Finite Group Databasebeta

← C3×#71566 #11773346

Subgroups — C3×#71566 #11773346

Order 1152.

1337 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4⋊C4C211yesmaximal, characteristic
2C22×C3×Dic3⋊C4C211yesmaximal, characteristic
3C2×C3×C3⋊#95C212yesmaximal
4C2×C3×C3⋊#95C211yesmaximal, characteristic
5C2×C3×C3⋊#95C212yesmaximal
6C3×#1441—31nomaximal
7#71566C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2212yes—
9C23×C32×C4C2211yescharacteristic
10C23×C3×Dic3C2211yescharacteristic
11C23×C3×Dic3C2212yes—
12C23×C3×Dic3C2211yescharacteristic
13C2×C32×C4⋊C4—24no—
14C2×C32×C4⋊C4—22no—
15C2×C3×Dic3⋊C4—22no—
16C2×C3×Dic3⋊C4—22no—
17C2×C3×Dic3⋊C4C414yescomplemented
18C3×C3⋊#95—22no—
19C3×C3⋊#95—24no—
20C3×C3⋊#95—24no—
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4—31no—
23C22×C3×C4⋊C4D311yescharacteristic
24C22×C3×C4⋊C4C2×C311yescharacteristic
25C22×Dic3⋊C4C2×C311yescharacteristic
26C2×C3×#95—32no—
27C2×C3×#95—31no—
28C2×C3×#95—32no—
29C2×C3⋊#95C2×C312yes—
30C2×C3⋊#95C2×C311yescharacteristic
31C2×C3⋊#95C2×C312yes—
32C24×C32C2311yescharacteristic
33C22×C32×C4—22no—
34C22×C32×C4—22no—
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4—22no—
38C22×C32×C4—22no—
39C22×C32×C4—22no—
40C22×C32×C4—22no—
41C22×C32×C4C2×C412yes—
42C22×C32×C4D414yes—
43C22×C32×C4D412yes—
44C22×C32×C4D412yes—
45C22×C3×Dic3—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—22no—
52C22×C3×Dic3—22no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—22no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—22no—
58C22×C3×Dic3—22no—
59C22×C3×Dic3C2×C412yes—
60C22×C3×Dic3C2×C412yes—
61C22×C3×Dic3D412yes—
62C22×C3×Dic3Q814yes—
63C22×C3×Dic3Q812yes—
64C32×C4⋊C4—28no—
65C3×Dic3⋊C4—28no—
66#1441—31noSylow
67C23×C3×C4—21no—
68C23×C3×C4—22no—
69C23×C3×C4—31no—
70C23×C3×C4—32no—
71C23×C3×C4—31no—
72C23×C3×C4C2×D312yes—
73C23×C3×C4C2×D311yescharacteristic
74C23×C3×C4C22×C312yes—
75C23×C3×C4C22×C311yescharacteristic
76C23×Dic3C22×C311yescharacteristic
77C23×Dic3C22×C312yes—
78C23×Dic3C22×C311yescharacteristic
79C2×C3×C4⋊C4—28no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—34no—
86C2×C3×C4⋊C4—62no—
87C2×C3×C4⋊C4—62no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4C3×C414yescomplemented
91C3×#95—64no—
92C3×#95—62no—
93C3×#95—64no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C23×C32C22×C411yescharacteristic
98C23×C32C2×D411yescharacteristic
99C23×C32C2×D411yescharacteristic
100C23×C32C2×D411yescharacteristic
101C23×C32C2×D412yes—
102C23×C32C2×Q811yescharacteristic
103C23×C32C2×Q812yes—
104C23×C32#4112yes—
105C23×C32#4111yescharacteristic
106C23×C32#4112yes—
107C23×C32#4111yescharacteristic
108C2×C32×C4—24no—
109C2×C32×C4—22no—
110C2×C32×C4—24no—
111C2×C32×C4—22no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—42no—
116C2×C32×C4—42no—
117C2×C32×C4—42no—
118C2×C32×C4—42no—
