Finite Group Databasebeta

← #13176512

Subgroups — #13176512

Order 1728.

1448 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723937C211yesmaximal, characteristic
2C3×#64098C211yesmaximal, characteristic, complemented
3C32×#890C211yesmaximal, characteristic, complemented
4#11723453C211yesmaximal, characteristic
5#11723446C211yesmaximal, characteristic
6#11723442C211yesmaximal, characteristic
7#11723413C211yesmaximal, characteristic
8#11723411C211yesmaximal, characteristic
9#11723411C211yesmaximal, characteristic
10#11723406C211yesmaximal, characteristic, complemented
11#11723405C211yesmaximal, characteristic, complemented
12#11723405C211yesmaximal, characteristic, complemented
13#11723401C211yesmaximal, characteristic, complemented
14#11723395C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10596167—31nomaximal
17#10594323—34nomaximal
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C33×C4⋊C4C2211yescharacteristic
22#88859C2211yescharacteristic
23#88859C2211yescharacteristic
24#88857C2211yescharacteristic
25#88857C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#3750C2211yescharacteristic
28C3×#3750C2211yescharacteristic
29C3×#3750C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×C4×C3⋊Dic3C2211yescharacteristic
32C32×C4⋊Dic3C2211yescharacteristic
33C32×C4⋊Dic3C2211yescharacteristic
34C32×C4⋊Dic3C2211yescharacteristic
35C32×Dic3⋊C4C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×C4×Dic3C2211yescharacteristic
38#88807C2211yescharacteristic
39#88806C2211yescharacteristic
40#88806C2211yescharacteristic
41#88806C2211yescharacteristic
42Dic3⋊(C3⋊Dic3)C2211yescharacteristic
43Dic3⋊(C3⋊Dic3)C2211yescharacteristic
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×#125—31no—
54#64107—31no—
55#64107—31no—
56#64107—31no—
57#64105—31no—
58#64105—31no—
59#64105—34no—
60#64105—64no—
61#64101—31no—
62#64099—31no—
63#64098—31no—
64#64098—31no—
65#64098D311yescharacteristic, complemented
66#64097—31no—
67#64096—31no—
68#64085—31no—
69C4×#634—31no—
70C3×#890D314yescomplemented
71C3×#889—34no—
72#63868—34no—
73#63868—34no—
74#63865—34no—
75#63849—34no—
76#63848—34no—
77#63848—34no—
78#63847—34no—
79#63847—34no—
80#63846—34no—
81#63844—34no—
82#63840—34no—
83#63837—34no—
84C22×C3⋊(C3⋊D3)C2311yescharacteristic
85C2×C33×C4C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C4×C3⋊(C3⋊D3)—22no—
92#6308—22no—
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101#6293—22no—
102#5696—94no—
103#5696—91no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—34no—
106C2×C4×C3⋊D3—64no—
107C2×#634—31no—
108C2×#634—34no—
109C2×#634—31no—
