Finite Group Databasebeta

← C22×#13049549 #72252910

Subgroups — C22×#13049549 #72252910

Order 5184.

21834 conjugacy classes of subgroups; only the 979 normal subgroups are shown.

normal subgroups only

This group has 21834 conjugacy classes of subgroups — too many to draw. Only its 979 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#6295C214yesmaximal, complemented
2C22×#10620150C211yesmaximal, characteristic, complemented
3C23×C3⋊D32C212yesmaximal, complemented
4C2×#13049549C2124yesmaximal, direct product factor
5C22×#88954—34nomaximal
6C23×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
7C23×C32×C3⋊D3C2214yescomplemented
8C22×C3⋊D32—216no—
9C22×C3⋊D32C22124yescomplemented
10C22×C32×C3⋊Dic3C2212yescomplemented
11C2×#10620150C22112yescomplemented
12C2×C3×#6295C22148yescomplemented
13#13049549C22164yesdirect product factor
14C23×D3×C3⋊D3—38no—
15C23×D3×C3⋊D3—64no—
16C22×#6295D314yescomplemented
17C22×#6289—34no—
18C22×#6289—62no—
19C22×C3×#626—38no—
20C22×C3×#625—34no—
21C2×#88954—396no—
22C23×C34C2311yescharacteristic
23C22×C3⋊(C3⋊(C3⋊D3))D418yescomplemented
24C22×C3⋊(C3⋊(C3⋊D3))C2316yes—
25C22×C32×C3⋊D3—216no—
26C22×C32×C3⋊D3C23124yescomplemented
27C2×C3⋊D32—296no—
28C2×C3⋊D32C23132yescomplemented
29C2×C32×C3⋊Dic3C23112yescomplemented
30#10620150C23116yescomplemented
31C3×#6295C23164yescomplemented
32C22×#3811—94no—
33C22×#3802—92no—
34C23×C3⋊(C3⋊D3)—34no—
35C23×C3⋊(C3⋊D3)—62no—
36C23×C3⋊(C3⋊D3)—64no—
37C23×C3⋊(C3⋊D3)—124no—
38C23×C3⋊(C3⋊D3)—122no—
39C23×C3×C3⋊D3—24no—
40C23×C3×C3⋊D3C2×D318yescomplemented
41C23×C32×D3—38no—
42C23×C32×D3—64no—
43C22×D3×C3⋊D3—396no—
44C22×D3×C3⋊D3—648no—
45C22×D3×C3⋊D3—664no—
46C22×D3×C3⋊D3—664no—
47C22×C3×C3⋊Dic3—22no—
48C22×C3×C3⋊Dic3C2×D314yescomplemented
49C22×C32×Dic3—34no—
50C22×C32×Dic3—62no—
51C2×#6295C2×D3148yescomplemented
52C2×#6289—348no—
53C2×#6289—624no—
54C2×C3×#626—396no—
55C2×C3×#625—348no—
56#88954—3256no—
57C22×C34C2×D414yescomplemented
58C22×C34C2413yes—
59C2×C3⋊(C3⋊(C3⋊D3))C2×D4124yescomplemented
60C2×C3⋊(C3⋊(C3⋊D3))C2414yes—
61C2×C32×C3⋊D3—248no—
62C2×C32×C3⋊D3C24116yescomplemented
63C3⋊D32—264no—
64C32×C3⋊Dic3C2418yescomplemented
65C24×C3⋊D3—94no—
66C23×D32—94no—
67C23×D32—94no—
68C23×D32—188no—
69C23×D32—184no—
70C22×C4×C3⋊D3—92no—
71C22×C3×#67—98no—
72C22×C3×D12—94no—
73C2×#3811—996no—
74C22×#626—38no—
