Finite Group Databasebeta

← #13148808

Subgroups — #13148808

Order 1728.

874 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic
2#11721748C211yesmaximal, characteristic
3C9×#887C211yesmaximal, characteristic, complemented
4C3×#63449C211yesmaximal, characteristic, complemented
5C4×#6189C212yesmaximal, complemented
6#11720931C211yesmaximal, characteristic
7Dic6×Dic9C211yesmaximal, characteristic, complemented
8Dic3×Dic18C211yesmaximal, characteristic, complemented
9#11720902C212yesmaximal, complemented
10#11720899C212yesmaximal
11#11720898C212yesmaximal
12#10594347—31nomaximal
13#10589674—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×Dic6C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C2×(C3⋊D9)⋊C4C2212yes—
18C2×#6189C2212yes—
19(C3⋊D9)⋊Q8C418yescomplemented
20C3×C9×C4⋊C4C2211yescharacteristic
21#88536C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4×C3⋊Dic9C2212yes—
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C3×Dic9⋊C4C2212yes—
28C3×C4×Dic9C2212yes—
29C3×C4×Dic9C2211yescharacteristic
30#88445C2211yescharacteristic
31#88445C2211yescharacteristic
32Dic3⋊Dic9C2212yes—
33#88442C2211yescharacteristic
34#88442C2211yescharacteristic
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3797—31no—
39#64101—31no—
40C3×#887—31no—
41C3×#887D311yescharacteristic
42C4×#627—32no—
43#63874—31no—
44Dic3×Dic6—31no—
45Dic3×Dic6—31no—
46#63838—32no—
47#63836—32no—
48#63836—32no—
49C2×Q8×D9—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52#63454—31no—
53#63454—32no—
54#63454—61no—
55#63449—32no—
56#63449D311yescharacteristic, complemented
57#63434—32no—
58#63434—31no—
59C4×Dic18—31no—
60C4×Dic18—32no—
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9C2×C414yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C9×Dic6C2×C414yes—
74C3×Dic18C2×C414yes—
75(C3⋊D9)⋊C4C2×C418yes—
76#6189C2×C418yes—
77#5572—91no—
78#5572—91no—
79C2×C4×C3⋊D3—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#627—32no—
83C2×#624—32no—
84#3797—38no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3752—32no—
87C4⋊(C3⋊Dic3)—31no—
88C4×C3⋊Dic3—32no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4C2×D312yes—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3C2×D312yes—
94C3×C4×Dic3C2×D311yescharacteristic
95#3724—31no—
96#3724—31no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99#3722—31no—
100#3722—31no—
101Dic32—32no—
102Dic32—32no—
103C2×Q8×C9—31no—
104Q8×D9—38no—
105C2×C4×D9—31no—
106C2×C4×D9—32no—
107C2×C4×D9—61no—
108C2×Dic18—32no—
109C2×Dic18C2×D311yescharacteristic
110C9×C4⋊C4—21no—
111C9×C4⋊C4—32no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—32no—
114C42×C9—31no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—62no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122Dic9⋊C4—32no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4C2×D312yes—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9—32no—
129C4×Dic9—62no—
130C4×Dic9C2×D311yescharacteristic
131C4×Dic9C2×D312yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9C22×C412yes—
136C3×C4×C9—22no—
137C3×C4×C9C22×C412yes—
138C3⋊Dic9—22no—
139C3⋊Dic9C22×C412yes—
140C9×Dic3—21no—
141C9×Dic3—21no—
142C9×Dic3C22×C414yes—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9C22×C414yes—
146C2×D3×Q8—91no—
147C2×D3×Q8—91no—
148C3×C4×Q8—31no—
