Finite Group Databasebeta

← #13147627

Subgroups — #13147627

Order 1728.

706 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
1C4×#6226C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11721025C211yesmaximal, characteristic
5#11721024C212yesmaximal, complemented
6#11721010C212yesmaximal, complemented
7#11720967C211yesmaximal, characteristic
8#11720965C212yesmaximal, complemented
9#11720963C212yesmaximal, complemented
10#11720930C211yesmaximal, characteristic, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10593942—31nomaximal
13#10589773—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×D3×Dic9C2212yes—
19C3×C42×C9C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2211yescharacteristic
22#88534C2211yescharacteristic
23C4×C3⋊Dic9C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C9×(C2×D3)⋊C4C2212yes—
26C9×(C2×D3)⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×C4×Dic9C2212yes—
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37C4×#634—31no—
38C3×#875D311yescharacteristic
39C3×C4×Dic6—31no—
40#63898—31no—
41#63896—32no—
42#63877—32no—
43#63873—31no—
44#63863—31no—
45#63861—32no—
46#63859—32no—
47#63843—31no—
48C9×#124—31no—
49Q8×Dic9—31no—
50#63506—31no—
51#63498—31no—
52D4×Dic9—31no—
53#63441—32no—
54#63441—32no—
55#63440—32no—
56#63439—32no—
57C4×Dic18—31no—
58C4×Dic18—61no—
59C4×Dic18D311yescharacteristic, complemented
60C22×D3×C9C2312yes—
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×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12—22no—
73C9×Dic6—22no—
74C3×Dic18—22no—
75D3×Dic9—24no—
76#5671—91no—
77#5568—91no—
78C2×#634—31no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×D3×Dic3—32no—
83C32×C42C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85#3750—31no—
86#3750—31no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—31no—
89C3×(C2×D3)⋊C4C2×D312yes—
90C3×(C2×D3)⋊C4C2×D312yes—
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—32no—
93C3×C4×Dic3—32no—
94#3724—32no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C2×D4×C9—31no—
103C22×Dic9—32no—
104C2×Dic18—31no—
105C2×Dic18—61no—
106C2×Dic18C2×D311yescharacteristic
107C9×C22⋊C4—32no—
108C9×C22⋊C4—32no—
109C42×C9—21no—
110C42×C9C2×D311yescharacteristic
111#3676—32no—
112#3676—32no—
113#3676—32no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—31no—
119Dic9⋊C4—31no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—61no—
123Dic9⋊C4—61no—
124Dic9⋊C4C2×D312yes—
125C4×Dic9—31no—
126C4×Dic9—31no—
127C4×Dic9—32no—
128C4×Dic9—61no—
129C4×Dic9—61no—
130C4×Dic9C2×D312yes—
131C22×C3×C9C2411yescharacteristic
132C2×D3×C9—22no—
133C2×D3×C9—22no—
134C2×D3×C9—22no—
135C3×C4×C9—21no—
136C3×C4×C9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—21no—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C3⋊Dic9#4112yes—
142C9×Dic3—22no—
143C3×Dic9—22no—
144C3×Dic9—22no—
145C3×#124—31no—
146C3×C4×Q8—91no—
147#947—91no—
148Q8×Dic3—91no—
149#944—91no—
150#944—91no—
151#936—91no—
152D4×Dic3—91no—
153#898—92no—
154#896—92no—
155#893—91no—
156#887—91no—
157#879—92no—
158#879—92no—
159#878—92no—
160#877—92no—
161#876—92no—
162#875D911yescharacteristic
163#872—92no—
164C4×Dic6—31no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C4×Dic6—181no—
168C22×C3×D3C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176#634—62no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3C22×D312yes—
180C3×D12—22no—
181C3×Dic6—22no—
182C3×Dic6—62no—
183D3×Dic3—64no—
184C23×C9—32no—
185Q8×C9—62no—
186D4×C9—62no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—21no—
190C2×C4×C9—32no—
191C2×C4×C9C22×D311yescharacteristic
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—31no—
195C2×Dic9—31no—
196C2×Dic9—31no—
197C2×Dic9—31no—
198C2×Dic9—61no—
199C2×Dic9—61no—
200C2×Dic9—61no—
201C2×Dic9—61no—
202C2×Dic9—64no—
203C2×Dic9C22×D312yes—
204C2×Dic9C22×D312yes—
205Dic18—22no—
206Dic18—62no—
207Dic18—64no—
208#556—271noSylow
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C92−1+411yescharacteristic
212D3×C9—42no—
213C2×C3×Q8—31no—
214C2×C3×Q8—91no—
215C2×C3×D4—31no—
216C22×Dic3—92no—
217C2×D12C2×D911yescharacteristic
218C2×C4×D3—92no—
219C2×Dic6—31no—
220C2×Dic6—91no—
221C2×Dic6—91no—
222C2×Dic6—181no—
223C2×Dic6C2×D911yescharacteristic
224C3×C4⋊C4—91no—
225C3×C4⋊C4—92no—
226C3×C22⋊C4—32no—
227C3×C22⋊C4—32no—
228C3×C42—21no—
229C3×C42—92no—
230C3×C42C2×D911yescharacteristic
231C3×C42D3211yescharacteristic
232#217—92no—
233#217—92no—
234#217—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4C2×D912yes—
239(C2×D3)⋊C4C2×D912yes—
240C4⋊Dic3—31no—
241C4⋊Dic3—92no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—181no—