119C2×C32×C4C22⋊C414yes—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—42no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—42no—
135C2×C3×Dic3—42no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—42no—
138C2×C3×Dic3C4⋊C414yescomplemented
139C22×C4⋊C4—31no—
140C22×C4⋊C4C3×D311yescharacteristic
141C2×#95—32no—
142C2×#95—31no—
143C2×#95—32no—
144C24×C3—21no—
145C24×C3C22×D311yescharacteristic
146C24×C3C23×C311yescharacteristic
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4—34no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—62no—
185C22×C3×C4—61no—
186C22×C3×C4—62no—
187C22×C3×C4—62no—
188C22×C3×C4—62no—
189C22×C3×C4—62no—
190C22×C3×C4—62no—
191C22×C3×C4—62no—
192C22×C3×C4—62no—
193C22×C3×C4—61no—
194C22×C3×C4C4×D312yes—
195C22×C3×C4D1212yes—
196C22×C3×C4#6714yes—
197C22×C3×C4#6712yes—
198C22×C3×C4C2×C3×C412yes—
199C22×C3×C4C3×D412yes—
200C22×C3×C4C3×D414yes—
201C22×C3×C4C3×D412yes—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3C2×C3×C412yes—
217C22×Dic3C2×C3×C412yes—
218C22×Dic3C3×D412yes—
219C22×Dic3C3×Q814yes—
220C22×Dic3C3×Q812yes—
221C3×C4⋊C4—216no—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—68no—
225Dic3⋊C4—28no—
226C22×C32C2×C22⋊C411yescharacteristic
227C22×C32C2×C4⋊C411yescharacteristic
228C22×C32#11711yescharacteristic
229C22×C32C4×D412yes—
230C22×C32C4×Q812yes—
231C22×C32#12011yescharacteristic
232C22×C32#12111yescharacteristic
233C22×C32#12112yes—
234C22×C32#12111yescharacteristic
235C22×C32#12112yes—
236C22×C32C22⋊Q812yes—
237C22×C32C22⋊Q811yescharacteristic
238C22×C32C22⋊Q812yes—
239C22×C32C22⋊Q811yescharacteristic
240C22×C32C22⋊Q812yes—
241C22×C32#12311yescharacteristic
242C22×C32#12312yes—
243C22×C32#12411yescharacteristic
244C22×C32#12512yes—
245C22×C32#12512yes—
246C22×C32#12612yes—
247C22×C32C4⋊Q812yes—
248C22×C32C4⋊Q811yescharacteristic
249C32×C4—24no—
250C32×C4—44no—
251C3×Dic3—24no—
252C3×Dic3—42no—
253C3×Dic3—44no—
254C23×C4—31no—
255C23×C4—31no—
256C23×C4—32no—
257C23×C4C2×C3×D312yes—
258C23×C4C2×C3×D311yescharacteristic
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264#95—64no—
265#95—62no—
266#95—64no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—21no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—22no—
278C23×C3C2×C4×D311yescharacteristic
279C23×C3C2×D1211yescharacteristic
280C23×C3#23512yes—
281C23×C3D3×D411yescharacteristic
282C23×C3#23712yes—
283C23×C3#23711yescharacteristic
284C23×C3D3×Q812yes—
285C23×C3D3×Q811yescharacteristic
286C23×C3#23911yescharacteristic
287C23×C3C2×#6711yescharacteristic
288C23×C3C2×#6712yes—
289C23×C3C22×C3×C411yescharacteristic
290C23×C3C2×C3×D411yescharacteristic
291C23×C3C2×C3×D411yescharacteristic
292C23×C3C2×C3×D411yescharacteristic
293C23×C3C2×C3×D412yes—
294C23×C3C2×C3×Q811yescharacteristic
295C23×C3C2×C3×Q812yes—
296C23×C3C3×#4112yes—
297C23×C3C3×#4111yescharacteristic
298C23×C3C3×#4112yes—
299C23×C3C3×#4111yescharacteristic
300C2×C3×C4—24no—
