Finite Group Databasebeta

← C2×#11724343 #13181143

Subgroups — C2×#11724343 #13181143

Order 1728.

1432 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 151 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×#157⋊D3C212yesmaximal, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C2×C4×#1113C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
5C2×#88939C212yesmaximal, complemented
6#11724343C218yesmaximal, direct product factor
7C2×#64230—31nomaximal
8C22×#1113C2212yes—
9C22×#1113C2211yescharacteristic
10C22×C3×#157C2212yes—
11C2×#157⋊D3—28no—
12C2×#157⋊D3C2218yescomplemented
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×#6295—21no—
17C2×#6294—21no—
18C2×#6294C2211yescharacteristic
19C4⋊#1113—22no—
20C4⋊#1113C2214yescomplemented
21C4×#1113—22no—
22C4×#1113C2214yescomplemented
23C3×C4⋊#157C2214yescomplemented
24#88939C2218yescomplemented
25C22×GO+(4,𝔽2)—32no—
26C2×#3802—31no—
27C2×D3×D12—61no—
28C2×C4×D32—31no—
29C2×C4×D32—61no—
30C2×C4⋊#157D311yescharacteristic, complemented
31C2×D32⋊C4—32no—
32#64230—38no—
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3C2311yescharacteristic
36C2×#1113—21no—
37C2×#1113—22no—
38C2×#1113—24no—
39C2×#1113—24no—
40C2×#1113—22no—
41C2×#1113D414yescomplemented
42C2×#1113C2314yes—
43C2×#1113C2312yes—
44C2×C3×#157—24no—
45C2×C3×#157C2314yes—
46#157⋊D3—216no—
47C2×C33×C4D411yescharacteristic, complemented
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic, complemented
50C3×C4⋊(C3⋊D3)—24no—
51C3×C4×C3⋊D3—24no—
52C3×C4×C3⋊D3D412yescomplemented
53C3×C4×C3⋊D3C2312yes—
54#6295—24no—
55#6294—21no—
56#6294—24no—
57#6294C2312yes—
58C2×#895—91nomaximal
59C22×D32—31no—
60C22×D32—32no—
61C22×D32—61no—
62C22×D32—62no—
63C22×#157C2×D312yes—
64C2×GO+(4,𝔽2)—38no—
65C2×GO+(4,𝔽2)—68no—
66C2×C4⋊(C3⋊D3)—21no—
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×D12—61no—
70C2×C3×D12—61no—
71C2×C3×D12—121no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—61no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—121no—
77C2×#626—62no—
78C2×#625—61no—
79C2×#624—31no—
80C2×#624—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—61no—
83C2×D3×Dic3—61no—
84#3802—34no—
85#3802—62no—
86D3×D12—64no—
87D3×D12—64no—
88C4×D32—34no—
89C4×D32—64no—
90C4×D32—64no—
91C4×D32—62no—
92C4⋊#157C2×D314yescomplemented
93D32⋊C4—38no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—24no—
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—24no—
98C2×C3×C3⋊D3—44no—
99C2×C3×C3⋊D3C2×D411yescharacteristic
100C2×C3×C3⋊D3C2×D412yes—
101C2×C3×C3⋊D3#4112yes—
102C2×C3×C3⋊D3C2411yescharacteristic
103#1113—24no—
104#1113—22no—
105#1113—24no—
106#1113—24no—
107#1113—44no—
108#1113C2×D414yescomplemented
109C3×#157—24no—
110C33×C4C2×D412yescomplemented
111C3×C3⋊Dic3—22no—
112C3×C3⋊Dic3C2×D412yescomplemented
113C2×D3×D4—181no—
114C22×D12—91no—
115C22×D12—92no—
116C22×C4×D3—91no—
117C22×C4×D3—181no—
118C2×C3×C4⋊C4—91no—
119C2×(C2×D3)⋊C4—92no—
120#895—98no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3C22×D311yescharacteristic
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—62no—
129C22×C3×D3—62no—
130C22×C3×D3—121no—
131C22×C3×D3—122no—
132C2×D32—34no—
133C2×D32—32no—
134C2×D32—34no—
135C2×D32—64no—
136C2×D32—62no—
137C2×D32—64no—
138C2×D32—64no—
139C2×D32—62no—
140C2×D32—61no—
141C2×D32—62no—
142C2×D32—64no—
143C2×D32—62no—
144C2×D32—64no—
145C2×D32—64no—
146C2×D32—124no—
147C2×D32—124no—
148C2×#157—24no—
149C2×#157C22×D314yes—
150GO+(4,𝔽2)—616no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4D1211yescharacteristic, complemented
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3D1211yescharacteristic, complemented
158C4⋊(C3⋊D3)—24no—
159C4×C3⋊D3—24no—
160C4×C3⋊D3D1212yescomplemented
