Finite Group Databasebeta

← #11803501

Subgroups — #11803501

Order 1152.

1273 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64106C211yesmaximal, characteristic
2C2×#63874C211yesmaximal, characteristic
3C2×#63844C211yesmaximal, characteristic, complemented
4C3×#4984C211yesmaximal, characteristic, complemented
5C3×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6#10591325C211yesmaximal, characteristic
7#10591271C212yesmaximal, complemented
8#10591266C212yesmaximal
9#10591213C211yesmaximal, characteristic, complemented
10#10591212C212yesmaximal, complemented
11#10591202C212yesmaximal
12#76197—31nomaximal
13#73762—31nomaximal
14C22×C4×C3⋊D3C2211yescharacteristic
15C22×#634C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×#3752C2212yes—
18C2×C4⋊(C3⋊Dic3)C2212yes—
19#64106C418yescomplemented
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic3C2212yes—
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2212yes—
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×#3724C2212yes—
27C2×#3724C2211yescharacteristic
28C2×#3724C2211yescharacteristic
29C2×#3722C2211yescharacteristic
30C2×#3722C2211yescharacteristic
31C2×#3722C2212yes—
32C2×Dic32C2212yes—
33#63874—24no—
34#63844—24no—
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39#63579C2212yes—
40C2×#897—31no—
41C2×#897—31no—
42C2×#897—61no—
43C2×#887—31no—
44C2×#872—31no—
45C2×#872—31no—
46C2×C4×Dic6—31no—
47C3×#389—31no—
48C3×C4×C4⋊C4—31no—
49#4990—32no—
50#4987—31no—
51#4987—31no—
52#4984D311yescharacteristic, complemented
53Dic3×C4⋊C4D311yescharacteristic, complemented
54#4973—31no—
55#4972—32no—
56#4940—32no—
57#4929—31no—
58#4928—32no—
59#4670—32no—
60#4669—32no—
61#4654—32no—
62#4653—32no—
63C23×C3⋊D3C2311yescharacteristic
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C22×C3⋊Dic3C2312yes—
67C22×C3⋊Dic3C2311yescharacteristic
68C2×C4×C3⋊D3—22no—
69C2×C4×C3⋊D3—22no—
70C2×C4×C3⋊D3C2×C414yes—
71C2×#634—22no—
72C2×#634—22no—
73C2×#634C2×C414yes—
74C22×C3×Dic3C2311yescharacteristic
75C22×C3×Dic3C2312yes—
76C22×C3×Dic3C2311yescharacteristic
77C22×C3×Dic3C2312yes—
78C22×C3×Dic3C2311yescharacteristic
79C22×C3×Dic3C2311yescharacteristic
80C32×C4⋊C4C2×C414yes—
81#3752C2×C418yes—
82C4⋊(C3⋊Dic3)C2×C418yes—
83C3×C4⋊Dic3—22no—
84C3×C4⋊Dic3—22no—
85C3×C4⋊Dic3—42no—
86C3×C4×Dic3—22no—
87C3×C4×Dic3—42no—
88C3×C4×Dic3D414yes—
89#3724—24no—
90#3724—22no—
91#3724—22no—
92#3722—22no—
93#3722—22no—
94#3722—42no—
95Dic32—24no—
96#2323—91noSylow
97C22×C4×D3—31no—
98C22×C4×D3—31no—
99C22×C4×D3—61no—
100C22×Dic6—31no—
101C22×Dic6—31no—
102C22×Dic6—61no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—31no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C4⋊C4C2×D311yescharacteristic
108C2×C3×C4⋊C4C2×D311yescharacteristic
109C2×C3×C42—31no—
110C2×C3×C42—31no—
111C2×C3×C42—32no—
112C2×(C2×D3)⋊C4—32no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4—31no—
115C2×(C2×D3)⋊C4—31no—
116C2×(C2×D3)⋊C4—31no—
117C2×(C2×D3)⋊C4—31no—
118C2×(C2×D3)⋊C4—32no—
119C2×(C2×D3)⋊C4—32no—
120C2×(C2×D3)⋊C4—62no—
121C2×C4⋊Dic3—32no—
122C2×C4⋊Dic3—32no—
123C2×C4⋊Dic3—62no—
124C2×C4⋊Dic3C2×D311yescharacteristic
125C2×C4⋊Dic3C2×D311yescharacteristic
126C2×C4⋊Dic3C2×D312yes—
127C2×Dic3⋊C4—31no—
