Finite Group Databasebeta

← C2×#11721242 #13151978

Subgroups — C2×#11721242 #13151978

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3C2×C9×#217C211yesmaximal, characteristic, complemented
4C2×C3×#3676C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×#88445C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11721242C218yesmaximal, direct product factor
9C2×#63953—31nomaximal
10C2×#63447—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C23×C3⋊D9C2211yescharacteristic
13C2×#6233—24no—
14C2×#6233C2214yes—
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6189—24no—
21C2×#6189C2214yescomplemented
22C9×#217C2214yescomplemented
23C3×#3676C2214yescomplemented
24#88445C2214yescomplemented
25#88445C2214yescomplemented
26Dic3×Dic9C2214yescomplemented
27C22×#638—31no—
28C22×#627—31no—
29C2×C3×#217—31no—
30C2×C3×#217D311yescharacteristic
31C2×#3722—31no—
32C2×#3722—31no—
33C2×Dic32—31no—
34#63953—38no—
35C22×#611—31no—
36C22×#611—61no—
37C22×Dic18—31no—
38C2×C9×C22⋊C4—31no—
39C2×#3676D311yescharacteristic, complemented
40C2×(C2×D9)⋊C4—31no—
41C2×(C2×D9)⋊C4—31no—
42C2×C4×Dic9—31no—
43#63447—38no—
44C23×C3×C9—22no—
45C23×C3×C9—22no—
46C23×C3×C9—22no—
47C23×C3×C9C2312yes—
48C23×C3×C9C2311yescharacteristic
49C22×C3⋊D9—22no—
50C22×C3⋊D9—22no—
51C22×C3⋊D9—22no—
52C22×C3⋊D9C2312yes—
53#6233—28no—
54C2×C3⋊Dic9D414yes—
55C2×C3⋊Dic9C2312yes—
56C2×C9×Dic3—22no—
57C2×C9×Dic3—22no—
58C2×C9×Dic3C2312yes—
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9—22no—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2312yes—
63C2×C3×Dic9C2312yes—
64#6189—28no—
65C2×#885—91no—
66C2×#885—91no—
67C24×C32C2×D311yescharacteristic
68C23×C3⋊D3—31no—
69C2×#638—34no—
70C2×#638—64no—
71C22×C3⋊Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#627—34no—
77C2×#627—64no—
78C24×C9—21no—
79C24×C9C2×D311yescharacteristic
80C23×D9—31no—
81C23×D9—61no—
82C3×#217—34no—
83C3×#217C2×D314yes—
84#3722—34no—
85#3722—34no—
86Dic32—34no—
87C22×C4×C9—31no—
88C22×C4×C9—31no—
89C2×#611—34no—
90C2×#611—64no—
91C2×#611—64no—
92C2×#611—68no—
93C22×Dic9—31no—
94C22×Dic9—61no—
95C22×Dic9C2×D311yescharacteristic
96C22×Dic9C2×D311yescharacteristic
97C2×Dic18—34no—
98C2×Dic18—64no—
99C9×C22⋊C4—34no—
100#3676C2×D314yescomplemented
101(C2×D9)⋊C4—34no—
102(C2×D9)⋊C4—34no—
103C4×Dic9—34no—
104C22×C3×C9—22no—
105C22×C3×C9—22no—
106C22×C3×C9—22no—
107C22×C3×C9—44no—
108C22×C3×C9C2×D412yes—
109C22×C3×C9#4112yes—
110C22×C3×C9#4112yes—
111C22×C3×C9C2411yescharacteristic
112C2×C3⋊D9—22no—
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9—44no—
116C3⋊Dic9C2×D414yes—
117C9×Dic3—22no—
118C9×Dic3—22no—
119C3×Dic9—22no—
120C3×Dic9—22no—
121C22×#67—91no—
122C22×#67—91no—
123C22×#67—181no—
124C22×Dic6—91no—
125C22×Dic6—91no—
126C2×C3×C22⋊C4—31no—
127C2×C3×C22⋊C4—91no—
128C2×#217—31no—
129C2×#217D911yescharacteristic, complemented
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×(C2×D3)⋊C4—91no—
133C2×(C2×D3)⋊C4—91no—
134C2×C4×Dic3—91no—
135C2×C4×Dic3—91no—
136#885—98no—
137#885—98no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32C22×D312yes—
142C23×C32C22×D311yescharacteristic
143C22×C3⋊D3—32no—
144C22×C3⋊D3—62no—
145C22×C3⋊D3—62no—
146C22×C3⋊D3—62no—
147#638—68no—
148C2×C3⋊Dic3—34no—
149C2×C3⋊Dic3—32no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3C22×D312yes—
157C2×C3×Dic3C22×D312yes—
158#627—68no—
159C23×C9—24no—
160C23×C9—22no—