161C4×C3⋊D3C22×D312yes—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—121no—
167C3×D12—64no—
168C3×D12—64no—
169C3×D12—124no—
170C3×C4×D3—62no—
171C3×C4×D3—62no—
172C3×C4×D3—62no—
173C3×C4×D3—62no—
174C3×C4×D3—64no—
175C3×C4×D3—64no—
176C3×C4×D3—124no—
177#626—68no—
178#625—64no—
179#624—32no—
180#624—64no—
181#624—61no—
182D3×Dic3—62no—
183D3×Dic3—64no—
184D3×Dic3—62no—
185D3×Dic3—64no—
186C2×#121—271noSylow
187C2×C33#12112yescomplemented
188C2×C33C22×D411yescharacteristic
189C3×C3⋊D3—24no—
190C3×C3⋊D3—44no—
191C3×C3⋊D3#12112yescomplemented
192C3×C3⋊D3C22×D411yescharacteristic
193C3×C3⋊D3C2×#4111yescharacteristic
194C23×D3—91no—
195C23×D3—92no—
196C23×D3—181no—
197C23×D3—182no—
198C2×C3×D4—181no—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C22×C3×C4—181no—
202C2×#67—182no—
203C22×Dic3—91no—
204C22×Dic3—181no—
205D3×D4—184no—
206D3×D4—184no—
207C2×D12—61no—
208C2×D12—61no—
209C2×D12—94no—
210C2×D12—98no—
211C2×D12—121no—
212C2×D12—182no—
213C2×D12—181no—
214C2×D12—188no—
215C2×C4×D3—61no—
216C2×C4×D3—61no—
217C2×C4×D3—61no—
218C2×C4×D3—61no—
219C2×C4×D3—94no—
220C2×C4×D3—121no—
221C2×C4×D3—184no—
222C2×C4×D3—181no—
223C2×C4×D3—182no—
224C2×C4×D3—184no—
225C2×C4×D3—181no—
226C2×C4×D3—181no—
227C3×C4⋊C4—94no—
228(C2×D3)⋊C4—98no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32C2×D1211yescharacteristic
234C2×C3⋊D3—22no—
235C2×C3⋊D3—24no—
236C2×C3⋊D3—24no—
237C2×C3⋊D3—44no—
238C2×C3⋊D3C2×D1211yescharacteristic
239C2×C3⋊D3D3×D412yes—
240C2×C3⋊D3#23912yes—
241C2×C3⋊D3C23×D311yescharacteristic
242C2×C3×D3—61no—
243C2×C3×D3—61no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—62no—
251C2×C3×D3—64no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—122no—
257C2×C3×D3—124no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C2×C3×D3—244no—
262D32—34no—
263D32—64no—
264D32—68no—
265D32—64no—
266D32—64no—
267D32—62no—
268D32—64no—
269D32—64no—
270D32—124no—
271D32—124no—
272D32—128no—
273D32—124no—
274#157—24no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—42no—
278C32×C4—42no—
279C32×C4C2×D1212yescomplemented
280C3⋊Dic3—22no—
281C3⋊Dic3C2×D1212yescomplemented
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—122no—
287C22×D4—271no—
288C22×D4—272no—
289C23×C4—271no—
290#121—278no—
291C2×C4⋊C4—91no—
292C2×C22⋊C4—272no—
293C33C2×#12111yescharacteristic, complemented, Sylow
294C23×C3—91no—
295C23×C3—181no—
296C23×C3—182no—
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—61no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—92no—
304C22×D3—94no—
305C22×D3—94no—
306C22×D3—121no—
307C22×D3—122no—
308C22×D3—182no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—181no—
312C22×D3—182no—
313C22×D3—182no—
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—91no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—94no—
330C2×C3×C4—184no—
331C2×C3×C4—181no—
332C2×C3×C4—184no—
333C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
334#67—188no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—92no—
340C2×Dic3—121no—
341C2×Dic3—181no—
342C2×Dic3—184no—
343D12—64no—
344D12—64no—
345D12—124no—
346D12—184no—
347D12—1816no—
348C4×D3—62no—
349C4×D3—62no—
350C4×D3—62no—
351C4×D3—62no—
352C4×D3—64no—
353C4×D3—64no—
354C4×D3—124no—
355C4×D3—182no—
356C4×D3—184no—
357C4×D3—184no—
358C4×D3—182no—
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#89512yescomplemented
368C2×C32C22×D1211yescharacteristic
369C3⋊D3—24no—
370C3⋊D3—44no—
371C3⋊D3#89512yescomplemented
372C3⋊D3C2×D3×D411yescharacteristic
373C3⋊D3C2×#23911yescharacteristic
374C3×D3—62no—
375C3×D3—62no—
376C3×D3—61no—
377C3×D3—61no—
378C3×D3—61no—
379C3×D3—61no—
380C3×D3—64no—
381C3×D3—64no—
382C3×D3—124no—
383C3×D3—124no—
