Finite Group Databasebeta

← D3×2−1+4⋊C9 #13164637

Subgroups — D3×2−1+4⋊C9 #13164637

Order 1728.

776 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C3×2−1+4⋊C9C211yesmaximal, characteristic, complemented
2#1010⋊C9C211yesmaximal, characteristic, complemented
3D3×#41⋊C9C211yesmaximal, characteristic, complemented
4#1007⋊C9C212yesmaximal, complemented
5C2×D3×Q8⋊C9C212yesmaximal, complemented
6C3×D3×2−1+4C311yesmaximal, characteristic
7C2×2−1+4⋊C9—31nomaximal
8D3×D4×C9—41nomaximal
9C3×#41⋊C9C2211yescharacteristic, complemented
10C2×C3×Q8⋊C9C2212yescomplemented
11D3×Q8⋊C9C2211yescharacteristic
12D3×Q8⋊C9C2212yescomplemented
13C3⋊(#41⋊C9)C2211yescharacteristic, complemented
14C32×2−1+4C2×C311yescharacteristic
15C3×#1010—32no—
16C3×#1010—31no—
17C3×#1010C2×C311yescharacteristic
18C3×D3×#41—31no—
19C3×D3×#41—32no—
20C3×D3×#41C2×C311yescharacteristic
21C3×#1007—31no—
22C3×#1007C2×C312yes—
23C2×C3×D3×Q8—31no—
24C2×C3×D3×Q8C2×C312yes—
252−1+4⋊C9—21no—
262−1+4⋊C9—31no—
272−1+4⋊C9—32no—
282−1+4⋊C9D311yescharacteristic, direct product factor
29C2×#41⋊C9—31no—
30C22×Q8⋊C9—32no—
31C22×D3×C9—42no—
32C3×D4×C9—41no—
33C9×#67—42no—
34C9×D12—41no—
35C4×D3×C9—41no—
36C3×Q8⋊C9C2311yescharacteristic
37C2×C3×2−1+4—31no—
38D3×2−1+4C911yescharacteristic, complemented
39C32×#41—31no—
40C32×#41—32no—
41C32×#41C22×C311yescharacteristic
42C2×C32×Q8—31no—
43C2×C32×Q8C22×C312yes—
44C3×#239—31no—
45C3×#239—32no—
46C3×#239C22×C311yescharacteristic
47C3×D3×Q8—32no—
48C3×D3×Q8—31no—
49C3×D3×Q8—32no—
50C3×D3×Q8—32no—
51C3×D3×Q8—32no—
52C3×D3×Q8C22×C311yescharacteristic
53C3×D3×Q8C22×C312yes—
54C3×#237—34no—
55C3×#237—31no—
56C3×#237—32no—
57C3×#237—32no—
58C3×#237—31no—
59C3×D3×D4—32no—
60C3×D3×D4—31no—
61C3×D3×D4A411yescharacteristic
62C3×#235—34no—
63C3×#235—32no—
64C3×#235—32no—
65C3×#235—31no—
66C3×#235—31no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—31no—
70C2×C3×Dic6—32no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—31no—
73C2×D4×C9—121no—
74#41⋊C9—21no—
75#41⋊C9—31no—
76#41⋊C9C2×D311yescharacteristic, complemented
77C2×Q8⋊C9—22no—
78C2×Q8⋊C9—31no—
79C2×Q8⋊C9—32no—
80C2×Q8⋊C9C2×D312yescomplemented
81C22×C3×C9—42no—
82C2×D3×C9—41no—
83C2×D3×C9—42no—
84C2×D3×C9—84no—
85C3×C4×C9—41no—
86C9×Dic3—41no—
87C3×2−1+4—21no—
88C3×2−1+4—31no—
89C3×2−1+4—32no—
90C3×2−1+4—91no—
91C3×2−1+4—91no—
92C3×2−1+4—92no—
93C3×2−1+4C2×C911yescharacteristic, complemented
94C3×2−1+4C3×D311yescharacteristic
95C2×C3×#41—31no—
96C2×C3×#41—91no—
97C2×C3×#41—92no—
98C22×C3×Q8—32no—
99C22×C3×Q8—91no—
100#1010—32no—
101#1010—31no—
102#1010C2×C911yescharacteristic, complemented
103D3×#41—31no—
104D3×#41—32no—
105D3×#41C2×C911yescharacteristic, complemented
