Finite Group Databasebeta

← C3×#71708 #11775111

Subgroups — C3×#71708 #11775111

Order 1152.

833 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#525C211yesmaximal, characteristic
2C2×C3×C3⋊#95C211yesmaximal, characteristic, complemented
3C3×#5207C211yesmaximal, characteristic, complemented
4C3×#4538C212yesmaximal, complemented
5C3×#4537C212yesmaximal, complemented
6C3×#1504—31nomaximal
7#71708C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C411yescharacteristic, complemented
9C23×C32×C4C2211yescharacteristic
10C23×C3×Dic3—21no—
11C32×#123—21no—
12C32×C22⋊Q8C411yescharacteristic, complemented
13C32×C22⋊Q8C2211yescharacteristic
14C32×#121C411yescharacteristic, complemented
15C32×#121C2211yescharacteristic
16C32×#120—21no—
17C32×C4×D4—21no—
18C32×C4×D4—21no—
19C32×#117C411yescharacteristic, complemented
20C2×C3×#208—21no—
21C3×C3⋊#95—21no—
22C3×C3⋊#95C2212yes—
23C3×#830—21no—
24C3×#830C2212yes—
25C3×#525—21no—
26C3×#525D311yescharacteristic
27C3×#525C2×C311yescharacteristic
28C3×#407—31no—
29C2×C3×#95—31no—
30C2×C3⋊#95C2×C311yescharacteristic, complemented
31#5207C2×C311yescharacteristic, complemented
32C3×#328—32no—
33C3×#327—32no—
34#4538C2×C312yescomplemented
35#4537C2×C312yescomplemented
36C24×C32D411yescharacteristic
37C32×#41—42no—
38C2×C32×Q8C2×C411yescharacteristic, complemented
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4C2×C411yescharacteristic, complemented
43C22×C32×C4—21no—
44C22×C32×C4—42no—
45C22×C32×C4C2×C411yescharacteristic
46C22×C32×C4C2×C411yescharacteristic
47C22×C32×C4D411yescharacteristic
48C22×C32×C4D411yescharacteristic
49C22×C32×C4D411yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3—24no—
52C22×C3×Dic3—22no—
53C22×C3×Dic3—42no—
54C22×C3×Dic3—42no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×C411yescharacteristic, complemented
58C32×C4⋊C4C2×C411yescharacteristic, complemented
59C32×C22⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C42—21no—
65C3×#208—42no—
66C2×C3×C3⋊C8—22no—
67#1504—31noSylow
68C2×C3×#41—21no—
69C2×C3×#41Dic311yescharacteristic, complemented
70C2×C3×#41C3×C411yescharacteristic, complemented
71C23×C3×C4—21no—
72C23×C3×C4—61no—
73C23×C3×C4C2×D311yescharacteristic
74C23×C3×C4C22×C311yescharacteristic
75C23×Dic3—21no—
76C2×C3×M16—61no—
77C3×#123—22no—
78C3×#123—21no—
79C3×#123—21no—
80C3×C22⋊Q8—21no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8Dic311yescharacteristic, complemented
83C3×C22⋊Q8C3×C411yescharacteristic, complemented
84C3×C22⋊Q8C2×D311yescharacteristic
85C3×C22⋊Q8C22×C311yescharacteristic
86C3×#121—21no—
87C3×#121—21no—
88C3×#121Dic311yescharacteristic, complemented
89C3×#121C3×C411yescharacteristic, complemented
90C3×#121C2×D311yescharacteristic
91C3×#121C22×C311yescharacteristic
92C3×#120—22no—
93C3×#120—21no—
94C3×#120—21no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×#117—21no—
102C3×#117Dic311yescharacteristic, complemented
103C3×#117C3×C411yescharacteristic, complemented
104C2×#208—21no—
105C3×C22⋊C8—32no—
106C3×C22⋊C8—61no—
107C3×#95—32no—
108C3×#95—61no—
109C3⋊#95—21no—
110C3⋊#95C22×C312yes—
111#830—21no—
112#830C22×C312yes—
113C23×C32—21no—
114C23×C32—22no—
115C23×C32—22no—
116C23×C32—21no—
117C23×C32—21no—
118C23×C32C22⋊C411yescharacteristic
119C23×C32C22⋊C411yescharacteristic
120C23×C32C2×D411yescharacteristic
121C32×Q8—41no—
122C32×D4—41no—
123C32×D4—42no—
124C32×D4—42no—
125C32×D4—41no—
126C32×D4—41no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4—42no—
137C2×C32×C4C22⋊C411yescharacteristic
138C2×C32×C4C22⋊C411yescharacteristic
139C2×C32×C4C22×C411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C3×Dic3—24no—
142C2×C3×Dic3—48no—
143C2×C3×Dic3—42no—
144C2×C3×Dic3—42no—
145C3×C3⋊C8—42no—
146#525C3×D311yescharacteristic
147#407—31no—
148C2×#95—31no—
149#328—32no—
150#327—32no—
151C24×C3—21no—
152C24×C3#6711yescharacteristic
153C24×C3C3×D411yescharacteristic
154C3×#41—44no—
155C3×#41—42no—
156C3×#41—42no—
157C2×C3×Q8—21no—
158C2×C3×Q8C2×Dic311yescharacteristic, complemented
159C2×C3×Q8C2×C3×C411yescharacteristic, complemented
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4C2×Dic311yescharacteristic, complemented
171C2×C3×D4C2×C3×C411yescharacteristic, complemented
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—44no—
184C22×C3×C4—64no—
185C22×C3×C4—62no—
186C22×C3×C4—122no—
187C22×C3×C4—122no—
188C22×C3×C4C2×Dic311yescharacteristic
189C22×C3×C4C2×Dic311yescharacteristic
190C22×C3×C4#6711yescharacteristic
191C22×C3×C4#6711yescharacteristic
192C22×C3×C4#6711yescharacteristic
193C22×C3×C4C2×C3×C411yescharacteristic
194C22×C3×C4C2×C3×C411yescharacteristic
195C22×C3×C4C3×D411yescharacteristic
196C22×C3×C4C3×D411yescharacteristic
197C22×C3×C4C3×D411yescharacteristic
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C23×C311yescharacteristic
200C22×Dic3—24no—
201C22×Dic3—22no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C3×M16—122no—
205C2×C3×C8—62no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×Dic311yescharacteristic, complemented
215C3×C4⋊C4C2×Dic311yescharacteristic, complemented
216C3×C4⋊C4C2×C3×C411yescharacteristic, complemented
217C3×C4⋊C4C2×C3×C411yescharacteristic, complemented
