Finite Group Databasebeta

← C22⋊C9×#235 #13164606

Subgroups — C22⋊C9×#235 #13164606

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
1C2×C3×C4×C22⋊C9C211yesmaximal, characteristic, complemented
2#67×C22⋊C9C212yesmaximal, complemented
3D12×C22⋊C9C211yesmaximal, characteristic, complemented
4C4×D3×C22⋊C9C212yesmaximal, complemented
5Dic6×C22⋊C9C211yesmaximal, characteristic, complemented
6C22×C3×#235C311yesmaximal, characteristic
7#41×C22⋊C9—31nomaximal
8C22×C3×C22⋊C9C2211yescharacteristic
9C2×D3×C22⋊C9C2212yes—
10C9×#235—41nomaximal
11C3×C4×C22⋊C9C2211yescharacteristic
12C3×C4×C22⋊C9C2211yescharacteristic
13Dic3×C22⋊C9C2212yes—
14C23×C32×C4C2×C311yescharacteristic
15C22×C3×#67C2×C312yes—
16C2×C3×#235—31no—
17C2×C3×#235—31no—
18C2×C3×#235—32no—
19C2×C3×#235—31no—
20C2×C3×#235—31no—
21C2×C3×#235—32no—
22C22×C3×D12C2×C311yescharacteristic
23C22×C3×C4×D3C2×C312yes—
24C22×C3×Dic6C2×C311yescharacteristic
25Q8×C22⋊C9—31no—
26D4×C22⋊C9—32no—
27D4×C22⋊C9—31no—
28C2×C4×C22⋊C9—21no—
29C2×C4×C22⋊C9—32no—
30C2×C4×C22⋊C9D311yescharacteristic, complemented
31C2×C3×C22⋊C9—21no—
32C2×C3×C22⋊C9C2311yescharacteristic
33D3×C22⋊C9—22no—
34C2×C3×C4×C9—41no—
35C9×#67—42no—
36C9×D12—41no—
37C4×D3×C9—42no—
38C9×Dic6—41no—
39C22×C3×#41—31no—
40C22×#235C911yescharacteristic, complemented
41C24×C32C22×C311yescharacteristic
42C23×C3×D3C22×C312yes—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C22×C32×C4—31no—
46C22×C32×C4—31no—
47C22×C32×C4C22×C311yescharacteristic
48C22×C32×C4C22×C311yescharacteristic
49C2×C3×#67—32no—
50C2×C3×#67—32no—
51C2×C3×#67—32no—
52C2×C3×#67—32no—
53C22×C3×Dic3C22×C312yes—
54C3×#235—32no—
55C3×#235—34no—
56C3×#235—32no—
57C3×#235—32no—
58C3×#235—34no—
59C3×#235—31no—
60C3×#235—32no—
61C3×#235—31no—
62C3×#235—32no—
63C3×#235—31no—
64C3×#235A411yescharacteristic
65C2×C3×D12—31no—
66C2×C3×D12—32no—
67C2×C3×D12—31no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—32no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—32no—
75C22×C22⋊C9—21no—
76C22×C22⋊C9—32no—
77C22×C22⋊C9C2×D311yescharacteristic
78C9×#41—121no—
79C4×C22⋊C9—21no—
80C4×C22⋊C9—21no—
81C4×C22⋊C9—32no—
82C4×C22⋊C9C2×D311yescharacteristic
83C4×C22⋊C9C2×D311yescharacteristic
84C22×C3×C9—41no—
85C2×D3×C9—42no—
86C3×C22⋊C9#4111yescharacteristic, complemented
87C3×C4×C9—41no—
88C3×C4×C9—41no—
89C9×Dic3—42no—
90C2×C3×#41—91no—
91C2×C3×#41—91no—
92C2×C3×#41—92no—
93C2×C3×#41—91no—
94C2×C3×#41—92no—
95C2×C3×#41—91no—
96C22×C3×Q8—31no—
97C22×C3×D4—32no—
98C22×C3×D4—31no—
99C23×C3×C4—21no—
100C23×C3×C4—32no—
101C23×C3×C4C2×C911yescharacteristic, complemented
102C23×C3×C4C3×D311yescharacteristic
103C22×#67C2×C912yescomplemented
104C2×#235—31no—
105C2×#235—31no—
106C2×#235—32no—
107C2×#235—31no—
108C2×#235—31no—
109C2×#235—32no—
110C22×D12C2×C911yescharacteristic, complemented
111C22×C4×D3C2×C912yescomplemented
112C22×Dic6C2×C911yescharacteristic, complemented
