Finite Group Databasebeta

← #11803128

Subgroups — #11803128

Order 1152.

1209 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63894C211yesmaximal, characteristic
2#10592638C211yesmaximal, characteristic
3C3×#4932C211yesmaximal, characteristic, complemented
4C3×#4928C211yesmaximal, characteristic, complemented
5#10591371C212yesmaximal, complemented
6#10591367C212yesmaximal, complemented
7#10591352C211yesmaximal, characteristic, complemented
8#10591320C211yesmaximal, characteristic
9Dic3×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#10591259C212yesmaximal, complemented
11#10591258C212yesmaximal, complemented
12#75969—31nomaximal
13#73415—31nomaximal
14C22×C4⋊(C3⋊D3)C2211yescharacteristic
15C22×#624C2212yes—
16C2×C32×C42C2211yescharacteristic
17C2×#3752C2212yes—
18C2×#3752C2212yes—
19C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C2×Dic3⋊Dic3C2212yes—
26C2×#3722C2212yes—
27C2×#3722C2212yes—
28C2×#3722C2212yes—
29C2×Dic32C2212yes—
30#63894C418yescomplemented
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2211yescharacteristic
33C3×C3⋊#95C2211yescharacteristic
34C3×C3⋊#95C2211yescharacteristic
35C3×C3⋊#95C2211yescharacteristic
36#63579C2212yes—
37#63579C2212yes—
38C2×#893—31no—
39C2×C4×D12—31no—
40C3×#389—31no—
41C3×#376—31no—
42#4990—32no—
43#4988—31no—
44#4987—32no—
45#4984—31no—
46Dic3×C4⋊C4—31no—
47#4939—31no—
48#4939—31no—
49#4939—31no—
50#4939—61no—
51C4×C4⋊Dic3—31no—
52#4932D311yescharacteristic, complemented
53#4928—31no—
54#4928D311yescharacteristic, complemented
55#4670—32no—
56#4669—32no—
57#4669—32no—
58#4667—32no—
59#4664—32no—
60#4664—32no—
61C23×C3⋊D3C2312yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C3⋊Dic3C2312yes—
66C2×C4⋊(C3⋊D3)—22no—
67C2×C4⋊(C3⋊D3)—22no—
68C2×C4⋊(C3⋊D3)C2×C414yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2311yescharacteristic
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2312yes—
75C2×#624—42no—
76C2×#624—42no—
77C2×#624C2×C418yes—
78C32×C42—22no—
79#3752—42no—
80#3752—42no—
81C4⋊(C3⋊Dic3)—22no—
82C3×C4⋊Dic3—22no—
83C3×C4⋊Dic3C2×C414yes—
84C3×Dic3⋊C4—42no—
85C3×C4×Dic3—22no—
86C3×C4×Dic3C2×C414yes—
87Dic3⋊Dic3—42no—
88#3722—42no—
89#3722—42no—
90#3722C2×C418yes—
91Dic32—42no—
92#2319—91noSylow
93C22×D12—31no—
94C22×D12—31no—
95C22×D12—61no—
96C22×C4×D3—32no—
97C22×C4×D3—32no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—32no—
101C2×C3×C42—21no—
102C2×C3×C42—31no—
103C2×C3×C42—31no—
104C2×C3×C42C2×D311yescharacteristic
105C2×C3×C42C2×D311yescharacteristic
106C2×(C2×D3)⋊C4—32no—
107C2×(C2×D3)⋊C4—32no—
108C2×(C2×D3)⋊C4—32no—
109C2×(C2×D3)⋊C4—32no—
110C2×(C2×D3)⋊C4—32no—
111C2×(C2×D3)⋊C4—32no—
112C2×(C2×D3)⋊C4—32no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—62no—
117C2×(C2×D3)⋊C4—62no—
118C2×C4⋊Dic3—31no—
119C2×C4⋊Dic3—31no—
120C2×C4⋊Dic3—32no—
121C2×C4⋊Dic3—61no—
122C2×C4⋊Dic3C2×D311yescharacteristic
123C2×C4⋊Dic3C2×D311yescharacteristic
124C2×Dic3⋊C4—32no—
125C2×Dic3⋊C4C2×D312yes—
126C2×C4×Dic3—32no—
127C2×C4×Dic3—32no—