301C2×C3×C4—28no—
302C2×C3×C4—24no—
303C2×C3×C4—28no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—28no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—34no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—64no—
342C2×C3×C4—64no—
343C2×C3×C4—62no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4—122no—
347C2×C3×C4—122no—
348C2×C3×C4—121no—
349C2×C3×C4—122no—
350C2×C3×C4—122no—
351C2×C3×C4—122no—
352C2×C3×C4—121no—
353C2×C3×C4(C2×D3)⋊C414yes—
354C2×C3×C4C3×C22⋊C414yes—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—41no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3C3×C4⋊C414yescomplemented
374C2×C32#38011yescharacteristic
375C2×C32#38611yescharacteristic
376C2×C32#38712yes—
377C2×C32#38912yes—
378C2×C32#39111yescharacteristic
379C2×C32#39411yescharacteristic
380C2×C32#39511yescharacteristic
381C2×C32#39612yes—
382C2×C32#39812yes—
383C2×C32#39811yescharacteristic
384C2×C32#40011yescharacteristic
385C24C22×C3×D311yescharacteristic
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—34no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4C3×C4×D312yes—
414C22×C4C3×D1212yes—
415C22×C4C3×#6714yes—
416C22×C4C3×#6712yes—
417C4⋊C4—28no—
418C4⋊C4—68no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—21no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3#88712yes—
443C22×C3#88912yes—
444C22×C3#88912yes—
445C22×C3D3×C4⋊C411yescharacteristic
446C22×C3#89211yescharacteristic
447C22×C3#89511yescharacteristic
448C22×C3#89612yes—
449C22×C3#89712yes—
450C22×C3#89711yescharacteristic
451C22×C3#89812yes—
452C22×C3C2×(C2×D3)⋊C411yescharacteristic
453C22×C3C4×#6712yes—
454C22×C3#92911yescharacteristic
455C22×C3#93012yes—
456C22×C3#93512yes—
457C22×C3#93611yescharacteristic
458C22×C3#93811yescharacteristic
459C22×C3#93912yes—
460C22×C3#94411yescharacteristic
461C22×C3#94412yes—
462C22×C3#94612yes—
463C22×C3#94611yescharacteristic
464C22×C3#95311yescharacteristic
465C22×C3C2×C3×C22⋊C411yescharacteristic
466C22×C3C2×C3×C4⋊C411yescharacteristic
467C22×C3C3×#11711yescharacteristic
468C22×C3C3×C4×D412yes—
469C22×C3C3×C4×Q812yes—
470C22×C3C3×#12011yescharacteristic
471C22×C3C3×#12111yescharacteristic
472C22×C3C3×#12112yes—
473C22×C3C3×#12111yescharacteristic
474C22×C3C3×#12112yes—
475C22×C3C3×C22⋊Q812yes—
476C22×C3C3×C22⋊Q811yescharacteristic
477C22×C3C3×C22⋊Q812yes—
478C22×C3C3×C22⋊Q811yescharacteristic
479C22×C3C3×C22⋊Q812yes—
480C22×C3C3×#12311yescharacteristic
481C22×C3C3×#12312yes—
482C22×C3C3×#12411yescharacteristic
483C22×C3C3×#12512yes—
484C22×C3C3×#12512yes—
485C22×C3C3×#12612yes—
486C22×C3C3×C4⋊Q812yes—
487C22×C3C3×C4⋊Q811yescharacteristic
488C3×C4—28no—
489C3×C4—24no—
490C3×C4—24no—
491C3×C4—48no—
492C3×C4—44no—
493C3×C4—44no—
494C3×C4—64no—
495C3×C4—122no—
496C3×C4—124no—
497Dic3—24no—
498Dic3—42no—
499Dic3—44no—
500C32#144111yescharacteristic, complemented, Sylow
501C23C2×C3×C4×D311yescharacteristic