75C22×#625—34no—
76C22×#624—94no—
77C22×#624—184no—
78C22×#624—182no—
79C2×#3802—948no—
80C23×C33—24no—
81C23×C33—22no—
82C23×C33—42no—
83C23×C33—44no—
84C23×C33C22×D314yes—
85C22×C3⋊(C3⋊D3)—332no—
86C22×C3⋊(C3⋊D3)—324no—
87C22×C3⋊(C3⋊D3)—632no—
88C22×C3⋊(C3⋊D3)—624no—
89C22×C3⋊(C3⋊D3)—616no—
90C22×C3⋊(C3⋊D3)—612no—
91C22×C3⋊(C3⋊D3)—1232no—
92C22×C3⋊(C3⋊D3)—1216no—
93C22×C3⋊(C3⋊D3)—1224no—
94C22×C3⋊(C3⋊D3)—1212no—
95C22×C3×C3⋊D3—232no—
96C22×C3×C3⋊D3—224no—
97C22×C3×C3⋊D3—232no—
98C22×C3×C3⋊D3C22×D3148yescomplemented
99C22×C32×D3—348no—
100C22×C32×D3—624no—
101C22×C32×D3—632no—
102C22×C32×D3—632no—
103C2×D3×C3⋊D3—3128no—
104C2×D3×C3⋊D3—6384no—
105C2×D3×C3⋊D3—664no—
106C2×D3×C3⋊D3—6384no—
107C2×C3×C3⋊Dic3—212no—
108C2×C3×C3⋊Dic3C22×D3124yescomplemented
109C2×C32×Dic3—324no—
110C2×C32×Dic3—612no—
111#6295C22×D3164yescomplemented
112#6289—364no—
113#6289—632no—
114C3×#626—3128no—
115C3×#625—364no—
116C22×D3×D4—274no—
117C2×C34C22×D416yescomplemented
118C2×C34C2511yescharacteristic
119C3⋊(C3⋊(C3⋊D3))C22×D418yescomplemented
120C32×C3⋊D3—216no—
121C24×C32—94no—
122C23×C3⋊D3—94no—
123C23×C3⋊D3—92no—
124C23×C3⋊D3—948no—
125C23×C3⋊D3—1832no—
126C23×C3⋊D3—184no—
127C23×C3⋊D3—188no—
128C23×C3⋊D3—182no—
129C23×C3⋊D3—184no—
130C23×C3⋊D3—364no—
131C23×C3⋊D3—364no—
132C23×C3⋊D3—364no—
133C23×C3⋊D3—368no—
134C23×C3⋊D3—362no—
135C23×C3⋊D3D3214yescomplemented
136C23×C3×D3—38no—
137C23×C3×D3—38no—
138C23×C3×D3—68no—
139C23×C3×D3—68no—
140C23×C3×D3—68no—
141C22×D32—948no—
142C22×D32—948no—
143C22×D32—1832no—
144C22×D32—1832no—
145C22×D32—1896no—
146C22×D32—18128no—
147C22×D32—1848no—
148C22×D32—1864no—
149C22×C32×C4—92no—
150C22×C3⋊Dic3D3212yescomplemented
151C2×C4×C3⋊D3—924no—
152C2×C3×#67—996no—
153C22×C3×Dic3—38no—
154C22×C3×Dic3—64no—
155C22×C3×Dic3—64no—
156C22×C3×Dic3—64no—
157C2×C3×D12—948no—
158#3811—9256no—
159C2×#626—396no—
160C2×#625—348no—
161C2×#624—948no—
162C2×#624—1848no—
163C2×#624—1824no—
164#3802—9128no—
165C22×C33—26no—
166C22×C33—212no—
167C22×C33—28no—
168C22×C33—216no—
169C22×C33—48no—
170C22×C33—416no—
171C22×C33—46no—
172C22×C33—412no—
173C22×C33D3×D4116yescomplemented
174C22×C33C23×D3112yes—
175C2×C3⋊(C3⋊D3)—396no—
176C2×C3⋊(C3⋊D3)—316no—
177C2×C3⋊(C3⋊D3)—68no—
178C2×C3⋊(C3⋊D3)—696no—
179C2×C3⋊(C3⋊D3)—648no—
180C2×C3⋊(C3⋊D3)—616no—
181C2×C3⋊(C3⋊D3)—1296no—
