Finite Group Databasebeta

← #13150148

Subgroups — #13150148

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#11721813C211yesmaximal, characteristic, complemented
2C3×C4×#611C211yesmaximal, characteristic
3C9×#926C211yesmaximal, characteristic, complemented
4#11721257C212yesmaximal
5#11721247C212yesmaximal, complemented
6#11720969C211yesmaximal, characteristic, complemented
7#11720947C212yesmaximal, complemented
8#11720945C211yesmaximal, characteristic, complemented
9#11720913C211yesmaximal, characteristic, complemented
10#11720907C211yesmaximal, characteristic, complemented
11#11720900C212yesmaximal, complemented
12#10594840—31nomaximal
13#10589635—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×#6189C2212yes—
19#88551C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C9×#217C2212yes—
23C3×#3676C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C4×C9×Dic3C2212yes—
28C3×Dic9⋊C4C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×C4×#67—31no—
39C3×#926D311yescharacteristic
40#63959—32no—
41#63956—32no—
42#63918—31no—
43#63859—32no—
44#63857—31no—
45#63844—31no—
46#63841—31no—
47#63837—32no—
48C9×#117—31no—
49C4×#611D311yescharacteristic
50#63486—31no—
51#63486—61no—
52#63459—31no—
53#63454—31no—
54#63450—31no—
55#63449—31no—
56#63447—32no—
57#63441—32no—
58#63438—32no—
59C4×Dic18—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C3×#611—22no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75#6189—24no—
76#5542—91no—
77#5533—91no—
78C22×C32×C4C2×D311yescharacteristic
79C2×#634—31no—
80C2×C3×#67—31no—
81C2×C3×C4×D3—31no—
82C2×#627—32no—
83#3757—32no—
84C4⋊(C3⋊Dic3)—31no—
85#3750—32no—
86C3×#217—31no—
87C3×#217C2×D312yes—
88C3×(C2×D3)⋊C4—31no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3C2×D312yes—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99Dic32—32no—
100Dic32—32no—
101C22×C4×C9—21no—
102C22×C4×C9C2×D311yescharacteristic
103C2×#611C2×D311yescharacteristic
104C2×C4×D9C2×D311yescharacteristic
105C2×Dic18—31no—
106C2×Dic18—32no—
107C2×Dic18—61no—
108C9×C4⋊C4—31no—
109C9×C4⋊C4—31no—
110C9×C22⋊C4—32no—
111C42×C9—32no—
112#3676—32no—
113#3676—62no—
114#3676C2×D311yescharacteristic
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4C2×D311yescharacteristic
118C4⋊Dic9—31no—
119C4⋊Dic9—32no—
120C4⋊Dic9—61no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125Dic9⋊C4C2×D311yescharacteristic
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C3×C4×C9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—22no—
139C3⋊Dic9—22no—
140C9×Dic3—22no—
141C9×Dic3—22no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9#4112yes—
145C3×C4×D4—91no—
146C3×#117—31no—
147#936—92no—
148C4×#67—31no—
149#926—91no—
150#926D911yescharacteristic, complemented
151#924—91no—
152#924—91no—
153#924—91no—
154#924—181no—
155#897—91no—
156#892—91no—
157#888—91no—
158#887—91no—
159#887—92no—
160#885—92no—
161#879—92no—
162#879—92no—
163#879—92no—
164#876—92no—
165#869—91no—
166C4×Dic6—92no—
167C4×Dic6—91no—
168C23×C32C22×D311yescharacteristic
169C22×C3×D3—31no—
170C2×C32×C4—22no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175#634—62no—
176C3×#67—62no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C22×D312yes—
182C3×C4×D3—62no—
183#627—64no—
184C23×C9—21no—
185C23×C9C22×D311yescharacteristic
186C22×D9C22×D311yescharacteristic
187C2×C4×C9—22no—
188C2×C4×C9—21no—
189C2×C4×C9—21no—
190C2×C4×C9—32no—
191C2×C4×C9—32no—
192C2×C4×C9—42no—
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195#611—22no—
196C2×Dic9—32no—
197C2×Dic9—32no—
198C2×Dic9—62no—
199C2×Dic9—62no—
200C2×Dic9C22×D311yescharacteristic
201C2×Dic9C22×D311yescharacteristic
202C2×Dic9C22×D311yescharacteristic
203C4×D9—22no—
204Dic18—62no—
205Dic18—64no—
206Dic18—122no—
207#556—271noSylow
208C2×C3×C9—41no—
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C92−1+411yescharacteristic
212C3×D9—41no—
213C2×C3×D4—91no—
214C22×C3×C4—21no—
215C22×C3×C4—91no—
216C22×C3×C4C2×D911yescharacteristic
217C22×C3×C4D3211yescharacteristic
218C2×#67—31no—
219C2×C4×D3—31no—
220C2×Dic6—92no—
221C2×Dic6—91no—
222C2×Dic6—91no—
223C2×Dic6—92no—
224C2×Dic6—181no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—91no—
228C3×C22⋊C4—32no—
229C3×C22⋊C4—91no—
230C3×C22⋊C4—91no—
231C3×C42—32no—
232C3×C42—91no—
233#217—31no—
234#217—92no—
235#217—92no—
236#217—92no—
237#217—92no—
238#217—182no—
239#217C2×D912yes—
240(C2×D3)⋊C4—31no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—181no—
