Finite Group Databasebeta

← C2×#11722918 #13171984

Subgroups — C2×#11722918 #13171984

Order 1728.

1954 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic, complemented
3C2×#88804C212yesmaximal
4C2×C3×#3724C211yesmaximal, characteristic, complemented
5C2×C3×Dic32C212yesmaximal, complemented
6#11722918C218yesmaximal, direct product factor
7C2×#63874—31nomaximal
8C2×#63836—32nomaximal
9C23×C3⋊(C3⋊D3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C2×#6289C418yescomplemented
15C2×#6289C2214yes—
16#88815C2214yescomplemented
17#88804C2218yes—
18C3×#3724C2214yescomplemented
19C3×Dic32C2218yescomplemented
20C22×C4×C3⋊D3—31no—
21C22×#624—32no—
22C22×#624—61no—
23C2×#3752—32no—
24C2×C3×Dic3⋊C4—32no—
25C2×C3×C4×Dic3—32no—
26C2×C3×C4×Dic3—32no—
27C2×#3724—31no—
28C2×#3724D311yescharacteristic, complemented
29C2×Dic3⋊Dic3—32no—
30C2×#3722—32no—
31C2×#3722—32no—
32C2×#3722—62no—
33C2×Dic32D312yescomplemented
34#63874—38no—
35#63836—316no—
36C23×C33C2311yescharacteristic
37C22×C3⋊(C3⋊D3)—22no—
38C22×C3⋊(C3⋊D3)—22no—
39C22×C3⋊(C3⋊D3)C2×C414yes—
40C22×C3⋊(C3⋊D3)C2312yes—
41C2×C3×C3⋊Dic3—24no—
42C2×C3×C3⋊Dic3C2×C414yes—
43C2×C3×C3⋊Dic3C2314yes—
44C2×C3×C3⋊Dic3C2312yes—
45C2×C32×Dic3—24no—
46C2×C32×Dic3C2×C414yes—
47C2×C32×Dic3C2314yes—
48C2×C32×Dic3C2312yes—
49#6289C2×C4116yescomplemented
50C2×#892—91no—
51C2×#872—92no—
52C23×C3⋊D3—32no—
53C23×C3⋊D3—31no—
54C23×C3⋊D3—61no—
55C23×C3⋊D3—62no—
56C23×C3⋊D3—121no—
57C22×C32×C4—31no—
58C22×C32×C4—32no—
59C22×C3⋊Dic3C2×D312yes—
60C22×C3⋊Dic3C2×D311yescharacteristic
61C2×C4×C3⋊D3—38no—
62C2×C4×C3⋊D3—34no—
63C22×C3×Dic3—21no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3—62no—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C2×#624—316no—
74C2×#624—38no—
75C2×#624—68no—
76C2×#624—64no—
77#3752—38no—
78C3×Dic3⋊C4—38no—
79C3×C4×Dic3—38no—
80C3×C4×Dic3—38no—
81#3724—34no—
82#3724C2×D314yescomplemented
83Dic3⋊Dic3—38no—
84#3722—38no—
85#3722—38no—
86#3722—68no—
87Dic32C2×D318yescomplemented
88C22×C33C22×C412yes—
89C22×C33#4112yes—
90C22×C33#4112yes—
91C22×C33C2411yescharacteristic
92C2×C3⋊(C3⋊D3)—28no—
93C2×C3⋊(C3⋊D3)—22no—
94C2×C3⋊(C3⋊D3)—22no—
95C2×C3⋊(C3⋊D3)C22×C414yes—
96C3×C3⋊Dic3—24no—
97C3×C3⋊Dic3C22×C414yes—
98C32×Dic3—24no—
99C32×Dic3C22×C414yes—
100C22×C4×D3—92no—
101C22×C4×D3—91no—
102C22×C4×D3—181no—
103C2×C3×C4⋊C4—91no—
104C2×C3×C42—92no—
105C2×(C2×D3)⋊C4—92no—
106C2×(C2×D3)⋊C4—92no—
107C2×(C2×D3)⋊C4—92no—
108C2×(C2×D3)⋊C4—182no—
109C2×C4⋊Dic3—91no—
110C2×Dic3⋊C4—32no—
111C2×Dic3⋊C4—92no—
112C2×C4×Dic3—32no—
113C2×C4×Dic3—32no—
114#892—98no—
115#872—916no—
