Finite Group Databasebeta

← #13148909

Subgroups — #13148909

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721748C211yesmaximal, characteristic, complemented
2C3×#63459C211yesmaximal, characteristic, complemented
3C9×#890C211yesmaximal, characteristic, complemented
4C4×#6192C211yesmaximal, characteristic
5#11720988C212yesmaximal
6D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#11720975C212yesmaximal, complemented
8#11720937C212yesmaximal, complemented
9#11720936C211yesmaximal, characteristic, complemented
10#11720921C212yesmaximal, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594457—31nomaximal
13#10589796—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C2×#6192C2211yescharacteristic
18C2×Dic3×D9C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2212yes—
23C3×(C2×D9)⋊C4C2212yes—
24C9×C4⋊Dic3C2211yescharacteristic
25C9×C4⋊Dic3C2212yes—
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29#88445C2212yes—
30#88445C2211yescharacteristic
31(C2×D9)⋊Dic3C2211yescharacteristic
32Dic3⋊Dic9C2212yes—
33Dic3⋊Dic9C2211yescharacteristic
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64101—31no—
38C3×#897—31no—
39C3×#890D311yescharacteristic
40C4×#626—31no—
41#63890—32no—
42D3×C4⋊Dic3—31no—
43#63878—32no—
44#63872—32no—
45#63871—31no—
46#63848—32no—
47Dic3×Dic6—31no—
48C9×#125—31no—
49#63460—32no—
50#63459—31no—
51#63459D311yescharacteristic, complemented
52#63456—32no—
53#63454—31no—
54#63454—32no—
55#63454—61no—
56C4⋊C4×D9—31no—
57#63452—32no—
58C4×D36—31no—
59C4×Dic18—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9—22no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9—22no—
73C3×Dic18—22no—
74#6192—22no—
75Dic3×D9—24no—
76#5694—91no—
77#5642—91no—
78C2×C4×C3⋊D3—31no—
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—31no—
81C2×#626—31no—
82C2×D3×Dic3—32no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3752—32no—
85C4⋊(C3⋊Dic3)—31no—
86C4×C3⋊Dic3—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×C4⋊Dic3—32no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D312yes—
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3C2×D311yescharacteristic
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—31no—
97#3722—31no—
98#3722—32no—
99(C2×D3)⋊Dic3—31no—
100Dic32—32no—
101C2×D36—31no—
102C2×C4×D9—32no—
103C2×C4×D9—31no—
104C2×C4×D9—61no—
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18C2×D311yescharacteristic
107C9×C4⋊C4—21no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—31no—
111C9×C4⋊C4C2×D311yescharacteristic
112C42×C9—31no—
113(C2×D9)⋊C4—31no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—62no—
118(C2×D9)⋊C4C2×D312yes—
119C4⋊Dic9—31no—
120C4⋊Dic9—31no—
121C4⋊Dic9—32no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D312yes—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9—62no—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—21no—
131C2×C3⋊D9—21no—
132C2×C3⋊D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9#4112yes—
136C3×C4×C9—22no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—21no—
139C3⋊Dic9—22no—
140C9×Dic3—22no—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C3×Dic9—22no—
144C3×Dic9—21no—
145C3×#125—31no—
146C3×C22⋊Q8—91no—
147#946—91no—
148Q8×Dic3—91no—
149C4×#67—91no—
150C2×C4⋊Dic3—91no—
151#898—92no—
152#898—92no—
153#897—31no—
154#897—91no—
155#894—92no—
156#892—92no—
