Finite Group Databasebeta

← C22⋊C9×#237 #13164612

Subgroups — C22⋊C9×#237 #13164612

Order 1728.

846 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×D4×C22⋊C9C211yesmaximal, characteristic, complemented
2#67×C22⋊C9C212yesmaximal, complemented
3C2×Dic3×C22⋊C9C212yesmaximal, complemented
4C4×D3×C22⋊C9C211yesmaximal, characteristic, complemented
5Dic6×C22⋊C9C211yesmaximal, characteristic, complemented
6C22×C3×#237C311yesmaximal, characteristic
7#41×C22⋊C9—31nomaximal
8C22×C3×C22⋊C9C2212yes—
9C2×D3×C22⋊C9C2211yescharacteristic
10C9×#237—41nomaximal
11C3×C4×C22⋊C9C2211yescharacteristic
12Dic3×C22⋊C9C2212yes—
13Dic3×C22⋊C9C2211yescharacteristic
14C22×C32×D4C2×C311yescharacteristic
15C22×C3×#67C2×C312yes—
16C23×C3×Dic3C2×C312yes—
17C2×C3×#237—31no—
18C2×C3×#237—31no—
19C2×C3×#237—31no—
20C2×C3×#237—31no—
21C2×C3×#237—32no—
22C2×C3×#237—32no—
23C22×C3×C4×D3C2×C311yescharacteristic
24C22×C3×Dic6C2×C311yescharacteristic
25Q8×C22⋊C9—31no—
26D4×C22⋊C9—21no—
27D4×C22⋊C9—32no—
28D4×C22⋊C9D311yescharacteristic, complemented
29C2×C4×C22⋊C9—32no—
30C2×C4×C22⋊C9—31no—
31C2×C3×C22⋊C9—22no—
32C2×C3×C22⋊C9C2311yescharacteristic
33D3×C22⋊C9—21no—
34C3×D4×C9—41no—
35C9×#67—42no—
36C2×C9×Dic3—42no—
37C4×D3×C9—41no—
38C9×Dic6—41no—
39C22×C3×#41—31no—
40C22×#237C911yescharacteristic, complemented
41C24×C32C22×C312yes—
42C23×C3×D3C22×C311yescharacteristic
43C2×C32×D4—31no—
44C2×C32×D4—31no—
45C2×C32×D4—32no—
46C22×C32×C4C22×C311yescharacteristic
47C2×C3×#67—32no—
48C2×C3×#67—32no—
49C2×C3×#67—32no—
50C2×C3×#67—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3C22×C312yes—
56C22×C3×Dic3C22×C311yescharacteristic
57C3×#237—34no—
58C3×#237—32no—
59C3×#237—32no—
60C3×#237—32no—
61C3×#237—34no—
62C3×#237—31no—
63C3×#237—32no—
64C3×#237—32no—
65C3×#237—31no—
66C3×#237—31no—
67C3×#237A411yescharacteristic
68C2×C3×C4×D3—31no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—32no—
75C22×C22⋊C9—22no—
76C22×C22⋊C9—31no—
77C22×C22⋊C9C2×D312yes—
78C9×#41—121no—
79C4×C22⋊C9—21no—
80C4×C22⋊C9—31no—
81C4×C22⋊C9—32no—
82C4×C22⋊C9C2×D311yescharacteristic
83C22×C3×C9—42no—
84C2×D3×C9—41no—
85C3×C22⋊C9#4111yescharacteristic, complemented
86C3×C4×C9—41no—
87C9×Dic3—42no—
88C9×Dic3—41no—
89C2×C3×#41—91no—
90C2×C3×#41—91no—
91C2×C3×#41—92no—
92C2×C3×#41—91no—
93C2×C3×#41—92no—
94C2×C3×#41—91no—
95C22×C3×Q8—31no—
96C22×C3×D4—21no—
97C22×C3×D4—32no—
98C22×C3×D4C2×C911yescharacteristic, complemented
99C22×C3×D4C3×D311yescharacteristic
100C23×C3×C4—32no—
101C23×C3×C4—31no—
102C22×#67C2×C912yescomplemented
103C23×Dic3C2×C912yescomplemented
104C2×#237—31no—
105C2×#237—31no—
106C2×#237—31no—
107C2×#237—31no—
108C2×#237—32no—
109C2×#237—32no—
110C22×C4×D3C2×C911yescharacteristic, complemented
111C22×Dic6C2×C911yescharacteristic, complemented
112C23×C32—22no—
113C23×C32—32no—
114C23×C32—32no—
115C23×C32—62no—
116C23×C32C23×C311yescharacteristic