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—32no—
130C2×Dic3⋊C4—31no—
131C2×Dic3⋊C4—32no—
132C2×Dic3⋊C4—31no—
133C2×C4×Dic3—32no—
134C2×C4×Dic3—32no—
135C2×C4×Dic3C2×D311yescharacteristic
136C2×C4×Dic3C2×D312yes—
137C2×C4×Dic3C2×D311yescharacteristic
138#897—38no—
139#897—38no—
140#897—68no—
141#887—64no—
142#872—64no—
143#872—64no—
144C4×Dic6—64no—
145C3×#95—32no—
146C3×#95—32no—
147C3⋊#95—32no—
148C3⋊#95—32no—
149C3⋊#95—32no—
150C3⋊#95—32no—
151C3⋊#95—32no—
152C3⋊#95—32no—
153C3⋊#95C2×D312yes—
154C3⋊#95C2×D312yes—
155C23×C32C2411yescharacteristic
156C22×C3⋊D3—21no—
157C22×C3⋊D3—21no—
158C22×C3⋊D3—21no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3—21no—
161C22×C3⋊D3—21no—
162C22×C3⋊D3C22×C412yes—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4C22×C414yes—
166C2×C32×C4C22×C412yes—
167C2×C32×C4C2×D412yes—
168C2×C32×C4#4112yes—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3C22×C412yes—
174C2×C3⋊Dic3C22×C414yes—
175C4×C3⋊D3—28no—
176#634—28no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3C2×D412yes—
194C2×C3×Dic3C2×D412yes—
195C2×C22⋊Q8—91no—
196C2×C4×Q8—91no—
197C2×#117—91no—
198#391—91no—
199#389—31no—
200#389—92no—
201#388—92no—
202#388—92no—
203#387—92no—
204#380—91no—
205C4×C4⋊C4—31no—
206C23×D3—31no—
207C23×D3—31no—
208C23×D3—61no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—32no—
214C22×C3×C4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—32no—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D312yes—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×Dic3—32no—
222C22×Dic3—32no—
223C22×Dic3—31no—
224C22×Dic3—31no—
225C22×Dic3—61no—
226C22×Dic3—62no—
227C22×Dic3C22×D312yes—
228C22×Dic3C22×D311yescharacteristic
229C22×Dic3C22×D311yescharacteristic
230C22×Dic3C22×D312yes—
231C22×Dic3C22×D311yescharacteristic
232C22×Dic3C22×D311yescharacteristic
233C2×C4×D3—34no—
234C2×C4×D3—34no—
235C2×C4×D3—64no—
236C2×C4×D3—62no—
237C2×C4×D3—62no—
238C2×C4×D3—62no—
239C2×C4×D3—62no—
240C2×C4×D3—122no—
241C2×C4×D3—122no—
242C2×Dic6—34no—
243C2×Dic6—34no—
244C2×Dic6—64no—
245C2×Dic6—62no—
246C2×Dic6—62no—
247C2×Dic6—62no—
248C2×Dic6—62no—
249C2×Dic6—122no—
250C2×Dic6—122no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—62no—
254C3×C4⋊C4—122no—
255C3×C4⋊C4C4×D314yes—
256C3×C4⋊C4C4×D314yes—
257C3×C42—34no—
258C3×C42—62no—
259C3×C42—122no—
260(C2×D3)⋊C4—38no—
261(C2×D3)⋊C4—38no—
262(C2×D3)⋊C4—68no—
263(C2×D3)⋊C4—62no—
264(C2×D3)⋊C4—62no—
265(C2×D3)⋊C4—62no—
266(C2×D3)⋊C4—62no—
267(C2×D3)⋊C4—122no—
268(C2×D3)⋊C4—122no—
269C4⋊Dic3—22no—
270C4⋊Dic3—22no—
271C4⋊Dic3—38no—
272C4⋊Dic3—38no—
273C4⋊Dic3—42no—
274C4⋊Dic3—68no—
275Dic3⋊C4—64no—
276Dic3⋊C4—62no—
277Dic3⋊C4—62no—
278Dic3⋊C4—62no—
279Dic3⋊C4—62no—
280Dic3⋊C4—64no—
281C4×Dic3—22no—
282C4×Dic3—42no—
283C4×Dic3—64no—
284C4×Dic3—64no—
285C4×Dic3D1214yes—
286C22×C32C23×C411yescharacteristic
287C22×C32C22×D411yescharacteristic
288C22×C32C2×#4111yescharacteristic
289C22×C32H(5,𝔽2)11yescharacteristic
290C22×C322−1+411yescharacteristic
291C22×C322−1+411yescharacteristic
292C22×C322−1+411yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3—22no—
296C2×C3⋊D3—41no—
297C2×C3⋊D3—41no—
298C2×C3⋊D3—41no—
299C2×C3⋊D3—41no—
300C32×C4—24no—
301C32×C4C4×D414yes—
302C3⋊Dic3—24no—
303C3⋊Dic3—22no—
