Finite Group Databasebeta

← C3×#82745 #11803910

Subgroups — C3×#82745 #11803910

Order 1152.

1217 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C3×#5229C212yesmaximal, complemented
3C3×#5212C211yesmaximal, characteristic
4C2×C3×C3⋊#95C211yesmaximal, characteristic
5C3×#4984C212yesmaximal, complemented
6C3×#4980C212yesmaximal, complemented
7C3×#4980C212yesmaximal, complemented
8C3×#4950C212yesmaximal
9C3×Dic3×C22⋊C4C212yesmaximal
10C3×#2329—31nomaximal
11#82745C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C23×C32×C4C2211yescharacteristic
14C23×C3×Dic3C2211yescharacteristic
15C23×C3×Dic3C2211yescharacteristic
16C32×C22⋊Q8C418yescomplemented
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C22⋊C4C2212yes—
20C2×C3×#217C2212yes—
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×C4⋊Dic3C2212yes—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4×Dic3C2212yes—
26C2×C3×C4×Dic3C2212yes—
27C3×C3⋊#95C2212yes—
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C2×C3×C22⋊Q8—21no—
35C2×C3×C22⋊Q8D311yescharacteristic
36C2×C3×C22⋊Q8C2×C311yescharacteristic
37C3×#391—32no—
38C3×#389—32no—
39C3×#387—32no—
40C3×#387—32no—
41C3×#385—32no—
42C3×#380—31no—
43C3×C4×C22⋊C4—32no—
44C2×C3×#95—31no—
45#5229C2×C312yescomplemented
46#5212C2×C311yescharacteristic
47C2×C3⋊#95C2×C311yescharacteristic
48#4984C2×C312yescomplemented
49#4980C2×C312yescomplemented
50#4980C2×C312yescomplemented
51#4950C2×C312yes—
52Dic3×C22⋊C4C2×C312yes—
53C24×C32C2311yescharacteristic
54C2×C32×Q8—41no—
55C2×C32×Q8—41no—
56C2×C32×Q8C2×C414yescomplemented
57C22×C32×C4—41no—
58C22×C32×C4—41no—
59C22×C32×C4C2×C414yes—
60C22×C32×C4C2312yes—
61C22×C32×C4C2312yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C22×C3×Dic3—22no—
65C22×C3×Dic3—41no—
66C22×C3×Dic3—41no—
67C22×C3×Dic3—41no—
68C22×C3×Dic3—41no—
69C22×C3×Dic3Q814yescomplemented
70C22×C3×Dic3C2312yes—
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C32×C4⋊C4C2×C414yescomplemented
75C32×C4⋊C4C2×C418yescomplemented
76C32×C22⋊C4C2×C418yes—
77C3×#217—24no—
78C3×C4⋊Dic3—41no—
79C3×C4⋊Dic3—42no—
80C3×C4⋊Dic3—41no—
81C3×C4⋊Dic3—42no—
82C3×C4×Dic3—42no—
83C3×C4×Dic3—42no—
84#2329—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C23×C3×C4—21no—
89C23×C3×C4—31no—
90C23×C3×C4—31no—
91C23×C3×C4C2×D311yescharacteristic
92C23×C3×C4C22×C311yescharacteristic
93C23×Dic3C22×C311yescharacteristic
94C23×Dic3C22×C311yescharacteristic
95C3×C22⋊Q8—28no—
96C3×C22⋊Q8Dic318yescomplemented
97C3×C22⋊Q8C3×C418yescomplemented
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—32no—
103C2×C3×C4⋊C4—32no—
104C2×C3×C4⋊C4C2×D311yescharacteristic
105C2×C3×C4⋊C4C2×D312yes—
106C2×C3×C4⋊C4C22×C311yescharacteristic
107C2×C3×C4⋊C4C22×C312yes—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4C2×D312yes—
111C2×C3×C22⋊C4C22×C312yes—
112C2×C3×C42—32no—
113C2×C3×C42—32no—
114C2×#217C22×C312yes—
115C2×C4⋊Dic3C22×C312yes—
116C2×C4⋊Dic3C22×C312yes—
117C2×C4⋊Dic3C22×C311yescharacteristic
118C2×C4⋊Dic3C22×C311yescharacteristic
119C2×C4×Dic3C22×C312yes—
120C2×C4×Dic3C22×C312yes—