128C2×C4×Dic3C2×D311yescharacteristic
129C2×C4×Dic3C2×D311yescharacteristic
130#893—38no—
131C4×D12—38no—
132C3×#95—31no—
133C3×#95—32no—
134C3×#95—31no—
135C3×#95—31no—
136C3×#95—31no—
137C3⋊#95—32no—
138C3⋊#95—32no—
139C3⋊#95—32no—
140C3⋊#95—32no—
141C3⋊#95C2×D311yescharacteristic
142C3⋊#95C2×D312yes—
143C3⋊#95C2×D311yescharacteristic
144C3⋊#95C2×D311yescharacteristic
145C3⋊#95C2×D311yescharacteristic
146C23×C32C2411yescharacteristic
147C22×C3⋊D3—22no—
148C22×C3⋊D3—22no—
149C22×C3⋊D3—22no—
150C22×C3⋊D3—22no—
151C22×C3⋊D3—22no—
152C22×C3⋊D3—22no—
153C22×C3⋊D3C22×C414yes—
154C2×C32×C4—21no—
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×C3⋊Dic3—22no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—22no—
166C4⋊(C3⋊D3)—28no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3C22×C412yes—
183C2×C3×Dic3C22×C414yes—
184C2×C3×Dic3C22×C412yes—
185#624—48no—
186C2×C4×D4—91no—
187#390—91no—
188#390—91no—
189#389—31no—
190#389—91no—
191#388—92no—
192#388—92no—
193#381—92no—
194#380—92no—
195C4×C4⋊C4—91no—
196#376—31no—
197C23×D3—32no—
198C23×D3—32no—
199C23×D3—62no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—31no—
204C22×C3×C4—32no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—32no—
208C22×C3×C4—31no—
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×Dic3—32no—
216C22×Dic3—32no—
217C22×Dic3—62no—
218C22×Dic3C22×D311yescharacteristic
219C22×Dic3C22×D312yes—
220C22×Dic3C22×D311yescharacteristic
221C22×Dic3C22×D311yescharacteristic
222C22×Dic3C22×D312yes—
223C22×Dic3C22×D311yescharacteristic
224C2×D12—34no—
225C2×D12—34no—
226C2×D12—64no—
227C2×D12—62no—
228C2×D12—62no—
229C2×D12—62no—
230C2×D12—62no—
231C2×D12—122no—
232C2×D12—122no—
233C2×C4×D3—38no—
234C2×C4×D3—38no—
235C2×C4×D3—122no—
236C2×C4×D3—122no—
237C2×C4×D3—122no—
238C2×C4×D3—122no—
239C3×C4⋊C4—34no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—122no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42—34no—
245C3×C42—42no—
246C3×C42—62no—
247(C2×D3)⋊C4—38no—
248(C2×D3)⋊C4—38no—
249(C2×D3)⋊C4—122no—
250(C2×D3)⋊C4—122no—
251(C2×D3)⋊C4—122no—
252(C2×D3)⋊C4—122no—
253(C2×D3)⋊C4—122no—
254(C2×D3)⋊C4—122no—
255(C2×D3)⋊C4—122no—
256(C2×D3)⋊C4—122no—
257(C2×D3)⋊C4—124no—
258(C2×D3)⋊C4—124no—
259C4⋊Dic3—22no—
260C4⋊Dic3—62no—
261C4⋊Dic3—62no—
262C4⋊Dic3—122no—
263C4⋊Dic3—122no—
264C4⋊Dic3C4×D314yes—
265Dic3⋊C4—42no—
266Dic3⋊C4—122no—
267C4×Dic3—22no—
268C4×Dic3—122no—
269C4×Dic3—122no—
270C4×Dic3C4×D314yes—
271C22×C32C23×C411yescharacteristic
272C22×C32C2×#4111yescharacteristic
273C22×C32C2×#4111yescharacteristic
274C22×C32H(5,𝔽2)11yescharacteristic
275C22×C32H(5,𝔽2)11yescharacteristic
276C22×C32H(5,𝔽2)11yescharacteristic
277C22×C322−1+411yescharacteristic
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—41no—
286C32×C4—41no—
287C32×C4#11714yes—
288C3⋊Dic3—42no—
289C3×Dic3—22no—
290C3×Dic3—24no—
291C3×Dic3—22no—
292C3×Dic3—41no—
293C3×Dic3—42no—
294C3×Dic3—41no—
295C22×D4—91no—
296C23×C4—92no—
297C4×D4—98no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—91no—
