Finite Group Databasebeta

← C2×#11724314 #13181084

Subgroups — C2×#11724314 #13181084

Order 1728.

1432 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 159 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×D32⋊D3C212yesmaximal, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C2×C3×C4×D32C211yesmaximal, characteristic, complemented
4C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
5C2×#88932C212yesmaximal, complemented
6#11724314C218yesmaximal, direct product factor
7C2×#64230—31nomaximal
8C22×#1113C2212yes—
9C22×C3×D32C2211yescharacteristic
10C22×C3⋊#157C2212yes—
11C2×D32⋊D3—28no—
12C2×D32⋊D3C2218yescomplemented
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×C4×D3—21no—
16C2×#6295—21no—
17C4⋊#1113—22no—
18C4⋊#1113C2214yescomplemented
19C2×C3×#624C2211yescharacteristic
20C2×C3×D3×Dic3—21no—
21C3×C4×D32—22no—
22C3×C4×D32C2214yescomplemented
23C4⋊(C3⋊#157)C2214yescomplemented
24#88932C2218yescomplemented
25C22×GO+(4,𝔽2)—32no—
26C22×C3×C4×D3—61no—
27C2×#3802—31no—
28C2×D3×D12—61no—
29C2×C4×D32D311yescharacteristic, complemented
30C2×C4⋊#157—31no—
31C2×D32⋊C4—32no—
32#64230—38no—
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3C2311yescharacteristic
35C22×C32×D3—21no—
36C2×#1113—24no—
37C2×#1113—24no—
38C2×#1113—22no—
39C2×#1113C2314yes—
40C2×C3×D32—21no—
41C2×C3×D32—22no—
42C2×C3×D32D414yescomplemented
43C2×C3×D32C2312yes—
44C2×C3⋊#157—24no—
45C2×C3⋊#157C2314yes—
46D32⋊D3—216no—
47C2×C33×C4D411yescharacteristic, complemented
48C2×C3×C3⋊Dic3D411yescharacteristic, complemented
49C3×C4⋊(C3⋊D3)—24no—
50C3×C4×C3⋊D3D412yescomplemented
51C3×C4×C3⋊D3C2312yes—
52C2×C32×Dic3—21no—
53C32×C4×D3—24no—
54#6295—24no—
55C3×#624—21no—
56C3×#624C2312yes—
57C3×D3×Dic3—24no—
58C2×#930—91nomaximal
59C23×C3×D3—61no—
60C22×D32—32no—
61C22×D32—62no—
62C22×D32C2×D311yescharacteristic
63C22×#157—32no—
64C2×GO+(4,𝔽2)—38no—
65C2×GO+(4,𝔽2)—68no—
66C22×C32×C4—61no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C22×C3×Dic3—61no—
70C2×C3×D12—61no—
71C2×C3×D12—61no—
72C2×C3×D12—121no—
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3—21no—
75C2×C3×C4×D3—41no—
76C2×C3×C4×D3—64no—
77C2×C3×C4×D3—64no—
78C2×C3×C4×D3—61no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81C2×C3×C4×D3—61no—
82C2×#626—62no—
83C2×#625—61no—
84C2×#624C2×D311yescharacteristic
85C2×D3×Dic3—21no—
86#3802—34no—
87#3802—62no—
88D3×D12—64no—
89D3×D12—64no—
90C4×D32—22no—
91C4×D32C2×D314yescomplemented
92C4⋊#157—34no—
93D32⋊C4—38no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—24no—
96C2×C3×C3⋊D3—44no—
97C2×C3×C3⋊D3C2×D411yescharacteristic
98C2×C3×C3⋊D3C2×D412yes—
99C2×C3×C3⋊D3#4112yes—
100C2×C3×C3⋊D3C2411yescharacteristic
101C2×C32×D3—24no—
102C2×C32×D3—22no—
103#1113—24no—
104#1113—24no—
105#1113—44no—
106C3×D32—24no—
107C3×D32—22no—
108C3×D32C2×D414yescomplemented
109C3⋊#157—24no—
110C33×C4C2×D412yescomplemented
111C3×C3⋊Dic3C2×D412yescomplemented
112C32×Dic3—22no—
113C23×C3×C4—91no—
114C22×#67—92no—