161C23×C9—22no—
162C23×C9—22no—
163C23×C9—21no—
164C23×C9—22no—
165C23×C9—42no—
166C23×C9—42no—
167C23×C9C22×D312yes—
168C23×C9C22×D311yescharacteristic
169C22×D9—32no—
170C22×D9—62no—
171C22×D9—64no—
172C22×D9—62no—
173C22×D9—62no—
174C22×D9—62no—
175C22×D9—122no—
176C22×D9—122no—
177C2×C4×C9—32no—
178C2×C4×C9—32no—
179C2×C4×C9—62no—
180C2×C4×C9—62no—
181#611—68no—
182#611—616no—
183C2×Dic9—22no—
184C2×Dic9—22no—
185C2×Dic9—34no—
186C2×Dic9—32no—
187C2×Dic9—64no—
188C2×Dic9—62no—
189C2×Dic9C22×D312yes—
190C2×Dic9C22×D312yes—
191Dic18—68no—
192C2×#124—271noSylow
193C2×C3×C9—42no—
194C2×C3×C9#12414yescomplemented
195C2×C3×C9C22×D411yescharacteristic
196C2×C3×C9C2×#4111yescharacteristic
197C2×C3×C9C2×#4111yescharacteristic
198C3⋊D9—42no—
199C24×C3—21no—
200C24×C3C2×D911yescharacteristic
201C24×C3D3211yescharacteristic
202C23×D3—91no—
203C23×D3—91no—
204C23×D3—181no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C2×#67—94no—
210C2×#67—94no—
211C2×#67—184no—
212C2×#67—184no—
213C2×#67—184no—
214C2×#67—188no—
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—181no—
220C22×Dic3C2×D911yescharacteristic
221C22×Dic3C2×D911yescharacteristic
222C2×Dic6—94no—
223C2×Dic6—94no—
224C2×Dic6—184no—
225C2×Dic6—184no—
226C3×C22⋊C4—34no—
227C3×C22⋊C4—94no—
228#217—34no—
229#217C2×D914yescomplemented
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234C4×Dic3—94no—
235C4×Dic3—94no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—44no—
240C22×C32#23512yes—
241C22×C32D3×D412yes—
242C22×C32#23712yes—
243C22×C32C23×D311yescharacteristic
244C2×C3⋊D3—62no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3—62no—
247C2×C3⋊D3—124no—
248C3⋊Dic3—34no—
249C3×Dic3—22no—
250C3×Dic3—22no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C22×C9—24no—
254C22×C9—22no—
255C22×C9—22no—
256C22×C9—22no—
257C22×C9—21no—
258C22×C9—22no—
259C22×C9—22no—
260C22×C9—22no—
261C22×C9—42no—
262C22×C9—42no—
263C22×C9—48no—
264C22×C9—44no—
265C22×C9#23512yes—
266C22×C9D3×D412yes—
267C22×C9#23712yes—
268C22×C9C23×D311yescharacteristic
269C2×D9—64no—
270C2×D9—62no—
271C2×D9—62no—
272C2×D9—62no—
273C2×D9—122no—
274C2×D9—122no—
275C2×D9—128no—
276C2×D9—124no—
277C4×C9—62no—
278C4×C9—62no—
279Dic9—22no—
280Dic9—22no—
281Dic9—34no—
282Dic9—64no—
283C22×Q8—271no—
284C22×D4—271no—
285#124—278no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—91no—
288C2×C22⋊C4—271no—
289C2×C22⋊C4—271no—
290C2×C42—271no—
291C3×C9C2×#12411yescharacteristic, complemented, Sylow
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3C22×D912yes—
304C23×C3C22×D911yescharacteristic
305C23×C3C2×D3212yes—
306C23×C3C2×D3211yescharacteristic
307C22×D3—92no—
308C22×D3—92no—
309C22×D3—182no—
310C22×D3—184no—
311C22×D3—182no—
312C22×D3—182no—
313C22×D3—182no—
314C22×D3—182no—
315C22×D3—182no—
316C22×D3—182no—
317C22×D3—362no—
318C22×D3—362no—
319C2×C3×C4—32no—
320C2×C3×C4—32no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4—182no—
326C2×C3×C4—182no—
327#67—188no—
328#67—188no—
329#67—1816no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—92no—
337C2×Dic3—94no—
338C2×Dic3—92no—
339C2×Dic3—94no—
340C2×Dic3—182no—
341C2×Dic3—184no—
342C2×Dic3C22×D912yes—
343C2×Dic3C22×D912yes—
344Dic6—188no—
345Dic6—188no—
346C2×C32—42no—
347C2×C32#88514yes—
348C2×C32C2×#23511yescharacteristic
349C2×C32C2×D3×D411yescharacteristic
350C2×C32C2×#23711yescharacteristic