106#1007—31no—
107#1007C2×C912yescomplemented
108C2×D3×Q8—31no—
109C2×D3×Q8C2×C912yescomplemented
110C22×C3×D3—31no—
111C22×C3×D3C2×A412yes—
112C32×Q8—32no—
113C32×Q8—31no—
114C32×Q8C23×C311yescharacteristic
115C32×D4—31no—
116C32×D4—32no—
117C32×D4C2×A411yescharacteristic
118C2×C32×C4—32no—
119C2×C32×C4—32no—
120C2×C32×C4—31no—
121C3×#67—32no—
122C3×#67—32no—
123C3×#67—32no—
124C3×#67C2×A412yes—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—31no—
128C3×D12—31no—
129C3×D12—32no—
130C3×D12C2×A411yescharacteristic
131C3×C4×D3—34no—
132C3×C4×D3—32no—
133C3×C4×D3—31no—
134C3×C4×D3—32no—
135C3×C4×D3—32no—
136C3×C4×D3—31no—
137C3×C4×D3—31no—
138C3×C4×D3C2×A411yescharacteristic
139C3×Dic6—34no—
140C3×Dic6—31no—
141C3×Dic6—32no—
142C3×Dic6—31no—
143C3×Dic6—32no—
144C23×C9—122no—
145D4×C9—41no—
146D4×C9—81no—
147D4×C9—121no—
148D4×C9—122no—
149C2×C4×C9—121no—
150Q8⋊C9—21no—
151Q8⋊C9C22×D311yescharacteristic
152C2×2−1+4—31noSylow
153C2×C3×C9—41no—
154C2×C3×C9—82no—
155D3×C9—42no—
156D3×C9—82no—
157C3×#41—21no—
158C3×#41—31no—
159C3×#41—31no—
160C3×#41—32no—
161C3×#41—32no—
162C3×#41—31no—
163C3×#41—61no—
164C3×#41—62no—
165C3×#41—92no—
166C3×#41—91no—
167C3×#41—94no—
168C3×#41—92no—
169C3×#41—91no—
170C3×#41—92no—
171C3×#41—91no—
172C3×#41—94no—
173C3×#41—92no—
174C3×#41—91no—
175C3×#41—92no—
176C3×#41—91no—
177C3×#41C22×C911yescharacteristic, complemented
178C3×#41C2×C3×D311yescharacteristic
179C2×C3×Q8—22no—
180C2×C3×Q8—31no—
181C2×C3×Q8—31no—
182C2×C3×Q8—31no—
183C2×C3×Q8—32no—
184C2×C3×Q8—61no—
185C2×C3×Q8—91no—
186C2×C3×Q8—91no—
187C2×C3×Q8—92no—
188C2×C3×Q8—92no—
189C2×C3×Q8—92no—
190C2×C3×Q8—92no—
191C2×C3×Q8—92no—
192C2×C3×Q8—92no—
193C2×C3×Q8C22×C912yescomplemented
194C2×C3×Q8C2×C3×D312yes—
195C2×C3×D4—31no—
196C2×C3×D4—92no—
197C2×C3×D4—91no—
198C22×C3×C4—91no—
199C22×C3×C4—92no—
200C22×C3×C4—92no—
201#239—31no—
202#239—32no—
203#239C22×C911yescharacteristic, complemented
204D3×Q8—32no—
205D3×Q8—31no—
206D3×Q8—32no—
207D3×Q8—32no—
208D3×Q8—32no—
209D3×Q8C22×C911yescharacteristic
210D3×Q8C22×C912yescomplemented
211#237—34no—
212#237—31no—
213#237—32no—
214#237—32no—
215#237—31no—
216D3×D4—31no—
217D3×D4—32no—
218D3×D4C22⋊C911yescharacteristic
219#235—34no—
220#235—32no—
221#235—32no—
222#235—31no—
223#235—31no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—31no—
227C2×Dic6—32no—
228C2×Dic6—32no—
229C2×Dic6—31no—
230C22×C32—31no—
231C22×C32C22×A412yes—
232C2×C3×D3—24no—
233C2×C3×D3—31no—
234C2×C3×D3—62no—
235C2×C3×D3C22×A412yes—
236C2×C3×D3C22×A411yescharacteristic
237C32×C4—32no—
238C32×C4—31no—
239C32×C4C22×A411yescharacteristic
240C3×Dic3—31no—
241C3×Dic3—32no—
242C3×Dic3C22×A411yescharacteristic