302C2×C4⋊C4—92no—
303C2×C22⋊C4—92no—
304C2×C22⋊C4—92no—
305C2×C22⋊C4—92no—
306C2×C22⋊C4—92no—
307C2×C22⋊C4—92no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—92no—
311C2×C42D3211yescharacteristic
312#95—31no—
313#95—31no—
314#95—31no—
315#95—31no—
316#95—32no—
317#95—92no—
318#95—92no—
319C23×C3—21no—
320C23×C3C23×D311yescharacteristic
321C23×C3C23×D311yescharacteristic
322C22×D3—34no—
323C22×D3—34no—
324C22×D3—64no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—62no—
328C22×D3—62no—
329C22×D3—62no—
330C22×D3—62no—
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—122no—
338C22×D3—122no—
339C22×D3—122no—
340C22×D3—122no—
341C22×D3—122no—
342C22×D3—122no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—32no—
361C2×C3×C4—34no—
362C2×C3×C4—32no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—62no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—62no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—62no—
382C2×C3×C4C2×C4×D312yes—
383C2×C3×C4C2×C4×D312yes—
384C2×C3×C4#23512yes—
385C2×C3×C4#23512yes—
386C2×C3×C4#23712yes—
387C2×C3×C4#23912yes—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—62no—
404C2×Dic3—62no—
405C2×Dic3—62no—
406C2×Dic3—62no—
407C2×Dic3—62no—
408C2×Dic3—62no—
409C2×Dic3—122no—
410C2×Dic3—122no—
411C2×Dic3—122no—
412C2×Dic3C2×C4×D312yes—
413C2×Dic3C2×C4×D312yes—
414C2×Dic3C2×C4×D314yes—
415D12—68no—
416D12—68no—
417D12—128no—
418C4×D3—128no—
419C4×D3—128no—
420C2×C32C2×#11711yescharacteristic
421C2×C32#51811yescharacteristic
422C2×C32#52011yescharacteristic
423C2×C32#54011yescharacteristic
424C2×C32#54011yescharacteristic
425C2×C32#54211yescharacteristic
426C2×C32#54211yescharacteristic
427C3⋊D3—44no—
428C24—92no—
429C2×D4—94no—
430C2×D4—182no—
431C2×D4—182no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—32no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—32no—
438C22×C4—98no—
439C22×C4—92no—
440C22×C4—362no—
441C22×C4—362no—
442C22×C4C2×D3211yescharacteristic
443C22×C4C2×D3211yescharacteristic
444C22×C4C2×D3211yescharacteristic
445C4⋊C4—34no—
446C4⋊C4—62no—
447C4⋊C4—122no—
448C4⋊C4—182no—
449C4⋊C4—362no—
450C22⋊C4—98no—
451C22⋊C4—362no—
452C22⋊C4—362no—
453C22⋊C4—362no—
454C22⋊C4—362no—
455C42—22no—
456C42—34no—
457C42—62no—
458C42—362no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3C22×C4×D311yescharacteristic
467C22×C3C22×C4×D311yescharacteristic
468C22×C3C2×#23511yescharacteristic
469C22×C3C2×#23511yescharacteristic
470C22×C3C2×#23711yescharacteristic
471C22×C3#100411yescharacteristic
472C22×C3C2×#23911yescharacteristic
473C22×C3#100711yescharacteristic
474C22×C3#100911yescharacteristic
475C22×C3#100911yescharacteristic
476C22×C3#100911yescharacteristic
477C22×C3#100911yescharacteristic
478C22×C3#100911yescharacteristic
479C22×C3#101011yescharacteristic
480C2×D3—64no—
481C2×D3—64no—
482C2×D3—64no—
483C2×D3—64no—
484C2×D3—64no—
485C2×D3—64no—
486C2×D3—124no—
487C2×D3—124no—
488C2×D3—124no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—122no—
492C2×D3—122no—