121C3×#95—32no—
122C3×#95—32no—
123C3×#95—32no—
124C3×#95—32no—
125C3×#95—32no—
126C3×#95—32no—
127C3×#95—32no—
128C3⋊#95C22×C312yes—
129C3⋊#95C22×C312yes—
130C3⋊#95C22×C312yes—
131C3⋊#95C22×C312yes—
132C3⋊#95C22×C312yes—
133C3⋊#95C22×C312yes—
134C3⋊#95C22×C312yes—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32C22×C412yes—
140C23×C32C2×Q812yes—
141C23×C32#4112yes—
142C23×C32C2411yescharacteristic
143C32×Q8—44no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—44no—
153C2×C32×C4C22×C414yes—
154C2×C32×C4C22×C414yes—
155C2×C32×C4C22×C412yes—
156C2×C32×C4C22×C412yes—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—44no—
169C2×C3×Dic3—44no—
170C2×C3×Dic3C2×Q814yes—
171C2×C22⋊Q8C3×D311yescharacteristic
172#391—32no—
173#389—32no—
174#387—32no—
175#387—32no—
176#385—32no—
177#380—31no—
178C4×C22⋊C4—32no—
179C2×#95—31no—
180C24×C3—21no—
181C24×C3C22×D311yescharacteristic
182C24×C3C23×C311yescharacteristic
183C2×C3×Q8—24no—
184C2×C3×Q8—41no—
185C2×C3×Q8—41no—
186C2×C3×Q8—41no—
187C2×C3×Q8—41no—
188C2×C3×Q8—42no—
189C2×C3×Q8—42no—
190C2×C3×Q8C2×Dic314yescomplemented
191C2×C3×Q8C2×C3×C414yescomplemented
192C22×C3×C4—24no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—32no—
198C22×C3×C4—34no—
199C22×C3×C4—32no—
200C22×C3×C4—32no—
201C22×C3×C4—32no—
202C22×C3×C4—41no—
203C22×C3×C4—41no—
204C22×C3×C4—41no—
205C22×C3×C4—41no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—62no—
209C22×C3×C4—121no—
210C22×C3×C4—121no—
211C22×C3×C4—121no—
212C22×C3×C4—121no—
213C22×C3×C4C2×Dic314yes—
214C22×C3×C4C2×C3×C414yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D312yes—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D311yescharacteristic
219C22×C3×C4C23×C311yescharacteristic
220C22×C3×C4C23×C312yes—
221C22×C3×C4C23×C312yes—
222C22×C3×C4C23×C311yescharacteristic
223C22×Dic3—22no—
224C22×Dic3—41no—
225C22×Dic3—41no—
226C22×Dic3—41no—
227C22×Dic3—41no—
228C22×Dic3C3×Q814yescomplemented
229C22×Dic3C23×C312yes—
230C22×Dic3C23×C312yes—
231C22×Dic3C23×C312yes—
232C22×Dic3C23×C312yes—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—121no—
236C3×C4⋊C4—121no—
237C3×C4⋊C4—122no—
238C3×C4⋊C4—122no—
239C3×C4⋊C4C2×Dic314yescomplemented
240C3×C4⋊C4C2×Dic318yescomplemented
241C3×C4⋊C4C2×C3×C414yescomplemented
242C3×C4⋊C4C2×C3×C418yescomplemented
243C3×C22⋊C4—28no—
244C3×C22⋊C4—64no—
245C3×C22⋊C4C2×Dic318yes—
246C3×C22⋊C4C2×C3×C418yes—
247C3×C42—122no—
248C3×C42—122no—
249#217—24no—
250C4⋊Dic3—41no—
251C4⋊Dic3—42no—
252C4⋊Dic3—41no—
253C4⋊Dic3—42no—
254C4×Dic3—42no—
255C4×Dic3—42no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32C4×Q814yes—
263C22×C32C23×C411yescharacteristic
264C22×C32C22×Q811yescharacteristic
265C22×C32C2×#4111yescharacteristic
266C22×C32H(5,𝔽2)11yescharacteristic
267C22×C32H(5,𝔽2)11yescharacteristic
268C22×C32H(5,𝔽2)11yescharacteristic
269C22×C322−1+411yescharacteristic
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—24no—
273C32×C4—24no—
274C3×Dic3—24no—
275C3×Dic3—42no—
276C3×Dic3—42no—
277C3×Dic3—42no—