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—21no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42—21no—
236#208—42no—
237C2×C3⋊C8—22no—
238C22×C32—24no—
239C22×C32—24no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32#9912yes—
250C22×C32#10412yes—
251C22×C32C2×C22⋊C411yescharacteristic
252C22×C32AGL(1,ℤ/8)11yescharacteristic
253C22×C32#13711yescharacteristic
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—41no—
259C32×C4—41no—
260C32×C4—41no—
261C3×Dic3—44no—
262C2×#41C3×Dic311yescharacteristic, complemented
263C23×C4—61no—
264C23×C4C2×C3×D311yescharacteristic
265C2×M16—61no—
266#123—21no—
267C22⋊Q8C3×Dic311yescharacteristic, complemented
268C22⋊Q8C2×C3×D311yescharacteristic
269#121C3×Dic311yescharacteristic, complemented
270#121C2×C3×D311yescharacteristic
271#120—21no—
272C4×D4—21no—
273C4×D4—21no—
274#117C3×Dic311yescharacteristic, complemented
275C22⋊C8—32no—
276C22⋊C8—61no—
277#95—32no—
278#95—61no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—22no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—42no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3#21711yescharacteristic
298C23×C3#21711yescharacteristic
299C23×C3C3×C22⋊C411yescharacteristic
300C23×C3C3×C22⋊C411yescharacteristic
301C23×C3C2×#6711yescharacteristic
302C23×C3C2×C3×D411yescharacteristic
303C3×Q8—41no—
304C3×Q8—41no—
305C3×Q8—42no—
306C3×D4—41no—
307C3×D4—41no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—44no—
312C3×D4—44no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—41no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—44no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—64no—
356C2×C3×C4—128no—
357C2×C3×C4—122no—
358C2×C3×C4—122no—
359C2×C3×C4#21711yescharacteristic
360C2×C3×C4#21711yescharacteristic
361C2×C3×C4C3×C22⋊C411yescharacteristic
362C2×C3×C4C3×C22⋊C411yescharacteristic
363C2×C3×C4C22×Dic311yescharacteristic
364C2×C3×C4C2×#6711yescharacteristic
365C2×C3×C4C22×C3×C411yescharacteristic
366C2×C3×C4C2×C3×D411yescharacteristic
367C2×Dic3—24no—
368C2×Dic3—48no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C3×C8—122no—
372C3⋊C8—42no—
373C2×C32—21no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—81no—
378C2×C32C2×#9911yescharacteristic
379C2×C32#41711yescharacteristic
380C2×C32C2×#10411yescharacteristic
381C24C3×#6711yescharacteristic
382#41—42no—
383C2×Q8C2×C3×Dic311yescharacteristic, complemented
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4C2×C3×Dic311yescharacteristic, complemented
388C22×C4—21no—
389C22×C4—42no—
390C22×C4—64no—
391C22×C4—62no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4C2×C3×Dic311yescharacteristic
395C22×C4C2×C3×Dic311yescharacteristic
396C22×C4C3×#6711yescharacteristic
397C22×C4C3×#6711yescharacteristic
398C22×C4C3×#6711yescharacteristic
399C22×C4C22×C3×D311yescharacteristic
400M16—122no—
401C2×C8—62no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4C2×C3×Dic311yescharacteristic, complemented
405C4⋊C4C2×C3×Dic311yescharacteristic, complemented
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C42—21no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3#83412yes—
451C22×C3#83712yes—
452C22×C3C3×#9912yes—
453C22×C3C3×#10412yes—
454C22×C3#94911yescharacteristic
455C22×C3#95111yescharacteristic
456C22×C3C2×#21711yescharacteristic
457C22×C3C2×C3×C22⋊C411yescharacteristic
458C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
459C22×C3C3×#13711yescharacteristic
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—124no—
482Dic3—44no—
483C32#150411yescharacteristic, complemented, Sylow
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23C3×#21711yescharacteristic
490C23C3×#21711yescharacteristic
491C23C2×C3×#6711yescharacteristic
492Q8—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—42no—
497D4—42no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—64no—
509C2×C4—128no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4C3×#21711yescharacteristic
513C2×C4C3×#21711yescharacteristic
514C2×C4C22×C3×Dic311yescharacteristic
515C2×C4C2×C3×#6711yescharacteristic
516C8—122no—
517C2×C3—21no—
518C2×C3—22no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—42no—
531C2×C3—42no—
532C2×C3—82no—
533C2×C3—81no—
534C2×C3—81no—
535C2×C3C2×#83411yescharacteristic
536C2×C3#523211yescharacteristic
537C2×C3C2×#83711yescharacteristic
538C2×C3C2×C3×#9911yescharacteristic
539C2×C3C3×#41711yescharacteristic
540C2×C3C2×C3×#10411yescharacteristic
541C22—24no—
542C22—24no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22C3×#83412yes—
553C22C3×#83712yes—
554C22C3×#94911yescharacteristic
555C22C3×#95111yescharacteristic
556C22C2×C3×#21711yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—124no—
565C3—21no—
566C3#7170811yescharacteristic, direct product factor
567C3C3×#150411yescharacteristic, complemented
568C2—21no—
569C2—21no—
570C2—22no—
571C2—22no—
572C2—81no—
573C2C2×C3×#83411yescharacteristic
574C2C3×#523211yescharacteristic
575C2C2×C3×#83711yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC32×#525C2
socleC22×C32C2×C22⋊C4
cosocleC22×C3×C4C23×C3
hypercenterC3×#525D3
hypocenterC3C3×#1504
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#525

