Finite Group Databasebeta

← #13175889

Subgroups — #13175889

Order 1728.

1648 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723922C211yesmaximal, characteristic
2C3×#64088C211yesmaximal, characteristic, complemented
3C32×#879C211yesmaximal, characteristic, complemented
4#11723642C211yesmaximal, characteristic, complemented
5#11723598C211yesmaximal, characteristic, complemented
6#11723592C211yesmaximal, characteristic
7#11723592C211yesmaximal, characteristic
8#11723536C211yesmaximal, characteristic, complemented
9#11723452C211yesmaximal, characteristic, complemented
10#11723442C211yesmaximal, characteristic, complemented
11#11723411C211yesmaximal, characteristic, complemented
12#11723403C211yesmaximal, characteristic, complemented
13#11723391C211yesmaximal, characteristic, complemented
14#11723390C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10596173—31nomaximal
17#10594250—34nomaximal
18C2×#6312C2211yescharacteristic
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
21C2×#6293C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88859C2211yescharacteristic
26#88857C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3757C2211yescharacteristic
29C3×#3757C2211yescharacteristic
30C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×C4×C3⋊Dic3C2211yescharacteristic
34C32×#217C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39C32×C4×Dic3C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45#88804C2211yescharacteristic
46#88804C2211yescharacteristic
47#88804C2211yescharacteristic
48#88804C2211yescharacteristic
49Dic3×C3⋊Dic3C2211yescharacteristic
50Dic3×C3⋊Dic3C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#126—31no—
54#64107—31no—
55#64107—31no—
56#64105—31no—
57#64097—31no—
58#64096—31no—
59#64096—31no—
60#64094—31no—
61#64094—31no—
62#64090—31no—
63#64090—34no—
64#64090—64no—
65#64088D311yescharacteristic, complemented
66#64087—31no—
67#64086—31no—
68#64081—31no—
69C4×#634—31no—
70C3×#879—34no—
71C3×#879D314yescomplemented
72#63982—34no—
73#63957—34no—
74#63953—34no—
75#63953—34no—
76#63919—34no—
77#63874—34no—
78#63865—34no—
79#63848—34no—
80#63848—34no—
81#63838—34no—
82#63837—34no—
83#63837—34no—
84C23×C33C2311yescharacteristic
85C22×C3⋊(C3⋊D3)C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88#6312—22no—
89C2×C3⋊(C3⋊Dic3)—22no—
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
93C4×C3⋊(C3⋊D3)—22no—
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102#6293—22no—
103#6293—22no—
104#6293—22no—
105#5702—94no—
106#5702—91no—
107C2×#638—31no—
108C2×#638—34no—
109C2×#638—64no—
110C22×C3⋊Dic3—31no—
111C22×C3⋊Dic3—34no—
112C22×C3⋊Dic3—64no—
113C2×C4×C3⋊D3—31no—
114C2×C4×C3⋊D3—34no—
115C2×C4×C3⋊D3—64no—
116C2×#634—31no—
117C2×#634—31no—
118C2×#634—31no—
119C2×#627—34no—
120C2×#627—34no—
121C2×#627—34no—
122C32×C4⋊C4—31no—
123C32×C4⋊C4—31no—
124C32×C4⋊C4—31no—
125C32×C22⋊C4—24no—
126C32×C22⋊C4—31no—
127C32×C22⋊C4—31no—
128C32×C22⋊C4C2×D314yes—
129C32×C22⋊C4C2×D311yescharacteristic
130C32×C42—31no—
131#3757C2×D311yescharacteristic
132#3757C2×D311yescharacteristic
133#3752—31no—
134#3752—34no—
135#3752—31no—
136#3752—31no—
137#3752—31no—
138#3752—31no—
139#3752—64no—
140C4⋊(C3⋊Dic3)—31no—
141C4⋊(C3⋊Dic3)C2×D311yescharacteristic
142#3750—31no—
143#3750—34no—
144#3750—31no—
145#3750—31no—
146#3750—31no—
147#3750—31no—
148#3750—64no—