149C3×C4×Q8—91no—
150Q8×Dic3—91no—
151Q8×Dic3—91no—
152#892—92no—
153#892—91no—
154#892—91no—
155#892—92no—
156#892—181no—
157#887—31no—
158#887—92no—
159#887—92no—
160#887D911yescharacteristic, complemented
161#872—91no—
162#872—92no—
163#872—92no—
164#872—91no—
165C4×Dic6—92no—
166C4×Dic6—91no—
167C4×Dic6—91no—
168C4×Dic6—92no—
169C22×C3⋊D3—31no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—32no—
174C4×C3⋊D3—34no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D312yes—
181C3×Dic6—34no—
182C3×Dic6C4×D314yes—
183#627—38no—
184#624—38no—
185C22×D9—31no—
186C22×D9—61no—
187Q8×C9—34no—
188C2×C4×C9—21no—
189C2×C4×C9—22no—
190C2×C4×C9—32no—
191C2×C4×C9—31no—
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×D311yescharacteristic
200C2×Dic9C22×D311yescharacteristic
201C2×Dic9C22×D312yes—
202C4×D9—34no—
203C4×D9—38no—
204C4×D9—64no—
205Dic18—38no—
206Dic18C4×D314yes—
207#519—271noSylow
208C2×C3×C9C23×C411yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211C3⋊D9—22no—
212C2×C3×Q8—31no—
213C2×C3×Q8—91no—
214D3×Q8—98no—
215D3×Q8—98no—
216C2×C4×D3—92no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—92no—
220C2×C4×D3—181no—
221C2×Dic6—31no—
222C2×Dic6—92no—
223C2×Dic6—92no—
224C2×Dic6C2×D911yescharacteristic
225C3×C4⋊C4—21no—
226C3×C4⋊C4—32no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4C2×D911yescharacteristic
229C3×C4⋊C4D3211yescharacteristic
230C3×C42—32no—
231C3×C42—31no—
232C3×C42—92no—
233C3×C42—91no—
234(C2×D3)⋊C4—91no—
235(C2×D3)⋊C4—91no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—91no—
239(C2×D3)⋊C4—91no—
240(C2×D3)⋊C4—182no—
241C4⋊Dic3—92no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—181no—
246Dic3⋊C4—32no—
247Dic3⋊C4—92no—
248Dic3⋊C4—91no—
249Dic3⋊C4—91no—
250Dic3⋊C4—92no—
251Dic3⋊C4—91no—
252Dic3⋊C4—91no—
253Dic3⋊C4C2×D912yes—
254C4×Dic3—32no—
255C4×Dic3—31no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—182no—
263C4×Dic3C2×D912yes—
264C4×Dic3C2×D911yescharacteristic
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—32no—
267C2×C3⋊D3—61no—
268C2×C3⋊D3—61no—
269C32×C4—22no—
270C32×C4C2×C4×D312yes—
271C3⋊Dic3—32no—
272C3⋊Dic3—62no—
273C3×Dic3—21no—
274C3×Dic3—21no—
275C3×Dic3—34no—
276C3×Dic3—61no—
277C3×Dic3—61no—
278C3×Dic3C2×C4×D314yes—
279C22×C9—21no—
280C22×C9C23×D311yescharacteristic
281C2×D9—32no—
282C2×D9—62no—
283C2×D9—61no—
284C2×D9—61no—
285C2×D9—121no—
286C2×D9—121no—
287C4×C9—22no—
288C4×C9—22no—
289C4×C9—34no—
290C4×C9—42no—
291C4×C9—61no—
292C4×C9—61no—
293C4×C9C2×C4×D312yes—
294Dic9—21no—
295Dic9—21no—
296Dic9—32no—
297Dic9—62no—
298Dic9—62no—
299Dic9—122no—
300Dic9C2×C4×D314yes—
301C22×Q8—271no—
302C4×Q8—31no—
303C4×Q8—91no—
304C4×Q8—272no—
305C4×Q8—271no—
306C4×Q8—271no—
307C4×Q8—272no—
308#117—271no—
309#117—272no—
310#117—272no—
311#117—271no—
312C3×C9#51911yescharacteristic, complemented, Sylow
313C22×D3—91no—
314C22×D3—91no—
315C22×D3—181no—
316C3×Q8—34no—
317C3×Q8—94no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—32no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—92no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4C22×D912yes—
327C2×C3×C4C22×D911yescharacteristic
328C2×C3×C4C2×D3212yes—
329C2×C3×C4C2×D3211yescharacteristic
330C2×Dic3—32no—
331C2×Dic3—31no—
332C2×Dic3—31no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—92no—