115C2×D3×D4—181no—
116C22×D12—91no—
117C22×C4×D3—61no—
118#930—98no—
119C2×(C2×D3)⋊C4—92no—
120C2×C4⋊Dic3—91no—
121C23×C32—61no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3C22×D311yescharacteristic
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3—41no—
127C22×C3×D3—64no—
128C22×C3×D3—62no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—64no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—122no—
136C2×D32—22no—
137C2×D32—21no—
138C2×D32—34no—
139C2×D32—64no—
140C2×D32—64no—
141C2×D32—62no—
142C2×D32—64no—
143C2×D32—64no—
144C2×D32—124no—
145C2×D32—124no—
146C2×D32D1214yescomplemented
147C2×D32C22×D312yes—
148C2×#157—34no—
149C2×#157—64no—
150GO+(4,𝔽2)—616no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4—64no—
156C2×C32×C4—61no—
157C2×C32×C4#6711yescharacteristic, complemented
158C2×C3⋊Dic3#6711yescharacteristic, complemented
159C4⋊(C3⋊D3)—24no—
160C4×C3⋊D3#6712yescomplemented
161C4×C3⋊D3C22×D312yes—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—64no—
168C3×D12—64no—
169C3×D12—64no—
170C3×D12—124no—
171C3×C4×D3—24no—
172C3×C4×D3—24no—
173C3×C4×D3—44no—
174C3×C4×D3—62no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—64no—
181#626—68no—
182#625—64no—
183#624—21no—
184#624C22×D312yes—
185D3×Dic3—24no—
186C2×#121—271noSylow
187C2×C33#12112yescomplemented
188C2×C33C22×D411yescharacteristic
189C3×C3⋊D3—44no—
190C3×C3⋊D3#12112yescomplemented
191C3×C3⋊D3C22×D411yescharacteristic
192C3×C3⋊D3C2×#4111yescharacteristic
193C32×D3—24no—
194C24×C3—91no—
195C23×D3—61no—
196C23×D3—92no—
197C23×D3—182no—
198C2×C3×D4—181no—
199C22×C3×C4—61no—
200C22×C3×C4—61no—
201C22×C3×C4—91no—
202C22×C3×C4—94no—
203C22×C3×C4—91no—
204C22×C3×C4—121no—
205C22×C3×C4—182no—
206C22×C3×C4—181no—
207C2×#67—98no—
208C2×#67—182no—
209C2×#67—188no—
210C22×Dic3—61no—
211C22×Dic3—92no—
212D3×D4—184no—
213D3×D4—184no—
214C2×D12—61no—
215C2×D12—61no—
216C2×D12—94no—
217C2×D12—121no—
218C2×D12—182no—
219C2×D12—181no—
220C2×C4×D3—21no—
221C2×C4×D3—64no—
222C2×C4×D3—61no—
223C2×C4×D3—61no—
224C2×C4×D3—61no—
225C2×C4×D3—64no—
226C2×C4×D3—61no—
227(C2×D3)⋊C4—98no—
228C4⋊Dic3—94no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—62no—
234C22×C32—64no—
235C22×C32C2×#6711yescharacteristic
236C2×C3⋊D3—24no—
237C2×C3⋊D3—44no—
238C2×C3⋊D3C2×D1212yes—
239C2×C3⋊D3#23512yes—
240C2×C3⋊D3C2×#6711yescharacteristic
241C2×C3⋊D3C23×D311yescharacteristic
242C2×C3×D3—22no—
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3—22no—
246C2×C3×D3—42no—
247C2×C3×D3—44no—
248C2×C3×D3—64no—
249C2×C3×D3—61no—
250C2×C3×D3—61no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—61no—
256C2×C3×D3—61no—
257C2×C3×D3—68no—
258C2×C3×D3—64no—
259C2×C3×D3—64no—
260C2×C3×D3—64no—
261C2×C3×D3—124no—
262C2×C3×D3—124no—
263C2×C3×D3—124no—
264C2×C3×D3—244no—
265D32—24no—
266D32—22no—
267D32—64no—
268D32—64no—
269D32—124no—
270D32—124no—
271D32—128no—
272D32—124no—
273D32C2×D1214yescomplemented