351C3⋊D3—122no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—24no—
356C2×C9—44no—
357C2×C9—42no—
358C2×C9#88514yescomplemented
359C2×C9C2×#23511yescharacteristic
360C2×C9C2×D3×D411yescharacteristic
361C2×C9C2×#23711yescharacteristic
362D9—124no—
363D9—122no—
364C24—271no—
365C24D3×D911yescharacteristic
366C2×Q8—274no—
367C2×Q8—544no—
368C2×D4—274no—
369C2×D4—544no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—271no—
375C22⋊C4—34no—
376C22⋊C4—94no—
377C22⋊C4—274no—
378C22⋊C4—274no—
379C42—274no—
380C22×C3—24no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—48no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3#369712yes—
397C22×C3D4×D912yes—
398C22×C3#369912yes—
399C22×C3C23×D911yescharacteristic
400C22×C3#380412yes—
401C22×C3#380412yes—
402C22×C3#381112yes—
403C22×C3C22×D3211yescharacteristic
404C2×D3—184no—
405C2×D3—182no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—182no—
411C2×D3—362no—
412C2×D3—362no—
413C2×D3—368no—
414C2×D3—364no—
415C2×D3—364no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—182no—
419C3×C4—182no—
420Dic3—22no—
421Dic3—22no—
422Dic3—62no—
423Dic3—62no—
424Dic3—94no—
425Dic3—94no—
426Dic3—184no—
427C32C2×#88511yescharacteristic
428C9—21no—
429C9C2×#88511yescharacteristic, complemented
430C23—22no—
431C23—22no—
432C23—22no—
433C23—272no—
434C23—542no—
435C23—542no—
436C23—542no—
437C23C2×D3×D912yes—
438C23C2×D3×D911yescharacteristic
439Q8—548no—
440D4—548no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—272no—
450C2×C4—274no—
451C2×C3—24no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—44no—
456C2×C3—42no—
457C2×C3—42no—
458C2×C3#6344714yescomplemented
459C2×C3C2×#369711yescharacteristic
460C2×C3C2×D4×D911yescharacteristic
461C2×C3C2×#369911yescharacteristic
462C2×C3#6395314yes—
463C2×C3C2×#380411yescharacteristic
464C2×C3C2×#380411yescharacteristic
465C2×C3C2×#381111yescharacteristic
466D3—364no—
467D3—362no—
468D3—362no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—44no—
473C22—542no—
474C22—542no—
475C22—542no—
476C22—1084no—
477C22#8865812yes—
478C22#8866212yes—
479C22#8866812yes—
480C22C22×D3×D911yescharacteristic
481C4—62no—
482C4—62no—
483C4—182no—
484C4—182no—
485C4—274no—
486C3—21no—
487C3C2×#6344711yescharacteristic, complemented
488C3C2×#6395311yescharacteristic
489C2—42no—
490C2—1082no—
491C2#1172124214yesdirect product factor
492C2C2×#8865811yescharacteristic
493C2C2×#8866211yescharacteristic
494C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×#6233C2
hypercenterC24D3×D9
hypocenterC3×C9C2×#124
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6189C2semidirect
3. C2×C9×#217C2semidirect
4. C2×C3×#3676C2semidirect
5. C2×#88445C2semidirect
6. C2×#88445C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11721242C2direct (N × H)
21. C2×#6189C22semidirect
22. C9×#217C22semidirect
23. C3×#3676C22semidirect
24. #88445C22semidirect
25. #88445C22semidirect
26. Dic3×Dic9C22semidirect
39. C2×#3676D3semidirect
100. #3676C2×D3semidirect
129. C2×#217D9semidirect
194. C2×C3×C9#124semidirect
229. #217C2×D9semidirect
291. C3×C9C2×#124semidirect
358. C2×C9#885semidirect
429. C9C2×#885semidirect
458. C2×C3#63447semidirect
487. C3C2×#63447semidirect
491. C2#11721242direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62332
C22×#61892
C2×C9×#2172
C2×C3×#36762
C2×#884452
C2×Dic3×Dic92
#117212422
C2×#639533
C2×#634473