Finite Group Databasebeta

← #13147539

Subgroups — #13147539

Order 1728.

754 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
1C4×#6226C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic
3C9×#870C211yesmaximal, characteristic, complemented
4#11720998C211yesmaximal, characteristic
5#11720998C211yesmaximal, characteristic
6#11720996C212yesmaximal, complemented
7#11720991C212yesmaximal, complemented
8D9×Dic3⋊C4C212yesmaximal, complemented
9#11720971C211yesmaximal, characteristic, complemented
10#11720939C212yesmaximal, complemented
11#11720937C211yesmaximal, characteristic, complemented
12#10593941—31nomaximal
13#10589792—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×Dic3×D9C2212yes—
18C2×Dic3×D9C2212yes—
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2211yescharacteristic
23#88534C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C9×C4⋊Dic3C2211yescharacteristic
29C9×Dic3⋊C4C2212yes—
30C9×Dic3⋊C4C2212yes—
31C3×C4⋊Dic9C2211yescharacteristic
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37C4×#634—31no—
38C3×C4×D12—31no—
39C3×#870D311yescharacteristic
40#63919—31no—
41#63898—31no—
42#63898—31no—
43#63896—32no—
44#63893—32no—
45D3×Dic3⋊C4—32no—
46#63872—31no—
47#63851—32no—
48C9×#125—31no—
49#63459—32no—
50#63458—32no—
51#63458—32no—
52#63455—31no—
53#63455—31no—
54C4⋊C4×D9—32no—
55#63452—31no—
56#63450—31no—
57C4×D36D311yescharacteristic
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×C3×D9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68#6226—22no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C3×D36Q814yescomplemented
73C3×C4×D9—42no—
74Dic3×D9—24no—
75Dic3×D9—42no—
76#6189—42no—
77#5690—91no—
78#5539—91no—
79C2×#634—31no—
80C2×C3×D12—31no—
81C2×C3×C4×D3—32no—
82C2×#627—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87#3750—31no—
88#3750—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4C2×D312yes—
96C3×Dic3⋊C4C2×D312yes—
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D36C2×D311yescharacteristic
103C2×C4×D9—32no—
104C2×C4×D9—32no—
105C2×C4×D9C2×D312yes—
106C2×Dic18—31no—
107C2×Dic18—32no—
108C2×Dic18—61no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—32no—
111C9×C4⋊C4—31no—
112C9×C4⋊C4—31no—
113C42×C9—21no—
114C42×C9C2×D311yescharacteristic
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4C2×D312yes—
118C4⋊Dic9—31no—
119C4⋊Dic9—32no—
120C4⋊Dic9—61no—
121C4⋊Dic9C2×D311yescharacteristic
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—61no—
127Dic9⋊C4—61no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C4×Dic9—61no—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—22no—
134C2×C3×D9—22no—
135C2×C3×D9C2×Q814yes—
136C3×C4×C9—21no—
137C3×C4×C9—21no—
138C3×C4×C9C2×Q812yes—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C3⋊Dic9—21no—
142C3⋊Dic9#4112yes—
143C9×Dic3—22no—
144C9×Dic3—22no—
145C3×Dic9—22no—
146C3×#125—31no—
147C3×C4×D4—91no—
148D4×Dic3—91no—
149D4×Dic3—91no—
150#924—92no—
151C2×Dic3⋊C4—92no—
152#897—92no—
153#896—92no—
154#896—92no—
155#893—91no—
156#893—91no—
157D3×C4⋊C4—92no—
158#890—91no—
159#890—91no—
160#888—91no—
161#888—91no—
162#878—92no—
163#878—92no—