116C23×C32—21no—
117C23×C32—22no—
118C23×C32—41no—
119C23×C32C22×D312yes—
120C23×C32C22×D311yescharacteristic
121C22×C3⋊D3—38no—
122C22×C3⋊D3—34no—
123C22×C3⋊D3—32no—
124C22×C3⋊D3—34no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—62no—
127C22×C3⋊D3—64no—
128C22×C3⋊D3—68no—
129C22×C3⋊D3—62no—
130C22×C3⋊D3—62no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—64no—
135C22×C3⋊D3—124no—
136C22×C3⋊D3—122no—
137C22×C3⋊D3—124no—
138C22×C3⋊D3—124no—
139C22×C3⋊D3—124no—
140C22×C3⋊D3—124no—
141C2×C32×C4—34no—
142C2×C32×C4—32no—
143C2×C32×C4—34no—
144C2×C32×C4—64no—
145C2×C3⋊Dic3—24no—
146C2×C3⋊Dic3C4×D314yes—
147C2×C3⋊Dic3C22×D314yes—
148C2×C3⋊Dic3C22×D312yes—
149C4×C3⋊D3—316no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—38no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—44no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—124no—
166C2×C3×Dic3C4×D318yes—
167C2×C3×Dic3C22×D314yes—
168C2×C3×Dic3C22×D314yes—
169C2×C3×Dic3C22×D314yes—
170#624—332no—
171#624—616no—
172C2×#117—271noSylow
173C2×C33#11714yescomplemented
174C2×C33C23×C411yescharacteristic
175C2×C33C2×#4111yescharacteristic
176C2×C33C2×#4111yescharacteristic
177C3⋊(C3⋊D3)—24no—
178C23×D3—92no—
179C23×D3—91no—
180C23×D3—181no—
181C23×D3—182no—
182C23×D3—361no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—32no—
186C22×C3×C4—61no—
187C22×C3×C4—62no—
188C22×C3×C4—91no—
189C22×C3×C4—92no—
190C22×Dic3—32no—
191C22×Dic3—32no—
192C22×Dic3—32no—
193C22×Dic3—61no—
194C22×Dic3—62no—
195C22×Dic3D3212yes—
196C22×Dic3D3211yescharacteristic
197C2×C4×D3—98no—
198C2×C4×D3—916no—
199C2×C4×D3—98no—
200C2×C4×D3—94no—
201C2×C4×D3—184no—
202C2×C4×D3—188no—
203C3×C4⋊C4—94no—
204C3×C42—98no—
205(C2×D3)⋊C4—98no—
206(C2×D3)⋊C4—98no—
207(C2×D3)⋊C4—98no—
208(C2×D3)⋊C4—188no—
209C4⋊Dic3—94no—
210Dic3⋊C4—38no—
211Dic3⋊C4—98no—
212C4×Dic3—38no—
213C4×Dic3—38no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—24no—
219C22×C32—24no—
220C22×C32—24no—
221C22×C32—22no—
222C22×C32—42no—
223C22×C32—42no—
224C22×C32—42no—
225C22×C32—41no—
226C22×C32C2×C4×D312yes—
227C22×C32C2×C4×D314yes—
228C22×C32#23514yes—
229C22×C32#23514yes—
230C22×C32#23712yes—
231C22×C32#23912yes—
232C22×C32C23×D312yes—
233C22×C32C23×D311yescharacteristic
234C2×C3⋊D3—38no—
235C2×C3⋊D3—34no—
236C2×C3⋊D3—64no—
237C2×C3⋊D3—68no—
238C2×C3⋊D3—68no—
239C2×C3⋊D3—62no—
240C2×C3⋊D3—62no—
241C2×C3⋊D3—64no—
242C2×C3⋊D3—64no—
243C2×C3⋊D3—616no—
244C2×C3⋊D3—616no—
245C2×C3⋊D3—64no—
246C2×C3⋊D3—64no—
247C2×C3⋊D3—124no—
248C2×C3⋊D3—124no—
249C2×C3⋊D3—124no—
250C2×C3⋊D3—1216no—
251C2×C3⋊D3—1216no—