110C2×#634—64no—
111C2×#627—34no—
112C32×C4⋊C4—24no—
113C32×C4⋊C4—31no—
114C32×C4⋊C4—31no—
115C32×C4⋊C4—31no—
116C32×C4⋊C4—31no—
117C32×C4⋊C4—31no—
118C32×C4⋊C4C2×D314yes—
119C32×C4⋊C4C2×D311yescharacteristic
120C32×C42—31no—
121#3752—31no—
122#3752—31no—
123#3752—34no—
124#3752—34no—
125#3752—31no—
126#3752—31no—
127#3752—31no—
128#3752—31no—
129#3752—64no—
130#3752—64no—
131C4⋊(C3⋊Dic3)—31no—
132C4⋊(C3⋊Dic3)—31no—
133C4⋊(C3⋊Dic3)—31no—
134C4⋊(C3⋊Dic3)C2×D311yescharacteristic
135#3750—31no—
136#3750—31no—
137#3750—34no—
138#3750—34no—
139#3750—31no—
140#3750—31no—
141#3750—31no—
142#3750—31no—
143#3750—64no—
144#3750—64no—
145#3750C2×D311yescharacteristic
146#3750C2×D311yescharacteristic
147#3750C2×D311yescharacteristic
148#3750C2×D311yescharacteristic
149C4×C3⋊Dic3—31no—
150C4×C3⋊Dic3—31no—
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—31no—
153C4×C3⋊Dic3C2×D311yescharacteristic
154C3×C4⋊Dic3—34no—
155C3×C4⋊Dic3C2×D314yes—
156C3×C4⋊Dic3C2×D314yes—
157C3×C4⋊Dic3C2×D314yes—
158C3×Dic3⋊C4—34no—
159C3×Dic3⋊C4—34no—
160C3×Dic3⋊C4—34no—
161C3×Dic3⋊C4—34no—
162C3×Dic3⋊C4C2×D314yes—
163C3×Dic3⋊C4C2×D314yes—
164C3×C4×Dic3—34no—
165C3×C4×Dic3C2×D314yes—
166#3724—34no—
167#3724—34no—
168#3724—34no—
169Dic3⋊Dic3—34no—
170Dic3⋊Dic3—34no—
171Dic3⋊Dic3—34no—
172Dic3⋊Dic3—34no—
173#3722—34no—
174#3722—34no—
175#3722—34no—
176#3722—34no—
177Dic32—34no—
178Dic32—34no—
179Dic32—34no—
180Dic32—34no—
181C22×C33C2411yescharacteristic
182C2×C3⋊(C3⋊D3)—21no—
183C2×C3⋊(C3⋊D3)—21no—
184C2×C3⋊(C3⋊D3)—21no—
185C33×C4—21no—
186C33×C4—21no—
187C33×C4—21no—
188C3⋊(C3⋊Dic3)—21no—
189C3⋊(C3⋊Dic3)—21no—
190C3⋊(C3⋊Dic3)#4112yes—
191C3×C3⋊Dic3—21no—
192C3×C3⋊Dic3—21no—
193C3×C3⋊Dic3—21no—
194C3×C3⋊Dic3—21no—
195C32×Dic3—21no—
196C32×Dic3—21no—
197C32×Dic3—21no—
198C32×Dic3—21no—
199C3×#125—34no—
200C3×#125—91no—
201#898—94no—
202#898—94no—
203#898—94no—
204#898—91no—
205#898—91no—
206#898—91no—
207#896—94no—
208#896—94no—
209#896—91no—
210#896—91no—
211#896—184no—
212#892—94no—
213#892—91no—
214#890—94no—
215#890C3⋊D311yescharacteristic, complemented
216#889—34no—
217#889—94no—
218#889—94no—
219#889—91no—
220#889—91no—
221#889—91no—
222#888—94no—
223#888—91no—
224#887—94no—
225#887—91no—
226#876—94no—
227#876—91no—
228C4×Dic6—94no—
229C4×Dic6—91no—
230C22×C3⋊D3—31no—
231C22×C3⋊D3—34no—
232C22×C3⋊D3—64no—
233C2×C32×C4—24no—
234C2×C32×C4—24no—
235C2×C32×C4—24no—
236C2×C32×C4—31no—
237C2×C32×C4—31no—
238C2×C32×C4—31no—
239C2×C32×C4—31no—
240C2×C32×C4C22×D314yes—
241C2×C32×C4C22×D314yes—
242C2×C32×C4C22×D314yes—
243C2×C32×C4C22×D311yescharacteristic
244C2×C32×C4C22×D311yescharacteristic