274#157—64no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—42no—
278C32×C4—42no—
279C32×C4—62no—
280C32×C4C2×#6712yescomplemented
281C3⋊Dic3C2×#6712yescomplemented
282C3×Dic3—22no—
283C3×Dic3—22no—
284C3×Dic3—42no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C22×D4—271no—
288C22×D4—272no—
289C23×C4—91no—
290#121—278no—
291C2×C4⋊C4—271no—
292C2×C22⋊C4—272no—
293C33C2×#12111yescharacteristic, complemented, Sylow
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—92no—
297C23×C3—94no—
298C23×C3—91no—
299C23×C3—121no—
300C23×C3—182no—
301C23×C3—181no—
302C23×C3—182no—
303C22×D3—21no—
304C22×D3—64no—
305C22×D3—62no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—62no—
309C22×D3—64no—
310C22×D3—62no—
311C22×D3—62no—
312C22×D3—94no—
313C22×D3—122no—
314C22×D3—184no—
315C22×D3—182no—
316C22×D3—184no—
317C22×D3—184no—
318C22×D3—184no—
319C22×D3—364no—
320C22×D3—364no—
321C3×D4—184no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—64no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—64no—
330C2×C3×C4—92no—
331C2×C3×C4—91no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—124no—
335C2×C3×C4—121no—
336C2×C3×C4—181no—
337C2×C3×C4—184no—
338C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
339#67—188no—
340#67—1816no—
341C2×Dic3—21no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—64no—
345C2×Dic3—94no—
346C2×Dic3—184no—
347D12—64no—
348D12—64no—
349D12—124no—
350D12—184no—
351C4×D3—24no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—62no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—62no—
358C4×D3—64no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—64no—
368C2×C32#93012yescomplemented
369C2×C32C22×#6711yescharacteristic
370C3⋊D3—44no—
371C3⋊D3#93012yescomplemented
372C3⋊D3C22×D1211yescharacteristic
373C3⋊D3C2×#23511yescharacteristic
374C3×D3—24no—
375C3×D3—24no—
376C3×D3—44no—
377C3×D3—62no—
378C3×D3—62no—
379C3×D3—61no—
380C3×D3—61no—
381C3×D3—61no—
382C3×D3—61no—
383C3×D3—124no—
384C3×D3—124no—
385C3×D3—244no—
386C24—91no—
387C24—272no—
388C2×D4—181no—
389C2×D4—274no—
390C2×D4—278no—
391C2×D4—542no—
392C2×D4—541no—
393C2×D4—548no—
394C22×C4—61no—
395C22×C4—91no—
396C22×C4—94no—
397C22×C4—91no—
398C22×C4—182no—
399C22×C4—181no—
400C22×C4—272no—
401C4⋊C4—274no—
402C22⋊C4—278no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—62no—
408C22×C3—64no—
409C22×C3—62no—
410C22×C3—64no—
411C22×C3—94no—
412C22×C3—91no—
413C22×C3—91no—
414C22×C3—92no—
415C22×C3—92no—
416C22×C3—122no—
417C22×C3—124no—
418C22×C3—184no—
419C22×C3—182no—
420C22×C3—184no—
421C22×C3—364no—
422C22×C3C2×GO+(4,𝔽2)11yescharacteristic
423C2×D3—22no—
424C2×D3—24no—
425C2×D3—64no—
426C2×D3—61no—
427C2×D3—61no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—68no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—124no—
439C2×D3—124no—
440C2×D3—124no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—244no—
444C2×D3—364no—
445C2×D3—364no—
446C2×D3—368no—
