Finite Group Databasebeta

← #11806059

Subgroups — #11806059

Order 1152.

1217 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D12C211yesmaximal, characteristic
2C4×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×#4939C211yesmaximal, characteristic
4C3×C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#10591368C212yesmaximal, complemented
6#10591362C212yesmaximal, complemented
7#10591318C211yesmaximal, characteristic
8#10591285C212yesmaximal, complemented
9#10591283C212yesmaximal, complemented
10Dic3×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#10591224C211yesmaximal, characteristic, complemented
12#76303—31nomaximal
13#73645—31nomaximal
14C22×C3×D12C2211yescharacteristic
15C22×D3×Dic3C2212yes—
16C2×C32×C42C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C4×C3⋊Dic3C2212yes—
20C2×C3×(C2×D3)⋊C4C2212yes—
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×Dic3⋊C4C2211yescharacteristic
24C2×C3×Dic3⋊C4C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×C4×Dic3C2211yescharacteristic
27C2×(C2×D3)⋊Dic3C2212yes—
28C2×(C2×D3)⋊Dic3C2212yes—
29C2×(C2×D3)⋊Dic3C2212yes—
30C2×Dic32C2212yes—
31Dic3×D12C418yescomplemented
32#63658C2212yes—
33C3×C3⋊#95C2211yescharacteristic
34C3×C3⋊#95C2211yescharacteristic
35#63579C2212yes—
36#63579C2212yes—
37#63579C2212yes—
38C2×D4×Dic3—31no—
39C2×C4×D12—31no—
40C3×#390—31no—
41C3×C4×C4⋊C4—31no—
42#5221—31no—
43#4990—32no—
44#4990—32no—
45#4977—31no—
46Dic3×C4⋊C4—31no—
47#4945—32no—
48#4944—32no—
49#4943—32no—
50Dic3×C22⋊C4—32no—
51#4939—31no—
52#4939D311yescharacteristic
53#4936—31no—
54C4×C4⋊Dic3—31no—
55C4×C4⋊Dic3—31no—
56C4×C4⋊Dic3—31no—
57C4×C4⋊Dic3—61no—
58C4×Dic3⋊C4D311yescharacteristic, complemented
59#4666—32no—
60#4664—32no—
61C23×C3×D3C2312yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C3⋊Dic3C2312yes—
66C22×C3⋊Dic3C2312yes—
67C22×C3×Dic3C2311yescharacteristic
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×D12—22no—
73C2×C3×D12—22no—
74C2×C3×D12C2×C414yes—
75C2×D3×Dic3—42no—
76C2×D3×Dic3—42no—
77C2×D3×Dic3C2×C418yes—
78C32×C42—22no—
79C4⋊(C3⋊Dic3)—22no—
80C4⋊(C3⋊Dic3)C2×C414yes—
81C4×C3⋊Dic3—42no—
82C3×(C2×D3)⋊C4—24no—
83C3×(C2×D3)⋊C4—42no—
84C3×C4⋊Dic3—22no—
85C3×Dic3⋊C4—22no—
86C3×Dic3⋊C4—22no—
87C3×C4×Dic3—22no—
88C3×C4×Dic3C2×C414yes—
89(C2×D3)⋊Dic3—24no—
90(C2×D3)⋊Dic3—42no—
91(C2×D3)⋊Dic3C2×C418yes—
92Dic32—42no—
93#2357—91noSylow
94C22×C3×D4—31no—
95C23×Dic3—32no—
96C22×D12C2×D311yescharacteristic
97C22×C4×D3—32no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C22⋊C4—32no—
102C2×C3×C22⋊C4—32no—
103C2×C3×C42—21no—
104C2×C3×C42—31no—
105C2×C3×C42—31no—
106C2×C3×C42C2×D311yescharacteristic
107C2×C3×C42C2×D311yescharacteristic
108C2×#217—32no—
109C2×#217—32no—
110C2×#217—32no—
111D4×Dic3—38no—
112C2×(C2×D3)⋊C4—32no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×(C2×D3)⋊C4C2×D312yes—
116C2×(C2×D3)⋊C4C2×D312yes—
117C2×C4⋊Dic3—31no—
118C2×C4⋊Dic3—31no—
119C2×C4⋊Dic3—31no—
120C2×C4⋊Dic3—31no—
121C2×C4⋊Dic3—61no—
122C2×C4⋊Dic3—61no—