245Dic3⋊C4—32no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—91no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—91no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—181no—
256Dic3⋊C4—181no—
257C4×Dic3—32no—
258C4×Dic3—92no—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—91no—
263C4×Dic3—92no—
264C4×Dic3—181no—
265C4×Dic3—181no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3—22no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4#23712yes—
273C3⋊Dic3—32no—
274C3⋊Dic3—61no—
275C3⋊Dic3—61no—
276C3⋊Dic3—61no—
277C3×Dic3—22no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C22×C9—21no—
281C22×C9—62no—
282C22×C9—62no—
283C22×C9—62no—
284C22×C9C23×D311yescharacteristic
285C4×C9—22no—
286C4×C9—21no—
287C4×C9—21no—
288C4×C9—22no—
289C4×C9—41no—
290C4×C9—62no—
291C4×C9#23512yes—
292Dic9—22no—
293Dic9—22no—
294Dic9—32no—
295Dic9—62no—
296Dic9—62no—
297Dic9—61no—
298Dic9—61no—
299Dic9—61no—
300Dic9—121no—
301Dic9—121no—
302C4⋊Q8—271no—
303#126—272no—
304#126—272no—
305#124—31no—
306#124—271no—
307C22⋊Q8—272no—
308C4×Q8—91no—
309C4×Q8—271no—
310C4×Q8—271no—
311C4×D4—271no—
312#117—272no—
313C3×C9#55611yescharacteristic, complemented, Sylow
314C23×C3—32no—
315C22×D3C22×D912yes—
316C3×Q8—62no—
317C3×Q8—182no—
318C3×D4—62no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—32no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4C22×D911yescharacteristic
326C2×C3×C4C22×D911yescharacteristic
327C2×C3×C4C22×D911yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—184no—
346C2×Dic3C22×D912yes—
347D12—22no—
348C4×D3—184no—
349Dic6—22no—
350Dic6—62no—
351Dic6—182no—
352Dic6—182no—
353Dic6—184no—
354C2×C32C2×#23711yescharacteristic
355C2×C32D3×#4111yescharacteristic
356C2×C32#101011yescharacteristic
357C3×D3—42no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—122no—
362C2×C9C2×#23511yescharacteristic
363C2×C9#100711yescharacteristic
364C2×C9D3×#4111yescharacteristic
365C2×Q8—31no—
366C2×Q8—91no—
367C2×Q8—271no—
368C2×D4—31no—
369C22×C4—272no—
370C4⋊C4—91no—
371C4⋊C4—92no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—271no—
376C4⋊C4—271no—
377C4⋊C4—271no—
378C22⋊C4—32no—
379C22⋊C4—32no—
380C22⋊C4—272no—
381C22⋊C4—272no—
382C22⋊C4—272no—
383C42—92no—
384C42—272no—
385C42—271no—
386C42—271no—
387C42D3×D911yescharacteristic
388C22×C3—21no—
389C22×C3—62no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3C23×D911yescharacteristic
393C22×C3C22×D3211yescharacteristic
394C2×D3—22no—
395C2×D3—22no—
396C2×D3—22no—
397C3×C4—22no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—22no—
403C3×C4—41no—
404C3×C4—62no—
405C3×C4—182no—
406C3×C4—182no—
407C3×C4#369912yes—
408C3×C4#379512yes—
409Dic3—22no—
410Dic3—62no—
411Dic3—62no—
412Dic3—92no—
413Dic3—92no—
414Dic3—182no—
415Dic3—182no—
416Dic3—181no—
417Dic3—181no—
418Dic3—181no—
419Dic3—181no—
420Dic3—181no—
421Dic3—181no—
422Dic3—361no—
423Dic3—361no—
424C32#567111yescharacteristic
425C9—21no—
426C9#556811yescharacteristic, complemented
427C23—32no—
428Q8—62no—
429Q8—182no—
430Q8—542no—
431D4—62no—
432C2×C4—32no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—271no—
436C2×C4—271no—
437C2×C4—271no—
438C2×C4—271no—
439C2×C4—544no—
440C2×C4C2×D3×D911yescharacteristic
441C2×C4C2×D3×D911yescharacteristic
442C2×C4C2×D3×D911yescharacteristic
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—122no—
447C2×C3C2×#369911yescharacteristic
448C2×C3#41×D911yescharacteristic
449C2×C3#6371611yescharacteristic
450C2×C3C2×#379511yescharacteristic
451C2×C3#6429711yescharacteristic
452C2×C3#6430611yescharacteristic
453D3—42no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22C22×D3×D911yescharacteristic
458C4—21no—
459C4—21no—
460C4—62no—
461C4—182no—
462C4—182no—
463C4—272no—
464C4—541no—
465C4—541no—
466C4—541no—
467C4#8863812yes—
468C3—21no—
469C3#1058977311yescharacteristic, complemented
470C3#1059394211yescharacteristic
471C2—122no—
472C2C2×#8863811yescharacteristic
473C2#1172246411yescharacteristic
474C2#1172247511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D12C2×D3
hypercenterC42D3×D9
hypocenterC3×C9#556
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. C3×C4×Dic18C2semidirect
5. #11721024C2semidirect
6. #11721010C2semidirect
8. #11720965C2semidirect
9. #11720963C2semidirect
10. #11720930C2semidirect
11. #11720911C2semidirect
59. C4×Dic18D3semidirect
313. C3×C9#556semidirect
426. C9#5568semidirect
469. C3#10589773semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C9×#8752
C3×C4×Dic182
#117210252
#117210242
#117210102
#117209672
#117209652
#117209632
#117209302
#117209112
#105939423
#105897733