493C2×D3—122no—
494C2×D3—122no—
495C2×D3—122no—
496C2×D3—122no—
497C2×D3—242no—
498C2×D3—242no—
499C2×D3—242no—
500C2×D3—242no—
501C3×C4—24no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—64no—
509C3×C4—62no—
510C3×C4—62no—
511C3×C4—121no—
512C3×C4—121no—
513C3×C4—122no—
514C3×C4#87214yes—
515C3×C4#89214yes—
516Dic3—22no—
517Dic3—22no—
518Dic3—24no—
519Dic3—41no—
520Dic3—41no—
521Dic3—42no—
522Dic3—122no—
523Dic3—122no—
524Dic3—242no—
525C32#231911yescharacteristic, complemented, Sylow
526C23—94no—
527C23—182no—
528C23—182no—
529C23—182no—
530C23—182no—
531C23—182no—
532C23—182no—
533C23C22×D3211yescharacteristic
534D4—188no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—34no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—182no—
560C2×C4—182no—
561C2×C4—182no—
562C2×C4—368no—
563C2×C4#379912yes—
564C2×C4C4×D3212yes—
565C2×C4#380412yes—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3C2×#87211yescharacteristic
574C2×C3C2×#89211yescharacteristic
575C2×C3#552111yescharacteristic
576C2×C3#552311yescharacteristic
577C2×C3#553411yescharacteristic
578C2×C3#553411yescharacteristic
579C2×C3#553511yescharacteristic
580C2×C3#553511yescharacteristic
581C2×C3#554511yescharacteristic
582C2×C3#557411yescharacteristic
583C2×C3#561911yescharacteristic
584C2×C3#564311yescharacteristic
585C2×C3#569711yescharacteristic
586C2×C3#573311yescharacteristic
587D3—124no—
588D3—124no—
589D3—244no—
590C22—184no—
591C22—184no—
592C22—184no—
593C22—362no—
594C22—362no—
595C22—362no—
596C22—362no—
597C22#6429711yescharacteristic
598C22C2×#379911yescharacteristic
599C22C2×C4×D3211yescharacteristic
600C22#6430711yescharacteristic
601C22#6430811yescharacteristic
602C22#6430811yescharacteristic
603C22C2×#380411yescharacteristic
604C4—41no—
605C4—41no—
606C4—62no—
607C4—62no—
608C4—64no—
609C4—121no—
610C4—122no—
611C4—121no—
612C4—362no—
613C4#6383614yes—
614C3—21no—
615C3#7341511yescharacteristic, complemented
616C3#7596911yescharacteristic, complemented
617C2—364no—
618C2#1059387711yescharacteristic
619C2#1059389011yescharacteristic
620C2#1059398911yescharacteristic
621C2#1059399411yescharacteristic
622C2C2×#6383611yescharacteristic
623C2#1059481411yescharacteristic
624C2#1059498511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, cosocleC23×C32C24
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C42D32
hypocenterC32#2319
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. C3×#4932C2semidirect
4. C3×#4928C2semidirect
5. #10591371C2semidirect
6. #10591367C2semidirect
7. #10591352C2semidirect
9. Dic3×C4⋊Dic3C2semidirect
10. #10591259C2semidirect
11. #10591258C2semidirect
30. #63894C4semidirect
52. #4932D3semidirect
54. #4928D3semidirect
525. C32#2319semidirect
615. C3#73415semidirect
616. C3#75969semidirect

Maximal subgroups

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

subgroupindex
C2×#638942
#105926382
C3×#49322
C3×#49282
#105913712
#105913672
#105913522
#105913202
Dic3×C4⋊Dic32
#105912592
#105912582
#759693
#734153