Finite Group Databasebeta

← C3×#82237 #11805063

Subgroups — C3×#82237 #11805063

Order 1152.

1229 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic
2C3×#5211C211yesmaximal, characteristic
3C2×C3×C3⋊#95C211yesmaximal, characteristic
4C3×#4980C212yesmaximal, complemented
5C3×#4980C212yesmaximal, complemented
6C3×#4950C212yesmaximal
7C3×Dic3×C22⋊C4C212yesmaximal
8C3×#4937C211yesmaximal, characteristic, complemented
9C3×#4936C211yesmaximal, characteristic, complemented
10C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
11C3×#4932C211yesmaximal, characteristic, complemented
12C3×#2343—31nomaximal
13#82237C311yesmaximal, characteristic, direct product factor
14C23×C32×C4C2211yescharacteristic
15C23×C3×Dic3C2212yes—
16C32×#117C418yescomplemented
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×C32×C22⋊C4C2211yescharacteristic
21C2×C32×C42C2211yescharacteristic
22C2×C32×C42C2211yescharacteristic
23C2×C3×#217C2212yes—
24C2×C3×C4⋊Dic3C2212yes—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×C4×Dic3C2212yes—
27C2×C3×C4×Dic3C2212yes—
28C3×C3⋊#95C2211yescharacteristic
29C3×C3⋊#95C2211yescharacteristic
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2211yescharacteristic
32C3×C3⋊#95C2211yescharacteristic
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38C2×C3×#117—21no—
39C2×C3×#117D311yescharacteristic
40C2×C3×#117C2×C311yescharacteristic
41C3×#387—32no—
42C3×#387—32no—
43C3×#385—32no—
44C3×#383—31no—
45C3×#382—31no—
46C3×#381—31no—
47C3×C4×C4⋊C4—31no—
48C3×C4×C22⋊C4—32no—
49C3×#376—31no—
50C2×C3×#95—31no—
51#5211C2×C311yescharacteristic
52C2×C3⋊#95C2×C311yescharacteristic
53#4980C2×C312yescomplemented
54#4980C2×C312yescomplemented
55#4950C2×C312yes—
56Dic3×C22⋊C4C2×C312yes—
57#4937C2×C311yescharacteristic, complemented
58#4936C2×C311yescharacteristic, complemented
59C4×C4⋊Dic3C2×C311yescharacteristic, complemented
60#4932C2×C311yescharacteristic, complemented
61C24×C32C2311yescharacteristic
62C22×C32×C4—24no—
63C22×C32×C4C2×C414yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2311yescharacteristic
70C22×C3×Dic3—24no—
71C22×C3×Dic3—24no—
72C22×C3×Dic3—42no—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2312yes—
76C22×C3×Dic3C2312yes—
77C32×C4⋊C4C2×C414yescomplemented
78C32×C4⋊C4C2×C414yescomplemented
79C32×C22⋊C4C2×C414yes—
80C32×C22⋊C4C2×C414yes—
81C32×C42C2×C414yescomplemented
82C32×C42C2×C414yescomplemented
83C3×#217—24no—
84C3×C4⋊Dic3—24no—
85C3×C4⋊Dic3—42no—
86C3×C4×Dic3—24no—
87C3×C4×Dic3—42no—
88#2343—31noSylow
89C23×C3×C4—21no—
90C23×C3×C4—32no—
91C23×C3×C4C2×D311yescharacteristic
92C23×C3×C4C22×C311yescharacteristic
93C23×Dic3C22×C312yes—
94C3×#117—28no—
95C3×#117Dic318yescomplemented
96C3×#117C3×C418yescomplemented
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—32no—
100C2×C3×C4⋊C4—32no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C2×D311yescharacteristic
103C2×C3×C4⋊C4C22×C311yescharacteristic
104C2×C3×C4⋊C4C22×C311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—32no—