182C2×C3⋊(C3⋊D3)—1248no—
183C2×C3⋊(C3⋊D3)—1216no—
184C2×C3⋊(C3⋊D3)—128no—
185C2×C3×C3⋊D3—296no—
186C2×C3×C3⋊D3—216no—
187C2×C3×C3⋊D3—296no—
188C2×C3×C3⋊D3C23×D3132yescomplemented
189C2×C32×D3—332no—
190C2×C32×D3—616no—
191C2×C32×D3—696no—
192C2×C32×D3—696no—
193D3×C3⋊D3—6256no—
194D3×C3⋊D3—6256no—
195C3×C3⋊Dic3—28no—
196C3×C3⋊Dic3C23×D3116yescomplemented
197C32×Dic3—316no—
198C32×Dic3—68no—
199C24×D3—278no—
200C24×D3—544no—
201C22×C3×D4—274no—
202C22×#67—98no—
203C2×D3×D4—2796no—
204C22×D12—94no—
205C22×C4×D3—274no—
206C22×C4×D3—542no—
207C34C23×D411yescharacteristic, complemented, Sylow
208C23×C32—24no—
209C23×C32—22no—
210C23×C32—28no—
211C23×C32—24no—
212C23×C32—44no—
213C23×C32—44no—
214C23×C32—48no—
215C23×C32—42no—
216C23×C32—44no—
217C23×C32—924no—
218C23×C32—1816no—
219C23×C32C2×D3212yes—
220C23×C32C2×D3214yes—
221C22×C3⋊D3—216no—
222C22×C3⋊D3—912no—
223C22×C3⋊D3—924no—
224C22×C3⋊D3—916no—
225C22×C3⋊D3—932no—
226C22×C3⋊D3—964no—
227C22×C3⋊D3—1848no—
228C22×C3⋊D3—1864no—
229C22×C3⋊D3—1832no—
230C22×C3⋊D3—1824no—
231C22×C3⋊D3—18192no—
232C22×C3⋊D3—1832no—
233C22×C3⋊D3—1824no—
234C22×C3⋊D3—1816no—
235C22×C3⋊D3—1812no—
236C22×C3⋊D3—3624no—
237C22×C3⋊D3—3632no—
238C22×C3⋊D3—3624no—
239C22×C3⋊D3—3632no—
240C22×C3⋊D3—3624no—
241C22×C3⋊D3—3632no—
242C22×C3⋊D3—3664no—
243C22×C3⋊D3—3648no—
244C22×C3⋊D3—3616no—
245C22×C3⋊D3—3612no—
246C22×C3⋊D3C2×D32124yescomplemented
247C22×C3×D3—348no—
248C22×C3×D3—348no—
249C22×C3×D3—648no—
250C22×C3×D3—648no—
251C22×C3×D3—648no—
252C22×C3×D3—664no—
253C22×C3×D3—664no—
254C22×C3×D3—664no—
255C22×C3×D3—632no—
256C22×C3×D3—632no—
257C2×D32—964no—
258C2×D32—964no—
259C2×D32—18128no—
260C2×D32—18192no—
261C2×D32—18192no—
262C2×D32—1864no—
263C2×D32—18768no—
264C2×D32—18384no—
265C2×C32×C4—912no—
266C2×C3⋊Dic3C2×D32112yescomplemented
267C4×C3⋊D3—932no—
268C3×#67—9128no—
269C2×C3×Dic3—348no—
270C2×C3×Dic3—624no—
271C2×C3×Dic3—624no—
272C2×C3×Dic3—624no—
273C3×D12—964no—
274#626—3128no—
275#625—364no—
276#624—964no—
277#624—1864no—
278#624—1832no—
279C23×D4—811noSylow
280C2×C33—212no—
281C2×C33—224no—
282C2×C33—22no—
283C2×C33—24no—
284C2×C33—412no—
285C2×C33—424no—
286C2×C33—42no—
287C2×C33—44no—
288C2×C33C2×D3×D4124yescomplemented
289C2×C33C24×D314yes—
290C3⋊(C3⋊D3)—332no—
291C3⋊(C3⋊D3)—632no—
292C3⋊(C3⋊D3)—616no—
293C3⋊(C3⋊D3)—1232no—