117C22×C3×D3—21no—
118C22×C3×D3—31no—
119C22×C3×D3—31no—
120C22×C3×D3—61no—
121C32×D4—32no—
122C32×D4—32no—
123C32×D4—31no—
124C32×D4C2×A411yescharacteristic
125C2×C32×C4—31no—
126C2×C32×C4—31no—
127C3×#67—34no—
128C3×#67—32no—
129C3×#67—32no—
130C3×#67—32no—
131C3×#67C2×A412yes—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3C2×A412yes—
141C3×C4×D3—32no—
142C3×C4×D3—31no—
143C3×C4×D3—31no—
144C3×C4×D3—31no—
145C3×C4×D3C2×A411yescharacteristic
146C3×Dic6—32no—
147C3×Dic6—31no—
148C3×Dic6—32no—
149C3×Dic6C2×A411yescharacteristic
150C2×C22⋊C9—22no—
151C2×C22⋊C9—21no—
152C2×C22⋊C9—42no—
153C2×C22⋊C9—61no—
154C2×C22⋊C9C22×D311yescharacteristic
155Q8×C9—121no—
156D4×C9—41no—
157D4×C9—81no—
158D4×C9—122no—
159C2×C4×C9—121no—
160C2×C4×C9—122no—
161C22×#41—31noSylow
162C2×C3×C9—41no—
163C2×C3×C9—82no—
164D3×C9—81no—
165C24×C3—22no—
166C24×C3—31no—
167C24×C3C22×C912yes—
168C24×C3C2×C3×D312yes—
169C23×D3C22×C911yescharacteristic
170C3×#41—31no—
171C3×#41—92no—
172C3×#41—91no—
173C3×#41—92no—
174C3×#41—91no—
175C3×#41—92no—
176C3×#41—92no—
177C3×#41—94no—
178C3×#41—94no—
179C3×#41—92no—
180C3×#41—91no—
181C2×C3×Q8—91no—
182C2×C3×Q8—91no—
183C2×C3×Q8—92no—
184C2×C3×D4—31no—
185C2×C3×D4—31no—
186C2×C3×D4—31no—
187C2×C3×D4—32no—
188C2×C3×D4—31no—
189C2×C3×D4—32no—
190C2×C3×D4—61no—
191C2×C3×D4—61no—
192C2×C3×D4—62no—
193C2×C3×D4—92no—
194C2×C3×D4—92no—
195C2×C3×D4—92no—
196C2×C3×D4—92no—
197C22×C3×C4—21no—
198C22×C3×C4—31no—
199C22×C3×C4—32no—
200C22×C3×C4—91no—
201C22×C3×C4—92no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—92no—
206C22×C3×C4—92no—
207C22×C3×C4—92no—
208C22×C3×C4C22×C911yescharacteristic
209C22×C3×C4C2×C3×D311yescharacteristic
210C2×#67—32no—
211C2×#67—32no—
212C2×#67—32no—
213C2×#67—32no—
214C22×Dic3—32no—
215C22×Dic3—32no—
216C22×Dic3—32no—
217C22×Dic3—32no—
218C22×Dic3C22×C912yes—
219C22×Dic3C22×C911yescharacteristic
220#237—34no—
221#237—32no—
222#237—32no—
223#237—32no—
224#237—34no—
225#237—31no—
226#237—32no—
227#237—32no—
228#237—31no—
229#237—31no—
230#237C22⋊C911yescharacteristic, direct product factor
231C2×C4×D3—31no—
232C2×C4×D3—31no—
233C2×C4×D3—31no—
234C2×C4×D3—31no—
235C2×Dic6—31no—
236C2×Dic6—31no—
237C2×Dic6—32no—
238C22×C32—31no—
239C22×C32—31no—
240C22×C32—32no—
241C22×C32—62no—
242C22×C32—62no—
243C22×C32—62no—
244C22×C32—62no—
245C22×C32C3×#4111yescharacteristic
246C22×C32C22×A412yes—
247C2×C3×D3—31no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—61no—
251C2×C3×D3—61no—
252C2×C3×D3C22×A411yescharacteristic
253C32×C4—31no—
254C32×C4C22×A411yescharacteristic
255C3×Dic3—32no—
256C3×Dic3—31no—
257C3×Dic3C22×A412yes—
258C3×Dic3C22×A411yescharacteristic
259C22×C9—42no—
260C22×C9—82no—
261C22×C9—121no—
262C22⋊C9—21no—
263C22⋊C9#23711yescharacteristic, direct product factor
264C4×C9—41no—
265C4×C9—81no—
266C4×C9—121no—
267C4×C9—122no—
268C2×#41—91no—
269C2×#41—91no—