149#3750C2×D311yescharacteristic
150#3750C2×D311yescharacteristic
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—34no—
153C4×C3⋊Dic3—31no—
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—31no—
157C4×C3⋊Dic3—64no—
158C4×C3⋊Dic3C2×D311yescharacteristic
159C3×#217—34no—
160C3×#217—34no—
161C3×#217C2×D314yes—
162C3×#217C2×D314yes—
163C3×C4⋊Dic3—34no—
164C3×C4⋊Dic3C2×D314yes—
165C3×Dic3⋊C4—34no—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4C2×D314yes—
168C3×Dic3⋊C4C2×D314yes—
169C3×C4×Dic3—34no—
170C3×C4×Dic3C2×D314yes—
171#3724—34no—
172#3724—34no—
173#3724—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176#3722—34no—
177#3722—34no—
178#3722—34no—
179#3722—34no—
180Dic32—34no—
181Dic32—34no—
182Dic32—34no—
183Dic32—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33—21no—
187C22×C33C2411yescharacteristic
188C2×C3⋊(C3⋊D3)—21no—
189C2×C3⋊(C3⋊D3)—21no—
190C2×C3⋊(C3⋊D3)—21no—
191C33×C4—21no—
192C33×C4—21no—
193C3⋊(C3⋊Dic3)—21no—
194C3⋊(C3⋊Dic3)#4112yes—
195C3⋊(C3⋊Dic3)#4112yes—
196C3×C3⋊Dic3—21no—
197C3×C3⋊Dic3—21no—
198C3×C3⋊Dic3—21no—
199C3×C3⋊Dic3—21no—
200C32×Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×#126—34no—
205C3×#126—91no—
206#898—94no—
207#898—94no—
208#898—91no—
209#898—91no—
210#896—94no—
211#896—91no—
212#888—94no—
213#888—91no—
214#887—94no—
215#887—94no—
216#887—91no—
217#887—91no—
218#885—94no—
219#885—94no—
220#885—91no—
221#885—91no—
222#67⋊C4—94no—
223#67⋊C4—91no—
224#67⋊C4—184no—
225#879—34no—
226#879C3⋊D311yescharacteristic, complemented
227#878—94no—
228#878—91no—
229#877—94no—
230#877—91no—
231#872—94no—
232#872—91no—
233C4×Dic6—94no—
234C4×Dic6—91no—
235C23×C32—24no—
236C23×C32C22×D314yes—
237C23×C32C22×D311yescharacteristic
238C22×C3⋊D3—31no—
239C22×C3⋊D3—34no—
240C22×C3⋊D3—64no—
241C2×C32×C4—24no—
242C2×C32×C4—24no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D311yescharacteristic
250C2×C32×C4C22×D311yescharacteristic
251#638—62no—
252#638—68no—
253#638—616no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—31no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—34no—
258C2×C3⋊Dic3—34no—
259C2×C3⋊Dic3—34no—
260C2×C3⋊Dic3—64no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—62no—
264C2×C3⋊Dic3—68no—
265C2×C3⋊Dic3—128no—
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270C4×C3⋊D3—62no—
271C4×C3⋊D3—68no—
272C4×C3⋊D3—128no—
273#634—62no—
274#634—62no—
275#634—62no—
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—
283C2×C3×Dic3C22×D314yes—
284#627—68no—
285#627—68no—
286#627—68no—
287#556—271noSylow
288C2×C33—41no—
289C2×C33C2×#4111yescharacteristic
290C2×C33C2×#4111yescharacteristic
291C2×C332−1+411yescharacteristic
292C3⋊(C3⋊D3)—41no—
293C2×#67—94no—
294C2×#67—91no—
295C2×#67—184no—
296C22×Dic3—94no—
297C22×Dic3—91no—
298C22×Dic3—184no—
299C2×C4×D3—94no—
300C2×C4×D3—91no—
301C2×C4×D3—184no—
302C2×Dic6—94no—
303C2×Dic6—94no—
304C2×Dic6—94no—
305C2×Dic6—91no—
306C2×Dic6—91no—
307C2×Dic6—91no—
308C3×C4⋊C4—34no—
309C3×C4⋊C4—34no—
310C3×C4⋊C4—34no—
311C3×C4⋊C4—91no—
312C3×C4⋊C4—91no—
313C3×C4⋊C4—91no—
314C3×C22⋊C4—24no—
315C3×C22⋊C4—34no—
316C3×C22⋊C4—34no—
317C3×C22⋊C4—91no—
318C3×C22⋊C4—91no—
319C3×C22⋊C4D3214yes—
320C3×C22⋊C4C2×C3⋊D311yescharacteristic
321C3×C42—34no—
322C3×C42—91no—
323#217—34no—
324#217—34no—
325#217C2×C3⋊D311yescharacteristic
326#217C2×C3⋊D311yescharacteristic
327(C2×D3)⋊C4—94no—