113C23×C32—21no—
114C23×C32—31no—
115C23×C32—31no—
116C23×C32—61no—
117C23×C32C23×C311yescharacteristic
118C22×C3×D3—22no—
119C22×C3×D3—32no—
120C22×C3×D3—32no—
121C22×C3×D3—62no—
122C2×C32×C4—32no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—31no—
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—31no—
130C2×C32×C4C2×A411yescharacteristic
131C3×#67—34no—
132C3×#67—32no—
133C3×#67—32no—
134C3×#67—32no—
135C3×#67C2×A412yes—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—32no—
138C3×D12—32no—
139C3×D12—32no—
140C3×D12—31no—
141C3×D12C2×A411yescharacteristic
142C3×C4×D3—34no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×C4×D3—32no—
146C3×C4×D3C2×A412yes—
147C3×Dic6—32no—
148C3×Dic6—31no—
149C3×Dic6—32no—
150C3×Dic6C2×A411yescharacteristic
151C2×C22⋊C9—22no—
152C2×C22⋊C9—21no—
153C2×C22⋊C9—21no—
154C2×C22⋊C9—62no—
155C2×C22⋊C9C22×D311yescharacteristic
156Q8×C9—121no—
157D4×C9—121no—
158D4×C9—122no—
159C2×C4×C9—41no—
160C2×C4×C9—81no—
161C2×C4×C9—122no—
162C22×#41—31noSylow
163C2×C3×C9—41no—
164C2×C3×C9—81no—
165D3×C9—82no—
166C24×C3—21no—
167C24×C3—32no—
168C24×C3C22×C911yescharacteristic
169C24×C3C2×C3×D311yescharacteristic
170C23×D3C22×C912yes—
171C3×#41—31no—
172C3×#41—92no—
173C3×#41—92no—
174C3×#41—91no—
175C3×#41—91no—
176C3×#41—94no—
177C3×#41—94no—
178C3×#41—92no—
179C3×#41—92no—
180C3×#41—92no—
181C3×#41—91no—
182C2×C3×Q8—91no—
183C2×C3×Q8—91no—
184C2×C3×Q8—92no—
185C2×C3×D4—91no—
186C2×C3×D4—91no—
187C2×C3×D4—92no—
188C2×C3×D4—92no—
189C2×C3×D4—92no—
190C2×C3×D4—92no—
191C2×C3×D4—92no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—32no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4—61no—
207C22×C3×C4—92no—
208C22×C3×C4—92no—
209C22×C3×C4—92no—
210C22×C3×C4—92no—
211C22×C3×C4C22×C911yescharacteristic
212C22×C3×C4C22×C911yescharacteristic
213C22×C3×C4C2×C3×D311yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C2×#67—32no—
216C2×#67—32no—
217C2×#67—32no—
218C2×#67—32no—
219C22×Dic3C22×C912yes—
220#235—32no—
221#235—34no—
222#235—32no—
223#235—32no—
224#235—34no—
225#235—31no—
226#235—32no—
227#235—31no—
228#235—32no—
229#235—31no—
230#235C22⋊C911yescharacteristic, direct product factor
231C2×D12—31no—
232C2×D12—32no—
233C2×D12—31no—
234C2×C4×D3—32no—
235C2×C4×D3—32no—
236C2×C4×D3—32no—
237C2×C4×D3—32no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—32no—
241C22×C32—31no—
242C22×C32—31no—
243C22×C32—31no—
244C22×C32—61no—
245C22×C32—61no—
246C22×C32—61no—
247C22×C32—61no—
248C22×C32C3×#4111yescharacteristic
249C22×C32C22×A411yescharacteristic
250C2×C3×D3—32no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3C22×A412yes—
256C32×C4—31no—
257C32×C4—31no—
258C32×C4C22×A411yescharacteristic
259C32×C4C22×A411yescharacteristic
260C3×Dic3—32no—
261C3×Dic3C22×A412yes—
262C22×C9—41no—
263C22×C9—81no—
264C22×C9—122no—
265C22⋊C9—21no—
266C22⋊C9#23511yescharacteristic, direct product factor
