Finite Group Databasebeta

← #13151007

Subgroups — #13151007

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D9C211yesmaximal, characteristic
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3C9×#936C211yesmaximal, characteristic, complemented
4C3×#63498C211yesmaximal, characteristic, complemented
5#11721274C214yesmaximal, complemented
6#11721242C214yesmaximal, complemented
7C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
8#11720953C211yesmaximal, characteristic, complemented
9#11720952C211yesmaximal, characteristic, complemented
10#10595170—31nomaximal
11D9×#124—31nomaximal
12C23×C3⋊D9C2212yes—
13C2×C3×D4×C9C2211yescharacteristic
14C2×#6233C2212yes—
15D4×C3⋊D9—24no—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic, complemented
19C2×C3×Dic18C2211yescharacteristic, complemented
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C2×#6189C2212yescomplemented
23(C3⋊D9)⋊Q8—24no—
24C9×#217C2214yescomplemented
25C3×#3676C2214yescomplemented
26C4×C9×Dic3C2211yescharacteristic, complemented
27C3×C4×Dic9C2211yescharacteristic, complemented
28#88445C2214yescomplemented
29#88445C2214yescomplemented
30Dic3×Dic9C2212yescomplemented
31C2×D4×C3⋊D3—31no—
32C2×#3797—31no—
33C3×#936—31no—
34C3×#936D311yescharacteristic
35#63974—34no—
36#63953—34no—
37C4×#624—31no—
38#63864—31no—
39#63864—31no—
40C2×Q8×D9—31no—
41C2×D4×D9—31no—
42C2×D4×D9—61no—
43C9×#124—31no—
44#63509—31no—
45#63498D311yescharacteristic, complemented
46#63447—34no—
47C22⋊C4×D9—34no—
48#63437—31no—
49C42×D9—31no—
50C23×C3×C9C2312yes—
51C22×C3⋊D9—24no—
52C22×C3⋊D9—28no—
53C22×C3⋊D9C2312yes—
54C22×C3⋊D9C2311yescharacteristic
55C3×D4×C9—22no—
56C2×C3×C4×C9C2311yescharacteristic
57#6233—24no—
58C2×C3⋊Dic9C2311yescharacteristic
59C4⋊(C3⋊D9)—22no—
60C4×C3⋊D9D414yes—
61C2×C9×Dic3C2312yes—
62C2×C9×Dic3C2312yes—
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65C9×Dic6—22no—
66C3×Dic18—22no—
67(C3⋊D9)⋊C4—24no—
68(C3⋊D9)⋊C4—24no—
69#6189—24no—
70D3×#124—91no—
71D3×#124—91no—
72C23×C3⋊D3—32no—
73C2×C32×D4C2×D311yescharacteristic
74C2×#638—32no—
75D4×C3⋊D3—64no—
76C2×C4⋊(C3⋊D3)—31no—
77C2×C4×C3⋊D3—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×#627—32no—
81C2×#624—32no—
82C2×#624—32no—
83#3797—64no—
84C23×D9—32no—
85C23×D9—62no—
86C3×#217—34no—
87C3×#217C2×D314yes—
88C3×C4×Dic3—31no—
89C3×C4×Dic3C2×D311yescharacteristic
90#3722—34no—
91#3722—34no—
92Dic32—32no—
93C2×Q8×C9—31no—
94C2×D4×C9—21no—
95C2×D4×C9C2×D311yescharacteristic
96C2×#611—32no—
97C2×#611—62no—
98Q8×D9—64no—
99D4×D9—64no—
100D4×D9—68no—
101C2×D36—31no—
102C2×D36—61no—
103C2×C4×D9—31no—
104C2×C4×D9—32no—
105C2×C4×D9—32no—
