Finite Group Databasebeta

← #13149010

Subgroups — #13149010

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721753C211yesmaximal, characteristic, complemented
2C3×#63454C211yesmaximal, characteristic, complemented
3C9×#888C211yesmaximal, characteristic, complemented
4#11721003C212yesmaximal
5C4×#6192C211yesmaximal, characteristic
6#11720949C212yesmaximal, complemented
7#11720945C211yesmaximal, characteristic, complemented
8#11720934C212yesmaximal, complemented
9#11720922C211yesmaximal, characteristic, complemented
10#11720912C211yesmaximal, characteristic, complemented
11Dic6×Dic9C212yesmaximal, complemented
12#10594556—31nomaximal
13#10589834—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2212yes—
18C2×#6192C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2212yes—
22C3×(C2×D9)⋊C4C2212yes—
23C9×C4⋊Dic3C2211yescharacteristic
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2211yescharacteristic
26C3×C4⋊Dic9C2211yescharacteristic
27C3×C4×Dic9C2212yes—
28#88445C2211yescharacteristic
29#88445C2212yes—
30(C2×D9)⋊Dic3C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2211yescharacteristic
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64106—31no—
38C3×#892—31no—
39C3×#888D311yescharacteristic
40#63907—32no—
41C4×#626—31no—
42#63868—32no—
43#63861—32no—
44#63857—31no—
45#63849—31no—
46#63844—31no—
47Dic3×Dic6—32no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50Q8×Dic9—32no—
51#63460—32no—
52#63460—32no—
53#63459—31no—
54#63459—31no—
55#63459—61no—
56#63454—31no—
57#63454D311yescharacteristic, complemented
58C4×D36—31no—
59C4×Dic18—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4×C3⋊D9—22no—
67#6226—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C3×C4×D9—22no—
74C9×Dic6—24no—
75#6192—22no—
76#5732—91no—
77#5533—91no—
78C2×C4×C3⋊D3—31no—
79C2×#634—31no—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6C2×D312yes—
82C2×#626—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3752—32no—
85C4⋊(C3⋊Dic3)—32no—
86C3×(C2×D3)⋊C4—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4C2×D312yes—
90C3×C4×Dic3—32no—
91C3×C4×Dic3C2×D311yescharacteristic
92#3724—32no—
93Dic3⋊Dic3—31no—
94Dic3⋊Dic3—32no—
95#3722—32no—
96#3722—31no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—31no—
99Dic32—32no—
100Dic32—32no—
101C2×Q8×C9—32no—
102C2×D36—31no—
103C2×C4×D9—31no—
104C2×C4×D9—61no—
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18—31no—
107C2×Dic18—61no—
108C9×C4⋊C4—21no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—31no—
111C9×C4⋊C4C2×D311yescharacteristic
112C42×C9—31no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—31no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—62no—
119(C2×D9)⋊C4C2×D312yes—
120C4⋊Dic9—32no—
121C4⋊Dic9—31no—
122C4⋊Dic9—32no—
123C4⋊Dic9—62no—
124C4⋊Dic9C2×D311yescharacteristic
125Dic9⋊C4—32no—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9C2×D312yes—
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—21no—
136C3×C4×C9—22no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—22no—
139C3⋊Dic9—21no—
140C9×Dic3—22no—
141C9×Dic3—21no—
142C9×Dic3#4112yes—
143C3×Dic9—21no—
144C3×Dic9—22no—
145C3×C4⋊Q8—31no—
146C3×#117—91no—
147#947—92no—
148Q8×Dic3—92no—
149C4×#67—91no—
150#924—91no—
151#898—92no—
152#898—92no—
153#897—91no—
154#897—91no—
155#897—91no—
156#897—181no—
157#892—31no—
158#892—91no—
159#892—91no—