108C2×C3×C22⋊C4C2×D311yescharacteristic
109C2×C3×C22⋊C4C2×D311yescharacteristic
110C2×C3×C22⋊C4C22×C311yescharacteristic
111C2×C3×C22⋊C4C22×C311yescharacteristic
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—32no—
115C2×C3×C42—32no—
116C2×C3×C42C2×D311yescharacteristic
117C2×C3×C42C2×D311yescharacteristic
118C2×C3×C42C22×C311yescharacteristic
119C2×C3×C42C22×C311yescharacteristic
120C2×#217C22×C312yes—
121C2×C4⋊Dic3C22×C312yes—
122C2×C4⋊Dic3C22×C312yes—
123C2×C4×Dic3C22×C312yes—
124C2×C4×Dic3C22×C312yes—
125C3×#95—31no—
126C3×#95—31no—
127C3×#95—32no—
128C3×#95—31no—
129C3×#95—32no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95—31no—
133C3×#95—32no—
134C3×#95—32no—
135C3⋊#95C22×C311yescharacteristic
136C3⋊#95C22×C311yescharacteristic
137C3⋊#95C22×C312yes—
138C3⋊#95C22×C311yescharacteristic
139C3⋊#95C22×C311yescharacteristic
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C312yes—
144C3⋊#95C22×C312yes—
145C23×C32—22no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32C22×C412yes—
149C23×C32#4114yes—
150C23×C32C2411yescharacteristic
151C2×C32×C4—28no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4C22×C412yes—
161C2×C32×C4C22×C412yes—
162C2×C32×C4C22×C412yes—
163C2×C32×C4C22×C412yes—
164C2×C32×C4C22×C412yes—
165C2×C32×C4C22×C412yes—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—48no—
178C2×C3×Dic3#4114yes—
179C2×#117C3×D311yescharacteristic
180#387—32no—
181#387—32no—
182#385—32no—
183#383—31no—
184#382—31no—
185#381—31no—
186C4×C4⋊C4—31no—
187C4×C22⋊C4—32no—
188#376—31no—
189C2×#95—31no—
190C24×C3—21no—
191C24×C3C22×D311yescharacteristic
192C24×C3C23×C311yescharacteristic
193C22×C3×C4—28no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—32no—
204C22×C3×C4—32no—
205C22×C3×C4—32no—
206C22×C3×C4—32no—
207C22×C3×C4—64no—
208C22×C3×C4—64no—
209C22×C3×C4—122no—
210C22×C3×C4C2×Dic314yes—
211C22×C3×C4C2×C3×C414yes—
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C22×C3×C4C23×C311yescharacteristic
220C22×C3×C4C23×C311yescharacteristic
221C22×C3×C4C23×C311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C22×C3×C4C23×C311yescharacteristic
224C22×Dic3—24no—
225C22×Dic3—24no—
226C22×Dic3—42no—
227C22×Dic3C23×C312yes—
228C22×Dic3C23×C312yes—
229C22×Dic3C23×C312yes—
230C22×Dic3C23×C312yes—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—64no—
234C3×C4⋊C4—122no—
235C3×C4⋊C4C2×Dic314yescomplemented
236C3×C4⋊C4C2×Dic314yescomplemented
237C3×C4⋊C4C2×C3×C414yescomplemented
238C3×C4⋊C4C2×C3×C414yescomplemented
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—64no—
242C3×C22⋊C4C2×Dic314yes—
243C3×C22⋊C4C2×Dic314yes—
244C3×C22⋊C4C2×C3×C414yes—
245C3×C22⋊C4C2×C3×C414yes—
246C3×C42—24no—
247C3×C42—24no—
248C3×C42—64no—
249C3×C42—122no—
250C3×C42C2×Dic314yescomplemented
251C3×C42C2×Dic314yescomplemented
252C3×C42C2×C3×C414yescomplemented
253C3×C42C2×C3×C414yescomplemented
254#217—24no—
255C4⋊Dic3—24no—
256C4⋊Dic3—42no—
257C4×Dic3—24no—
258C4×Dic3—42no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—24no—
