Finite Group Databasebeta

← C2×#11721323 #13152089

Subgroups — C2×#11721323 #13152089

Order 1728.

3065 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 3065 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C24×#272C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×#6200C212yesmaximal, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5#11721323C218yesmaximal, direct product factor
6C2×#88451C212yesmaximal, complemented
7C2×#63977—31nomaximal
8C2×#63976—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×#272—24no—
11C23×#272—28no—
12C23×#272C2212yes—
13C23×#272C2214yescomplemented
14C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
15C2×#638⋊C3—24no—
16C2×#638⋊C3C2214yescomplemented
17C22×(C3⋊Dic3)⋊C3C2212yescomplemented
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
19C2×#6200—28no—
20C2×#6200C2218yescomplemented
21#3757⋊C3C2214yescomplemented
22#88451C2218yescomplemented
23C24×C3×D3—31no—
24C24×C3×D3—31no—
25C24×C3×D3—61no—
26C22×#638D311yescharacteristic, complemented
27C22×C3×#67—31no—
28C22×#626—32no—
29C22×#625—32no—
30C2×#3757D311yescharacteristic, complemented
31C2×C3×#217—31no—
32#63977—38no—
33#63976—38no—
34C2×(C2×D3)⋊Dic3—32no—
35C2×(C2×D3)⋊Dic3—32no—
36C23×H(3,𝔽3)—24no—
37C23×H(3,𝔽3)D414yescomplemented
38C23×H(3,𝔽3)C2312yes—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×#272—232no—
41C22×#272—24no—
42C22×#272—24no—
43C22×#272—28no—
44C22×#272—216no—
45C22×#272D418yescomplemented
46C22×#272C2314yes—
47C22×(C3⋊D3)⋊C3—24no—
48C22×(C3⋊D3)⋊C3—22no—
49C22×(C3⋊D3)⋊C3C2312yescomplemented
50#638⋊C3—28no—
51C2×(C3⋊Dic3)⋊C3—24no—
52C2×(C3⋊Dic3)⋊C3—22no—
53C2×(C3⋊Dic3)⋊C3C2314yescomplemented
54C2×(C3⋊Dic3)⋊C3C2312yescomplemented
55#6200—216no—
56C2×#953—91no—
57C2×#937—91no—
58C2×#882—91no—
59C24×C32—21no—
60C24×C32C2×D311yescharacteristic
61C24×C32C2×D311yescharacteristic
62C23×C3⋊D3C2×D311yescharacteristic, complemented
63C23×C3×D3—32no—
64C23×C3×D3—32no—
65C23×C3×D3—31no—
66C23×C3×D3—34no—
67C23×C3×D3—34no—
68C23×C3×D3—68no—
69C23×C3×D3—68no—
70C23×C3×D3—62no—
71C23×C3×D3—64no—
72C23×C3×D3—616no—
73C23×C3×D3—68no—
74C23×C3×D3—64no—
75C23×C3×D3—64no—
76C2×#638—24no—
77C2×#638C2×D314yescomplemented
78C22×C3⋊Dic3C2×D312yescomplemented
79C22×C3⋊Dic3C2×D311yescharacteristic, complemented
80C2×C3×#67—34no—
81C2×C3×#67—64no—
82C22×C3×Dic3—32no—
83C22×C3×Dic3—31no—
84C2×#626—38no—
85C2×#626—68no—
86C2×#625—38no—
87C2×#625—68no—
88#3757C2×D314yescomplemented
89C3×#217—34no—
90(C2×D3)⋊Dic3—38no—
91(C2×D3)⋊Dic3—38no—
92C22×H(3,𝔽3)—28no—
93C22×H(3,𝔽3)—24no—
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)C2×D414yes—
96C22×H(3,𝔽3)C2×D414yescomplemented
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×#272—232no—
99C2×#272—216no—
100C2×#272—24no—
101C2×#272—24no—
102C2×#272C2×D418yescomplemented
103C2×(C3⋊D3)⋊C3—24no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—44no—
106(C3⋊Dic3)⋊C3—24no—
107(C3⋊Dic3)⋊C3—22no—
108C25×C3—91no—
109C24×D3—31no—
110C24×D3—31no—
111C24×D3—61no—