243C22×C9—42no—
244C22×C9—82no—
245C22×C9—121no—
246C22×C9—122no—
247C22×C9—244no—
248C4×C9—41no—
249C4×C9—81no—
250C4×C9—121no—
2512−1+4—31no—
2522−1+4—32no—
2532−1+4—91no—
2542−1+4—91no—
2552−1+4—92no—
2562−1+4D3×C911yescharacteristic, complemented
257C2×#41—31no—
258C2×#41—91no—
259C2×#41—92no—
260C22×Q8—32no—
261C22×Q8—91no—
262C3×C9—41noSylow
263C23×C3—32no—
264C23×C3—91no—
265C22×D3—31no—
266C22×D3C2×C22⋊C912yes—
267C3×Q8—21no—
268C3×Q8—32no—
269C3×Q8—32no—
270C3×Q8—31no—
271C3×Q8—31no—
272C3×Q8—61no—
273C3×Q8—62no—
274C3×Q8—92no—
275C3×Q8—91no—
276C3×Q8—92no—
277C3×Q8—94no—
278C3×Q8—91no—
279C3×Q8C23×C911yescharacteristic
280C3×Q8C22×C3×D311yescharacteristic
281C3×D4—21no—
282C3×D4—31no—
283C3×D4—31no—
284C3×D4—32no—
285C3×D4—32no—
286C3×D4—32no—
287C3×D4—31no—
288C3×D4—62no—
289C3×D4—61no—
290C3×D4—92no—
291C3×D4—91no—
292C3×D4—92no—
293C3×D4—92no—
294C3×D4—92no—
295C3×D4C2×C22⋊C911yescharacteristic
296C3×D4D3×A411yescharacteristic
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—32no—
300C2×C3×C4—32no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—61no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—91no—
308C2×C3×C4—91no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—91no—
313C2×C3×C4—94no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—91no—
317#67—32no—
318#67—32no—
319#67—32no—
320#67C2×C22⋊C912yes—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—31no—
324D12—31no—
325D12—32no—
326D12C2×C22⋊C911yescharacteristic
327C4×D3—34no—
328C4×D3—32no—
329C4×D3—31no—
330C4×D3—32no—
331C4×D3—32no—
332C4×D3—31no—
333C4×D3—31no—
334C4×D3C2×C22⋊C911yescharacteristic
335Dic6—34no—
336Dic6—31no—
337Dic6—32no—
338Dic6—31no—
339Dic6—32no—
340C2×C32—22no—
341C2×C32—61no—
342C2×C32C23×A411yescharacteristic
343C3×D3—22no—
344C3×D3—61no—
345C3×D32−1+4⋊C312yes—
346C2×C9—41no—
347C2×C9—82no—
348C2×C9—81no—
349C2×C9—122no—
350C2×C9—162no—
351C2×C9—242no—
352#41—31no—
353#41—32no—
354#41—31no—
355#41—92no—
356#41—91no—
357#41—94no—
358#41—92no—
359#41—91no—
360#41—92no—
361#41—91no—
362#41—94no—
363#41—92no—
364#41—91no—
365#41—92no—
366#41—91no—
367#41C2×D3×C911yescharacteristic, complemented
368C2×Q8—31no—
369C2×Q8—31no—
370C2×Q8—32no—
371C2×Q8—91no—
372C2×Q8—91no—
373C2×Q8—92no—
374C2×Q8—92no—
375C2×Q8—92no—
376C2×Q8—92no—
377C2×Q8—92no—
378C2×Q8—92no—
379C2×Q8C2×D3×C912yescomplemented
380C2×D4—31no—
381C2×D4—92no—
382C2×D4—91no—
383C22×C4—91no—
384C22×C4—92no—
385C22×C4—92no—
386C22×C3—22no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—32no—
390C22×C3—31no—
391C22×C3—61no—
392C22×C3—64no—
393C22×C3—91no—
394C22×C3—182no—
395C22×C3C22×C22⋊C912yes—
396C22×C3C2×D3×A412yes—