106C2×C4×D9—61no—
107C2×Dic18—32no—
108C2×Dic18C2×D311yescharacteristic, complemented
109C9×C22⋊C4—34no—
110C42×C9—31no—
111#3676C2×D314yescomplemented
112(C2×D9)⋊C4—34no—
113(C2×D9)⋊C4—34no—
114C4×Dic9—32no—
115C4×Dic9C2×D311yescharacteristic, complemented
116C22×C3×C9—22no—
117C22×C3×C9—24no—
118C22×C3×C9C2411yescharacteristic
119C2×C3⋊D9—22no—
120C2×C3⋊D9—24no—
121C2×C3⋊D9—48no—
122C2×C3⋊D9C2×D412yes—
123C2×C3⋊D9#4114yes—
124C3×C4×C9C2×D412yes—
125C3⋊Dic9C2×D412yes—
126C9×Dic3—22no—
127C9×Dic3—22no—
128C3×Dic9—22no—
129C3×Dic9—22no—
130C2×D3×Q8—91no—
131C2×D3×Q8—91no—
132C2×D3×D4—91no—
133C2×D3×D4—91no—
134C2×D3×D4—181no—
135C3×#124—31no—
136C3×#124—91no—
137#947—91no—
138#947—91no—
139#936—31no—
140#936D911yescharacteristic, complemented
141#885—94no—
142#885—94no—
143D3×C22⋊C4—94no—
144D3×C22⋊C4—94no—
145#875—91no—
146#875—91no—
147C42×D3—91no—
148C42×D3—91no—
149C23×C32C22×D312yes—
150C22×C3⋊D3—31no—
151C22×C3⋊D3—32no—
152C22×C3⋊D3—64no—
153C22×C3⋊D3—68no—
154C32×D4—22no—
155C2×C32×C4C22×D311yescharacteristic
156#638—64no—
157C2×C3⋊Dic3—31no—
158C4⋊(C3⋊D3)—62no—
159C4×C3⋊D3—34no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3C22×D312yes—
163C2×C3×Dic3C22×D312yes—
164C3×Dic6—22no—
165C3×Dic6—62no—
166#627—64no—
167#624—64no—
168#624—64no—
169C23×C9—22no—
170C23×C9C22×D312yes—
171C22×D9—31no—
172C22×D9—32no—
173C22×D9—61no—
174C22×D9—62no—
175C22×D9—64no—
176C22×D9—68no—
177C22×D9—68no—
178C22×D9—128no—
179Q8×C9—62no—
180D4×C9—24no—
181D4×C9—22no—
182C2×C4×C9—21no—
183C2×C4×C9—32no—
184C2×C4×C9—32no—
185C2×C4×C9C22×D311yescharacteristic
186#611—64no—
187#611—68no—
188C2×Dic9—31no—
189C2×Dic9—61no—
190C2×Dic9C22×D312yes—
191C2×Dic9C22×D312yes—
192D36—64no—
193D36—62no—
194C4×D9—34no—
195C4×D9—64no—
196C4×D9—64no—
197C4×D9—64no—
198Dic18—22no—
199Dic18—64no—
200C2×#124—271noSylow
201C2×C3×C9—42no—
202C2×C3×C9C22×D411yescharacteristic
203C2×C3×C9C2×#4112yes—
204C3⋊D9—42no—
205C3⋊D9#12414yescomplemented
206C23×D3—92no—
207C23×D3—92no—
208C23×D3—182no—
209C2×C3×Q8—31no—
210C2×C3×Q8—91no—
211C2×C3×D4—21no—
212C2×C3×D4C2×D911yescharacteristic
213C2×C3×D4D3211yescharacteristic
214C2×#67—92no—
215C2×#67—92no—
216C2×#67—182no—
217D3×Q8—184no—
218D3×Q8—184no—
219D3×D4—184no—
220D3×D4—184no—
221D3×D4—188no—
222C2×D12—91no—
223C2×D12—91no—
224C2×D12—181no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—181no—
232C2×Dic6—31no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C2×Dic6C2×D911yescharacteristic, complemented
236C3×C22⋊C4—34no—
237C3×C22⋊C4—94no—
238C3×C42—31no—
239C3×C42—91no—
240#217—34no—
241#217C2×D914yescomplemented
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—94no—
