Finite Group Databasebeta

← #13147511

Subgroups — #13147511

Order 1728.

712 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
4D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#11720981C212yesmaximal
6#11720975C212yesmaximal, complemented
7#11720950C212yesmaximal, complemented
8#11720945C211yesmaximal, characteristic, complemented
9#11720945C211yesmaximal, characteristic, complemented
10#11720916C212yesmaximal, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10593920—31nomaximal
13#10589796—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×Dic3×D9C2212yes—
19C3×C42×C9C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2212yes—
23C3×(C2×D9)⋊C4C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C9×Dic3⋊C4C2212yes—
29C3×C4⋊Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33Dic3⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C4×#634—31no—
38C3×C4×D12—31no—
39C3×#870D311yescharacteristic
40D3×C4⋊Dic3—31no—
41#63884—32no—
42#63878—32no—
43#63862—32no—
44#63857—31no—
45#63857—31no—
46#63846—32no—
47#63841—31no—
48C9×#125—31no—
49#63460—32no—
50#63459—31no—
51#63459—31no—
52#63456—32no—
53#63454—31no—
54#63454—32no—
55C4⋊C4×D9—31no—
56#63452—32no—
57C4×D36D311yescharacteristic
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×C3×D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C3×D36—22no—
73C3×C4×D9—22no—
74C3×C4×D9—22no—
75Dic3×D9—24no—
76#5694—91no—
77#5541—91no—
78C2×#634—31no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×D3×Dic3—32no—
83C32×C42C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85#3750—32no—
86C4×C3⋊Dic3—32no—
87C3×(C2×D3)⋊C4—31no—
88C3×(C2×D3)⋊C4—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4C2×D312yes—
93C3×Dic3⋊C4C2×D312yes—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101C2×D36C2×D311yescharacteristic
102C2×C4×D9—32no—
103C2×C4×D9C2×D311yescharacteristic
104C2×C4×D9C2×D311yescharacteristic
105C2×Dic18—31no—
106C2×Dic18—61no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4—31no—
110C9×C4⋊C4—31no—
111C42×C9—21no—
112C42×C9C2×D311yescharacteristic
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4C2×D311yescharacteristic
117(C2×D9)⋊C4C2×D311yescharacteristic
118C4⋊Dic9—31no—
119C4⋊Dic9—32no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—62no—
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—21no—
137C3×C4×C9—21no—
138C3×C4×C9#4112yes—
139C3⋊Dic9—22no—
140C3⋊Dic9—22no—
141C9×Dic3—22no—
142C9×Dic3—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×#125—31no—
146C3×C4×D4—91no—
147#935—92no—
148#926—91no—
149C2×C4⋊Dic3—91no—
150#924—91no—
151#924—91no—
152#898—92no—
153#897—91no—
154#897—91no—
155#894—92no—
156#892—92no—
157#892—91no—
158D3×C4⋊C4—91no—
159#890—92no—
160#889—92no—
161#879—92no—
162#877—92no—
163C4×D12—31no—
164#870D911yescharacteristic, complemented
165C4×Dic6—91no—
166C4×Dic6—91no—
167C4×Dic6—181no—
168C22×C3×D3—31no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175#634—62no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D312yes—
180C3×D12—62no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183D3×Dic3—64no—
184C22×D9C22×D311yescharacteristic
185C22×D9C22×D311yescharacteristic
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—32no—
190C2×C4×C9—32no—
191C2×C4×C9C22×D311yescharacteristic
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—32no—
195C2×Dic9—32no—
196C2×Dic9—62no—
197C2×Dic9—62no—
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D311yescharacteristic
200D36—22no—
201C4×D9—22no—
202C4×D9—22no—
203C4×D9—64no—
204Dic18—64no—
205Dic18—62no—
206#552—271noSylow
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C3×D9—22no—
211C3×D9—41no—
212C2×C3×D4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×Dic3—92no—
216C2×D12—31no—
217C2×C4×D3—31no—
218C2×C4×D3—31no—
219C2×C4×D3—92no—
220C2×Dic6—91no—
221C2×Dic6—91no—
222C2×Dic6—181no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—32no—
227C3×C4⋊C4—91no—
228C3×C22⋊C4—91no—
229C3×C22⋊C4—91no—
230C3×C42—21no—
231C3×C42C2×D911yescharacteristic
232C3×C42D3211yescharacteristic
233#217—92no—
234#217—92no—
235#217—92no—
236(C2×D3)⋊C4—31no—
237(C2×D3)⋊C4—31no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—92no—
241C4⋊Dic3—31no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—181no—
248C4⋊Dic3C2×D911yescharacteristic
249C4⋊Dic3C2×D911yescharacteristic
