Finite Group Databasebeta

← #13175890

Subgroups — #13175890

Order 1728.

1660 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×#64086C211yesmaximal, characteristic, complemented
3C32×#877C211yesmaximal, characteristic, complemented
4#11723642C211yesmaximal, characteristic, complemented
5#11723612C211yesmaximal, characteristic
6C2×#88806C211yesmaximal, characteristic, complemented
7#11723608C211yesmaximal, characteristic
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723453C211yesmaximal, characteristic, complemented
10#11723452C211yesmaximal, characteristic, complemented
11#11723448C211yesmaximal, characteristic, complemented
12#11723446C211yesmaximal, characteristic, complemented
13#11723445C211yesmaximal, characteristic, complemented
14#11723444C211yesmaximal, characteristic, complemented
15#11723442C211yesmaximal, characteristic, complemented
16#10596002—31nomaximal
17#10594046—34nomaximal
18C2×#6312C2211yescharacteristic
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
21C22×C3×C3⋊Dic3C2211yescharacteristic
22C22×C32×Dic3C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88859C2211yescharacteristic
26#88857C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3757C2211yescharacteristic
29C3×#3750C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×C4×C3⋊Dic3C2211yescharacteristic
32C3×C4×C3⋊Dic3C2211yescharacteristic
33C32×#217C2211yescharacteristic
34C32×Dic3⋊C4C2211yescharacteristic
35C32×Dic3⋊C4C2211yescharacteristic
36C32×C4×Dic3C2211yescharacteristic
37C32×C4×Dic3C2211yescharacteristic
38#88807C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45#88806C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47Dic3⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3⋊(C3⋊Dic3)C2211yescharacteristic
49#88804C2211yescharacteristic
50#88804C2211yescharacteristic
51#88804C2211yescharacteristic
52#88804C2211yescharacteristic
53C32×#117—31no—
54#64138—31no—
55C2×#3750—31no—
56#64107—31no—
57#64105—31no—
58#64098—31no—
59#64098—31no—
60#64095—31no—
61#64090—31no—
62#64090—34no—
63#64090—64no—
64#64087—31no—
65#64086D311yescharacteristic, complemented
66#64085—31no—
67#64081—31no—
68#64079—31no—
69C4×#634—31no—
70C3×#877—34no—
71C3×#877D314yescomplemented
72#63982—34no—
73C2×#3724—34no—
74#63966—34no—
75#63966—34no—
76C4×#627—34no—
77#63874—34no—
78#63870—34no—
79#63868—34no—
80#63868—34no—
81#63867—34no—
82#63867—34no—
83#63865—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⋊Dic3—22no—
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C2×C3×C3⋊Dic3C2311yescharacteristic
99C2×C32×Dic3—22no—
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C2×C32×Dic3C2311yescharacteristic
103C2×C32×Dic3C2311yescharacteristic
104#6293—22no—
105#5531—94no—
106#5531—91no—
107C22×C32×C4—31no—
108C2×#638—31no—
109C2×#638—34no—
110C2×#638—64no—
111C22×C3⋊Dic3—31no—
112C22×C3⋊Dic3—34no—
113C22×C3⋊Dic3—64no—
114C22×C3⋊Dic3C2×D311yescharacteristic
115C2×C4×C3⋊D3—31no—
116C2×C4×C3⋊D3—34no—
117C2×C4×C3⋊D3—64no—