112C22×C3×D4—91no—
113C22×#67—31no—
114C22×#67—91no—
115C22×#67—92no—
116C22×#67—92no—
117C22×D12—92no—
118C2×C3×C22⋊C4—91no—
119#953—98no—
120C2×#217—31no—
121C2×#217—91no—
122C2×#217—92no—
123#937—98no—
124C2×(C2×D3)⋊C4—92no—
125C2×(C2×D3)⋊C4—92no—
126#882—98no—
127C23×C32—28no—
128C23×C32—24no—
129C23×C32—24no—
130C23×C32—24no—
131C23×C32—22no—
132C23×C32—21no—
133C23×C32D1214yescomplemented
134C23×C32#6714yescomplemented
135C23×C32C22×D312yes—
136C23×C32C22×D312yes—
137C23×C32C22×D311yescharacteristic
138C23×C32C22×D311yescharacteristic
139C22×C3⋊D3—24no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3C22×D312yescomplemented
142C22×C3×D3—38no—
143C22×C3×D3—38no—
144C22×C3×D3—34no—
145C22×C3×D3—34no—
146C22×C3×D3—32no—
147C22×C3×D3—68no—
148C22×C3×D3—64no—
149C22×C3×D3—616no—
150C22×C3×D3—616no—
151C22×C3×D3—632no—
152C22×C3×D3—664no—
153C22×C3×D3—632no—
154C22×C3×D3—632no—
155C22×C3×D3—68no—
156C22×C3×D3—616no—
157C22×C3×D3—68no—
158C22×C3×D3—64no—
159C22×C3×D3—68no—
160C22×C3×D3—64no—
161C22×C3×D3—64no—
162C22×C3×D3—64no—
163C22×C3×D3—68no—
164C22×C3×D3—62no—
165C22×C3×D3—64no—
166#638—28no—
167C2×C3⋊Dic3—24no—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3C22×D312yescomplemented
170C2×C3⋊Dic3C22×D314yescomplemented
171C3×#67—68no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—64no—
176#626—616no—
177#625—616no—
178C2×#120—271noSylow
179C2×H(3,𝔽3)—24no—
180C2×H(3,𝔽3)#12014yescomplemented
181C2×H(3,𝔽3)C22×D411yescharacteristic
182C2×H(3,𝔽3)C22×D412yes—
183#272—28no—
184(C3⋊D3)⋊C3—42no—
185C24×C3—31no—
186C24×C3—31no—
187C24×C3—61no—
188C24×C3—91no—
189C24×C3—92no—
190C24×C3—94no—
191C24×C3—188no—
192C24×C3—184no—
193C24×C3D3211yescharacteristic
194C23×D3—32no—
195C23×D3—32no—
196C23×D3—34no—
197C23×D3—34no—
198C23×D3—31no—
199C23×D3—62no—
200C23×D3—64no—
201C23×D3—68no—
202C23×D3—68no—
203C23×D3—64no—
204C23×D3—64no—
205C23×D3—616no—
206C23×D3—68no—
207C23×D3—91no—
208C2×C3×D4—94no—
209C2×C3×D4—184no—
210C22×C3×C4—91no—
211C22×C3×C4—92no—
212C2×#67—34no—
213C2×#67—64no—
214C2×#67—94no—
215C2×#67—98no—
216C2×#67—98no—
217C2×#67—184no—
218C2×#67—188no—
219C2×#67—188no—
220C22×Dic3—32no—
221C22×Dic3—31no—
222C22×Dic3—91no—
223C22×Dic3—92no—
224C2×D12—98no—
225C2×D12—188no—
226C3×C22⋊C4—94no—
227#217—34no—
228#217—94no—
229#217—98no—
230(C2×D3)⋊C4—98no—
231(C2×D3)⋊C4—98no—
232C22×C32—28no—
233C22×C32—216no—
234C22×C32—28no—
235C22×C32—28no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—21no—
239C22×C32—22no—
240C22×C32—24no—
241C22×C32—24no—
242C22×C32C2×D1212yes—
243C22×C32C2×D1214yescomplemented
244C22×C32D3×D414yes—
245C22×C32D3×D414yes—
246C22×C32C2×#6712yes—
247C22×C32C2×#6714yescomplemented
248C22×C32C23×D311yescharacteristic
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—24no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3—44no—
253C2×C3×D3—38no—
254C2×C3×D3—38no—
255C2×C3×D3—68no—
256C2×C3×D3—664no—
257C2×C3×D3—632no—
258C2×C3×D3—632no—
259C2×C3×D3—68no—
260C2×C3×D3—68no—
261C2×C3×D3—632no—