294C3⋊(C3⋊D3)—1216no—
295C3×C3⋊D3—232no—
296C3×C3⋊D3—232no—
297C32×D3—632no—
298C32×D3—632no—
299C24×C3—98no—
300C24×C3—184no—
301C23×D3—38no—
302C23×D3—64no—
303C23×D3—2796no—
304C23×D3—274no—
305C23×D3—5448no—
306C23×D3—542no—
307C23×D3—544no—
308C23×D3—5464no—
309C23×D3—5464no—
310C23×D3—1084no—
311C23×D3—1082no—
312C2×C3×D4—2748no—
313C22×C3×C4—94no—
314C22×C3×C4—182no—
315C2×#67—996no—
316C22×Dic3—34no—
317C22×Dic3—62no—
318D3×D4—27256no—
319C2×D12—948no—
320C2×C4×D3—2748no—
321C2×C4×D3—5424no—
322C22×C32—26no—
323C22×C32—212no—
324C22×C32—216no—
325C22×C32—28no—
326C22×C32—212no—
327C22×C32—224no—
328C22×C32—232no—
329C22×C32—216no—
330C22×C32—416no—
331C22×C32—412no—
332C22×C32—412no—
333C22×C32—424no—
334C22×C32—432no—
335C22×C32—416no—
336C22×C32—46no—
337C22×C32—48no—
338C22×C32—412no—
339C22×C32—416no—
340C22×C32—916no—
341C22×C32—1848no—
342C22×C32#380218yescomplemented
343C22×C32#3811116yescomplemented
344C22×C32C22×D3216yes—
345C22×C32C22×D32112yes—
346C2×C3⋊D3—248no—
347C2×C3⋊D3—948no—
348C2×C3⋊D3—996no—
349C2×C3⋊D3—916no—
350C2×C3⋊D3—98no—
351C2×C3⋊D3—1816no—
352C2×C3⋊D3—18192no—
353C2×C3⋊D3—1832no—
354C2×C3⋊D3—1896no—
355C2×C3⋊D3—18128no—
356C2×C3⋊D3—188no—
357C2×C3⋊D3—1896no—
358C2×C3⋊D3—1816no—
359C2×C3⋊D3—1848no—
360C2×C3⋊D3—3696no—
361C2×C3⋊D3—3616no—
362C2×C3⋊D3—3616no—
363C2×C3⋊D3—3696no—
364C2×C3⋊D3—3616no—
365C2×C3⋊D3—36192no—
366C2×C3⋊D3—3632no—
367C2×C3⋊D3—3696no—
368C2×C3⋊D3—3648no—
369C2×C3⋊D3—368no—
370C2×C3⋊D3C22×D32116yescomplemented
371C2×C3×D3—332no—
372C2×C3×D3—332no—
373C2×C3×D3—632no—
374C2×C3×D3—632no—
375C2×C3×D3—632no—
376C2×C3×D3—6192no—
377C2×C3×D3—6192no—
378C2×C3×D3—6192no—
379C2×C3×D3—696no—
380C2×C3×D3—696no—
381D32—18128no—
382D32—18128no—
383D32—18512no—
384D32—18256no—
385C32×C4—98no—
386C3⋊Dic3C22×D3218yescomplemented
387C3×Dic3—332no—
388C3×Dic3—616no—
389C3×Dic3—616no—
390C3×Dic3—616no—
391C25—812no—
392C22×D4—274no—
393C22×D4—8124no—
394C23×C4—811no—
395C33—22no—
396C33—24no—
397C33—42no—
398C33—44no—
399C33C22×D3×D414yescomplemented
400C23×C3—22no—
401C23×C3—24no—
402C23×C3—42no—
403C23×C3—44no—
404C23×C3—948no—
405C23×C3—1824no—
406C23×C3—1832no—
407C23×C3—1832no—
408C23×C3D3314yes—
409C22×D3—348no—
410C22×D3—624no—
411C22×D3—632no—
412C22×D3—632no—
413C22×D3—27128no—
414C22×D3—2724no—
415C22×D3—2732no—
416C22×D3—5464no—