245(C2×D3)⋊C4—94no—
246C4×Dic3—31no—
247C4×Dic3—92no—
248C4×Dic3—92no—
249C4×Dic3C2×D911yescharacteristic, complemented
250C22×C32—22no—
251C22×C32—24no—
252C22×C32C23×D311yescharacteristic
253C2×C3⋊D3—32no—
254C2×C3⋊D3—34no—
255C2×C3⋊D3—62no—
256C2×C3⋊D3—64no—
257C2×C3⋊D3—128no—
258C32×C4D3×D412yes—
259C3⋊Dic3—32no—
260C3×Dic3—22no—
261C3×Dic3—22no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C22×C9—24no—
265C22×C9—22no—
266C22×C9—24no—
267C22×C9—21no—
268C22×C9—44no—
269C22×C9C23×D311yescharacteristic
270C2×D9—32no—
271C2×D9—34no—
272C2×D9—62no—
273C2×D9—64no—
274C2×D9—64no—
275C2×D9—62no—
276C2×D9—64no—
277C2×D9—124no—
278C2×D9—128no—
279C2×D9—1216no—
280C4×C9—22no—
281C4×C9—62no—
282C4×C9—62no—
283C4×C9D3×D412yes—
284Dic9—22no—
285Dic9—22no—
286Dic9—32no—
287Dic9—62no—
288C22×Q8—271no—
289C22×D4—271no—
290#124—31no—
291#124—91no—
292#124—271no—
293#124—271no—
294#124—274no—
295C2×C22⋊C4—274no—
296C2×C42—271no—
297C3×C9C2×#12411yescharacteristic, complemented, Sylow
298C23×C3—22no—
299C23×C3C22×D912yes—
300C23×C3C2×D3212yes—
301C22×D3—91no—
302C22×D3—91no—
303C22×D3—92no—
304C22×D3—92no—
305C22×D3—181no—
306C22×D3—182no—
307C22×D3—184no—
308C22×D3—188no—
309C22×D3—184no—
310C22×D3—188no—
311C22×D3—188no—
312C22×D3—368no—
313C3×Q8—62no—
314C3×Q8—182no—
315C3×D4—24no—
316C3×D4—22no—
317C3×D4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—32no—
320C2×C3×C4—32no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325#67—184no—
326#67—184no—
327#67—188no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—91no—
331C2×Dic3—91no—
332C2×Dic3—181no—
333C2×Dic3C22×D912yes—
334C2×Dic3C22×D912yes—
335D12—184no—
336D12—182no—
337D12—182no—
338C4×D3—94no—
339C4×D3—94no—
340C4×D3—184no—
341C4×D3—184no—
342C4×D3—184no—
343C4×D3—184no—
344C4×D3—184no—
345Dic6—22no—
346Dic6—62no—
347Dic6—184no—
348Dic6—184no—
349C2×C32—42no—
350C2×C32C2×D3×D411yescharacteristic
351C2×C32D3×#4112yes—
352C3⋊D3—34no—
353C3⋊D3—122no—
354C2×C9—22no—
355C2×C9—21no—
356C2×C9—44no—
357C2×C9—42no—
358C2×C9C2×D3×D411yescharacteristic
359C2×C9D3×#4112yes—
360D9—34no—
361D9—64no—
362D9—124no—
363D9—122no—
364C24—272no—
365C2×Q8—31no—
366C2×Q8—91no—
367C2×Q8—272no—
368C2×Q8—544no—
369C2×D4—272no—
370C2×D4—271no—
371C2×D4—544no—
372C2×D4D3×D911yescharacteristic
373C22×C4—272no—
374C22×C4—272no—
375C22×C4—271no—
376C22⋊C4—34no—
377C22⋊C4—94no—
378C22⋊C4—274no—
379C22⋊C4—274no—
380C42—31no—
381C42—91no—
382C42—272no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—21no—
389C22×C3—44no—
390C22×C3C23×D911yescharacteristic
391C22×C3C22×D3211yescharacteristic
392C2×D3—94no—