123C2×C4⋊Dic3C2×D311yescharacteristic
124C2×Dic3⋊C4C2×D311yescharacteristic
125C2×Dic3⋊C4C2×D311yescharacteristic
126C2×C4×Dic3—32no—
127C2×C4×Dic3—32no—
128C2×C4×Dic3—32no—
129C2×C4×Dic3—32no—
130C2×C4×Dic3—62no—
131C2×C4×Dic3C2×D311yescharacteristic
132C2×C4×Dic3C2×D311yescharacteristic
133C4×D12—38no—
134C3×#95—31no—
135C3×#95—31no—
136C3⋊#95—32no—
137C3⋊#95—32no—
138C3⋊#95—32no—
139C3⋊#95—32no—
140C3⋊#95—32no—
141C3⋊#95—32no—
142C3⋊#95—32no—
143C3⋊#95—32no—
144C3⋊#95—62no—
145C3⋊#95C2×D311yescharacteristic
146C3⋊#95C2×D311yescharacteristic
147C23×C32C2411yescharacteristic
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3C22×C414yes—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4C22×C412yes—
161C2×C32×C4#4112yes—
162C2×C32×C4#4112yes—
163C2×C32×C4#4112yes—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3C22×C414yes—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3C22×C412yes—
183C2×C3×Dic3C22×C412yes—
184C2×C3×Dic3#4112yes—
185C3×D12—28no—
186D3×Dic3—48no—
187C2×C4×D4—91no—
188#390—31no—
189#390—91no—
190#388—92no—
191#388—92no—
192#382—91no—
193#381—92no—
194C4×C4⋊C4—31no—
195C4×C4⋊C4—91no—
196C4×C4⋊C4—91no—
197C4×C22⋊C4—92no—
198C24×C3—32no—
199C23×D3C22×D312yes—
200C2×C3×D4—34no—
201C2×C3×D4—62no—
202C2×C3×D4—62no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—32no—
210C22×C3×C4—31no—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D311yescharacteristic
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—38no—
220C22×Dic3—32no—
221C22×Dic3—32no—
222C22×Dic3—62no—
223C22×Dic3—62no—
224C22×Dic3—122no—
225C22×Dic3—122no—
226C22×Dic3C22×D312yes—
227C22×Dic3C22×D311yescharacteristic
228C22×Dic3C22×D311yescharacteristic
229C22×Dic3C22×D311yescharacteristic
230C22×Dic3C22×D311yescharacteristic
231C2×D12—22no—
232C2×D12—22no—
233C2×D12C4×D314yes—
234C2×C4×D3—38no—
235C2×C4×D3—122no—
236C2×C4×D3—122no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—62no—
240C3×C22⋊C4—64no—
241C3×C22⋊C4—122no—
242C3×C42—24no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—34no—
246C3×C42—62no—
247#217—38no—
248#217—64no—
249#217—122no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—38no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—64no—
254(C2×D3)⋊C4—122no—
255C4⋊Dic3—22no—
256C4⋊Dic3—34no—
257C4⋊Dic3—34no—
258C4⋊Dic3—64no—
259C4⋊Dic3—64no—
260C4⋊Dic3—62no—
261C4⋊Dic3—62no—
262Dic3⋊C4—22no—
263Dic3⋊C4—22no—
264C4×Dic3—22no—
265C4×Dic3—122no—
266C4×Dic3—122no—
267C4×Dic3—122no—
268C4×Dic3—122no—
269C4×Dic3—124no—
270C4×Dic3C4×D314yes—
271C22×C32C23×C411yescharacteristic
272C22×C32C2×#4111yescharacteristic
273C22×C32C2×#4111yescharacteristic
274C22×C32C2×#4111yescharacteristic
275C22×C32C2×#4111yescharacteristic
276C22×C32H(5,𝔽2)11yescharacteristic
277C22×C32H(5,𝔽2)11yescharacteristic
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C2×C3×D3—42no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—42no—
285C32×C4—22no—
286C32×C4—41no—
287C32×C4#11714yes—