328(C2×D3)⋊C4—94no—
329(C2×D3)⋊C4—94no—
330(C2×D3)⋊C4—94no—
331(C2×D3)⋊C4—94no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—91no—
334(C2×D3)⋊C4—91no—
335(C2×D3)⋊C4—91no—
336(C2×D3)⋊C4—91no—
337(C2×D3)⋊C4—184no—
338C4⋊Dic3—34no—
339C4⋊Dic3—94no—
340C4⋊Dic3—91no—
341C4⋊Dic3C2×C3⋊D311yescharacteristic
342Dic3⋊C4—34no—
343Dic3⋊C4—34no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347Dic3⋊C4—94no—
348Dic3⋊C4—94no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—184no—
355Dic3⋊C4C2×C3⋊D311yescharacteristic
356Dic3⋊C4C2×C3⋊D311yescharacteristic
357C4×Dic3—34no—
358C4×Dic3—94no—
359C4×Dic3—94no—
360C4×Dic3—94no—
361C4×Dic3—94no—
362C4×Dic3—94no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3—184no—
369C4×Dic3C2×C3⋊D311yescharacteristic
370C22×C32—28no—
371C22×C32—21no—
372C22×C32—24no—
373C22×C32—21no—
374C22×C32—21no—
375C22×C32—24no—
376C22×C32—24no—
377C22×C32—24no—
378C22×C32—44no—
379C22×C32—44no—
380C22×C32C23×D314yes—
381C22×C32C23×D311yescharacteristic
382C2×C3⋊D3—68no—
383C2×C3⋊D3—61no—
384C2×C3⋊D3—64no—
385C2×C3⋊D3—61no—
386C2×C3⋊D3—61no—
387C2×C3⋊D3—64no—
388C2×C3⋊D3—64no—
389C2×C3⋊D3—124no—
390C2×C3⋊D3—124no—
391C32×C4—21no—
392C32×C4—21no—
393C32×C4—24no—
394C32×C4—24no—
395C32×C4—44no—
396C32×C4—44no—
397C32×C4—61no—
398C32×C4—61no—
399C32×C4—61no—
400C32×C4—61no—
401C3⋊Dic3—21no—
402C3⋊Dic3—21no—
403C3⋊Dic3—21no—
404C3⋊Dic3—21no—
405C3⋊Dic3—32no—
406C3⋊Dic3—32no—
407C3⋊Dic3—38no—
408C3⋊Dic3—38no—
409C3⋊Dic3—68no—
410C3⋊Dic3—68no—
411C3⋊Dic3—61no—
412C3⋊Dic3—64no—
413C3⋊Dic3—124no—
414C3×Dic3—24no—
415C3×Dic3—24no—
416C3×Dic3—24no—
417C3×Dic3—24no—
418C3×Dic3—64no—
419C3×Dic3—64no—
420C3×Dic3—64no—
421C3×Dic3—64no—
422C4⋊Q8—271no—
423#126—31no—
424#126—91no—
425#126—271no—
426#126—271no—
427#124—271no—
428#124—271no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C4×Q8—271no—
432C4×Q8—271no—
433C4×Q8—271no—
434C4×D4—271no—
435#117—271no—
436#117—271no—
437C33#55611yescharacteristic, complemented, Sylow
438C23×C3—24no—
439C23×C3C2×D3214yes—
440C23×C3C22×C3⋊D311yescharacteristic
441C22×D3—94no—
442C22×D3—91no—
443C22×D3—184no—
444C2×C3×C4—24no—
445C2×C3×C4—24no—
446C2×C3×C4—34no—
447C2×C3×C4—34no—
448C2×C3×C4—34no—
449C2×C3×C4—34no—
450C2×C3×C4—91no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4C2×D3214yes—
455C2×C3×C4C2×D3214yes—
456C2×C3×C4C22×C3⋊D311yescharacteristic
457C2×C3×C4C22×C3⋊D311yescharacteristic
458#67—188no—
459#67—182no—
460#67—1816no—
461C2×Dic3—34no—
462C2×Dic3—34no—
463C2×Dic3—34no—
464C2×Dic3—34no—
465C2×Dic3—94no—
466C2×Dic3—94no—
467C2×Dic3—91no—
468C2×Dic3—91no—
469C2×Dic3—94no—
470C2×Dic3—91no—
471C2×Dic3—184no—
472C2×Dic3—184no—
473C2×Dic3—184no—
474C2×Dic3—182no—
475C2×Dic3—188no—
476C2×Dic3—368no—
477C2×Dic3C22×C3⋊D311yescharacteristic
478C2×Dic3C22×C3⋊D311yescharacteristic
479C2×Dic3C22×C3⋊D311yescharacteristic
480C2×Dic3C22×C3⋊D311yescharacteristic
481C4×D3—182no—
482C4×D3—188no—
483C4×D3—368no—
484Dic6—182no—
485Dic6—182no—
486Dic6—182no—
487Dic6—188no—
488Dic6—188no—
489Dic6—188no—
490C2×C32—24no—
491C2×C32—24no—
492C2×C32—24no—
493C2×C32—48no—
494C2×C32—44no—
495C2×C32—41no—
496C2×C32D3×#4114yes—
497C2×C32D3×#4114yes—
498C2×C32D3×#4111yescharacteristic
499C2×C32D3×#4111yescharacteristic
500C2×C32#101014yes—
501C2×C32#101011yescharacteristic
502C3⋊D3—128no—
503C3⋊D3—124no—
504C3⋊D3—121no—
505C2×Q8—271no—