252C2×C3⋊D3—124no—
253C2×C3⋊D3—124no—
254C32×C4—34no—
255C32×C4—64no—
256C3⋊Dic3—24no—
257C3⋊Dic3C2×C4×D314yes—
258C3×Dic3—24no—
259C3×Dic3—24no—
260C3×Dic3—24no—
261C3×Dic3—38no—
262C3×Dic3—44no—
263C3×Dic3—64no—
264C3×Dic3—64no—
265C3×Dic3—64no—
266C3×Dic3—124no—
267C3×Dic3C2×C4×D318yes—
268C23×C4—271no—
269#117—278no—
270C2×C4⋊C4—91no—
271C2×C4⋊C4—271no—
272C2×C22⋊C4—272no—
273C2×C42—92no—
274C33C2×#11711yescharacteristic, complemented, Sylow
275C23×C3—21no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3C2×D3211yescharacteristic
279C23×C3C2×D3212yes—
280C22×D3—94no—
281C22×D3—92no—
282C22×D3—98no—
283C22×D3—94no—
284C22×D3—182no—
285C22×D3—184no—
286C22×D3—184no—
287C22×D3—188no—
288C22×D3—184no—
289C22×D3—184no—
290C22×D3—182no—
291C22×D3—182no—
292C22×D3—184no—
293C22×D3—184no—
294C22×D3—362no—
295C22×D3—364no—
296C22×D3—364no—
297C22×D3—364no—
298C22×D3—364no—
299C22×D3—364no—
300C2×C3×C4—34no—
301C2×C3×C4—38no—
302C2×C3×C4—34no—
303C2×C3×C4—34no—
304C2×C3×C4—62no—
305C2×C3×C4—64no—
306C2×C3×C4—64no—
307C2×C3×C4—64no—
308C2×C3×C4—64no—
309C2×C3×C4—92no—
310C2×C3×C4—94no—
311C2×C3×C4—94no—
312C2×C3×C4—124no—
313C2×C3×C4—184no—
314C2×Dic3—24no—
315C2×Dic3—34no—
316C2×Dic3—38no—
317C2×Dic3—34no—
318C2×Dic3—34no—
319C2×Dic3—64no—
320C2×Dic3—62no—
321C2×Dic3—64no—
322C2×Dic3—64no—
323C2×Dic3—64no—
324C2×Dic3—124no—
325C2×Dic3#62414yes—
326C2×Dic3C2×D3214yes—
327C2×Dic3C2×D3212yes—
328C4×D3—932no—
329C4×D3—916no—
330C4×D3—1816no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—28no—
338C2×C32—24no—
339C2×C32—44no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32#87218yescomplemented
344C2×C32#89214yescomplemented
345C2×C32C22×C4×D311yescharacteristic
346C2×C32C22×C4×D312yes—
347C2×C32C2×#23512yes—
348C2×C32C2×#23512yes—
349C2×C32C2×#23711yescharacteristic
350C2×C32C2×#23911yescharacteristic
351C3⋊D3—64no—
352C3⋊D3—68no—
353C3⋊D3—68no—
354C3⋊D3—128no—
355C3⋊D3—128no—
356C24—271no—
357C22×C4—31no—
358C22×C4—32no—
359C22×C4—91no—
360C22×C4—92no—
361C22×C4—274no—
362C22×C4—278no—
363C4⋊C4—94no—
364C4⋊C4—274no—
365C22⋊C4—278no—
366C42—98no—
367C22×C3—21no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—24no—
371C22×C3—24no—
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—22no—
375C22×C3—41no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3#379812yes—
380C22×C3#379812yes—
381C22×C3#379914yes—
382C22×C3C4×D3214yes—
383C22×C3#380414yes—
384C22×C3C2×#62412yes—
385C22×C3C22×D3211yescharacteristic
386C22×C3C22×D3212yes—
387C2×D3—98no—
388C2×D3—94no—
389C2×D3—184no—
390C2×D3—188no—
391C2×D3—184no—