288C3⋊Dic3—24no—
289C3⋊Dic3—42no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—22no—
293C3×Dic3—41no—
294C3×Dic3—42no—
295C22×D4—31no—
296C23×C4—92no—
297C4×D4—98no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—91no—
302C2×C4⋊C4—91no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4—92no—
306C2×C22⋊C4—92no—
307C2×C22⋊C4—92no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—92no—
311C2×C42—92no—
312C2×C42D3211yescharacteristic
313#95—31no—
314#95—31no—
315#95—92no—
316#95—92no—
317#95—92no—
318#95—92no—
319C23×C3—21no—
320C23×C3—34no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—62no—
324C23×C3—62no—
325C23×C3—62no—
326C23×C3—62no—
327C23×C3C23×D311yescharacteristic
328C23×C3C23×D311yescharacteristic
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—22no—
335C22×D3C2×C4×D314yes—
336C3×D4—68no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—32no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4C2×C4×D312yes—
372C2×C3×C4C2×C4×D312yes—
373C2×C3×C4#23512yes—
374C2×C3×C4#23712yes—
375C2×C3×C4#23712yes—
376C2×C3×C4#23712yes—
377C2×C3×C4#23712yes—
378C2×C3×C4#23912yes—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—21no—
391C2×Dic3—34no—
392C2×Dic3—34no—
393C2×Dic3—64no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—62no—
398C2×Dic3—64no—
399C2×Dic3—62no—
400C2×Dic3—62no—
401C2×Dic3—62no—
402C2×Dic3—62no—
403C2×Dic3—62no—
404C2×Dic3—62no—
405C2×Dic3—122no—
406C2×Dic3—122no—
407C2×Dic3—122no—
408C2×Dic3—122no—
409C2×Dic3—128no—
410C2×Dic3C2×C4×D312yes—
411C2×Dic3C2×C4×D312yes—
412C2×Dic3#23512yes—
413D12—28no—
414C4×D3—128no—
415C2×C32C2×#11711yescharacteristic
416C2×C32C4×#4111yescharacteristic
417C2×C32#51811yescharacteristic
418C2×C32#55311yescharacteristic
419C2×C32#55311yescharacteristic
420C2×C32#55911yescharacteristic
421C2×C32#55911yescharacteristic
422C3×D3—44no—
423C24—32no—
424C2×D4—34no—
425C2×D4—62no—
426C2×D4—62no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—32no—
432C22×C4—98no—
433C22×C4—92no—
434C22×C4—92no—
435C22×C4—362no—
436C22×C4—362no—
437C22×C4C2×D3211yescharacteristic
438C22×C4C2×D3211yescharacteristic
439C22×C4C2×D3211yescharacteristic
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—94no—
444C4⋊C4—182no—
445C22⋊C4—64no—
446C22⋊C4—98no—
447C22⋊C4—122no—
448C22⋊C4—184no—
449C22⋊C4—362no—
450C42—22no—
451C42—34no—
452C42—62no—
453C42—362no—
454C42—362no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—64no—
463C22×C3—64no—
464C22×C3—64no—
465C22×C3—122no—
466C22×C3—122no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3C22×C4×D311yescharacteristic
470C22×C3C22×C4×D311yescharacteristic
471C22×C3C2×#23511yescharacteristic
472C22×C3C2×#23511yescharacteristic
473C22×C3C2×#23711yescharacteristic
474C22×C3C2×#23711yescharacteristic
475C22×C3C2×#23711yescharacteristic
476C22×C3C2×#23711yescharacteristic
477C22×C3#100411yescharacteristic
478C22×C3C2×#23911yescharacteristic
479C22×C3D3×#4111yescharacteristic
480C22×C3#100911yescharacteristic