278C22×Q8C2×C3×D311yescharacteristic
279C23×C4—31no—
280C23×C4—31no—
281C23×C4C2×C3×D311yescharacteristic
282C22⋊Q8C3×Dic318yescomplemented
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4C2×C3×D311yescharacteristic
288C2×C4⋊C4C2×C3×D312yes—
289C2×C22⋊C4—32no—
290C2×C22⋊C4C2×C3×D312yes—
291C2×C42—32no—
292C2×C42—32no—
293#95—32no—
294#95—32no—
295#95—32no—
296#95—32no—
297#95—32no—
298#95—32no—
299#95—32no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3D3×Q812yes—
317C23×C3#23912yes—
318C23×C3C22×Dic312yes—
319C23×C3C22×C3×C412yes—
320C23×C3C2×C3×Q812yes—
321C23×C3C3×#4112yes—
322C23×C3C23×D311yescharacteristic
323C23×C3C24×C311yescharacteristic
324C3×Q8—44no—
325C3×Q8—44no—
326C3×Q8—48no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—34no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—44no—
358C2×C3×C4—48no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—124no—
371C2×C3×C4—124no—
372C2×C3×C4C22×Dic314yes—
373C2×C3×C4C22×Dic314yes—
374C2×C3×C4C22×Dic312yes—
375C2×C3×C4C22×Dic312yes—
376C2×C3×C4C22×C3×C414yes—
377C2×C3×C4C22×C3×C414yes—
378C2×C3×C4C22×C3×C412yes—
379C2×C3×C4C22×C3×C412yes—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—44no—
392C2×Dic3—44no—
393C2×Dic3C2×C3×Q814yes—
394C2×C32—24no—
395C2×C32C2×C4×Q811yescharacteristic
396C2×C32#51811yescharacteristic
397C2×C32#52011yescharacteristic
398C2×C32#53811yescharacteristic
399C2×C32#53911yescharacteristic
400C2×C32#54311yescharacteristic
401C2×C32#54411yescharacteristic
402C24C22×C3×D311yescharacteristic
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C2×C3×Dic314yescomplemented
406C22×C4—32no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—62no—
414C22×C4—121no—
415C22×C4—121no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4C2×C3×Dic314yes—
419C22×C4C22×C3×D311yescharacteristic
420C22×C4C22×C3×D312yes—
421C22×C4C22×C3×D312yes—
422C22×C4C22×C3×D311yescharacteristic
423C4⋊C4—121no—
424C4⋊C4—121no—
425C4⋊C4—122no—
426C4⋊C4—122no—
427C4⋊C4C2×C3×Dic314yescomplemented
428C4⋊C4C2×C3×Dic318yescomplemented
429C22⋊C4—64no—
430C22⋊C4C2×C3×Dic318yes—
431C42—122no—
432C42—122no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—24no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3Q8×Dic314yes—
460C22×C3C3×C4×Q814yes—
461C22×C3#100411yescharacteristic
462C22×C3#100411yescharacteristic
463C22×C3C2×D3×Q811yescharacteristic
464C22×C3C2×#23911yescharacteristic
465C22×C3#100911yescharacteristic
466C22×C3#101011yescharacteristic
467C22×C3C23×Dic311yescharacteristic
468C22×C3C23×C3×C411yescharacteristic
469C22×C3C22×C3×Q811yescharacteristic
470C22×C3C2×C3×#4111yescharacteristic
471C22×C3C3×H(5,𝔽2)11yescharacteristic
472C22×C3C3×H(5,𝔽2)11yescharacteristic
473C22×C3C3×H(5,𝔽2)11yescharacteristic
474C22×C3C3×2−1+411yescharacteristic
475C3×C4—24no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—24no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—24no—
483C3×C4—24no—
484C3×C4—42no—
485C3×C4—44no—
486C3×C4—44no—
487C3×C4—64no—
488C3×C4—122no—
489C3×C4—122no—
490C3×C4—122no—
491Dic3—24no—