502C23C2×C3×D1211yescharacteristic
503C23C3×#23512yes—
504C23C3×D3×D411yescharacteristic
505C23C3×#23712yes—
506C23C3×#23711yescharacteristic
507C23C3×D3×Q812yes—
508C23C3×D3×Q811yescharacteristic
509C23C3×#23911yescharacteristic
510C23C2×C3×#6711yescharacteristic
511C23C2×C3×#6712yes—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—34no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—121no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4C3×(C2×D3)⋊C414yes—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—22no—
554C2×C3#497311yescharacteristic
555C2×C3#497612yes—
556C2×C3#497812yes—
557C2×C3#498112yes—
558C2×C3#498311yescharacteristic
559C2×C3#498511yescharacteristic
560C2×C3#498711yescharacteristic
561C2×C3#499412yes—
562C2×C3#521311yescharacteristic
563C2×C3#522211yescharacteristic
564C2×C3#523011yescharacteristic
565C2×C3C3×#38011yescharacteristic
566C2×C3C3×#38611yescharacteristic
567C2×C3C3×#38712yes—
568C2×C3C3×#38912yes—
569C2×C3C3×#39111yescharacteristic
570C2×C3C3×#39411yescharacteristic
571C2×C3C3×#39511yescharacteristic
572C2×C3C3×#39612yes—
573C2×C3C3×#39812yes—
574C2×C3C3×#39811yescharacteristic
575C2×C3C3×#40011yescharacteristic
576C22C3×#88712yes—
577C22C3×#88912yes—
578C22C3×#88912yes—
579C22C3×D3×C4⋊C411yescharacteristic
580C22C3×#89211yescharacteristic
581C22C3×#89511yescharacteristic
582C22C3×#89612yes—
583C22C3×#89712yes—
584C22C3×#89711yescharacteristic
585C22C3×#89812yes—
586C22C2×C3×(C2×D3)⋊C411yescharacteristic
587C22C3×C4×#6712yes—
588C22C3×#92911yescharacteristic
589C22C3×#93012yes—
590C22C3×#93512yes—
591C22C3×#93611yescharacteristic
592C22C3×#93811yescharacteristic
593C22C3×#93912yes—
594C22C3×#94411yescharacteristic
595C22C3×#94412yes—
596C22C3×#94612yes—
597C22C3×#94611yescharacteristic
598C22C3×#95311yescharacteristic
599C4—24no—
600C4—44no—
601C4—64no—
602C4—122no—
603C4—124no—
604C3—21no—
605C3#7156611yescharacteristic, direct product factor
606C3C3×#144111yescharacteristic, complemented
607C2C3×#497311yescharacteristic
608C2C3×#497612yes—
609C2C3×#497812yes—
610C2C3×#498112yes—
611C2C3×#498311yescharacteristic
612C2C3×#498511yescharacteristic
613C2C3×#498711yescharacteristic
614C2C3×#499412yes—
615C2C3×#521311yescharacteristic
616C2C3×#522211yescharacteristic
617C2C3×#523011yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutatorC23×C3C22×C3×C4
FrattiniC24C22×C3×D3
FittingC22×C32×C4⋊C4C2
socle, prime radicalC24×C32C23
cosocleC24×C3C23×C3
hypercenterC22×C3×C4⋊C4D3
hypocenterC3C3×#1441
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #71566C3direct (N × H)
17. C2×C3×Dic3⋊C4C4semidirect
90. C2×Dic3⋊C4C3×C4semidirect
138. C2×C3×Dic3C4⋊C4semidirect
373. C2×Dic3C3×C4⋊C4semidirect
500. C32#1441semidirect
605. C3#71566direct (N × H)
606. C3C3×#1441semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C4⋊C42
C22×C3×Dic3⋊C42
C2×C3×C3⋊#952
C3×#14413
#715663