392C2×D3—184no—
393C2×D3—1816no—
394C2×D3—182no—
395C2×D3—182no—
396C2×D3—184no—
397C2×D3—184no—
398C2×D3—1816no—
399C2×D3—188no—
400C2×D3—364no—
401C2×D3—364no—
402C2×D3—364no—
403C2×D3—3616no—
404C2×D3—364no—
405C2×D3—364no—
406C2×D3—3616no—
407C3×C4—38no—
408C3×C4—64no—
409C3×C4—64no—
410C3×C4—64no—
411C3×C4—94no—
412C3×C4—124no—
413C3×C4—184no—
414Dic3—24no—
415Dic3—38no—
416Dic3—64no—
417Dic3—64no—
418Dic3—64no—
419Dic3—124no—
420Dic3C2×#62414yes—
421C32—21no—
422C32—22no—
423C32—41no—
424C32C2×#87212yescomplemented
425C32C2×#89211yescharacteristic, complemented
426C23—272no—
427C23—274no—
428C23—542no—
429C23—542no—
430C23D3311yescharacteristic
431C2×C4—32no—
432C2×C4—34no—
433C2×C4—34no—
434C2×C4—64no—
435C2×C4—92no—
436C2×C4—94no—
437C2×C4—94no—
438C2×C4—184no—
439C2×C4—2716no—
440C2×C3—21no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—24no—
447C2×C3—28no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—44no—
452C2×C3#6383618yescomplemented
453C2×C3#6387414yescomplemented
454C2×C3C2×#379811yescharacteristic
455C2×C3C2×#379811yescharacteristic
456C2×C3C2×#379912yes—
457C2×C3C2×C4×D3212yes—
458C2×C3C2×#380412yes—
459C2×C3C22×#62411yescharacteristic
460D3—188no—
461D3—184no—
462D3—188no—
463D3—368no—
464D3—368no—
465C22—274no—
466C22—542no—
467C22—542no—
468C22—548no—
469C22D3×#62412yes—
470C22#8895912yes—
471C22#8896512yes—
472C22C2×D3311yescharacteristic
473C4—34no—
474C4—64no—
475C4—94no—
476C4—184no—
477C3—21no—
478C3—22no—
479C3—41no—
480C3C2×#6383612yescomplemented
481C3C2×#6387411yescharacteristic, complemented
482C2—544no—
483C2#1172291814yesdirect product factor
484C2C2×D3×#62411yescharacteristic
485C2C2×#8895911yescharacteristic
486C2C2×#8896511yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
cosocleC22×C33C24
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88815C2semidirect
4. C2×C3×#3724C2semidirect
5. C2×C3×Dic32C2semidirect
6. #11722918C2direct (N × H)
14. C2×#6289C4semidirect
16. #88815C22semidirect
18. C3×#3724C22semidirect
19. C3×Dic32C22semidirect
28. C2×#3724D3semidirect
33. C2×Dic32D3semidirect
49. #6289C2×C4semidirect
82. #3724C2×D3semidirect
87. Dic32C2×D3semidirect
173. C2×C33#117semidirect
274. C33C2×#117semidirect
343. C2×C32#872semidirect
344. C2×C32#892semidirect
424. C32C2×#872semidirect
425. C32C2×#892semidirect
452. C2×C3#63836semidirect
453. C2×C3#63874semidirect
480. C3C2×#63836semidirect
481. C3C2×#63874semidirect
483. C2#11722918direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#888152
C2×#888042
C2×C3×#37242
C2×C3×Dic322
#117229182
C2×#638743
C2×#638363