118C2×#634—31no—
119C22×C3×Dic3—34no—
120C22×C3×Dic3C2×D314yes—
121C2×#627—34no—
122C32×C4⋊C4—31no—
123C32×C4⋊C4—31no—
124C32×C22⋊C4—24no—
125C32×C22⋊C4—31no—
126C32×C22⋊C4C2×D314yes—
127C32×C22⋊C4C2×D311yescharacteristic
128C32×C42—31no—
129C32×C42—31no—
130#3757C2×D311yescharacteristic
131#3752—31no—
132#3752—34no—
133#3752—31no—
134#3752—31no—
135#3752—31no—
136#3752—31no—
137#3752—64no—
138C4⋊(C3⋊Dic3)—31no—
139C4⋊(C3⋊Dic3)—31no—
140C4⋊(C3⋊Dic3)—31no—
141#3750—31no—
142#3750—34no—
143#3750—31no—
144#3750—31no—
145#3750—31no—
146#3750—31no—
147#3750—31no—
148#3750—31no—
149#3750—31no—
150#3750—31no—
151#3750—64no—
152#3750C2×D311yescharacteristic
153#3750C2×D311yescharacteristic
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—34no—
156C4×C3⋊Dic3—64no—
157C4×C3⋊Dic3C2×D311yescharacteristic
158C4×C3⋊Dic3C2×D311yescharacteristic
159C3×#217—34no—
160C3×#217C2×D314yes—
161C3×Dic3⋊C4—34no—
162C3×Dic3⋊C4—34no—
163C3×Dic3⋊C4C2×D314yes—
164C3×Dic3⋊C4C2×D314yes—
165C3×C4×Dic3—34no—
166C3×C4×Dic3—34no—
167C3×C4×Dic3C2×D314yes—
168C3×C4×Dic3C2×D314yes—
169#3724—34no—
170#3724—34no—
171#3724—34no—
172#3724—34no—
173#3724—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178Dic3⋊Dic3—34no—
179Dic3⋊Dic3—34no—
180#3722—34no—
181#3722—34no—
182#3722—34no—
183#3722—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33#4112yes—
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)—21no—
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×#117—34no—
205C3×#117—91no—
206#929—94no—
207#929—91no—
208C2×Dic3⋊C4—94no—
209C2×Dic3⋊C4—91no—
210#898—94no—
211#898—91no—
212#896—94no—
213#896—91no—
214#889—94no—
215#889—94no—
216#889—91no—
217#889—91no—
218#886—94no—
219#886—91no—
220#67⋊C4—94no—
221#67⋊C4—91no—
222#67⋊C4—184no—
223#878—94no—
224#878—91no—
225#877—34no—
226#877C3⋊D311yescharacteristic, complemented
227#876—94no—
228#876—91no—
229#872—94no—
230#872—91no—
231#870—94no—
232#870—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×C4—62no—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D311yescharacteristic
251C2×C32×C4C22×D311yescharacteristic
252#638—616no—
253#638—68no—
254#638—62no—
255C2×C3⋊Dic3—22no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—31no—
258C2×C3⋊Dic3—31no—
259C2×C3⋊Dic3—34no—
260C2×C3⋊Dic3—34no—
261C2×C3⋊Dic3—34no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—64no—
264C2×C3⋊Dic3—64no—
265C2×C3⋊Dic3—62no—
266C2×C3⋊Dic3—68no—
267C2×C3⋊Dic3—128no—
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270C2×C3⋊Dic3C22×D311yescharacteristic
271C2×C3⋊Dic3C22×D311yescharacteristic
272C4×C3⋊D3—62no—
273C4×C3⋊D3—68no—
274C4×C3⋊D3—128no—
275#634—62no—
276C2×C3×Dic3—28no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3—34no—
280C2×C3×Dic3—34no—
281C2×C3×Dic3—68no—
282C2×C3×Dic3C22×D314yes—
283C2×C3×Dic3C22×D314yes—
284C2×C3×Dic3C22×D314yes—
285C2×C3×Dic3C22×D314yes—
286#627—68no—
287#552—271noSylow
288C2×C33—22no—
289C2×C33C2×#4111yescharacteristic
290C2×C33C2×#4111yescharacteristic