270C2×#41—92no—
271C2×#41—91no—
272C2×#41—92no—
273C2×#41—91no—
274C22×Q8—31no—
275C22×D4—32no—
276C22×D4D3×C911yescharacteristic, complemented
277C23×C4—32no—
278C23×C4—31no—
279C3×C9—41noSylow
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—21no—
283C23×C3—32no—
284C23×C3—32no—
285C23×C3—32no—
286C23×C3—32no—
287C23×C3—42no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—61no—
293C23×C3—91no—
294C23×C3—91no—
295C23×C3—122no—
296C23×C3—181no—
297C23×C3C23×C911yescharacteristic
298C23×C3C22×C3×D311yescharacteristic
299C22×D3—21no—
300C22×D3—31no—
301C22×D3—31no—
302C22×D3—61no—
303C3×Q8—31no—
304C3×Q8—92no—
305C3×Q8—92no—
306C3×Q8—91no—
307C3×D4—21no—
308C3×D4—32no—
309C3×D4—32no—
310C3×D4—32no—
311C3×D4—32no—
312C3×D4—32no—
313C3×D4—31no—
314C3×D4—31no—
315C3×D4—61no—
316C3×D4—62no—
317C3×D4—62no—
318C3×D4—92no—
319C3×D4—92no—
320C3×D4—94no—
321C3×D4—92no—
322C3×D4C2×C22⋊C911yescharacteristic, complemented
323C3×D4D3×A411yescharacteristic
324C2×C3×C4—32no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—94no—
340C2×C3×C4—92no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4—92no—
344#67—34no—
345#67—32no—
346#67—32no—
347#67—32no—
348#67C2×C22⋊C912yescomplemented
349C2×Dic3—34no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—31no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—31no—
357C2×Dic3C2×C22⋊C912yescomplemented
358C4×D3—32no—
359C4×D3—31no—
360C4×D3—31no—
361C4×D3—31no—
362C4×D3C2×C22⋊C911yescharacteristic, complemented
363Dic6—32no—
364Dic6—31no—
365Dic6—32no—
366Dic6C2×C22⋊C911yescharacteristic, complemented
367C2×C32—22no—
368C2×C32—31no—
369C2×C32—31no—
370C2×C32—62no—
371C2×C32C23×A411yescharacteristic
372C3×D3—21no—
373C3×D3—61no—
374C2×C9—41no—
375C2×C9—82no—
376C2×C9—81no—
377C2×C9—162no—
378C2×C9—241no—
379C24—31no—
380C24C2×D3×C912yes—
381#41—31no—
382#41—92no—
383#41—91no—
384#41—92no—
385#41—91no—
386#41—92no—
387#41—92no—
388#41—94no—
389#41—94no—
390#41—92no—
391#41—91no—
392C2×Q8—91no—
393C2×Q8—91no—
394C2×Q8—92no—
395C2×D4—31no—
396C2×D4—31no—
397C2×D4—32no—
398C2×D4—92no—
399C2×D4—92no—
400C2×D4—92no—
401C2×D4—92no—
402C22×C4—31no—
403C22×C4—32no—
404C22×C4—91no—
405C22×C4—92no—
406C22×C4—91no—
407C22×C4—91no—
408C22×C4—91no—
409C22×C4—92no—
410C22×C4—92no—
411C22×C4—92no—
412C22×C4C2×D3×C911yescharacteristic
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—31no—
416C22×C3—31no—
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—31no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—61no—
423C22×C3—62no—
424C22×C3—61no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—91no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3C9×#4111yescharacteristic, complemented
443C22×C3C22×C22⋊C912yes—
444C22×C3C3×#23711yescharacteristic
445C22×C3C2×D3×A412yes—
446C2×D3—31no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3C22×C22⋊C911yescharacteristic