447C2×D3—364no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—92no—
455C3×C4—122no—
456C3×C4C2×GO+(4,𝔽2)12yescomplemented
457Dic3—22no—
458Dic3—62no—
459Dic3—62no—
460Dic3—184no—
461C32—21no—
462C32—21no—
463C32—41no—
464C32—41no—
465C32C2×#93011yescharacteristic, complemented
466C23—61no—
467C23—92no—
468C23—94no—
469C23—91no—
470C23—181no—
471C23—182no—
472C23—182no—
473C23—274no—
474C23—542no—
475C23—544no—
476C23—544no—
477D4—184no—
478D4—544no—
479D4—5416no—
480C2×C4—61no—
481C2×C4—64no—
482C2×C4—92no—
483C2×C4—91no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—181no—
487C2×C4—184no—
488C2×C4—274no—
489C2×C4—544no—
490C2×C4D32⋊D311yescharacteristic, complemented
491C2×C3—22no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—42no—
496C2×C3—42no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—64no—
500C2×C3—64no—
501C2×C3—91no—
502C2×C3—91no—
503C2×C3—92no—
504C2×C3—124no—
505C2×C3—364no—
506C2×C3#6423012yescomplemented
507C2×C3C22×GO+(4,𝔽2)11yescharacteristic
508D3—24no—
509D3—62no—
510D3—61no—
511D3—61no—
512D3—62no—
513D3—61no—
514D3—61no—
515D3—124no—
516D3—124no—
517D3—244no—
518C22—64no—
519C22—62no—
520C22—94no—
521C22—92no—
522C22—92no—
523C22—91no—
524C22—91no—
525C22—182no—
526C22—184no—
527C22—184no—
528C22—364no—
529C22—544no—
530C22—544no—
531C22—1084no—
532C22C2×D32⋊D311yescharacteristic
533C4—62no—
534C4—92no—
535C4—544no—
536C4C2×D32⋊D312yescomplemented
537C3—21no—
538C3—21no—
539C3—41no—
540C3—41no—
541C3C2×#6423011yescharacteristic, complemented
542C2—64no—
543C2—91no—
544C2—91no—
545C2—92no—
546C2—364no—
547C2#1172431412yesdirect product factor
548C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#121
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D32⋊D3C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×C3×C4×D32C2semidirect
4. C2×C4⋊(C3⋊#157)C2semidirect
5. C2×#88932C2semidirect
6. #11724314C2direct (N × H)
12. C2×D32⋊D3C22semidirect
18. C4⋊#1113C22semidirect
22. C3×C4×D32C22semidirect
23. C4⋊(C3⋊#157)C22semidirect
24. #88932C22semidirect
29. C2×C4×D32D3semidirect
42. C2×C3×D32D4semidirect
47. C2×C33×C4D4semidirect
48. C2×C3×C3⋊Dic3D4semidirect
50. C3×C4×C3⋊D3D4semidirect
91. C4×D32C2×D3semidirect
108. C3×D32C2×D4semidirect
110. C33×C4C2×D4semidirect
111. C3×C3⋊Dic3C2×D4semidirect
146. C2×D32D12semidirect
157. C2×C32×C4#67semidirect
158. C2×C3⋊Dic3#67semidirect
160. C4×C3⋊D3#67semidirect
187. C2×C33#121semidirect
190. C3×C3⋊D3#121semidirect
273. D32C2×D12semidirect
280. C32×C4C2×#67semidirect
281. C3⋊Dic3C2×#67semidirect
293. C33C2×#121semidirect
338. C2×C3×C4GO+(4,𝔽2)semidirect
368. C2×C32#930semidirect
371. C3⋊D3#930semidirect
456. C3×C4C2×GO+(4,𝔽2)semidirect
465. C32C2×#930semidirect
490. C2×C4D32⋊D3semidirect
506. C2×C3#64230semidirect
536. C4C2×D32⋊D3semidirect
541. C3C2×#64230semidirect
547. C2#11724314direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D32⋊D32
C2×C4⋊#11132
C2×C3×C4×D322
C2×C4⋊(C3⋊#157)2
C2×#889322
#117243142
C2×#642303
C2×#9309