304C3×Dic3—22no—
305C3×Dic3—22no—
306C3×Dic3—41no—
307C3×Dic3—42no—
308C3×Dic3—41no—
309C3×Dic3—42no—
310C22×Q8—91no—
311C23×C4—91no—
312C22⋊Q8—98no—
313C4×Q8—184no—
314#117—184no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—92no—
319C2×C4⋊C4—91no—
320C2×C4⋊C4—91no—
321C2×C4⋊C4—92no—
322C2×C4⋊C4D3211yescharacteristic
323C2×C22⋊C4—92no—
324C2×C22⋊C4—91no—
325C2×C22⋊C4—91no—
326C2×C22⋊C4—92no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—32no—
330C2×C42—92no—
331#95—32no—
332#95—32no—
333#95—92no—
334#95—92no—
335#95—92no—
336C23×C3—21no—
337C23×C3C23×D311yescharacteristic
338C23×C3C23×D311yescharacteristic
339C22×D3—32no—
340C22×D3—32no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—62no—
344C22×D3—61no—
345C22×D3—61no—
346C22×D3—61no—
347C22×D3—61no—
348C22×D3—61no—
349C22×D3—61no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—61no—
356C22×D3—121no—
357C22×D3—121no—
358C22×D3—121no—
359C22×D3—121no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—24no—
368C2×C3×C4—32no—
369C2×C3×C4—32no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—62no—
380C2×C3×C4—62no—
381C2×C3×C4—61no—
382C2×C3×C4—62no—
383C2×C3×C4—62no—
384C2×C3×C4—61no—
385C2×C3×C4—62no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4C2×C4×D314yes—
389C2×C3×C4C2×C4×D314yes—
390C2×C3×C4C2×C4×D312yes—
391C2×C3×C4C2×C4×D312yes—
392C2×C3×C4C2×D1212yes—
393C2×C3×C4#23512yes—
394C2×C3×C4D3×D412yes—
395C2×C3×C4#23912yes—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—32no—
413C2×Dic3—32no—
414C2×Dic3—34no—
415C2×Dic3—34no—
416C2×Dic3—62no—
417C2×Dic3—64no—
418C2×Dic3—62no—
419C2×Dic3—61no—
420C2×Dic3—62no—
421C2×Dic3—61no—
422C2×Dic3—62no—
423C2×Dic3—61no—
424C2×Dic3—62no—
425C2×Dic3—61no—
426C2×Dic3—122no—
427C2×Dic3—122no—
428C2×Dic3—121no—
429C2×Dic3—121no—
430C2×Dic3C2×D1212yes—
431C2×Dic3C2×D1212yes—
432C4×D3—68no—
433C4×D3—68no—
434C4×D3—128no—
435Dic6—68no—
436Dic6—68no—
437Dic6—128no—
438C2×C32C2×C4×D411yescharacteristic
439C2×C32#51911yescharacteristic
440C2×C32#52011yescharacteristic
441C2×C32#53611yescharacteristic
442C2×C32#53711yescharacteristic
443C2×C32#54111yescharacteristic
444C2×C32#54211yescharacteristic
445C3⋊D3—42no—
446C24—91no—
447C2×Q8—94no—
448C2×Q8—182no—
449C2×Q8—182no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—31no—
456C22×C4—92no—
457C22×C4—91no—
458C22×C4—94no—
459C22×C4—182no—
460C22×C4—182no—
461C22×C4C2×D3212yes—
462C22×C4C2×D3211yescharacteristic
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—98no—
466C4⋊C4—122no—
467C4⋊C4—182no—
468C4⋊C4—182no—
469C4⋊C4—184no—
470C4⋊C4#62414yes—
471C22⋊C4—98no—
472C22⋊C4—182no—
473C22⋊C4—182no—
474C22⋊C4—362no—
475C42—34no—
476C42—62no—
477C42—122no—
478C42—184no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3C22×C4×D311yescharacteristic
487C22×C3C22×C4×D311yescharacteristic
488C22×C3C22×D1211yescharacteristic
489C22×C3C2×#23511yescharacteristic
490C22×C3C2×D3×D411yescharacteristic
491C22×C3#100411yescharacteristic
492C22×C3#100411yescharacteristic
493C22×C3C2×#23911yescharacteristic
494C22×C3#100711yescharacteristic
495C22×C3#100711yescharacteristic
496C22×C3#101011yescharacteristic
497C22×C3#101011yescharacteristic
498C22×C3#101011yescharacteristic
499C22×C3#101011yescharacteristic