506C2×Q8—271no—
507C2×Q8—271no—
508C2×D4—271no—
509C22×C4—271no—
510C22×C4—271no—
511C4⋊C4—31no—
512C4⋊C4—31no—
513C4⋊C4—31no—
514C4⋊C4—91no—
515C4⋊C4—91no—
516C4⋊C4—91no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C22⋊C4—31no—
524C22⋊C4—31no—
525C22⋊C4—91no—
526C22⋊C4—91no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4D3×C3⋊D311yescharacteristic
533C42—31no—
534C42—91no—
535C42—271no—
536C42—271no—
537C42—271no—
538C42—271no—
539C42—271no—
540C22×C3—28no—
541C22×C3—24no—
542C22×C3—24no—
543C22×C3—24no—
544C22×C3—21no—
545C22×C3—21no—
546C22×C3—21no—
547C22×C3—24no—
548C22×C3—44no—
549C22×C3—44no—
550C22×C3C22×D3214yes—
551C22×C3C23×C3⋊D311yescharacteristic
552C2×D3—188no—
553C2×D3—184no—
554C2×D3—184no—
555C2×D3—184no—
556C2×D3—181no—
557C2×D3—181no—
558C2×D3—181no—
559C2×D3—364no—
560C2×D3—364no—
561C3×C4—24no—
562C3×C4—24no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—44no—
566C3×C4—44no—
567C3×C4—64no—
568C3×C4—64no—
569C3×C4—64no—
570C3×C4—64no—
571C3×C4—181no—
572C3×C4—181no—
573C3×C4—181no—
574C3×C4—181no—
575Dic3—21no—
576Dic3—21no—
577Dic3—21no—
578Dic3—21no—
579Dic3—64no—
580Dic3—64no—
581Dic3—64no—
582Dic3—64no—
583Dic3—98no—
584Dic3—98no—
585Dic3—92no—
586Dic3—92no—
587Dic3—188no—
588Dic3—188no—
589Dic3—184no—
590Dic3—181no—
591Dic3—364no—
592C32—24no—
593C32#570214yescomplemented
594C32#570211yescharacteristic, complemented
595C23—271no—
596C23C2×D3×C3⋊D311yescharacteristic
597Q8—542no—
598Q8—542no—
599Q8—542no—
600D4—542no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—271no—
610C2×C4—271no—
611C2×C4—271no—
612C2×C4—542no—
613C2×C4—542no—
614C2×C4C2×D3×C3⋊D311yescharacteristic
615C2×C4C2×D3×C3⋊D311yescharacteristic
616C2×C3—24no—
617C2×C3—24no—
618C2×C3—24no—
619C2×C3—48no—
620C2×C3—41no—
621C2×C3—44no—
622C2×C3#6430614yes—
623C2×C3#6430914yes—
624C2×C3#6431214yes—
625C2×C3#41×C3⋊D311yescharacteristic
626C2×C3#41×C3⋊D311yescharacteristic
627C2×C3#6436711yescharacteristic
628D3—368no—
629D3—361no—
630D3—364no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—541no—
635C22—541no—
636C22—541no—
637C22C22×D3×C3⋊D311yescharacteristic
638C4—21no—
639C4—21no—
640C4—61no—
641C4—61no—
642C4—61no—
643C4—61no—
644C4—181no—
645C4—181no—
646C4—181no—
647C4—181no—
648C4—272no—
649C4—272no—
650C4—541no—
651C3—24no—
652C3#1059425014yescomplemented
653C3#1059617311yescharacteristic, complemented
654C2—41no—
655C2—1081no—
656C2#1172457811yescharacteristic
657C2#1172458211yescharacteristic
658C2#1172458611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×#6312C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64088C2semidirect
3. C32×#879C2semidirect
4. #11723642C2semidirect
5. #11723598C2semidirect
8. #11723536C2semidirect
9. #11723452C2semidirect
10. #11723442C2semidirect
11. #11723411C2semidirect
12. #11723403C2semidirect
13. #11723391C2semidirect
14. #11723390C2semidirect
15. #11723389C2semidirect
65. #64088D3semidirect
71. C3×#879D3semidirect
226. #879C3⋊D3semidirect
437. C33#556semidirect
593. C32#5702semidirect
594. C32#5702semidirect
652. C3#10594250semidirect
653. C3#10596173semidirect

Maximal subgroups

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

subgroupindex
#117239222
C3×#640882
C32×#8792
#117236422
#117235982
#117235922
#117235362
#117234522
#117234422
#117234112
#117234032
#117233912
#117233902
#117233892
#105961733
#105942503