452C3×C4—21no—
453C3×C4—31no—
454C3×C4—31no—
455C3×C4—32no—
456C3×C4—31no—
457C3×C4—61no—
458C3×C4—91no—
459C3×C4—92no—
460C3×C4C22×C22⋊C911yescharacteristic
461C3×C4C2×D3×A411yescharacteristic
462Dic3—32no—
463Dic3—31no—
464Dic3C22×C22⋊C912yes—
465Dic3C22×C22⋊C911yescharacteristic
466C32A4×#4111yescharacteristic
467C9—41no—
468C9—81no—
469C23—22no—
470C23—32no—
471C23—32no—
472C23—62no—
473C23—61no—
474C23—91no—
475C23—91no—
476C23—181no—
477C23C22×D3×C911yescharacteristic
478Q8—31no—
479Q8—92no—
480Q8—92no—
481Q8—91no—
482D4—32no—
483D4—32no—
484D4—32no—
485D4—31no—
486D4—92no—
487D4—92no—
488D4—94no—
489D4—92no—
490D4D3×C22⋊C911yescharacteristic, complemented
491C2×C4—32no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—91no—
496C2×C4—91no—
497C2×C4—92no—
498C2×C4—91no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—92no—
502C2×C4—94no—
503C2×C4—92no—
504C2×C4—91no—
505C2×C4—91no—
506C2×C4—92no—
507C2×C3—22no—
508C2×C3—22no—
509C2×C3—21no—
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—42no—
515C2×C3—61no—
516C2×C3—61no—
517C2×C3—62no—
518C2×C3—61no—
519C2×C3—62no—
520C2×C3—122no—
521C2×C3—181no—
522C2×C3C23×C22⋊C911yescharacteristic
523C2×C3C22×D3×A411yescharacteristic
524D3—21no—
525D3—61no—
526C22—31no—
527C22—31no—
528C22—31no—
529C22—32no—
530C22—62no—
531C22—62no—
532C22—62no—
533C22—62no—
534C22—91no—
535C22—181no—
536C22—181no—
537C22—181no—
538C22—181no—
539C22C9×#23711yescharacteristic, complemented
540C22C2×D3×C22⋊C912yes—
541C4—31no—
542C4—31no—
543C4—32no—
544C4—91no—
545C4—92no—
546C4C2×D3×C22⋊C911yescharacteristic
547C3—21no—
548C3#41×C22⋊C911yescharacteristic, complemented
549C3A4×#23711yescharacteristic
550C2—22no—
551C2—31no—
552C2—31no—
553C2—62no—
554C2—61no—
555C2—181no—
556C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×D3×A4
commutatorC23×C3C23×C9
FittingC22×C32×D4C2×C3
socle, cosocleC23×C32C23×C3
prime radicalC22×C3×#237C3
hypercenterC3×D4D3×A4
hypocenterC22×C3C9×#41
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×C22⋊C9C2semidirect
2. #67×C22⋊C9C2semidirect
3. C2×Dic3×C22⋊C9C2semidirect
4. C4×D3×C22⋊C9C2semidirect
5. Dic6×C22⋊C9C2semidirect
28. D4×C22⋊C9D3semidirect
40. C22×#237C9semidirect
85. C3×C22⋊C9#41semidirect
98. C22×C3×D4C2×C9semidirect
102. C22×#67C2×C9semidirect
103. C23×Dic3C2×C9semidirect
110. C22×C4×D3C2×C9semidirect
111. C22×Dic6C2×C9semidirect
230. #237C22⋊C9direct (N × H)
263. C22⋊C9#237direct (N × H)
276. C22×D4D3×C9semidirect
322. C3×D4C2×C22⋊C9semidirect
348. #67C2×C22⋊C9semidirect
357. C2×Dic3C2×C22⋊C9semidirect
362. C4×D3C2×C22⋊C9semidirect
366. Dic6C2×C22⋊C9semidirect
442. C22×C3C9×#41semidirect
490. D4D3×C22⋊C9semidirect
539. C22C9×#237semidirect
548. C3#41×C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C22⋊C92
#67×C22⋊C92
C2×Dic3×C22⋊C92
C4×D3×C22⋊C92
Dic6×C22⋊C92
C22×C3×#2373
#41×C22⋊C93
C9×#2374