393C2×D3—92no—
394C2×D3—92no—
395C2×D3—94no—
396C2×D3—182no—
397C2×D3—184no—
398C2×D3—184no—
399C2×D3—184no—
400C2×D3—182no—
401C2×D3—182no—
402C2×D3—184no—
403C2×D3—364no—
404C2×D3—368no—
405C2×D3—368no—
406C2×D3—3616no—
407C3×C4—22no—
408C3×C4—62no—
409C3×C4—62no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4D4×D912yes—
413C3×C4#381112yes—
414Dic3—22no—
415Dic3—22no—
416Dic3—62no—
417Dic3—62no—
418Dic3—92no—
419Dic3—92no—
420Dic3—182no—
421C32D3×#12411yescharacteristic
422C9—21no—
423C9D3×#12411yescharacteristic, complemented
424C23—271no—
425C23—272no—
426C23—544no—
427C23—548no—
428C23C2×D3×D912yes—
429Q8—62no—
430Q8—182no—
431Q8—544no—
432D4—22no—
433D4—544no—
434D4—542no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—271no—
440C2×C4—274no—
441C2×C4—544no—
442C2×C4—544no—
443C2×C4C2×D3×D911yescharacteristic
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—44no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3C2×D4×D911yescharacteristic
450C2×C3#41×D912yes—
451C2×C3#6431212yes—
452C2×C3C2×#381111yescharacteristic
453D3—94no—
454D3—94no—
455D3—184no—
456D3—364no—
457D3—362no—
458D3—362no—
459C22—24no—
460C22—22no—
461C22—274no—
462C22—272no—
463C22—544no—
464C22—542no—
465C22—1088no—
466C22C22×D3×D911yescharacteristic
467C4—62no—
468C4—62no—
469C4—182no—
470C4—182no—
471C4—272no—
472C4#8866812yes—
473C3—21no—
474C3D9×#12411yescharacteristic, complemented
475C3#1059517011yescharacteristic
476C2—42no—
477C2—274no—
478C2—1082no—
479C2#1172249812yes—
480C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×D4×C3⋊D9C2
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#124
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊D9)⋊Q8C2semidirect
3. C9×#936C2semidirect
4. C3×#63498C2semidirect
5. #11721274C2semidirect
6. #11721242C2semidirect
7. C4×(C3⋊D9)⋊C4C2semidirect
8. #11720953C2semidirect
9. #11720952C2semidirect
18. C2×C9×Dic6C22semidirect
19. C2×C3×Dic18C22semidirect
22. C2×#6189C22semidirect
24. C9×#217C22semidirect
25. C3×#3676C22semidirect
26. C4×C9×Dic3C22semidirect
27. C3×C4×Dic9C22semidirect
28. #88445C22semidirect
29. #88445C22semidirect
30. Dic3×Dic9C22semidirect
45. #63498D3semidirect
108. C2×Dic18C2×D3semidirect
111. #3676C2×D3semidirect
115. C4×Dic9C2×D3semidirect
140. #936D9semidirect
205. C3⋊D9#124semidirect
235. C2×Dic6C2×D9semidirect
241. #217C2×D9semidirect
249. C4×Dic3C2×D9semidirect
297. C3×C9C2×#124semidirect
423. C9D3×#124semidirect
474. C3D9×#124semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C2×(C3⋊D9)⋊Q82
C9×#9362
C3×#634982
#117212742
#117212422
C4×(C3⋊D9)⋊C42
#117209532
#117209522
#105951703
D9×#1243