
406C3⋊Dic3—64no—
407C3⋊Dic3—64no—
408C3⋊Dic3—64no—
409C3⋊Dic3—124no—
410C3⋊Dic3—124no—
411C3⋊Dic3—124no—
412C3×Dic3—24no—
413C3×Dic3—24no—
414C3×Dic3—24no—
415C3×Dic3—64no—
416C3×Dic3—64no—
417C3×Dic3—64no—
418C3×Dic3—64no—
419C4⋊Q8—271no—
420C4⋊Q8—271no—
421#126—91no—
422#126—271no—
423#126—271no—
424#126—271no—
425#125—271no—
426#125—271no—
427#124—271no—
428#123—31no—
429#123—271no—
430C22⋊Q8—271no—
431C22⋊Q8—271no—
432C22⋊Q8—271no—
433C22⋊Q8—271no—
434C33#56311yescharacteristic, complemented, Sylow
435C23×C3—24no—
436C23×C3—34no—
437C23×C3C2×D3214yes—
438C23×C3C22×C3⋊D311yescharacteristic
439C22×D3C22×C3⋊D311yescharacteristic
440C3×D4—124no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4—124no—
451C2×C3×C4C2×D3214yes—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C22×C3⋊D311yescharacteristic
454C2×C3×C4C22×C3⋊D311yescharacteristic
455#67—41no—
456C2×Dic3—34no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—34no—
460C2×Dic3—94no—
461C2×Dic3—94no—
462C2×Dic3—94no—
463C2×Dic3—94no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3—184no—
472C2×Dic3—364no—
473C2×Dic3—361no—
474C2×Dic3—368no—
475C2×Dic3C22×C3⋊D311yescharacteristic
476C2×Dic3C22×C3⋊D311yescharacteristic
477C2×Dic3C22×C3⋊D311yescharacteristic
478C4×D3—41no—
479Dic6—361no—
480Dic6—361no—
481Dic6—364no—
482Dic6—364no—
483Dic6—364no—
484Dic6—361no—
485Dic6—368no—
486C2×C32—24no—
487C2×C32—24no—
488C2×C32—24no—
489C2×C32—48no—
490C2×C32—44no—
491C2×C32—41no—
492C2×C32—121no—
493C2×C32#100414yes—
494C2×C32#100911yescharacteristic
495C2×C32#101014yes—
496C2×C32#101014yes—
497C2×C32#101011yescharacteristic
498C2×C32#101011yescharacteristic
499C3×D3—44no—
500C2×Q8—271no—
501C2×Q8—271no—
502C2×Q8—271no—
503C2×D4—31no—
504C22×C4—31no—
505C22×C4—271no—
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—
522C22⋊C4—31no—
523C22⋊C4—31no—
524C22⋊C4—91no—
525C22⋊C4—91no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4D3×C3⋊D311yescharacteristic
532C42—91no—
533C42—271no—
534C42—271no—
535C22×C3—28no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—24no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—24no—
543C22×C3—44no—
544C22×C3—44no—
545C22×C3—64no—
546C22×C3—64no—
547C22×C3—64no—
548C22×C3C22×D3214yes—
549C22×C3C23×C3⋊D311yescharacteristic
550C2×D3—21no—
551C2×D3—21no—
552C2×D3—21no—
553C3×C4—24no—
554C3×C4—24no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—44no—
558C3×C4—44no—
559C3×C4—64no—
560C3×C4—64no—
561C3×C4—64no—
562C3×C4—181no—
563C3×C4—181no—
564C3×C4—181no—
565C3×C4—181no—
566Dic3—21no—
567Dic3—21no—
568Dic3—21no—
569Dic3—64no—
570Dic3—64no—
571Dic3—64no—
572Dic3—64no—
573Dic3—188no—
574Dic3—184no—
575Dic3—184no—
576Dic3—184no—
577Dic3—184no—
578Dic3—181no—
579Dic3—181no—
580Dic3—181no—
581Dic3—181no—
582Dic3—364no—
583Dic3—364no—
584Dic3—364no—
585C32—24no—
586C32#565114yescomplemented
587C32#571211yescharacteristic, complemented
588C23—31no—
589C23C2×D3×C3⋊D311yescharacteristic
590Q8—1081no—
591Q8—1081no—
592Q8—1081no—
593D4—121no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—91no—
601C2×C4—121no—
602C2×C4—271no—
603C2×C4—271no—
604C2×C4—271no—
605C2×C4—271no—
606C2×C4—1081no—
607C2×C4C2×D3×C3⋊D311yescharacteristic
608C2×C4C2×D3×C3⋊D311yescharacteristic
609C2×C3—24no—
610C2×C3—24no—
611C2×C3—24no—
612C2×C3—48no—
613C2×C3—41no—
614C2×C3—44no—
615C2×C3—124no—
616C2×C3#6429814yes—
617C2×C3#6430914yes—
618C2×C3#6431414yes—
619C2×C3#6436111yescharacteristic
620C2×C3#6436711yescharacteristic
621C2×C3#6436711yescharacteristic
622D3—41no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—61no—
627C22—61no—
628C22—61no—
629C22C22×D3×C3⋊D311yescharacteristic
630C4—21no—
631C4—21no—
632C4—61no—
633C4—61no—
634C4—61no—
635C4—181no—
636C4—181no—
637C4—181no—
638C4—181no—
639C4—541no—
640C4—541no—
641C4—541no—
642C4—541no—
643C3—24no—
644C3#1059404414yescomplemented
645C3#1059612211yescharacteristic, complemented
646C2—41no—
647C2—121no—
648C2#1172456311yescharacteristic
649C2#1172458211yescharacteristic
650C2#1172458811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C32×#67C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723919C2semidirect
2. C3×#64088C2semidirect
4. #11723642C2semidirect
7. #11723597C2semidirect
8. #11723535C2semidirect
9. #11723430C2semidirect
10. #11723428C2semidirect
11. #11723426C2semidirect
12. #11723415C2semidirect
13. #11723412C2semidirect
14. #11723402C2semidirect
15. #11723395C2semidirect
65. #64088D3semidirect
434. C33#563semidirect
586. C32#5651semidirect
587. C32#5712semidirect
644. C3#10594044semidirect
645. C3#10596122semidirect

Maximal subgroups

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

subgroupindex
#117239192
C3×#640882
C32×#8832
#117236422
#117236032
#117236002
#117235972
#117235352
#117234302
#117234282
#117234262
#117234152
#117234122
#117234022
#117233952
#105961223
#105940443