267C4×C9—41no—
268C4×C9—41no—
269C4×C9—81no—
270C4×C9—81no—
271C4×C9—122no—
272C2×#41—91no—
273C2×#41—91no—
274C2×#41—92no—
275C2×#41—91no—
276C2×#41—92no—
277C2×#41—91no—
278C22×Q8—31no—
279C22×D4—32no—
280C22×D4—31no—
281C23×C4—32no—
282C23×C4D3×C911yescharacteristic, complemented
283C3×C9—41noSylow
284C23×C3—22no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—31no—
289C23×C3—31no—
290C23×C3—31no—
291C23×C3—31no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—92no—
299C23×C3—92no—
300C23×C3—182no—
301C23×C3C23×C911yescharacteristic
302C23×C3C22×C3×D311yescharacteristic
303C22×D3—22no—
304C22×D3—32no—
305C22×D3—32no—
306C22×D3—62no—
307C3×Q8—31no—
308C3×Q8—92no—
309C3×Q8—92no—
310C3×Q8—91no—
311C3×D4—32no—
312C3×D4—31no—
313C3×D4—92no—
314C3×D4—92no—
315C3×D4—92no—
316C3×D4—92no—
317C3×D4—94no—
318C3×D4—92no—
319C3×D4—91no—
320C2×C3×C4—21no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—32no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—92no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—94no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4C2×C22⋊C911yescharacteristic, complemented
353C2×C3×C4D3×A411yescharacteristic
354#67—34no—
355#67—32no—
356#67—32no—
357#67—32no—
358#67C2×C22⋊C912yescomplemented
359C2×Dic3—32no—
360C2×Dic3—32no—
361D12—32no—
362D12—32no—
363D12—31no—
364D12C2×C22⋊C911yescharacteristic, complemented
365C4×D3—34no—
366C4×D3—32no—
367C4×D3—32no—
368C4×D3—32no—
369C4×D3C2×C22⋊C912yescomplemented
370Dic6—32no—
371Dic6—31no—
372Dic6—32no—
373Dic6C2×C22⋊C911yescharacteristic, complemented
374C2×C32—21no—
375C2×C32—31no—
376C2×C32—31no—
377C2×C32—61no—
378C2×C32C23×A411yescharacteristic
379C3×D3—22no—
380C3×D3—62no—
381C2×C9—41no—
382C2×C9—81no—
383C2×C9—82no—
384C2×C9—81no—
385C2×C9—242no—
386C24—32no—
387C24C2×D3×C911yescharacteristic
388#41—31no—
389#41—92no—
390#41—92no—
391#41—91no—
392#41—91no—
393#41—94no—
394#41—94no—
395#41—92no—
396#41—92no—
397#41—92no—
398#41—91no—
399C2×Q8—91no—
400C2×Q8—91no—
401C2×Q8—92no—
402C2×D4—91no—
403C2×D4—91no—
404C2×D4—92no—
405C2×D4—92no—
406C2×D4—92no—
407C2×D4—92no—
408C2×D4—92no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—92no—
415C22×C4—92no—
416C22×C4—92no—
417C22×C4—92no—
418C22×C4C2×D3×C911yescharacteristic
419C22×C4C2×D3×C911yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—32no—
428C22×C3—31no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—61no—
433C22×C3—61no—
434C22×C3—61no—
435C22×C3—61no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3—61no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—92no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3C9×#4111yescharacteristic, complemented
450C22×C3C22×C22⋊C911yescharacteristic
451C22×C3C3×#23511yescharacteristic
452C22×C3C2×D3×A411yescharacteristic