262C22×C32—24no—
263C22×C32#11714yes—
264C22×C32C23×C411yescharacteristic
265C22×C32C2×#4112yes—
266C22×C32C2×#4112yes—
267C22×C32H(5,𝔽2)11yescharacteristic
268C22×C322−1+411yescharacteristic
269C32×C4—22no—
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—22no—
275C3×Dic3—24no—
276C3×Dic3—42no—
277C3×Dic3—42no—
278C3×Dic3—42no—
279C23×C4—32no—
280C23×C4C2×C3×D311yescharacteristic
281#117C3×Dic318yescomplemented
282C2×C4⋊C4—32no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4C2×C3×D311yescharacteristic
285C2×C4⋊C4C2×C3×D311yescharacteristic
286C2×C22⋊C4—32no—
287C2×C22⋊C4C2×C3×D311yescharacteristic
288C2×C22⋊C4C2×C3×D311yescharacteristic
289C2×C42—32no—
290C2×C42—32no—
291C2×C42C2×C3×D311yescharacteristic
292C2×C42C2×C3×D311yescharacteristic
293#95—31no—
294#95—31no—
295#95—32no—
296#95—31no—
297#95—32no—
298#95—32no—
299#95—32no—
300#95—31no—
301#95—32no—
302#95—32no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3#23514yes—
316C23×C3C22×Dic312yes—
317C23×C3C22×C3×C412yes—
318C23×C3C3×#4114yes—
319C23×C3C23×D311yescharacteristic
320C23×C3C24×C311yescharacteristic
321C2×C3×C4—216no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—28no—
325C2×C3×C4—28no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—34no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
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—128no—
367C2×C3×C4C22×Dic312yes—
368C2×C3×C4C22×Dic312yes—
369C2×C3×C4C22×Dic312yes—
370C2×C3×C4C22×Dic312yes—
371C2×C3×C4C22×Dic312yes—
372C2×C3×C4C22×Dic312yes—
373C2×C3×C4C22×C3×C412yes—
374C2×C3×C4C22×C3×C412yes—
375C2×C3×C4C22×C3×C412yes—
376C2×C3×C4C22×C3×C412yes—
377C2×C3×C4C22×C3×C412yes—
378C2×C3×C4C22×C3×C412yes—
379C2×Dic3—22no—
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—48no—
391C2×Dic3C3×#4114yes—
392C2×C32—24no—
393C2×C32C2×#11711yescharacteristic
394C2×C32C4×#4111yescharacteristic
395C2×C32#52011yescharacteristic
396C2×C32#55112yes—
397C2×C32#55212yes—
398C24C22×C3×D311yescharacteristic
399C22×C4—24no—
400C22×C4—32no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—122no—
407C22×C4C2×C3×Dic314yes—
408C22×C4C22×C3×D311yescharacteristic
409C22×C4C22×C3×D311yescharacteristic
410C22×C4C22×C3×D311yescharacteristic
411C22×C4C22×C3×D311yescharacteristic
412C22×C4C22×C3×D311yescharacteristic
413C22×C4C22×C3×D311yescharacteristic
414C4⋊C4—64no—
415C4⋊C4—122no—
416C4⋊C4C2×C3×Dic314yescomplemented
417C4⋊C4C2×C3×Dic314yescomplemented
418C22⋊C4—64no—
419C22⋊C4C2×C3×Dic314yes—
420C22⋊C4C2×C3×Dic314yes—
421C42—64no—
422C42—122no—
423C42C2×C3×Dic314yescomplemented
424C42C2×C3×Dic314yescomplemented
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3#92614yes—
444C22×C3C3×#11714yes—
445C22×C3C2×#23512yes—
446C22×C3D3×#4112yes—
447C22×C3#100911yescharacteristic
448C22×C3#101011yescharacteristic
449C22×C3C23×Dic311yescharacteristic
450C22×C3C23×C3×C411yescharacteristic
451C22×C3C2×C3×#4112yes—
452C22×C3C2×C3×#4112yes—
453C22×C3C3×H(5,𝔽2)11yescharacteristic
454C22×C3C3×2−1+411yescharacteristic