157#892—91no—
158#892—91no—
159#892—181no—
160D3×C4⋊C4—91no—
161#890—92no—
162#890—92no—
163#890D911yescharacteristic, complemented
164#886—92no—
165#877—92no—
166C4×D12—91no—
167C4×Dic6—91no—
168C22×C3⋊D3—31no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—31no—
174C4×C3⋊D3—62no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D311yescharacteristic
180C3×C4×D3—62no—
181C3×Dic6—62no—
182#626—62no—
183D3×Dic3—64no—
184C22×D9—31no—
185C22×D9—61no—
186C22×D9C22×D311yescharacteristic
187C2×C4×C9—21no—
188C2×C4×C9—22no—
189C2×C4×C9—31no—
190C2×C4×C9—31no—
191C2×C4×C9—32no—
192C2×C4×C9C22×D312yes—
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—32no—
195C2×Dic9—31no—
196C2×Dic9—61no—
197C2×Dic9—62no—
198C2×Dic9C22×D312yes—
199C2×Dic9C22×D311yescharacteristic
200D36—62no—
201C4×D9—22no—
202C4×D9—64no—
203C4×D9—64no—
204C4×D9—62no—
205Dic18—22no—
206#552—271noSylow
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C3⋊D9—41no—
211C3×D9—22no—
212C2×C3×Q8—91no—
213C22×C3×C4—91no—
214C2×#67—91no—
215C22×Dic3—92no—
216C2×D12—91no—
217C2×C4×D3—31no—
218C2×C4×D3—92no—
219C2×C4×D3—91no—
220C2×C4×D3—91no—
221C2×C4×D3—181no—
222C2×Dic6—31no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—32no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4C2×D911yescharacteristic
229C3×C4⋊C4D3211yescharacteristic
230C3×C22⋊C4—92no—
231C3×C42—31no—
232#217—91no—
233(C2×D3)⋊C4—32no—
234(C2×D3)⋊C4—91no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—91no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—91no—
241(C2×D3)⋊C4—182no—
242C4⋊Dic3—32no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—91no—
246C4⋊Dic3—91no—
247C4⋊Dic3—92no—
248C4⋊Dic3—92no—
249C4⋊Dic3—181no—
250C4⋊Dic3C2×D912yes—
251C4⋊Dic3C2×D911yescharacteristic
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—91no—
256Dic3⋊C4C2×D912yes—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—182no—
262C4×Dic3C2×D911yescharacteristic
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—61no—
265C2×C3⋊D3—61no—
266C2×C3⋊D3—61no—
267C2×C3×D3—32no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—22no—
271C32×C4#23912yes—
272C3⋊Dic3—61no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C3×Dic3—21no—
276C3×Dic3—21no—
277C3×Dic3—62no—
278C3×Dic3—61no—
279C22×C9—21no—
280C22×C9C23×D311yescharacteristic
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—62no—
284C2×D9—61no—
285C2×D9—61no—
286C2×D9—61no—
287C2×D9—121no—
288C2×D9—121no—
289C2×D9#23912yes—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—42no—
293C4×C9—62no—
294C4×C9—61no—
295C4×C9—61no—
296C4×C9#23712yes—
297Dic9—22no—
298Dic9—21no—
299Dic9—62no—
300Dic9—61no—
301Dic9—62no—
302Dic9—122no—
303#126—272no—
304#125—31no—
305#125—272no—
306#123—272no—
307C22⋊Q8—91no—
308C22⋊Q8—271no—
309C4×Q8—271no—
310C4×D4—271no—
311#117—272no—
312#117—271no—
313C2×C4⋊C4—271no—
314C3×C9#55211yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C22×D3—31no—
317C22×D3—91no—
318C22×D3—91no—
319C22×D3—181no—
320C3×Q8—182no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—92no—
327C2×C3×C4—91no—
328C2×C3×C4—182no—
329C2×C3×C4C22×D912yes—
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C2×D3212yes—
332C2×C3×C4C2×D3211yescharacteristic
333#67—182no—
334C2×Dic3—32no—
335C2×Dic3—31no—
336C2×Dic3—92no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—92no—