160#888D911yescharacteristic, complemented
161#887—91no—
162#885—92no—
163#879—92no—
164#876—92no—
165C4×D12—91no—
166C4×Dic6—91no—
167C4×Dic6—92no—
168C22×C3⋊D3—31no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—32no—
174C4×C3⋊D3—62no—
175#634—62no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C3×C4×D3—62no—
182C3×Dic6—24no—
183#626—62no—
184C22×D9—31no—
185C22×D9—61no—
186C22×D9C22×D311yescharacteristic
187Q8×C9—64no—
188C2×C4×C9—21no—
189C2×C4×C9—22no—
190C2×C4×C9—31no—
191C2×C4×C9—32no—
192C2×C4×C9—31no—
193C2×C4×C9C22×D312yes—
194C2×C4×C9C22×D311yescharacteristic
195C2×Dic9—31no—
196C2×Dic9—32no—
197C2×Dic9—61no—
198C2×Dic9—62no—
199C2×Dic9C22×D312yes—
200C2×Dic9C22×D311yescharacteristic
201D36—62no—
202C4×D9—22no—
203C4×D9—62no—
204C4×D9—122no—
205Dic18—62no—
206Dic18—122no—
207#556—271noSylow
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211C3⋊D9—41no—
212C3×D9—41no—
213C2×C3×Q8—32no—
214C22×C3×C4—91no—
215C2×#67—91no—
216C2×D12—91no—
217C2×C4×D3—31no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—181no—
221C2×Dic6—91no—
222C2×Dic6—91no—
223C2×Dic6—181no—
224C2×Dic6C2×D912yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4C2×D911yescharacteristic
230C3×C4⋊C4D3211yescharacteristic
231C3×C22⋊C4—92no—
232C3×C42—31no—
233C3×C42—92no—
234#217—91no—
235#217—92no—
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—91no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—91no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—91no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—182no—
246C4⋊Dic3—31no—
247C4⋊Dic3—91no—
248C4⋊Dic3—92no—
249C4⋊Dic3—92no—
250C4⋊Dic3—92no—
251C4⋊Dic3—182no—
252C4⋊Dic3C2×D911yescharacteristic
253Dic3⋊C4—92no—
254Dic3⋊C4—91no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4C2×D912yes—
258C4×Dic3—32no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3C2×D911yescharacteristic
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3—61no—
266C2×C3⋊D3—61no—
267C2×C3⋊D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—22no—
272C32×C4#23912yes—
273C3⋊Dic3—62no—
274C3⋊Dic3—61no—
275C3×Dic3—22no—
276C3×Dic3—21no—
277C3×Dic3—61no—
278C3×Dic3—62no—
279C3×Dic3#23712yes—
280C22×C9—21no—
281C22×C9C23×D311yescharacteristic
282C2×D9—21no—
283C2×D9—21no—
284C2×D9—21no—
285C2×D9—62no—
286C2×D9—61no—
287C2×D9—61no—
288C2×D9—61no—
289C2×D9—121no—
290C2×D9—121no—
291C4×C9—22no—
292C4×C9—22no—
293C4×C9—32no—
294C4×C9—42no—
295C4×C9—62no—
296C4×C9—61no—
297C4×C9#23512yes—
298Dic9—21no—
299Dic9—22no—
300Dic9—62no—
301Dic9—61no—
302Dic9—122no—
303Dic9—121no—
304C4⋊Q8—31no—
305#126—272no—
306#126—272no—
307#124—272no—
308C22⋊Q8—271no—
309C22⋊Q8—271no—
310C4×Q8—271no—
311C4×Q8—272no—
312C4×D4—271no—
313#117—91no—
314#117—271no—
315C3×C9#55611yescharacteristic, complemented, Sylow
316C23×C3—91no—
317C22×D3—31no—
318C22×D3—91no—
319C22×D3—91no—
320C22×D3—181no—
321C3×Q8—64no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—31no—
327C2×C3×C4—92no—
328C2×C3×C4—91no—
329C2×C3×C4—182no—
330C2×C3×C4C22×D912yes—
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C2×D3212yes—
333C2×C3×C4C2×D3211yescharacteristic
334#67—182no—
335C2×Dic3—31no—
336C2×Dic3—32no—
337C2×Dic3—91no—
338C2×Dic3—92no—
339C2×Dic3—91no—
340C2×Dic3—92no—
341C2×Dic3—181no—
342C2×Dic3—182no—
343C2×Dic3C22×D912yes—
344C2×Dic3C22×D911yescharacteristic
345C2×Dic3C22×D911yescharacteristic