291C2×C332−1+411yescharacteristic
292C3⋊(C3⋊D3)—41no—
293C22×C3×C4—34no—
294C22×C3×C4—91no—
295C2×#67—94no—
296C2×#67—91no—
297C2×#67—184no—
298C22×Dic3—34no—
299C22×Dic3—94no—
300C22×Dic3—91no—
301C22×Dic3—184no—
302C22×Dic3C2×C3⋊D311yescharacteristic
303C2×C4×D3—94no—
304C2×C4×D3—91no—
305C2×C4×D3—184no—
306C2×Dic6—94no—
307C2×Dic6—91no—
308C3×C4⋊C4—34no—
309C3×C4⋊C4—34no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C22⋊C4—24no—
313C3×C22⋊C4—34no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4D3214yes—
316C3×C22⋊C4C2×C3⋊D311yescharacteristic
317C3×C42—34no—
318C3×C42—34no—
319C3×C42—91no—
320C3×C42—91no—
321#217—34no—
322#217C2×C3⋊D311yescharacteristic
323(C2×D3)⋊C4—94no—
324(C2×D3)⋊C4—94no—
325(C2×D3)⋊C4—94no—
326(C2×D3)⋊C4—94no—
327(C2×D3)⋊C4—94no—
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—184no—
334C4⋊Dic3—94no—
335C4⋊Dic3—94no—
336C4⋊Dic3—94no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—91no—
340Dic3⋊C4—34no—
341Dic3⋊C4—34no—
342Dic3⋊C4—94no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347Dic3⋊C4—94no—
348Dic3⋊C4—94no—
349Dic3⋊C4—94no—
350Dic3⋊C4—94no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—184no—
361Dic3⋊C4C2×C3⋊D311yescharacteristic
362Dic3⋊C4C2×C3⋊D311yescharacteristic
363C4×Dic3—34no—
364C4×Dic3—34no—
365C4×Dic3—94no—
366C4×Dic3—91no—
367C4×Dic3—184no—
368C4×Dic3C2×C3⋊D311yescharacteristic
369C4×Dic3C2×C3⋊D311yescharacteristic
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—24no—
373C22×C32—28no—
374C22×C32—24no—
375C22×C32—24no—
376C22×C32—44no—
377C22×C32—44no—
378C22×C32#23518yes—
379C22×C32#23512yes—
380C22×C32C23×D314yes—
381C22×C32C23×D311yescharacteristic
382C2×C3⋊D3—68no—
383C2×C3⋊D3—64no—
384C2×C3⋊D3—61no—
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—38no—
407C3⋊Dic3—68no—
408C3⋊Dic3—68no—
409C3⋊Dic3—64no—
410C3⋊Dic3—61no—
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—
422#126—271no—
423#126—271no—
424#125—271no—
425#125—271no—
426#125—271no—
427#123—271no—
428#123—271no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C4×Q8—271no—
432C4×D4—271no—
433#117—31no—
434#117—91no—
435#117—271no—
436C2×C4⋊C4—271no—
437C33#55211yescharacteristic, 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—68no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—91no—
455C2×C3×C4—182no—
456C2×C3×C4C2×D3214yes—
457C2×C3×C4C2×D3214yes—
458C2×C3×C4C22×C3⋊D311yescharacteristic
459C2×C3×C4C22×C3⋊D311yescharacteristic
460#67—1816no—
461#67—188no—
462#67—182no—
463C2×Dic3—22no—
464C2×Dic3—34no—
465C2×Dic3—34no—
466C2×Dic3—34no—
467C2×Dic3—34no—
468C2×Dic3—68no—
469C2×Dic3—94no—
470C2×Dic3—91no—
471C2×Dic3—94no—
472C2×Dic3—94no—
473C2×Dic3—91no—
474C2×Dic3—91no—
475C2×Dic3—184no—
476C2×Dic3—184no—
477C2×Dic3—184no—
478C2×Dic3—182no—
479C2×Dic3—188no—
480C2×Dic3—368no—
481C2×Dic3C22×C3⋊D311yescharacteristic
482C2×Dic3C22×C3⋊D311yescharacteristic
483C2×Dic3C22×C3⋊D311yescharacteristic
484C2×Dic3C22×C3⋊D311yescharacteristic
485C4×D3—182no—
486C4×D3—188no—
487C4×D3—368no—