245C2×C32×C4C22×D311yescharacteristic
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C2×C3⋊Dic3—34no—
250C2×C3⋊Dic3—34no—
251C2×C3⋊Dic3—34no—
252C2×C3⋊Dic3—64no—
253C2×C3⋊Dic3—64no—
254C2×C3⋊Dic3—64no—
255C2×C3⋊Dic3C22×D311yescharacteristic
256C2×C3⋊Dic3C22×D311yescharacteristic
257C2×C3⋊Dic3C22×D311yescharacteristic
258C2×C3⋊Dic3C22×D311yescharacteristic
259C4×C3⋊D3—62no—
260C4×C3⋊D3—68no—
261C4×C3⋊D3—128no—
262#634—62no—
263#634—62no—
264#634—68no—
265#634—128no—
266C2×C3×Dic3—34no—
267C2×C3×Dic3—34no—
268C2×C3×Dic3—34no—
269C2×C3×Dic3—34no—
270C2×C3×Dic3C22×D314yes—
271C2×C3×Dic3C22×D314yes—
272C2×C3×Dic3C22×D314yes—
273C2×C3×Dic3C22×D314yes—
274#627—68no—
275#541—271noSylow
276C2×C33C2×#4111yescharacteristic
277C2×C332−1+411yescharacteristic
278C2×C332−1+411yescharacteristic
279C3⋊(C3⋊D3)—41no—
280C2×C4×D3—94no—
281C2×C4×D3—91no—
282C2×C4×D3—184no—
283C2×Dic6—94no—
284C2×Dic6—94no—
285C2×Dic6—91no—
286C2×Dic6—91no—
287C2×Dic6—184no—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—34no—
290C3×C4⋊C4—34no—
291C3×C4⋊C4—34no—
292C3×C4⋊C4—34no—
293C3×C4⋊C4—34no—
294C3×C4⋊C4—91no—
295C3×C4⋊C4—91no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—91no—
298C3×C4⋊C4—91no—
299C3×C4⋊C4D3214yes—
300C3×C4⋊C4C2×C3⋊D311yescharacteristic
301C3×C42—34no—
302C3×C42—91no—
303(C2×D3)⋊C4—94no—
304(C2×D3)⋊C4—94no—
305(C2×D3)⋊C4—94no—
306(C2×D3)⋊C4—94no—
307(C2×D3)⋊C4—94no—
308(C2×D3)⋊C4—94no—
309(C2×D3)⋊C4—91no—
310(C2×D3)⋊C4—91no—
311(C2×D3)⋊C4—91no—
312(C2×D3)⋊C4—91no—
313(C2×D3)⋊C4—91no—
314(C2×D3)⋊C4—91no—
315(C2×D3)⋊C4—184no—
316(C2×D3)⋊C4—184no—
317C4⋊Dic3—34no—
318C4⋊Dic3—94no—
319C4⋊Dic3—94no—
320C4⋊Dic3—94no—
321C4⋊Dic3—91no—
322C4⋊Dic3C2×C3⋊D311yescharacteristic
323C4⋊Dic3C2×C3⋊D311yescharacteristic
324C4⋊Dic3C2×C3⋊D311yescharacteristic
325Dic3⋊C4—34no—
326Dic3⋊C4—34no—
327Dic3⋊C4—34no—
328Dic3⋊C4—34no—
329Dic3⋊C4—94no—
330Dic3⋊C4—94no—
331Dic3⋊C4—94no—
332Dic3⋊C4—94no—
333Dic3⋊C4—94no—
334Dic3⋊C4—94no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340Dic3⋊C4—91no—
341Dic3⋊C4—91no—
342Dic3⋊C4—91no—
343Dic3⋊C4—184no—
344Dic3⋊C4—184no—
345Dic3⋊C4C2×C3⋊D311yescharacteristic
346Dic3⋊C4C2×C3⋊D311yescharacteristic
347C4×Dic3—34no—
348C4×Dic3—94no—
349C4×Dic3—94no—
350C4×Dic3—94no—
351C4×Dic3—94no—
352C4×Dic3—91no—
353C4×Dic3—91no—
354C4×Dic3—91no—
355C4×Dic3—91no—
356C4×Dic3C2×C3⋊D311yescharacteristic
357C22×C32—24no—
358C22×C32C23×D314yes—
359C22×C32C23×D311yescharacteristic
360C2×C3⋊D3—68no—
361C2×C3⋊D3—61no—
362C2×C3⋊D3—61no—
363C2×C3⋊D3—64no—
364C2×C3⋊D3—61no—
365C2×C3⋊D3—64no—
366C2×C3⋊D3—64no—
367C2×C3⋊D3—124no—
368C2×C3⋊D3—124no—