346D12—182no—
347C4×D3—62no—
348C4×D3—182no—
349C4×D3—182no—
350C4×D3—362no—
351Dic6—24no—
352Dic6—182no—
353Dic6—182no—
354Dic6—362no—
355C2×C32C2×#23711yescharacteristic
356C2×C32C2×#23911yescharacteristic
357C2×C32#100711yescharacteristic
358C3⋊D3—121no—
359C3×D3—121no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9C2×#23511yescharacteristic
364C2×C9D3×#4111yescharacteristic
365C2×C9#101011yescharacteristic
366D9—41no—
367D9—122no—
368D9—121no—
369C2×Q8—32no—
370C2×Q8—271no—
371C2×D4—271no—
372C22×C4—91no—
373C22×C4—271no—
374C4⋊C4—31no—
375C4⋊C4—32no—
376C4⋊C4—91no—
377C4⋊C4—271no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4D3×D911yescharacteristic
382C22⋊C4—92no—
383C22⋊C4—272no—
384C22⋊C4—271no—
385C22⋊C4—271no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C42—31no—
389C42—92no—
390C42—272no—
391C42—272no—
392C22×C3—21no—
393C22×C3—181no—
394C22×C3—181no—
395C22×C3—181no—
396C22×C3C23×D911yescharacteristic
397C22×C3C22×D3211yescharacteristic
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—182no—
402C2×D3—181no—
403C2×D3—181no—
404C2×D3—181no—
405C2×D3—181no—
406C2×D3—181no—
407C2×D3—181no—
408C2×D3—361no—
409C2×D3—361no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—32no—
414C3×C4—42no—
415C3×C4—62no—
416C3×C4—61no—
417C3×C4—181no—
418C3×C4—182no—
419C3×C4#370112yes—
420C3×C4#379912yes—
421Dic3—22no—
422Dic3—21no—
423Dic3—61no—
424Dic3—62no—
425Dic3—182no—
426Dic3—181no—
427Dic3—182no—
428Dic3—181no—
429Dic3—362no—
430Dic3—361no—
431Dic3#369912yes—
432C32#573211yescharacteristic
433C9—21no—
434C9#553311yescharacteristic, complemented
435C23—91no—
436C23—271no—
437Q8—64no—
438Q8—542no—
439D4—542no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—32no—
443C2×C4—91no—
444C2×C4—92no—
445C2×C4—182no—
446C2×C4—271no—
447C2×C4—272no—
448C2×C4—542no—
449C2×C4C2×D3×D912yes—
450C2×C4C2×D3×D911yescharacteristic
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—361no—
455C2×C3C2×#369911yescharacteristic
456C2×C3C2×#370111yescharacteristic
457C2×C3#6371311yescharacteristic
458C2×C3C2×#379911yescharacteristic
459C2×C3D3×#23711yescharacteristic
460C2×C3#6431511yescharacteristic
461D3—121no—
462D3—362no—
463D3—361no—
464D3—361no—
465C22—181no—
466C22—181no—
467C22—181no—
468C22—541no—
469C22—541no—
470C22—541no—
471C22C22×D3×D911yescharacteristic
472C4—22no—
473C4—32no—
474C4—62no—
475C4—61no—
476C4—182no—
477C4—181no—
478C4—541no—
479C4—542no—
480C4#8863512yes—
481C3—21no—
482C3#1058983411yescharacteristic, complemented
483C3#1059455611yescharacteristic
484C2—361no—
485C2—1081no—
486C2C2×#8863511yescharacteristic
487C2D3×#369911yescharacteristic
488C2#1172251011yescharacteristic
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#556
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. #11721753C2semidirect
2. C3×#63454C2semidirect
3. C9×#888C2semidirect
6. #11720949C2semidirect
7. #11720945C2semidirect
8. #11720934C2semidirect
9. #11720922C2semidirect
10. #11720912C2semidirect
11. Dic6×Dic9C2semidirect
57. #63454D3semidirect
160. #888D9semidirect
315. C3×C9#556semidirect
434. C9#5533semidirect
482. C3#10589834semidirect

Maximal subgroups

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

subgroupindex
#117217532
C3×#634542
C9×#8882
#117210032
C4×#61922
#117209492
#117209452
#117209342
#117209222
#117209122
Dic6×Dic92
#105945563
#105898343