164C4×D12—31no—
165#870D911yescharacteristic, complemented
166C4×Dic6—91no—
167C4×Dic6—91no—
168C4×Dic6—181no—
169C22×C3×D3—32no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177#634—62no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3C22×D312yes—
180C2×C3×Dic3C22×D312yes—
181C3×D12—34no—
182C3×C4×D3—122no—
183#627—122no—
184D3×Dic3—64no—
185D3×Dic3—122no—
186C22×D9C22×D312yes—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—21no—
190C2×C4×C9—32no—
191C2×C4×C9—32no—
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195C2×Dic9—31no—
196C2×Dic9—31no—
197C2×Dic9—31no—
198C2×Dic9—31no—
199C2×Dic9—61no—
200C2×Dic9—61no—
201C2×Dic9—61no—
202C2×Dic9—61no—
203C2×Dic9C22×D312yes—
204D36Dic614yescomplemented
205C4×D9—42no—
206C4×D9—64no—
207C4×D9—122no—
208Dic18—62no—
209Dic18—64no—
210Dic18—122no—
211#554—271noSylow
212C2×C3×C9C22×Q811yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9H(5,𝔽2)11yescharacteristic
215C3×D9—24no—
216C2×C3×D4—91no—
217C22×C3×C4—92no—
218C22×Dic3—92no—
219C22×Dic3—92no—
220C2×D12—31no—
221C2×C4×D3—32no—
222C2×C4×D3—92no—
223C2×C4×D3—92no—
224C2×Dic6—91no—
225C2×Dic6—91no—
226C2×Dic6—92no—
227C2×Dic6—92no—
228C2×Dic6—181no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—91no—
234C3×C22⋊C4—92no—
235C3×C42—21no—
236C3×C42C2×D911yescharacteristic
237C3×C42D3211yescharacteristic
238#217—92no—
239#217—92no—
240(C2×D3)⋊C4—32no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243C4⋊Dic3—31no—
244C4⋊Dic3—92no—
245C4⋊Dic3—91no—
246C4⋊Dic3—91no—
247C4⋊Dic3—92no—
248C4⋊Dic3—181no—
249C4⋊Dic3C2×D911yescharacteristic
250C4⋊Dic3C2×D911yescharacteristic
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—91no—
254Dic3⋊C4—91no—
255Dic3⋊C4—91no—
256Dic3⋊C4—91no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—181no—
260Dic3⋊C4—181no—
261Dic3⋊C4C2×D912yes—
262Dic3⋊C4C2×D912yes—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—34no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4D3×Q812yes—
276C3⋊Dic3—32no—
277C3⋊Dic3—61no—
278C3⋊Dic3—61no—
279C3⋊Dic3—61no—
280C3×Dic3—22no—
281C3×Dic3—22no—
282C3×Dic3—62no—
283C22×C9—21no—
284C22×C9C23×D311yescharacteristic
285C2×D9—22no—
286C2×D9—22no—
287C2×D9C2×Dic614yes—
288C4×C9—22no—
289C4×C9—21no—
290C4×C9—21no—
291C4×C9—22no—
292C4×C9—41no—
293C4×C9—62no—
294C4×C9—62no—
295C4×C9C2×Dic612yes—
296Dic9—22no—
297Dic9—32no—
298Dic9—62no—
299Dic9—62no—
300Dic9—61no—
301Dic9—61no—
302Dic9—61no—
303Dic9—121no—
304Dic9—121no—
305C4⋊Q8—271no—
306#125—31no—
307#125—271no—
308C22⋊Q8—272no—
309C22⋊Q8—272no—
310C22⋊Q8—272no—
311C4×Q8—271no—
312C4×D4—91no—
313C4×D4—271no—
314C4×D4—271no—
315C2×C4⋊C4—272no—
316C3×C9#55411yescharacteristic, complemented, Sylow
317C23×C3—92no—
318C22×D3—32no—
319C3×D4—94no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—92no—
326C2×C3×C4—362no—
327C2×C3×C4C22×D911yescharacteristic
328C2×C3×C4C22×D911yescharacteristic
329C2×C3×C4C22×D911yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×Dic3—32no—
334C2×Dic3—91no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341C2×Dic3—91no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—184no—