340C2×Dic3—181no—
341C2×Dic3—182no—
342C2×Dic3—184no—
343C2×Dic3C22×D911yescharacteristic
344C2×Dic3C22×D911yescharacteristic
345C2×Dic3C22×D912yes—
346D12—182no—
347C4×D3—62no—
348C4×D3—184no—
349C4×D3—184no—
350C4×D3—182no—
351C4×D3—182no—
352Dic6—62no—
353C2×C32C2×#23911yescharacteristic
354C2×C32D3×#4111yescharacteristic
355C2×C32#101011yescharacteristic
356C3⋊D3—121no—
357C3×D3—62no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9C2×#23711yescharacteristic
362C2×C9C2×#23911yescharacteristic
363C2×C9#101011yescharacteristic
364D9—22no—
365D9—122no—
366D9—121no—
367C2×Q8—91no—
368C2×D4—271no—
369C22×C4—91no—
370C22×C4—272no—
371C22×C4—271no—
372C4⋊C4—31no—
373C4⋊C4—32no—
374C4⋊C4—32no—
375C4⋊C4—92no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4—272no—
380C4⋊C4—271no—
381C4⋊C4D3×D911yescharacteristic
382C22⋊C4—92no—
383C22⋊C4—271no—
384C22⋊C4—272no—
385C22⋊C4—271no—
386C22⋊C4—272no—
387C42—31no—
388C42—272no—
389C42—272no—
390C22×C3—21no—
391C22×C3—92no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×D911yescharacteristic
395C22×C3C22×D3211yescharacteristic
396C2×D3—32no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—182no—
400C2×D3—181no—
401C2×D3—181no—
402C2×D3—181no—
403C2×D3—181no—
404C2×D3—181no—
405C2×D3—181no—
406C2×D3—361no—
407C2×D3—361no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—61no—
413C3×C4—62no—
414C3×C4—61no—
415C3×C4—181no—
416C3×C4—182no—
417C3×C4#370112yes—
418C3×C4#379612yes—
419Dic3—22no—
420Dic3—21no—
421Dic3—21no—
422Dic3—62no—
423Dic3—61no—
424Dic3—182no—
425Dic3—181no—
426Dic3—182no—
427Dic3—181no—
428Dic3—182no—
429Dic3—362no—
430C32#569411yescharacteristic
431C9—21no—
432C9#564211yescharacteristic, complemented
433C23—91no—
434C23—271no—
435Q8—182no—
436D4—542no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—32no—
440C2×C4—92no—
441C2×C4—91no—
442C2×C4—182no—
443C2×C4—272no—
444C2×C4—271no—
445C2×C4—544no—
446C2×C4—542no—
447C2×C4C2×D3×D912yes—
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—182no—
453C2×C3C2×#370111yescharacteristic
454C2×C3#41×D911yescharacteristic
455C2×C3#6371611yescharacteristic
456C2×C3C2×#379611yescharacteristic
457C2×C3#6430911yescharacteristic
458C2×C3D3×#23911yescharacteristic
459D3—62no—
460D3—362no—
461D3—361no—
462D3—361no—
463C22—92no—
464C22—181no—
465C22—181no—
466C22—541no—
467C22—541no—
468C22—541no—
469C22C22×D3×D911yescharacteristic
470C4—22no—
471C4—62no—
472C4—61no—
473C4—61no—
474C4—181no—
475C4—182no—
476C4—542no—
477C4—541no—
478C4#8863412yes—
479C3—21no—
480C3#1058979611yescharacteristic, complemented
481C3#1059445711yescharacteristic
482C2—182no—
483C2—1081no—
484C2C2×#8863411yescharacteristic
485C2#1172249411yescharacteristic
486C2D9×#23911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721748C2semidirect
2. C3×#63459C2semidirect
3. C9×#890C2semidirect
6. D9×C4⋊Dic3C2semidirect
7. #11720975C2semidirect
8. #11720937C2semidirect
9. #11720936C2semidirect
10. #11720921C2semidirect
11. Dic3×Dic18C2semidirect
51. #63459D3semidirect
163. #890D9semidirect
314. C3×C9#552semidirect
432. C9#5642semidirect
480. C3#10589796semidirect

Maximal subgroups

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

subgroupindex
#117217482
C3×#634592
C9×#8902
C4×#61922
#117209882
D9×C4⋊Dic32
#117209752
#117209372
#117209362
#117209212
Dic3×Dic182
#105944573
#105897963