488Dic6—182no—
489Dic6—188no—
490C2×C32—22no—
491C2×C32—24no—
492C2×C32—24no—
493C2×C32—24no—
494C2×C32—28no—
495C2×C32—48no—
496C2×C32C2×#23514yes—
497C2×C32C2×#23511yescharacteristic
498C2×C32D3×#4114yes—
499C2×C32D3×#4111yescharacteristic
500C2×C32#101014yes—
501C2×C32#101011yescharacteristic
502C3⋊D3—128no—
503C3⋊D3—124no—
504C3⋊D3—121no—
505C2×Q8—271no—
506C2×D4—271no—
507C22×C4—31no—
508C22×C4—91no—
509C22×C4—271no—
510C22×C4—271no—
511C4⋊C4—31no—
512C4⋊C4—31no—
513C4⋊C4—91no—
514C4⋊C4—91no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C4⋊C4—271no—
524C4⋊C4—271no—
525C4⋊C4—271no—
526C4⋊C4—271no—
527C22⋊C4—31no—
528C22⋊C4—91no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C22⋊C4—271no—
534C22⋊C4D3×C3⋊D311yescharacteristic
535C42—31no—
536C42—31no—
537C42—91no—
538C42—91no—
539C42—271no—
540C22×C3—24no—
541C22×C3—24no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—24no—
545C22×C3—28no—
546C22×C3—44no—
547C22×C3—44no—
548C22×C3#379818yes—
549C22×C3#382812yes—
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—92no—
585Dic3—188no—
586Dic3—188no—
587Dic3—184no—
588Dic3—184no—
589Dic3—181no—
590Dic3—181no—
591Dic3—364no—
592C32—24no—
593C32#553114yescomplemented
594C32#553111yescharacteristic, complemented
595C23—271no—
596C23C2×D3×C3⋊D311yescharacteristic
597Q8—542no—
598D4—542no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—62no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—182no—
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—22no—
617C2×C3—28no—
618C2×C3—24no—
619C2×C3—24no—
620C2×C3—24no—
621C2×C3—48no—
622C2×C3C2×#379814yes—
623C2×C3#6430914yes—
624C2×C3#6431214yes—
625C2×C3C2×#382811yescharacteristic
626C2×C3#41×C3⋊D311yescharacteristic
627C2×C3#6436711yescharacteristic
628D3—368no—
629D3—361no—
630D3—364no—
631C22—21no—
632C22—21no—
633C22—541no—
634C22—541no—
635C22—541no—
636C22#8902012yes—
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—541no—
650C4—541no—
651C3—24no—
652C3#1059404614yescomplemented
653C3#1059600211yescharacteristic, complemented
654C2—22no—
655C2—1081no—
656C2C2×#8902011yescharacteristic
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#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64086C2semidirect
3. C32×#877C2semidirect
4. #11723642C2semidirect
6. C2×#88806C2semidirect
8. C4×#6293C2semidirect
9. #11723453C2semidirect
10. #11723452C2semidirect
11. #11723448C2semidirect
12. #11723446C2semidirect
13. #11723445C2semidirect
14. #11723444C2semidirect
15. #11723442C2semidirect
65. #64086D3semidirect
71. C3×#877D3semidirect
226. #877C3⋊D3semidirect
437. C33#552semidirect
593. C32#5531semidirect
594. C32#5531semidirect
652. C3#10594046semidirect
653. C3#10596002semidirect

Maximal subgroups

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

subgroupindex
#117239222
C3×#640862
C32×#8772
#117236422
#117236122
C2×#888062
#117236082
C4×#62932
#117234532
#117234522
#117234482
#117234462
#117234452
#117234442
#117234422
#105960023
#105940463