336C2×Dic3—92no—
337C2×Dic3—181no—
338C2×Dic3—182no—
339C2×Dic3C22×D911yescharacteristic
340C2×Dic3C22×D911yescharacteristic
341C2×Dic3C22×D912yes—
342C4×D3—98no—
343C4×D3—94no—
344C4×D3—94no—
345C4×D3—98no—
346C4×D3—184no—
347Dic6—34no—
348Dic6—98no—
349Dic6—98no—
350Dic6C4×D914yes—
351C2×C32C22×C4×D311yescharacteristic
352C2×C32#100711yescharacteristic
353C2×C32#101011yescharacteristic
354C3⋊D3—62no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9C22×C4×D311yescharacteristic
359C2×C9#100711yescharacteristic
360C2×C9#101011yescharacteristic
361D9—62no—
362D9—122no—
363C2×Q8—31no—
364C2×Q8—91no—
365C2×Q8—272no—
366C2×Q8—278no—
367C22×C4—272no—
368C22×C4—271no—
369C4⋊C4—32no—
370C4⋊C4—92no—
371C4⋊C4—271no—
372C4⋊C4—271no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—271no—
376C4⋊C4D3×D911yescharacteristic
377C22⋊C4—271no—
378C22⋊C4—271no—
379C22⋊C4—272no—
380C42—32no—
381C42—31no—
382C42—92no—
383C42—91no—
384C42—272no—
385C42—272no—
386C42—272no—
387C22×C3—21no—
388C22×C3C23×D911yescharacteristic
389C22×C3C22×D3211yescharacteristic
390C2×D3—92no—
391C2×D3—92no—
392C2×D3—182no—
393C2×D3—181no—
394C2×D3—181no—
395C2×D3—181no—
396C2×D3—181no—
397C2×D3—361no—
398C2×D3—361no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—34no—
403C3×C4—42no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—94no—
407C3×C4—181no—
408C3×C4—181no—
409C3×C4C2×C4×D912yes—
410C3×C4C4×D3212yes—
411Dic3—21no—
412Dic3—21no—
413Dic3—34no—
414Dic3—61no—
415Dic3—61no—
416Dic3—92no—
417Dic3—92no—
418Dic3—182no—
419Dic3—182no—
420Dic3—182no—
421Dic3—362no—
422Dic3C2×C4×D914yes—
423C32#557211yescharacteristic
424C9—21no—
425C9#557211yescharacteristic, complemented
426C23—271no—
427Q8—34no—
428Q8—94no—
429Q8—278no—
430C2×C4—32no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—92no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—278no—
437C2×C4—271no—
438C2×C4—274no—
439C2×C4—272no—
440C2×C4C2×D3×D912yes—
441C2×C4C2×D3×D911yescharacteristic
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3C22×C4×D911yescharacteristic
446C2×C3#6371311yescharacteristic
447C2×C3#6371611yescharacteristic
448C2×C3C2×C4×D3211yescharacteristic
449C2×C3#6430911yescharacteristic
450C2×C3#6431611yescharacteristic
451D3—182no—
452D3—182no—
453D3—362no—
454C22—272no—
455C22—541no—
456C22—541no—
457C22C22×D3×D911yescharacteristic
458C4—22no—
459C4—34no—
460C4—61no—
461C4—61no—
462C4—94no—
463C4—181no—
464C4—181no—
465C4—272no—
466C4—542no—
467C4C4×D3×D912yes—
468C3—21no—
469C3#1058967411yescharacteristic, complemented
470C3#1059434711yescharacteristic
471C2—542no—
472C2C2×C4×D3×D911yescharacteristic
473C2#1172249411yescharacteristic
474C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
cosocleC22×C3×C9C24
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#519
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C9×#887C2semidirect
4. C3×#63449C2semidirect
5. C4×#6189C2semidirect
7. Dic6×Dic9C2semidirect
8. Dic3×Dic18C2semidirect
9. #11720902C2semidirect
19. (C3⋊D9)⋊Q8C4semidirect
56. #63449D3semidirect
160. #887D9semidirect
312. C3×C9#519semidirect
425. C9#5572semidirect
469. C3#10589674semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
#117217482
C9×#8872
C3×#634492
C4×#61892
#117209312
Dic6×Dic92
Dic3×Dic182
#117209022
#117208992
#117208982
#105943473
#105896743