369C32×C4—21no—
370C32×C4—21no—
371C32×C4—21no—
372C32×C4—24no—
373C32×C4—24no—
374C32×C4—24no—
375C32×C4—44no—
376C32×C4—44no—
377C32×C4—44no—
378C32×C4—61no—
379C32×C4—61no—
380C32×C4—61no—
381C32×C4—61no—
382C3⋊Dic3—21no—
383C3⋊Dic3—21no—
384C3⋊Dic3—21no—
385C3⋊Dic3—21no—
386C3⋊Dic3—32no—
387C3⋊Dic3—38no—
388C3⋊Dic3—68no—
389C3⋊Dic3—61no—
390C3⋊Dic3—61no—
391C3⋊Dic3—64no—
392C3⋊Dic3—64no—
393C3⋊Dic3—124no—
394C3⋊Dic3—124no—
395C3×Dic3—24no—
396C3×Dic3—24no—
397C3×Dic3—24no—
398C3×Dic3—24no—
399C3×Dic3—64no—
400C3×Dic3—64no—
401C3×Dic3—64no—
402C3×Dic3—64no—
403C4⋊Q8—271no—
404#126—271no—
405#126—271no—
406#126—271no—
407#126—271no—
408#125—31no—
409#125—91no—
410#125—271no—
411#125—271no—
412#125—271no—
413C22⋊Q8—271no—
414C22⋊Q8—271no—
415C4×Q8—271no—
416C4×Q8—271no—
417#117—271no—
418C33#54111yescharacteristic, complemented, Sylow
419C22×D3—94no—
420C22×D3—91no—
421C22×D3—184no—
422C2×C3×C4—24no—
423C2×C3×C4—24no—
424C2×C3×C4—24no—
425C2×C3×C4—34no—
426C2×C3×C4—34no—
427C2×C3×C4—34no—
428C2×C3×C4—34no—
429C2×C3×C4—91no—
430C2×C3×C4—91no—
431C2×C3×C4—91no—
432C2×C3×C4—91no—
433C2×C3×C4C2×D3214yes—
434C2×C3×C4C2×D3214yes—
435C2×C3×C4C2×D3214yes—
436C2×C3×C4C22×C3⋊D311yescharacteristic
437C2×C3×C4C22×C3⋊D311yescharacteristic
438C2×C3×C4C22×C3⋊D311yescharacteristic
439C2×Dic3—34no—
440C2×Dic3—34no—
441C2×Dic3—34no—
442C2×Dic3—34no—
443C2×Dic3—94no—
444C2×Dic3—91no—
445C2×Dic3—94no—
446C2×Dic3—94no—
447C2×Dic3—91no—
448C2×Dic3—91no—
449C2×Dic3—184no—
450C2×Dic3—184no—
451C2×Dic3—184no—
452C2×Dic3C22×C3⋊D311yescharacteristic
453C2×Dic3C22×C3⋊D311yescharacteristic
454C2×Dic3C22×C3⋊D311yescharacteristic
455C2×Dic3C22×C3⋊D311yescharacteristic
456C4×D3—182no—
457C4×D3—188no—
458C4×D3—368no—
459Dic6—182no—
460Dic6—188no—
461Dic6—182no—
462Dic6—188no—
463Dic6—368no—
464C2×C32—24no—
465C2×C32—24no—
466C2×C32—24no—
467C2×C32#100714yes—
468C2×C32#100711yescharacteristic
469C2×C32D3×#4114yes—
470C2×C32D3×#4111yescharacteristic
471C2×C32#101014yes—
472C2×C32#101011yescharacteristic
473C3⋊D3—128no—
474C3⋊D3—124no—
475C3⋊D3—121no—
476C2×Q8—271no—
477C2×Q8—271no—
478C22×C4—271no—
479C4⋊C4—31no—
480C4⋊C4—31no—
481C4⋊C4—31no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—91no—
485C4⋊C4—91no—
486C4⋊C4—91no—
487C4⋊C4—91no—
488C4⋊C4—91no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4—271no—
492C4⋊C4—271no—
493C4⋊C4—271no—
494C4⋊C4—271no—
495C4⋊C4—271no—
496C4⋊C4—271no—
497C4⋊C4—271no—
498C4⋊C4D3×C3⋊D311yescharacteristic
499C22⋊C4—271no—
500C22⋊C4—271no—
501C22⋊C4—271no—
502C22⋊C4—271no—
503C22⋊C4—271no—
504C22⋊C4—271no—
505C42—31no—
506C42—91no—