262C2×C3×D3—616no—
263C2×C3×D3—616no—
264C2×C3×D3—64no—
265C2×C3×D3—64no—
266C2×C3×D3—64no—
267C2×C3×D3—64no—
268C2×C3×D3—62no—
269C2×C3×D3—64no—
270C2×C3×D3—124no—
271C3⋊Dic3—24no—
272C3⋊Dic3—22no—
273C3×Dic3—62no—
274C3×Dic3—64no—
275C25—91no—
276C22×D4—91no—
277C22×D4—272no—
278#120—278no—
279C2×C22⋊C4—91no—
280C2×C22⋊C4—272no—
281H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
282C23×C3—24no—
283C23×C3—34no—
284C23×C3—34no—
285C23×C3—32no—
286C23×C3—32no—
287C23×C3—31no—
288C23×C3—31no—
289C23×C3—64no—
290C23×C3—64no—
291C23×C3—68no—
292C23×C3—64no—
293C23×C3—62no—
294C23×C3—61no—
295C23×C3—92no—
296C23×C3—94no—
297C23×C3—98no—
298C23×C3—1816no—
299C23×C3—1832no—
300C23×C3—184no—
301C23×C3—182no—
302C23×C3—184no—
303C23×C3—188no—
304C23×C3—184no—
305C23×C3#62614yes—
306C23×C3C2×D3212yes—
307C23×C3C2×D3211yescharacteristic
308C22×D3—34no—
309C22×D3—34no—
310C22×D3—38no—
311C22×D3—38no—
312C22×D3—32no—
313C22×D3—64no—
314C22×D3—68no—
315C22×D3—616no—
316C22×D3—616no—
317C22×D3—64no—
318C22×D3—64no—
319C22×D3—68no—
320C22×D3—68no—
321C22×D3—64no—
322C22×D3—64no—
323C22×D3—632no—
324C22×D3—632no—
325C22×D3—632no—
326C22×D3—664no—
327C22×D3—68no—
328C22×D3—616no—
329C22×D3—68no—
330C22×D3—62no—
331C22×D3—64no—
332C22×D3—92no—
333C22×D3—184no—
334C22×D3—182no—
335C3×D4—188no—
336C2×C3×C4—92no—
337C2×C3×C4—94no—
338C2×C3×C4—184no—
339C2×C3×C4—182no—
340#67—68no—
341#67—188no—
342#67—1816no—
343#67—1816no—
344C2×Dic3—34no—
345C2×Dic3—32no—
346C2×Dic3—62no—
347C2×Dic3—64no—
348C2×Dic3—92no—
349C2×Dic3—94no—
350C2×Dic3—182no—
351C2×Dic3—184no—
352D12—1816no—
353C2×C32—28no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—21no—
357C2×C32—22no—
358C2×C32—24no—
359C2×C32#88214yescomplemented
360C2×C32#93714yescomplemented
361C2×C32C22×D1211yescharacteristic
362C2×C32C2×D3×D412yes—
363C2×C32C2×D3×D412yes—
364C2×C32C22×#6711yescharacteristic
365C3⋊D3—42no—
366C3×D3—616no—
367C3×D3—68no—
368C3×D3—68no—
369C3×D3—122no—
370C24—92no—
371C24—94no—
372C24—91no—
373C24—188no—
374C24—184no—
375C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C2×D4—94no—
377C2×D4—184no—
378C2×D4—278no—
379C2×D4—548no—
380C22×C4—92no—
381C22×C4—91no—
382C22⋊C4—94no—
383C22⋊C4—278no—
384C22×C3—24no—
385C22×C3—28no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—32no—
389C22×C3—32no—
390C22×C3—34no—
391C22×C3—34no—
392C22×C3—34no—
393C22×C3—34no—
394C22×C3—64no—
395C22×C3—64no—
396C22×C3—68no—
397C22×C3—68no—
398C22×C3—68no—
399C22×C3—616no—
400C22×C3—61no—
401C22×C3—62no—
402C22×C3—64no—
403C22×C3—64no—
404C22×C3—98no—
405C22×C3—1832no—
406C22×C3—184no—
407C22×C3—182no—
408C22×C3—184no—
409C22×C3—184no—
410C22×C3—1816no—
411C22×C3—364no—
412C22×C3C2×#62612yes—
413C22×C3C2×#62614yes—
414C22×C3#381114yes—
415C22×C3C22×D3211yescharacteristic
416C2×D3—38no—
417C2×D3—38no—
418C2×D3—68no—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—64no—
423C2×D3—616no—
424C2×D3—616no—
425C2×D3—632no—
426C2×D3—632no—
427C2×D3—664no—
428C2×D3—68no—
429C2×D3—68no—