397C2×D3—24no—
398C2×D3—31no—
399C2×D3—62no—
400C2×D3C22×C22⋊C912yes—
401C2×D3C22×C22⋊C911yescharacteristic
402C3×C4—21no—
403C3×C4—31no—
404C3×C4—32no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—32no—
408C3×C4—61no—
409C3×C4—62no—
410C3×C4—91no—
411C3×C4—92no—
412C3×C4C22×C22⋊C911yescharacteristic
413C3×C4C2×D3×A411yescharacteristic
414Dic3—31no—
415Dic3—32no—
416Dic3C22×C22⋊C911yescharacteristic
417C32C2×2−1+4⋊C311yescharacteristic
418C9—41no—
419C9—81no—
420C23—32no—
421C23—91no—
422Q8—32no—
423Q8—31no—
424Q8—92no—
425Q8—91no—
426Q8—92no—
427Q8—94no—
428Q8—91no—
429Q8C22×D3×C911yescharacteristic
430D4—31no—
431D4—32no—
432D4—32no—
433D4—31no—
434D4—92no—
435D4—91no—
436D4—92no—
437D4—92no—
438D4—92no—
439D4D3×C22⋊C911yescharacteristic
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—91no—
449C2×C4—92no—
450C2×C4—94no—
451C2×C4—92no—
452C2×C4—92no—
453C2×C4—91no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—32no—
458C2×C3—42no—
459C2×C3—62no—
460C2×C3—61no—
461C2×C3—61no—
462C2×C3—121no—
463C2×C3—181no—
464C2×C3C23×C22⋊C911yescharacteristic
465C2×C3C22×D3×A411yescharacteristic
466D3—22no—
467D3—61no—
468D32−1+4⋊C912yesdirect product factor
469C22—31no—
470C22—31no—
471C22—32no—
472C22—64no—
473C22—91no—
474C22—182no—
475C22C2×D3×C22⋊C912yes—
476C4—31no—
477C4—32no—
478C4—31no—
479C4—91no—
480C4—92no—
481C4C2×D3×C22⋊C911yescharacteristic
482C3—21no—
483C3C2×2−1+4⋊C911yescharacteristic, complemented
484C3D3×2−1+4⋊C311yescharacteristic
485C2—22no—
486C2—32no—
487C2—62no—
488C2—61no—
489C2—181no—
490C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×D3×A4
commutator, hypocenterC3×Q8C23×C9
FittingC32×2−1+4C2×C3
socleC2×C32C23×A4
prime radicalC3×D3×2−1+4C3
cosocleC32×Q8C23×C3
hypercenterC3×D4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×2−1+4⋊C9C2semidirect
2. #1010⋊C9C2semidirect
3. D3×#41⋊C9C2semidirect
4. #1007⋊C9C2semidirect
5. C2×D3×Q8⋊C9C2semidirect
9. C3×#41⋊C9C22semidirect
10. C2×C3×Q8⋊C9C22semidirect
12. D3×Q8⋊C9C22semidirect
13. C3⋊(#41⋊C9)C22semidirect
28. 2−1+4⋊C9D3direct (N × H)
38. D3×2−1+4C9semidirect
76. #41⋊C9C2×D3semidirect
80. C2×Q8⋊C9C2×D3semidirect
93. C3×2−1+4C2×C9semidirect
102. #1010C2×C9semidirect
105. D3×#41C2×C9semidirect
107. #1007C2×C9semidirect
109. C2×D3×Q8C2×C9semidirect
177. C3×#41C22×C9semidirect
193. C2×C3×Q8C22×C9semidirect
203. #239C22×C9semidirect
210. D3×Q8C22×C9semidirect
256. 2−1+4D3×C9semidirect
367. #41C2×D3×C9semidirect
379. C2×Q8C2×D3×C9semidirect
468. D32−1+4⋊C9direct (N × H)
483. C3C2×2−1+4⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×2−1+4⋊C92
#1010⋊C92
D3×#41⋊C92
#1007⋊C92
C2×D3×Q8⋊C92
C3×D3×2−1+43
C2×2−1+4⋊C93
D3×D4×C94