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#95C2semidirect
3. C3×#5207C2semidirect
4. C3×#4538C2semidirect
5. C3×#4537C2semidirect
7. #71708C3direct (N × H)
8. C2×C32×#41C4semidirect
12. C32×C22⋊Q8C4semidirect
14. C32×#121C4semidirect
19. C32×#117C4semidirect
30. C2×C3⋊#95C2×C3semidirect
31. #5207C2×C3semidirect
34. #4538C2×C3semidirect
35. #4537C2×C3semidirect
38. C2×C32×Q8C2×C4semidirect
42. C2×C32×D4C2×C4semidirect
57. C32×C4⋊C4C2×C4semidirect
58. C32×C4⋊C4C2×C4semidirect
69. C2×C3×#41Dic3semidirect
70. C2×C3×#41C3×C4semidirect
82. C3×C22⋊Q8Dic3semidirect
83. C3×C22⋊Q8C3×C4semidirect
88. C3×#121Dic3semidirect
89. C3×#121C3×C4semidirect
102. C3×#117Dic3semidirect
103. C3×#117C3×C4semidirect
158. C2×C3×Q8C2×Dic3semidirect
159. C2×C3×Q8C2×C3×C4semidirect
170. C2×C3×D4C2×Dic3semidirect
171. C2×C3×D4C2×C3×C4semidirect
214. C3×C4⋊C4C2×Dic3semidirect
215. C3×C4⋊C4C2×Dic3semidirect
216. C3×C4⋊C4C2×C3×C4semidirect
217. C3×C4⋊C4C2×C3×C4semidirect
262. C2×#41C3×Dic3semidirect
267. C22⋊Q8C3×Dic3semidirect
269. #121C3×Dic3semidirect
274. #117C3×Dic3semidirect
383. C2×Q8C2×C3×Dic3semidirect
387. C2×D4C2×C3×Dic3semidirect
404. C4⋊C4C2×C3×Dic3semidirect
405. C4⋊C4C2×C3×Dic3semidirect
483. C32#1504semidirect
566. C3#71708direct (N × H)
567. C3C3×#1504semidirect

Maximal subgroups

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

subgroupindex
C32×#5252
C2×C3×C3⋊#952
C3×#52072
C3×#45382
C3×#45372
C3×#15043
#717083