248C4⋊Dic3C2×D911yescharacteristic
249C4⋊Dic3C2×D911yescharacteristic
250Dic3⋊C4—31no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—182no—
258C4×Dic3—31no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3C2×D912yes—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—21no—
272C32×C4#23512yes—
273C3⋊Dic3—62no—
274C3⋊Dic3—62no—
275C3×Dic3—22no—
276C3×Dic3—22no—
277C3×Dic3—32no—
278C3×Dic3—61no—
279C3×Dic3—61no—
280C22×C9—22no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—41no—
286C22×C9—41no—
287C22×C9C23×D311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—21no—
291C4×C9—21no—
292C4×C9—22no—
293C4×C9—41no—
294C4×C9—62no—
295C4×C9—62no—
296C4×C9#23712yes—
297Dic9—21no—
298Dic9—21no—
299Dic9—62no—
300Dic9—62no—
301Dic9—122no—
302Dic9—122no—
303Dic9#23512yes—
304C4⋊Q8—271no—
305#126—272no—
306#126—272no—
307#124—272no—
308C22⋊Q8—271no—
309C22⋊Q8—271no—
310C4×Q8—272no—
311C4×Q8—271no—
312C4×D4—91no—
313#117—31no—
314#117—271no—
315C3×C9#55611yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—91no—
318C23×C3C22×D911yescharacteristic
319C23×C3C2×D3211yescharacteristic
320C22×D3—31no—
321C3×D4—182no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—42no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—182no—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337#67—62no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—182no—
346C2×Dic3—182no—
347C2×Dic3C22×D912yes—
348C2×Dic3C22×D912yes—
349C4×D3—62no—
350Dic6—184no—
351Dic6—184no—
352Dic6—182no—
353Dic6—182no—
354Dic6—362no—
355C2×C32—41no—
356C2×C32C2×#23511yescharacteristic
357C2×C32D3×#4111yescharacteristic
358C2×C32#101011yescharacteristic
359C3×D3—121no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—42no—
364C2×C9—41no—
365C2×C9C2×#23511yescharacteristic
366C2×C9C2×#23711yescharacteristic
367C2×C9#101011yescharacteristic
368D9—41no—
369C2×Q8—272no—
370C2×Q8—271no—
371C2×D4—91no—
372C22×C4—91no—
373C22×C4D3×D911yescharacteristic
374C4⋊C4—31no—
375C4⋊C4—31no—
376C4⋊C4—91no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—271no—
381C4⋊C4—272no—
382C22⋊C4—32no—
383C22⋊C4—91no—
384C22⋊C4—91no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C42—32no—
389C42—91no—
390C42—272no—
391C42—272no—
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—181no—
403C22×C3—181no—
404C22×C3—181no—
405C22×C3C23×D911yescharacteristic
406C22×C3C22×D3211yescharacteristic
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—92no—
417C3×C4—181no—
418C3×C4—181no—
419C3×C4#369712yes—
420C3×C4#380412yes—
421Dic3—22no—
422Dic3—22no—
423Dic3—32no—
424Dic3—61no—
425Dic3—61no—
426Dic3—182no—
427Dic3—182no—
428Dic3—182no—
429Dic3—182no—
430Dic3—362no—
431Dic3—362no—
432C32#553311yescharacteristic
433C9—21no—
434C9#554211yescharacteristic, complemented
435C23—91no—
436C23C2×D3×D911yescharacteristic
437Q8—544no—
438Q8—542no—
439D4—182no—
440C2×C4—22no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—182no—
447C2×C4—272no—
448C2×C4—272no—
449C2×C4C2×D3×D911yescharacteristic
450C2×C4C2×D3×D911yescharacteristic
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—42no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—361no—
458C2×C3C2×#369711yescharacteristic
459C2×C3#41×D911yescharacteristic
460C2×C3#6371611yescharacteristic
461C2×C3#6429811yescharacteristic
462C2×C3D3×#23511yescharacteristic
463C2×C3C2×#380411yescharacteristic
464D3—121no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—181no—
469C22—181no—
470C22—181no—
471C22C22×D3×D911yescharacteristic
472C4—21no—
473C4—62no—
474C4—62no—
475C4—92no—
476C4—181no—
477C4—181no—
478C4—542no—
479C4—542no—
480C4#8865812yes—
481C3—21no—
482C3#1058963511yescharacteristic, complemented
483C3#1059484011yescharacteristic
484C2—41no—
485C2—361no—
486C2#1172246511yescharacteristic
487C2D9×#23511yescharacteristic
488C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721813C2semidirect
3. C9×#926C2semidirect
5. #11721247C2semidirect
6. #11720969C2semidirect
7. #11720947C2semidirect
8. #11720945C2semidirect
9. #11720913C2semidirect
10. #11720907C2semidirect
11. #11720900C2semidirect
150. #926D9semidirect
315. C3×C9#556semidirect
434. C9#5542semidirect
482. C3#10589635semidirect

Maximal subgroups

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

subgroupindex
#117218132
C3×C4×#6112
C9×#9262
#117212572
#117212472
#117209692
#117209472
#117209452
#117209132
#117209072
#117209002
#105948403
#105896353