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—182no—
256Dic3⋊C4C2×D912yes—
257Dic3⋊C4C2×D912yes—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—182no—
263C22×C32C23×D311yescharacteristic
264C2×C3×D3—32no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4#23912yes—
273C3⋊Dic3—62no—
274C3⋊Dic3—62no—
275C3×Dic3—22no—
276C3×Dic3—22no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C22×C9—21no—
280C22×C9C23×D311yescharacteristic
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—21no—
284C2×D9—21no—
285C2×D9—21no—
286C2×D9#23512yes—
287C4×C9—22no—
288C4×C9—21no—
289C4×C9—21no—
290C4×C9—22no—
291C4×C9—41no—
292C4×C9—62no—
293C4×C9—62no—
294C4×C9#23712yes—
295Dic9—21no—
296Dic9—21no—
297Dic9—64no—
298Dic9—62no—
299Dic9—62no—
300Dic9—122no—
301#126—272no—
302#125—31no—
303#125—272no—
304#123—272no—
305C22⋊Q8—271no—
306C22⋊Q8—271no—
307C4×Q8—271no—
308C4×D4—91no—
309#117—272no—
310#117—271no—
311C2×C4⋊C4—271no—
312C3×C9#55211yescharacteristic, complemented, Sylow
313C23×C3—91no—
314C23×C3—91no—
315C22×D3—31no—
316C22×D3—31no—
317C3×D4—182no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—182no—
326C2×C3×C4—182no—
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—31no—
334C2×Dic3—31no—
335C2×Dic3—92no—
336C2×Dic3—92no—
337C2×Dic3—92no—
338C2×Dic3—92no—
339C2×Dic3—182no—
340C2×Dic3—182no—
341C2×Dic3—184no—
342C2×Dic3C22×D912yes—
343C2×Dic3C22×D912yes—
344D12—62no—
345C4×D3—62no—
346C4×D3—62no—
347C4×D3—184no—
348Dic6—184no—
349Dic6—182no—
350Dic6—182no—
351C2×C32C2×#23911yescharacteristic
352C2×C32D3×#4111yescharacteristic
353C2×C32#101011yescharacteristic
354C3×D3—62no—
355C3×D3—121no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9C2×#23511yescharacteristic
360C2×C9C2×#23711yescharacteristic
361C2×C9#101011yescharacteristic
362D9—22no—
363D9—41no—
364C2×Q8—271no—
365C2×D4—91no—
366C22×C4—91no—
367C22×C4—91no—
368C22×C4—272no—
369C4⋊C4—31no—
370C4⋊C4—31no—
371C4⋊C4—32no—
372C4⋊C4—32no—
373C4⋊C4—91no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—271no—
378C4⋊C4—272no—
379C22⋊C4—91no—
380C22⋊C4—91no—
381C22⋊C4—272no—
382C22⋊C4—272no—
383C22⋊C4—272no—
384C42—272no—
385C42—272no—
386C42D3×D911yescharacteristic
387C22×C3—21no—
388C22×C3—92no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×D911yescharacteristic
395C22×C3C22×D3211yescharacteristic
396C2×D3—32no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
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—181no—
412C3×C4—181no—
413C3×C4#370112yes—
414C3×C4#379612yes—
415Dic3—22no—
416Dic3—22no—
417Dic3—61no—
418Dic3—61no—
419Dic3—184no—
420Dic3—182no—
421Dic3—182no—
422Dic3—182no—
423Dic3—182no—
424Dic3—362no—
425C32#569411yescharacteristic
426C9—21no—
427C9#554111yescharacteristic, complemented
428C23—91no—
429C23—91no—
430Q8—542no—
431D4—182no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—182no—
437C2×C4—182no—
438C2×C4—272no—
439C2×C4—272no—
440C2×C4—544no—
441C2×C4C2×D3×D911yescharacteristic
442C2×C4C2×D3×D911yescharacteristic
443C2×C4C2×D3×D911yescharacteristic
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—182no—
448C2×C3—361no—
449C2×C3C2×#370111yescharacteristic
450C2×C3#41×D911yescharacteristic
451C2×C3#6371611yescharacteristic
452C2×C3C2×#379611yescharacteristic
453C2×C3#6429811yescharacteristic
454C2×C3D3×#23511yescharacteristic
455D3—62no—
456D3—121no—
457C22—92no—
458C22—181no—
459C22—181no—
460C22—181no—
461C22—181no—
462C22—181no—
463C22C22×D3×D911yescharacteristic
464C4—21no—
465C4—21no—
466C4—62no—
467C4—62no—
468C4—181no—
469C4—181no—
470C4—542no—
471C4—542no—
472C4#8863412yes—
473C3—21no—
474C3#1058979611yescharacteristic, complemented
475C3#1059392011yescharacteristic
476C2—182no—
477C2—361no—
478C2C2×#8863411yescharacteristic
479C2#1172246511yescharacteristic
480C2D9×#23511yescharacteristic
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#552
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
4. D9×C4⋊Dic3C2semidirect
6. #11720975C2semidirect
7. #11720950C2semidirect
8. #11720945C2semidirect
9. #11720945C2semidirect
10. #11720916C2semidirect
11. #11720907C2semidirect
164. #870D9semidirect
312. C3×C9#552semidirect
427. C9#5541semidirect
474. C3#10589796semidirect

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
D9×C4⋊Dic32
#117209812
#117209752
#117209502
#117209452
#117209162
#117209072
#105939203
#105897963