453C2×D3—32no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3C22×C22⋊C912yes—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—31no—
462C3×C4—31no—
463C3×C4—32no—
464C3×C4—31no—
465C3×C4—31no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—92no—
469C3×C4C22×C22⋊C911yescharacteristic
470C3×C4C22×C22⋊C911yescharacteristic
471C3×C4C2×D3×A411yescharacteristic
472C3×C4C2×D3×A411yescharacteristic
473Dic3—32no—
474Dic3C22×C22⋊C912yes—
475C32A4×#4111yescharacteristic
476C9—41no—
477C9—81no—
478C23—21no—
479C23—31no—
480C23—31no—
481C23—61no—
482C23—62no—
483C23—92no—
484C23—92no—
485C23—182no—
486C23C22×D3×C911yescharacteristic
487Q8—31no—
488Q8—92no—
489Q8—92no—
490Q8—91no—
491D4—32no—
492D4—31no—
493D4—92no—
494D4—92no—
495D4—92no—
496D4—92no—
497D4—94no—
498D4—92no—
499D4—91no—
500C2×C4—32no—
501C2×C4—31no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—32no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—92no—
512C2×C4—94no—
513C2×C4—92no—
514C2×C4—92no—
515C2×C4D3×C22⋊C911yescharacteristic, complemented
516C2×C3—22no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—62no—
527C2×C3—61no—
528C2×C3—62no—
529C2×C3—61no—
530C2×C3—182no—
531C2×C3C23×C22⋊C911yescharacteristic
532C2×C3C22×D3×A411yescharacteristic
533D3—22no—
534D3—62no—
535C22—31no—
536C22—31no—
537C22—31no—
538C22—32no—
539C22—61no—
540C22—61no—
541C22—61no—
542C22—61no—
543C22—92no—
544C22—182no—
545C22—182no—
546C22—182no—
547C22—182no—
548C22C9×#23511yescharacteristic, complemented
549C22C2×D3×C22⋊C911yescharacteristic
550C4—31no—
551C4—31no—
552C4—32no—
553C4—92no—
554C4C2×D3×C22⋊C911yescharacteristic
555C4C2×D3×C22⋊C911yescharacteristic
556C3—21no—
557C3#41×C22⋊C911yescharacteristic, complemented
558C3A4×#23511yescharacteristic
559C2—21no—
560C2—31no—
561C2—31no—
562C2—61no—
563C2—62no—
564C2—182no—
565C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC3×C4C2×D3×A4
commutatorC23×C3C23×C9
FrattiniC2×C3C22×D3×A4
FittingC23×C32×C4C2×C3
socle, cosocleC23×C32C23×C3
prime radicalC22×C3×#235C3
hypercenterC2×C3×C4D3×A4
hypocenterC22×C3C9×#41
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C22⋊C9C2semidirect
2. #67×C22⋊C9C2semidirect
3. D12×C22⋊C9C2semidirect
4. C4×D3×C22⋊C9C2semidirect
5. Dic6×C22⋊C9C2semidirect
30. C2×C4×C22⋊C9D3semidirect
40. C22×#235C9semidirect
86. C3×C22⋊C9#41semidirect
101. C23×C3×C4C2×C9semidirect
103. C22×#67C2×C9semidirect
110. C22×D12C2×C9semidirect
111. C22×C4×D3C2×C9semidirect
112. C22×Dic6C2×C9semidirect
230. #235C22⋊C9direct (N × H)
266. C22⋊C9#235direct (N × H)
282. C23×C4D3×C9semidirect
352. C2×C3×C4C2×C22⋊C9semidirect
358. #67C2×C22⋊C9semidirect
364. D12C2×C22⋊C9semidirect
369. C4×D3C2×C22⋊C9semidirect
373. Dic6C2×C22⋊C9semidirect
449. C22×C3C9×#41semidirect
515. C2×C4D3×C22⋊C9semidirect
548. C22C9×#235semidirect
557. C3#41×C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
#67×C22⋊C92
D12×C22⋊C92
C4×D3×C22⋊C92
Dic6×C22⋊C92
C22×C3×#2353
#41×C22⋊C93
C9×#2354