492Dic3—42no—
493Dic3—42no—
494Dic3—42no—
495C32#232911yescharacteristic, complemented, Sylow
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C3×D3×Q812yes—
501C23C3×#23912yes—
502C23C22×C3×Dic312yes—
503C23C23×C3×D311yescharacteristic
504Q8—44no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—34no—
514C2×C4—44no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—124no—
527C2×C4—124no—
528C2×C4C22×C3×Dic314yes—
529C2×C4C22×C3×Dic314yes—
530C2×C4C22×C3×Dic312yes—
531C2×C4C22×C3×Dic312yes—
532C2×C3—24no—
533C2×C3—24no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—44no—
542C2×C3#562311yescharacteristic
543C2×C3#563411yescharacteristic
544C2×C3#564411yescharacteristic
545C2×C3#564511yescharacteristic
546C2×C3#579811yescharacteristic
547C2×C3C2×Q8×Dic311yescharacteristic
548C2×C3#582711yescharacteristic
549C2×C3C2×C3×C4×Q811yescharacteristic
550C2×C3C3×#51811yescharacteristic
551C2×C3C3×#52011yescharacteristic
552C2×C3C3×#53811yescharacteristic
553C2×C3C3×#53911yescharacteristic
554C2×C3C3×#54311yescharacteristic
555C2×C3C3×#54411yescharacteristic
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22C3×Q8×Dic314yes—
563C22C3×#100411yescharacteristic
564C22C3×#100411yescharacteristic
565C22C2×C3×D3×Q811yescharacteristic
566C22C2×C3×#23911yescharacteristic
567C22C3×#100911yescharacteristic
568C22C3×#101011yescharacteristic
569C22C23×C3×Dic311yescharacteristic
570C4—22no—
571C4—22no—
572C4—24no—
573C4—24no—
574C4—64no—
575C4—122no—
576C4—122no—
577C4—122no—
578C3—21no—
579C3#8274511yescharacteristic, direct product factor
580C3C3×#232911yescharacteristic, complemented
581C2—24no—
582C2C3×#562311yescharacteristic
583C2C3×#563411yescharacteristic
584C2C3×#564411yescharacteristic
585C2C3×#564511yescharacteristic
586C2C3×#579811yescharacteristic
587C2C2×C3×Q8×Dic311yescharacteristic
588C2C3×#582711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C24
prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC2×C3×C22⋊Q8D3
hypocenterC3C3×#2329
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5229C2semidirect
5. C3×#4984C2semidirect
6. C3×#4980C2semidirect
7. C3×#4980C2semidirect
11. #82745C3direct (N × H)
16. C32×C22⋊Q8C4semidirect
45. #5229C2×C3semidirect
48. #4984C2×C3semidirect
49. #4980C2×C3semidirect
50. #4980C2×C3semidirect
56. C2×C32×Q8C2×C4semidirect
69. C22×C3×Dic3Q8semidirect
74. C32×C4⋊C4C2×C4semidirect
75. C32×C4⋊C4C2×C4semidirect
96. C3×C22⋊Q8Dic3semidirect
97. C3×C22⋊Q8C3×C4semidirect
190. C2×C3×Q8C2×Dic3semidirect
191. C2×C3×Q8C2×C3×C4semidirect
228. C22×Dic3C3×Q8semidirect
239. C3×C4⋊C4C2×Dic3semidirect
240. C3×C4⋊C4C2×Dic3semidirect
241. C3×C4⋊C4C2×C3×C4semidirect
242. C3×C4⋊C4C2×C3×C4semidirect
282. C22⋊Q8C3×Dic3semidirect
405. C2×Q8C2×C3×Dic3semidirect
427. C4⋊C4C2×C3×Dic3semidirect
428. C4⋊C4C2×C3×Dic3semidirect
495. C32#2329semidirect
579. C3#82745direct (N × H)
580. C3C3×#2329semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C3×#52292
C3×#52122
C2×C3×C3⋊#952
C3×#49842
C3×#49802
C3×#49502
C3×Dic3×C22⋊C42
C3×#23293
#827453