500C2×D3—64no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—122no—
508C2×D3—122no—
509C2×D3—122no—
510C2×D3—122no—
511C2×D3—122no—
512C2×D3—122no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—121no—
517C2×D3—121no—
518C2×D3—121no—
519C2×D3—121no—
520C2×D3—121no—
521C3×C4—24no—
522C3×C4—24no—
523C3×C4—24no—
524C3×C4—44no—
525C3×C4—62no—
526C3×C4—62no—
527C3×C4—122no—
528C3×C4—122no—
529C3×C4—121no—
530C3×C4—121no—
531C3×C4C4×D1214yes—
532C3×C4#89314yes—
533Dic3—22no—
534Dic3—22no—
535Dic3—42no—
536Dic3—41no—
537Dic3—42no—
538Dic3—41no—
539Dic3—64no—
540Dic3—62no—
541Dic3—64no—
542Dic3—62no—
543Dic3—124no—
544Dic3—122no—
545C32#232311yescharacteristic, complemented, Sylow
546C23—92no—
547C23—181no—
548C23—181no—
549C23—181no—
550C23—181no—
551C23—181no—
552C23—181no—
553C23C22×D3211yescharacteristic
554Q8—188no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—62no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—61no—
575C2×C4—94no—
576C2×C4—92no—
577C2×C4—181no—
578C2×C4—182no—
579C2×C4—182no—
580C2×C4—181no—
581C2×C4—188no—
582C2×C4#379912yes—
583C2×C4D3×D1212yes—
584C2×C4C2×#62414yes—
585C2×C4C2×#62412yes—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3C2×C4×D1211yescharacteristic
594C2×C3C2×#89311yescharacteristic
595C2×C3#551011yescharacteristic
596C2×C3#552611yescharacteristic
597C2×C3#554611yescharacteristic
598C2×C3#554611yescharacteristic
599C2×C3#556111yescharacteristic
600C2×C3#557211yescharacteristic
601C2×C3#557211yescharacteristic
602C2×C3#558311yescharacteristic
603C2×C3#560811yescharacteristic
604C2×C3#562811yescharacteristic
605C2×C3#569811yescharacteristic
606C2×C3#573311yescharacteristic
607D3—124no—
608D3—122no—
609D3—122no—
610C22—182no—
611C22—182no—
612C22—182no—
613C22—361no—
614C22—361no—
615C22—361no—
616C22—361no—
617C22C2×#379911yescharacteristic
618C22C2×D3×D1211yescharacteristic
619C22#6430911yescharacteristic
620C22#6431311yescharacteristic
621C22#6431511yescharacteristic
622C22#6431511yescharacteristic
623C22C22×#62411yescharacteristic
624C4—24no—
625C4—62no—
626C4—62no—
627C4—122no—
628C4—121no—
629C4—121no—
630C4—122no—
631C4—182no—
632C4—184no—
633C4#6389414yes—
634C3—21no—
635C3#7376211yescharacteristic, complemented
636C3#7619711yescharacteristic, complemented
637C2—362no—
638C2#1059434111yescharacteristic
639C2#1059435511yescharacteristic
640C2#1059438011yescharacteristic
641C2#1059452211yescharacteristic
642C2C2×#6389411yescharacteristic
643C2#1059521911yescharacteristic
644C2#1059533911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, cosocleC23×C32C24
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32#2323
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#63844C2semidirect
4. C3×#4984C2semidirect
5. C3×Dic3×C4⋊C4C2semidirect
7. #10591271C2semidirect
9. #10591213C2semidirect
10. #10591212C2semidirect
19. #64106C4semidirect
52. #4984D3semidirect
53. Dic3×C4⋊C4D3semidirect
545. C32#2323semidirect
635. C3#73762semidirect
636. C3#76197semidirect

Maximal subgroups

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

subgroupindex
C2×#641062
C2×#638742
C2×#638442
C3×#49842
C3×Dic3×C4⋊C42
#105913252
#105912712
#105912662
#105912132
#105912122
#105912022
#761973
#737623