Finite Group Databasebeta

← #13150952

Subgroups — #13150952

Order 1728.

724 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721836C211yesmaximal, characteristic, complemented
2C3×#63500C211yesmaximal, characteristic
3D4×C9×Dic3C211yesmaximal, characteristic, complemented
4#11721259C212yesmaximal, complemented
5#11721257C212yesmaximal, complemented
6#11721249C212yesmaximal, complemented
7#11721246C212yesmaximal, complemented
8#11720971C211yesmaximal, characteristic, complemented
9#11720945C211yesmaximal, characteristic, complemented
10#11720943C211yesmaximal, characteristic, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10595124—31nomaximal
13#10589663—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×#611C2212yes—
17C22×C9×Dic3C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×#6189C2212yes—
20#88551C2212yes—
21#88551C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C9×#217C2212yes—
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28(C2×D9)⋊Dic3C2211yescharacteristic
29(C2×D9)⋊Dic3C2212yes—
30(C2×D9)⋊Dic3C2211yescharacteristic
31Dic3⋊Dic9C2212yes—
32#88442C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2211yescharacteristic
35Dic9⋊Dic3C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64145—31no—
38C3×#938—31no—
39C3×D4×Dic3D311yescharacteristic
40#63964—32no—
41#63961—32no—
42#63959—32no—
43#63956—32no—
44#63919—31no—
45#63857—31no—
46#63855—31no—
47#63844—31no—
48C4×D4×C9—31no—
49#63500D311yescharacteristic
50#63498—31no—
51#63498—61no—
52#63491—32no—
53#63486—32no—
54#63459—31no—
55#63450—31no—
56#63447—32no—
57#63441—32no—
58#63434—31no—
59C4×Dic18—31no—
60C23×C3×C9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C3×D4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67#6226—22no—
68C3×#611—42no—
69C2×C9×Dic3—42no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C3×C4×D9—22no—
76#6189—42no—
77#5603—91no—
78#5561—91no—
79C2×C32×D4C2×D311yescharacteristic
80C2×#634—31no—
81C2×C3×#67—32no—
82C22×C3×Dic3C2×D312yes—
83C2×C3×C4×D3—31no—
84C2×#627—32no—
85#3757—32no—
86#3757—32no—
87C4×C3⋊Dic3—31no—
88C3×#217—32no—
89C3×#217C2×D312yes—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D311yescharacteristic
93#3724—32no—
94#3724—32no—
95Dic3⋊Dic3—31no—
96Dic3⋊Dic3—31no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—31no—
99(C2×D3)⋊Dic3—31no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C2×D4×C9—21no—
103C2×D4×C9C2×D311yescharacteristic
104C22×C4×C9—32no—
105C2×#611C2×D312yes—
106C2×C4×D9C2×D311yescharacteristic
107C2×Dic18—31no—
108C2×Dic18—32no—
109C2×Dic18—61no—
110C9×C4⋊C4—31no—
111C9×C22⋊C4—32no—
112C42×C9—31no—
113#3676—32no—
114#3676—32no—
115#3676—62no—
116#3676—62no—
117#3676C2×D312yes—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—31no—
121C4⋊Dic9—32no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—31no—
126Dic9⋊C4—31no—
127C4×Dic9—31no—
128C4×Dic9—32no—
129C4×Dic9—61no—
130C22×C3×C9—22no—
131C22×C3×C9—22no—
132C22×C3×C9—22no—
133C22×C3×C9C2411yescharacteristic
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—22no—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C9×Dic3—22no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×#121—91no—
147C3×C4×D4—31no—
148#938—31no—
149#936—91no—
150#936—91no—
151#936—92no—
152#936—181no—
153#935—92no—
154D4×Dic3D911yescharacteristic, complemented
155#929—92no—
156#926—91no—
157#924—92no—
158#924—91no—
159#897—91no—
160#888—91no—
161#888—91no—
162#887—91no—
163#885—92no—
164#879—92no—
165#879—92no—
166#878—92no—
167#872—91no—
168C4×Dic6—91no—
169C23×C32C22×D312yes—
170C22×C3×D3—31no—
171C32×D4—22no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176#634—62no—
177C3×#67—122no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—42no—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D312yes—
184C3×C4×D3—62no—
185#627—122no—
186C23×C9—22no—
187C23×C9C22×D312yes—
188C22×D9C22×D311yescharacteristic
189D4×C9—22no—
190D4×C9—42no—
191C2×C4×C9—21no—
192C2×C4×C9—31no—
193C2×C4×C9—31no—
194C2×C4×C9—32no—
195C2×C4×C9—122no—
196C2×C4×C9C22×D311yescharacteristic
197#611—42no—
198C2×Dic9—31no—
199C2×Dic9—32no—
200C2×Dic9—31no—
201C2×Dic9—61no—
202C2×Dic9—61no—
203C2×Dic9—62no—
204C2×Dic9C22×D312yes—
205C2×Dic9C22×D311yescharacteristic
206C4×D9—22no—
207Dic18—62no—
208Dic18—122no—
209Dic18—122no—
210#542—271noSylow
211C2×C3×C9—42no—