455C3×C4—24no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—64no—
474C3×C4—122no—
475C3×C4—122no—
476C3×C4—122no—
477Dic3—24no—
478Dic3—42no—
479Dic3—42no—
480Dic3—42no—
481C32#234311yescharacteristic, complemented, Sylow
482C23—22no—
483C23—21no—
484C23—21no—
485C23C3×#23514yes—
486C23C22×C3×Dic312yes—
487C23C23×C3×D311yescharacteristic
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—34no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—128no—
510C2×C4C22×C3×Dic312yes—
511C2×C4C22×C3×Dic312yes—
512C2×C4C22×C3×Dic312yes—
513C2×C4C22×C3×Dic312yes—
514C2×C4C22×C3×Dic312yes—
515C2×C4C22×C3×Dic312yes—
516C2×C3—24no—
517C2×C3—24no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—44no—
524C2×C3#552712yes—
525C2×C3#553112yes—
526C2×C3C2×#92611yescharacteristic
527C2×C3Dic3×#4111yescharacteristic
528C2×C3#582711yescharacteristic
529C2×C3C2×C3×#11711yescharacteristic
530C2×C3C3×C4×#4111yescharacteristic
531C2×C3C3×#52011yescharacteristic
532C2×C3C3×#55112yes—
533C2×C3C3×#55212yes—
534C22—24no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22C3×#92614yes—
539C22C2×C3×#23512yes—
540C22C3×D3×#4112yes—
541C22C3×#100911yescharacteristic
542C22C3×#101011yescharacteristic
543C22C23×C3×Dic311yescharacteristic
544C4—22no—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—64no—
551C4—122no—
552C4—122no—
553C4—122no—
554C3—21no—
555C3#8223711yescharacteristic, direct product factor
556C3C3×#234311yescharacteristic, complemented
557C2—24no—
558C2C3×#552712yes—
559C2C3×#553112yes—
560C2C2×C3×#92611yescharacteristic
561C2C3×Dic3×#4111yescharacteristic
562C2C3×#582711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC2×C32×#117C2
socleC23×C32C24
prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC2×C3×#117D3
hypocenterC3C3×#2343
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
4. C3×#4980C2semidirect
5. C3×#4980C2semidirect
8. C3×#4937C2semidirect
9. C3×#4936C2semidirect
10. C3×C4×C4⋊Dic3C2semidirect
11. C3×#4932C2semidirect
13. #82237C3direct (N × H)
16. C32×#117C4semidirect
53. #4980C2×C3semidirect
54. #4980C2×C3semidirect
57. #4937C2×C3semidirect
58. #4936C2×C3semidirect
59. C4×C4⋊Dic3C2×C3semidirect
60. #4932C2×C3semidirect
77. C32×C4⋊C4C2×C4semidirect
78. C32×C4⋊C4C2×C4semidirect
81. C32×C42C2×C4semidirect
82. C32×C42C2×C4semidirect
95. C3×#117Dic3semidirect
96. C3×#117C3×C4semidirect
235. C3×C4⋊C4C2×Dic3semidirect
236. C3×C4⋊C4C2×Dic3semidirect
237. C3×C4⋊C4C2×C3×C4semidirect
238. C3×C4⋊C4C2×C3×C4semidirect
250. C3×C42C2×Dic3semidirect
251. C3×C42C2×Dic3semidirect
252. C3×C42C2×C3×C4semidirect
253. C3×C42C2×C3×C4semidirect
281. #117C3×Dic3semidirect
416. C4⋊C4C2×C3×Dic3semidirect
417. C4⋊C4C2×C3×Dic3semidirect
423. C42C2×C3×Dic3semidirect
424. C42C2×C3×Dic3semidirect
481. C32#2343semidirect
555. C3#82237direct (N × H)
556. C3C3×#2343semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C3×#52112
C2×C3×C3⋊#952
C3×#49802
C3×#49502
C3×Dic3×C22⋊C42
C3×#49372
C3×#49362
C3×C4×C4⋊Dic32
C3×#49322
C3×#23433
#822373