481C22×C3#100911yescharacteristic
482C22×C3#100911yescharacteristic
483C2×D3—24no—
484C2×D3—24no—
485C2×D3—24no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C3×C4—24no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—24no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—62no—
500C3×C4—122no—
501C3×C4—121no—
502C3×C4#87714yes—
503C3×C4#89214yes—
504Dic3—22no—
505Dic3—22no—
506Dic3—22no—
507Dic3—42no—
508Dic3—41no—
509Dic3—64no—
510Dic3—64no—
511Dic3—124no—
512Dic3—124no—
513Dic3—122no—
514Dic3—122no—
515C32#235711yescharacteristic, complemented, Sylow
516C23—34no—
517C23—62no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23C22×D3211yescharacteristic
524D4—68no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—94no—
546C2×C4—182no—
547C2×C4—182no—
548C2×C4—182no—
549C2×C4—182no—
550C2×C4—182no—
551C2×C4—368no—
552C2×C4#379512yes—
553C2×C4#379612yes—
554C2×C4C4×D3212yes—
555C2×C4#380512yes—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—124no—
564C2×C3C4×#23511yescharacteristic
565C2×C3C2×#87711yescharacteristic
566C2×C3C2×#89211yescharacteristic
567C2×C3#552111yescharacteristic
568C2×C3C4×#23711yescharacteristic
569C2×C3#554511yescharacteristic
570C2×C3#556511yescharacteristic
571C2×C3#556511yescharacteristic
572C2×C3#558511yescharacteristic
573C2×C3#558511yescharacteristic
574C2×C3#561411yescharacteristic
575C2×C3#561811yescharacteristic
576C2×C3#568111yescharacteristic
577C2×C3#571311yescharacteristic
578D3—44no—
579C22—64no—
580C22—64no—
581C22—64no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22C2×#379511yescharacteristic
587C22C2×#379611yescharacteristic
588C22C2×C4×D3211yescharacteristic
589C22D3×#23511yescharacteristic
590C22#6430711yescharacteristic
591C22#6430811yescharacteristic
592C22C2×#380511yescharacteristic
593C4—22no—
594C4—41no—
595C4—62no—
596C4—62no—
597C4—62no—
598C4—122no—
599C4—121no—
600C4—184no—
601C4—362no—
602C4#6387814yes—
603C3—21no—
604C3#7364511yescharacteristic, complemented
605C3#7630311yescharacteristic, complemented
606C2—124no—
607C2C4×#379511yescharacteristic
608C2#1059393911yescharacteristic
609C2#1059395111yescharacteristic
610C2#1059399411yescharacteristic
611C2C2×#6387811yescharacteristic
612C2#1059494511yescharacteristic
613C2#1059499511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, cosocleC23×C32C24
prime radicalC22×C3×D12C22
hypercenterC2×C42D32
hypocenterC32#2357
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊(C3⋊Dic3)C2semidirect
4. C3×C4×Dic3⋊C4C2semidirect
5. #10591368C2semidirect
6. #10591362C2semidirect
8. #10591285C2semidirect
9. #10591283C2semidirect
10. Dic3×C4⋊Dic3C2semidirect
11. #10591224C2semidirect
31. Dic3×D12C4semidirect
58. C4×Dic3⋊C4D3semidirect
515. C32#2357semidirect
604. C3#73645semidirect
605. C3#76303semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D122
C4×C4⋊(C3⋊Dic3)2
C3×#49392
C3×C4×Dic3⋊C42
#105913682
#105913622
#105913182
#105912852
#105912832
Dic3×C4⋊Dic32
#105912242
#763033
#736453