507C42—271no—
508C42—271no—
509C42—271no—
510C42—271no—
511C22×C3—24no—
512C22×C3C22×D3214yes—
513C22×C3C23×C3⋊D311yescharacteristic
514C2×D3—188no—
515C2×D3—184no—
516C2×D3—184no—
517C2×D3—184no—
518C2×D3—181no—
519C2×D3—181no—
520C2×D3—181no—
521C2×D3—364no—
522C2×D3—364no—
523C3×C4—24no—
524C3×C4—24no—
525C3×C4—24no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—44no—
530C3×C4—44no—
531C3×C4—44no—
532C3×C4—64no—
533C3×C4—64no—
534C3×C4—64no—
535C3×C4—64no—
536C3×C4—181no—
537C3×C4—181no—
538C3×C4—181no—
539C3×C4—181no—
540Dic3—21no—
541Dic3—21no—
542Dic3—21no—
543Dic3—21no—
544Dic3—64no—
545Dic3—64no—
546Dic3—64no—
547Dic3—64no—
548Dic3—98no—
549Dic3—92no—
550Dic3—188no—
551Dic3—184no—
552Dic3—184no—
553Dic3—181no—
554Dic3—181no—
555Dic3—364no—
556Dic3—364no—
557C32—24no—
558C32#569614yescomplemented
559C32#569611yescharacteristic, complemented
560C23—271no—
561Q8—542no—
562Q8—542no—
563C2×C4—31no—
564C2×C4—31no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—91no—
568C2×C4—91no—
569C2×C4—91no—
570C2×C4—91no—
571C2×C4—271no—
572C2×C4—271no—
573C2×C4—271no—
574C2×C4—542no—
575C2×C4C2×D3×C3⋊D311yescharacteristic
576C2×C4C2×D3×C3⋊D311yescharacteristic
577C2×C4C2×D3×C3⋊D311yescharacteristic
578C2×C3—24no—
579C2×C3—24no—
580C2×C3—24no—
581C2×C3#6429714yes—
582C2×C3#6431214yes—
583C2×C3#6431514yes—
584C2×C3#6436411yescharacteristic
585C2×C3#41×C3⋊D311yescharacteristic
586C2×C3#6436711yescharacteristic
587D3—368no—
588D3—364no—
589D3—361no—
590C22—541no—
591C22—541no—
592C22—541no—
593C22C22×D3×C3⋊D311yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—61no—
598C4—61no—
599C4—61no—
600C4—61no—
601C4—181no—
602C4—181no—
603C4—181no—
604C4—181no—
605C4—272no—
606C4—541no—
607C4—541no—
608C3—24no—
609C3#1059432314yescomplemented
610C3#1059616711yescharacteristic, complemented
611C2—1081no—
612C2#1172456611yescharacteristic
613C2#1172458611yescharacteristic
614C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64098C2semidirect
3. C32×#890C2semidirect
10. #11723406C2semidirect
11. #11723405C2semidirect
12. #11723405C2semidirect
13. #11723401C2semidirect
14. #11723395C2semidirect
15. #11723389C2semidirect
65. #64098D3semidirect
70. C3×#890D3semidirect
215. #890C3⋊D3semidirect
418. C33#541semidirect
558. C32#5696semidirect
559. C32#5696semidirect
609. C3#10594323semidirect
610. C3#10596167semidirect

Maximal subgroups

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

subgroupindex
#117239372
C3×#640982
C32×#8902
#117234532
#117234462
#117234422
#117234132
#117234112
#117234062
#117234052
#117234012
#117233952
#117233892
#105961673
#105943233