384C3×D3—124no—
385C3×D3—244no—
386C24—271no—
387C24—272no—
388C2×D4—181no—
389C2×D4—274no—
390C2×D4—278no—
391C2×D4—542no—
392C2×D4—541no—
393C2×D4—548no—
394C22×C4—91no—
395C22×C4—92no—
396C22×C4—181no—
397C22×C4—274no—
398C22×C4—271no—
399C22×C4—542no—
400C22×C4—541no—
401C4⋊C4—94no—
402C22⋊C4—278no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—91no—
408C22×C3—91no—
409C22×C3—92no—
410C22×C3—92no—
411C22×C3—182no—
412C22×C3—184no—
413C22×C3—184no—
414C22×C3—364no—
415C22×C3C2×GO+(4,𝔽2)11yescharacteristic
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—61no—
419C2×D3—61no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—64no—
426C2×D3—62no—
427C2×D3—64no—
428C2×D3—64no—
429C2×D3—64no—
430C2×D3—94no—
431C2×D3—122no—
432C2×D3—124no—
433C2×D3—124no—
434C2×D3—124no—
435C2×D3—124no—
436C2×D3—184no—
437C2×D3—182no—
438C2×D3—184no—
439C2×D3—188no—
440C2×D3—184no—
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—92no—
453C3×C4—182no—
454C3×C4—184no—
455C3×C4C2×GO+(4,𝔽2)12yescomplemented
456Dic3—62no—
457Dic3—62no—
458Dic3—62no—
459Dic3—62no—
460Dic3—122no—
461C32—21no—
462C32—21no—
463C32—41no—
464C32—41no—
465C32C2×#89511yescharacteristic, complemented
466C23—91no—
467C23—181no—
468C23—182no—
469C23—272no—
470C23—274no—
471C23—274no—
472C23—541no—
473C23—542no—
474C23—542no—
475C23—544no—
476C23—544no—
477D4—184no—
478D4—544no—
479D4—5416no—
480C2×C4—91no—
481C2×C4—92no—
482C2×C4—92no—
483C2×C4—94no—
484C2×C4—184no—
485C2×C4—181no—
486C2×C4—184no—
487C2×C4—272no—
488C2×C4—544no—
489C2×C4—541no—
490C2×C4#157⋊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—91no—
500C2×C3—91no—
501C2×C3—92no—
502C2×C3—184no—
503C2×C3—364no—
504C2×C3#6423012yescomplemented
505C2×C3C22×GO+(4,𝔽2)11yescharacteristic
506D3—62no—
507D3—61no—
508D3—61no—
509D3—62no—
510D3—61no—
511D3—61no—
512D3—64no—
513D3—64no—
514D3—124no—
515D3—124no—
516D3—124no—
517D3—244no—
518C22—92no—
519C22—92no—
520C22—91no—
521C22—91no—
522C22—184no—
523C22—182no—
524C22—184no—
525C22—274no—
526C22—364no—
527C22—542no—
528C22—544no—
529C22—544no—
530C22—544no—
531C22—1084no—
532C22C2×#157⋊D311yescharacteristic
533C4—92no—
534C4—184no—
535C4—182no—
536C4C2×#157⋊D312yescomplemented
537C3—21no—
538C3—21no—
539C3—41no—
540C3—41no—
541C3C2×#6423011yescharacteristic, complemented
542C2—91no—
543C2—91no—
544C2—92no—
545C2—184no—
546C2—364no—
547C2#1172434312yesdirect product factor
548C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊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×#157⋊D3C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×C4×#1113C2semidirect
4. C2×C3×C4⋊#157C2semidirect
5. C2×#88939C2semidirect
6. #11724343C2direct (N × H)
12. C2×#157⋊D3C22semidirect
20. C4⋊#1113C22semidirect
22. C4×#1113C22semidirect
23. C3×C4⋊#157C22semidirect
24. #88939C22semidirect
30. C2×C4⋊#157D3semidirect
41. C2×#1113D4semidirect
47. C2×C33×C4D4semidirect
49. C2×C3×C3⋊Dic3D4semidirect
52. C3×C4×C3⋊D3D4semidirect
92. C4⋊#157C2×D3semidirect
108. #1113C2×D4semidirect
110. C33×C4C2×D4semidirect
112. C3×C3⋊Dic3C2×D4semidirect
155. C2×C32×C4D12semidirect
157. C2×C3⋊Dic3D12semidirect
160. C4×C3⋊D3D12semidirect
187. C2×C33#121semidirect
191. C3×C3⋊D3#121semidirect
279. C32×C4C2×D12semidirect
281. C3⋊Dic3C2×D12semidirect
293. C33C2×#121semidirect
333. C2×C3×C4GO+(4,𝔽2)semidirect
367. C2×C32#895semidirect
371. C3⋊D3#895semidirect
455. C3×C4C2×GO+(4,𝔽2)semidirect
465. C32C2×#895semidirect
490. C2×C4#157⋊D3semidirect
504. C2×C3#64230semidirect
536. C4C2×#157⋊D3semidirect
541. C3C2×#64230semidirect
547. C2#11724343direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C2×C4⋊#11132
C2×C4×#11132
C2×C3×C4⋊#1572
C2×#889392
#117243432
C2×#642303
C2×#8959