417C22×D3—5412no—
418C22×D3—5416no—
419C22×D3—5424no—
420C22×D3—5432no—
421C22×D3—54384no—
422C22×D3—54384no—
423C22×D3—10824no—
424C22×D3—10812no—
425C22×D3—10832no—
426C22×D3—10816no—
427C3×D4—2764no—
428C2×C3×C4—924no—
429C2×C3×C4—1812no—
430#67—9128no—
431C2×Dic3—324no—
432C2×Dic3—612no—
433D12—964no—
434C4×D3—2764no—
435C4×D3—5432no—
436C2×C32—22no—
437C2×C32—224no—
438C2×C32—24no—
439C2×C32—212no—
440C2×C32—248no—
441C2×C32—224no—
442C2×C32—24no—
443C2×C32—28no—
444C2×C32—424no—
445C2×C32—44no—
446C2×C32—448no—
447C2×C32—424no—
448C2×C32—44no—
449C2×C32—48no—
450C2×C32—412no—
451C2×C32—42no—
452C2×C32—424no—
453C2×C32—44no—
454C2×C32—1816no—
455C2×C32C2×#3802112yescomplemented
456C2×C32C2×#3811124yescomplemented
457C2×C32C23×D3212yes—
458C2×C32C23×D3214yes—
459C3⋊D3—216no—
460C3⋊D3—916no—
461C3⋊D3—932no—
462C3⋊D3—1864no—
463C3⋊D3—1832no—
464C3⋊D3—1832no—
465C3⋊D3—1816no—
466C3⋊D3—3632no—
467C3⋊D3—3632no—
468C3⋊D3—3664no—
469C3⋊D3—3632no—
470C3⋊D3—3616no—
471C3×D3—664no—
472C3×D3—664no—
473C3×D3—664no—
474C3×D3—632no—
475C3×D3—632no—
476C24—94no—
477C24—8124no—
478C24—811no—
479C24—16216no—
480C2×D4—2748no—
481C2×D4—8164no—
482C22×C4—92no—
483C22×C4—8112no—
484C22×C3—26no—
485C22×C3—212no—
486C22×C3—216no—
487C22×C3—28no—
488C22×C3—46no—
489C22×C3—412no—
490C22×C3—48no—
491C22×C3—416no—
492C22×C3—932no—
493C22×C3—1816no—
494C22×C3—1896no—
495C22×C3—1896no—
496C22×C3#88954116yescomplemented
497C22×C3C2×D33112yes—
498C2×D3—332no—
499C2×D3—616no—
500C2×D3—696no—
501C2×D3—696no—
502C2×D3—2796no—
503C2×D3—2716no—
504C2×D3—548no—
505C2×D3—5496no—
506C2×D3—5416no—
507C2×D3—5448no—
508C2×D3—54256no—
509C2×D3—54256no—
510C2×D3—10896no—
511C2×D3—10848no—
512C2×D3—10816no—
513C2×D3—1088no—
514C3×C4—916no—
515C3×C4—188no—
516Dic3—316no—
517Dic3—68no—
518C32—24no—
519C32—22no—
520C32—28no—
521C32—24no—
522C32—44no—
523C32—48no—
524C32—44no—
525C32—42no—
526C32—44no—
527C32C22×#380212yescomplemented
528C32C22×#381114yescomplemented
529C23—924no—
530C23—1816no—
531C23—8132no—
532C23—816no—
533C23—818no—
534C23—16296no—
535C23#1062021511yescharacteristic
536D4—2764no—
537C2×C4—912no—
538C2×C4—8116no—
539C2×C3—224no—
540C2×C3—212no—
541C2×C3—22no—
542C2×C3—24no—
543C2×C3—412no—
544C2×C3—424no—
545C2×C3—42no—
546C2×C3—44no—
547C2×C3—1832no—
548C2×C3—1832no—
549C2×C3C2×#88954124yescomplemented