347C2×Dic3—362no—
348C2×Dic3C22×D912yes—
349C2×Dic3C22×D912yes—
350D12—34no—
351C4×D3—122no—
352C4×D3—184no—
353C4×D3—362no—
354Dic6—182no—
355Dic6—182no—
356Dic6—184no—
357Dic6—362no—
358Dic6—362no—
359C2×C32C2×D3×Q811yescharacteristic
360C2×C32D3×#4111yescharacteristic
361C2×C32#100911yescharacteristic
362C3×D3—64no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9C22×Dic611yescharacteristic
367C2×C9#100411yescharacteristic
368C2×C9D3×#4111yescharacteristic
369D9—24no—
370C2×Q8—272no—
371C2×Q8—271no—
372C2×D4—91no—
373C22×C4—92no—
374C22×C4—272no—
375C22×C4—272no—
376C4⋊C4—32no—
377C4⋊C4—31no—
378C4⋊C4—31no—
379C4⋊C4—32no—
380C4⋊C4—91no—
381C4⋊C4—272no—
382C4⋊C4—272no—
383C4⋊C4—272no—
384C4⋊C4—271no—
385C4⋊C4—271no—
386C4⋊C4—271no—
387C22⋊C4—92no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C42—271no—
391C42—271no—
392C42D3×D911yescharacteristic
393C22×C3—21no—
394C22×C3—94no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3C23×D911yescharacteristic
398C22×C3C22×D3211yescharacteristic
399C2×D3—34no—
400C2×D3—62no—
401C2×D3—62no—
402C3×C4—22no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—41no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—182no—
412C3×C4Q8×D912yes—
413C3×C4D3×Dic612yes—
414Dic3—22no—
415Dic3—22no—
416Dic3—62no—
417Dic3—92no—
418Dic3—92no—
419Dic3—182no—
420Dic3—182no—
421Dic3—181no—
422Dic3—181no—
423Dic3—181no—
424Dic3—181no—
425Dic3—181no—
426Dic3—181no—
427Dic3—361no—
428Dic3—361no—
429C32#569011yescharacteristic
430C9—21no—
431C9#553911yescharacteristic, complemented
432C23—92no—
433Q8—542no—
434Q8—1082no—
435D4—94no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—92no—
439C2×C4—271no—
440C2×C4—271no—
441C2×C4—271no—
442C2×C4—271no—
443C2×C4—362no—
444C2×C4—544no—
445C2×C4—1082no—
446C2×C4C2×D3×D911yescharacteristic
447C2×C4C2×D3×D911yescharacteristic
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—184no—
453C2×C3C2×Q8×D911yescharacteristic
454C2×C3#41×D911yescharacteristic
455C2×C3#6371511yescharacteristic
456C2×C3C2×D3×Dic611yescharacteristic
457C2×C3#6430611yescharacteristic
458C2×C3#6430711yescharacteristic
459D3—64no—
460C22—94no—
461C22—182no—
462C22—182no—
463C22C22×D3×D911yescharacteristic
464C4—21no—
465C4—21no—
466C4—62no—
467C4—62no—
468C4—182no—
469C4—272no—
470C4—541no—
471C4—541no—
472C4—541no—
473C4D9×Dic612yes—
474C3—21no—
475C3#1058979211yescharacteristic, complemented
476C3#1059394111yescharacteristic
477C2—184no—
478C2C2×D9×Dic611yescharacteristic
479C2#1172247511yescharacteristic
480C2#1172247711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC42D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
3. C9×#870C2semidirect
6. #11720996C2semidirect
7. #11720991C2semidirect
8. D9×Dic3⋊C4C2semidirect
9. #11720971C2semidirect
10. #11720939C2semidirect
11. #11720937C2semidirect
72. C3×D36Q8semidirect
165. #870D9semidirect
204. D36Dic6semidirect
316. C3×C9#554semidirect
431. C9#5539semidirect
475. C3#10589792semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C3×C4×D362
C9×#8702
#117209982
#117209962
#117209912
D9×Dic3⋊C42
#117209712
#117209392
#117209372
#105939413
#105897923