430C2×D3—632no—
431C2×D3—62no—
432C2×D3—64no—
433C2×D3—124no—
434C2×D3—184no—
435C2×D3—182no—
436C2×D3—364no—
437C3×C4—184no—
438C3×C4—182no—
439Dic3—62no—
440Dic3—64no—
441Dic3—182no—
442Dic3—184no—
443C32—21no—
444C32C2×#88211yescharacteristic, complemented
445C32C2×#93711yescharacteristic, complemented
446C23—24no—
447C23—94no—
448C23—98no—
449C23—92no—
450C23—1816no—
451C23—1832no—
452C23—184no—
453C23—188no—
454C23—184no—
455C23—182no—
456C23—184no—
457C23#620014yescomplemented
458C23C2×((C3⋊D3)⋊C3)⋊C212yes—
459C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460D4—188no—
461D4—5416no—
462C2×C4—94no—
463C2×C4—92no—
464C2×C4—182no—
465C2×C4—184no—
466C2×C3—24no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—32no—
470C2×C3—32no—
471C2×C3—34no—
472C2×C3—34no—
473C2×C3—64no—
474C2×C3—64no—
475C2×C3—68no—
476C2×C3—61no—
477C2×C3—62no—
478C2×C3—64no—
479C2×C3—188no—
480C2×C3—362no—
481C2×C3#6398114yes—
482C2×C3C22×#62611yescharacteristic
483C2×C3C2×#381112yes—
484D3—68no—
485D3—68no—
486D3—616no—
487D3—122no—
488D3—362no—
489C22—24no—
490C22—28no—
491C22—98no—
492C22—1832no—
493C22—184no—
494C22—184no—
495C22—1816no—
496C22—182no—
497C22—184no—
498C22—364no—
499C22C2×#620012yes—
500C22C2×#620014yescomplemented
501C22#8867714yes—
502C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
503C4—182no—
504C4—184no—
505C3—31no—
506C3—31no—
507C3—61no—
508C3C2×#6398111yescharacteristic
509C2—24no—
510C2—188no—
511C2—362no—
512C2#1172132314yesdirect product factor
513C2C22×#620011yescharacteristic
514C2C2×#8867712yes—
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×#272C2semidirect
2. C22×#638⋊C3C2semidirect
3. C22×#6200C2semidirect
4. C2×#3757⋊C3C2semidirect
5. #11721323C2direct (N × H)
6. C2×#88451C2semidirect
13. C23×#272C22semidirect
14. C23×(C3⋊D3)⋊C3C22semidirect
16. C2×#638⋊C3C22semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
18. C22×(C3⋊Dic3)⋊C3C22semidirect
20. C2×#6200C22semidirect
21. #3757⋊C3C22semidirect
22. #88451C22semidirect
26. C22×#638D3semidirect
30. C2×#3757D3semidirect
37. C23×H(3,𝔽3)D4semidirect
45. C22×#272D4semidirect
49. C22×(C3⋊D3)⋊C3C23semidirect
53. C2×(C3⋊Dic3)⋊C3C23semidirect
54. C2×(C3⋊Dic3)⋊C3C23semidirect
62. C23×C3⋊D3C2×D3semidirect
77. C2×#638C2×D3semidirect
78. C22×C3⋊Dic3C2×D3semidirect
79. C22×C3⋊Dic3C2×D3semidirect
88. #3757C2×D3semidirect
96. C22×H(3,𝔽3)C2×D4semidirect
102. C2×#272C2×D4semidirect
133. C23×C32D12semidirect
134. C23×C32#67semidirect
141. C22×C3⋊D3C22×D3semidirect
169. C2×C3⋊Dic3C22×D3semidirect
170. C2×C3⋊Dic3C22×D3semidirect
180. C2×H(3,𝔽3)#120semidirect
243. C22×C32C2×D12semidirect
247. C22×C32C2×#67semidirect
281. H(3,𝔽3)C2×#120semidirect
359. C2×C32#882semidirect
360. C2×C32#937semidirect
444. C32C2×#882semidirect
445. C32C2×#937semidirect
457. C23#6200semidirect
500. C22C2×#6200semidirect
512. C2#11721323direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#2722
C22×#638⋊C32
C22×#62002
C2×#3757⋊C32
#117213232
C2×#884512
C2×#639773
C2×#639763