550C2×C3C22×D3314yes—
551D3—632no—
552D3—632no—
553D3—2732no—
554D3—5432no—
555D3—5416no—
556D3—10832no—
557D3—10816no—
558C22—916no—
559C22—1848no—
560C22—8124no—
561C22—814no—
562C22—16264no—
563C22#1304954914yesdirect product factor
564C22C2×#1062021513yes—
565C4—98no—
566C3—24no—
567C3—22no—
568C3—42no—
569C3—44no—
570C3C22×#8895414yescomplemented
571C2—1816no—
572C2—818no—
573C2C2×#1304954916yesdirect product factor
574C2C22×#1062021511yescharacteristic
subgroupG/H
center, hypercenterC23#10620215
commutator, cosocleC2×C34C25
FrattiniC2C22×#10620215
Fitting, socleC23×C34C23
hypocenterC34C23×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3×#6295C2semidirect
2. C22×#10620150C2semidirect
3. C23×C3⋊D32C2semidirect
4. C2×#13049549C2direct (N × H)
7. C23×C32×C3⋊D3C22semidirect
9. C22×C3⋊D32C22semidirect
10. C22×C32×C3⋊Dic3C22semidirect
11. C2×#10620150C22semidirect
12. C2×C3×#6295C22semidirect
13. #13049549C22direct (N × H)
16. C22×#6295D3semidirect
23. C22×C3⋊(C3⋊(C3⋊D3))D4semidirect
26. C22×C32×C3⋊D3C23semidirect
28. C2×C3⋊D32C23semidirect
29. C2×C32×C3⋊Dic3C23semidirect
30. #10620150C23semidirect
31. C3×#6295C23semidirect
40. C23×C3×C3⋊D3C2×D3semidirect
48. C22×C3×C3⋊Dic3C2×D3semidirect
51. C2×#6295C2×D3semidirect
57. C22×C34C2×D4semidirect
59. C2×C3⋊(C3⋊(C3⋊D3))C2×D4semidirect
62. C2×C32×C3⋊D3C24semidirect
64. C32×C3⋊Dic3C24semidirect
98. C22×C3×C3⋊D3C22×D3semidirect
108. C2×C3×C3⋊Dic3C22×D3semidirect
111. #6295C22×D3semidirect
117. C2×C34C22×D4semidirect
119. C3⋊(C3⋊(C3⋊D3))C22×D4semidirect
135. C23×C3⋊D3D32semidirect
150. C22×C3⋊Dic3D32semidirect
173. C22×C33D3×D4semidirect
188. C2×C3×C3⋊D3C23×D3semidirect
196. C3×C3⋊Dic3C23×D3semidirect
207. C34C23×D4semidirect
246. C22×C3⋊D3C2×D32semidirect
266. C2×C3⋊Dic3C2×D32semidirect
288. C2×C33C2×D3×D4semidirect
342. C22×C32#3802semidirect
343. C22×C32#3811semidirect
370. C2×C3⋊D3C22×D32semidirect
386. C3⋊Dic3C22×D32semidirect
399. C33C22×D3×D4semidirect
455. C2×C32C2×#3802semidirect
456. C2×C32C2×#3811semidirect
496. C22×C3#88954semidirect
527. C32C22×#3802semidirect
528. C32C22×#3811semidirect
549. C2×C3C2×#88954semidirect
563. C22#13049549direct (N × H)
570. C3C22×#88954semidirect
573. C2C2×#13049549direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#62952
C22×#106201502
C23×C3⋊D322
C2×#130495492
C22×#889543