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9H(5,𝔽2)11yescharacteristic
214C2×C3×C92−1+411yescharacteristic
215C3×D9—41no—
216C2×C3×D4—21no—
217C2×C3×D4—92no—
218C2×C3×D4C2×D911yescharacteristic
219C2×C3×D4D3211yescharacteristic
220C22×C3×C4—32no—
221C22×C3×C4—91no—
222C2×#67—32no—
223C22×Dic3C2×D912yes—
224C2×C4×D3—31no—
225C2×Dic6—91no—
226C2×Dic6—91no—
227C2×Dic6—92no—
228C2×Dic6—92no—
229C2×Dic6—181no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—91no—
232C3×C22⋊C4—32no—
233C3×C22⋊C4—92no—
234C3×C42—31no—
235#217—32no—
236#217—92no—
237#217—92no—
238#217—92no—
239#217—92no—
240#217—92no—
241#217—91no—
242#217—91no—
243#217—182no—
244#217—182no—
245#217C2×D912yes—
246(C2×D3)⋊C4—91no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—92no—
249C4⋊Dic3—31no—
250C4⋊Dic3—92no—
251C4⋊Dic3—91no—
252C4⋊Dic3—91no—
253C4⋊Dic3C2×D911yescharacteristic
254Dic3⋊C4—92no—
255Dic3⋊C4—91no—
256Dic3⋊C4—91no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261C4×Dic3—92no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—92no—
265C4×Dic3—181no—
266C4×Dic3C2×D911yescharacteristic
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32C23×D311yescharacteristic
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—61no—
274C32×C4#23512yes—
275C3⋊Dic3—62no—
276C3⋊Dic3—61no—
277C3⋊Dic3—61no—
278C3×Dic3—22no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—61no—
282C3×Dic3—62no—
283C22×C9—22no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9—21no—
287C22×C9—42no—
288C22×C9—42no—
289C22×C9—42no—
290C22×C9C23×D311yescharacteristic
291C2×D9—21no—
292C2×D9—21no—
293C2×D9—21no—
294C4×C9—22no—
295C4×C9—62no—
296C4×C9—61no—
297C4×C9—61no—
298C4×C9#23712yes—
299Dic9—21no—
300Dic9—22no—
301Dic9—62no—
302Dic9—61no—
303Dic9—61no—
304Dic9—122no—
305Dic9—121no—
306Dic9—121no—
307C4⋊Q8—271no—
308#126—272no—
309#124—272no—
310#124—271no—
311#123—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314#121—91no—
315C4×Q8—271no—
316C4×D4—31no—
317#117—271no—
318C3×C9#54211yescharacteristic, complemented, Sylow
319C23×C3—22no—
320C23×C3—91no—
321C23×C3C22×D912yes—
322C23×C3C2×D3212yes—
323C22×D3—31no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—42no—
327C3×D4—362no—
328C2×C3×C4—21no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—32no—
332C2×C3×C4—92no—
333C2×C3×C4—91no—
334C2×C3×C4—122no—
335C2×C3×C4—182no—
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338#67—122no—
339C2×Dic3—32no—
340C2×Dic3—31no—
341C2×Dic3—42no—
342C2×Dic3—91no—
343C2×Dic3—92no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—92no—
347C2×Dic3—91no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—182no—
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353C2×Dic3C22×D912yes—
354C4×D3—62no—
355Dic6—182no—
356Dic6—182no—
357Dic6—362no—
358Dic6—362no—
359Dic6—362no—
360C2×C32—42no—
361C2×C32C2×#23511yescharacteristic
362C2×C32#100411yescharacteristic
363C2×C32#101011yescharacteristic
364C3×D3—121no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—42no—
369C2×C9—82no—
370C2×C9C2×#23711yescharacteristic
371C2×C9#100411yescharacteristic
372C2×C9#101011yescharacteristic
373D9—41no—
374C2×Q8—272no—
375C2×Q8—271no—
376C2×D4—92no—
377C2×D4D3×D911yescharacteristic
378C22×C4—32no—
379C22×C4—91no—
380C4⋊C4—31no—
381C4⋊C4—91no—
382C4⋊C4—271no—
383C4⋊C4—271no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C4⋊C4—272no—
387C22⋊C4—32no—
388C22⋊C4—92no—
389C22⋊C4—271no—
390C22⋊C4—271no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—31no—
395C42—272no—
396C42—271no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—181no—
408C22×C3—181no—
409C22×C3—181no—
410C22×C3C23×D911yescharacteristic
411C22×C3C22×D3211yescharacteristic
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—61no—
415C3×C4—22no—
416C3×C4—61no—
417C3×C4—62no—
418C3×C4—61no—
419C3×C4—181no—
420C3×C4—182no—
421C3×C4#369712yes—
422C3×C4#380412yes—
423Dic3—22no—
424Dic3—21no—
425Dic3—21no—
426Dic3—62no—
427Dic3—61no—
428Dic3—181no—
429Dic3—181no—
430Dic3—182no—
431Dic3—182no—
432Dic3—181no—
433Dic3—181no—
434Dic3—362no—
435Dic3—361no—
436Dic3—361no—
437C32#556111yescharacteristic
438C9—21no—
439C9#560311yescharacteristic, complemented
440C23—91no—
441C23C2×D3×D912yes—
442Q8—542no—
443Q8—1082no—
444D4—22no—
445D4—362no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—92no—
450C2×C4—91no—
451C2×C4—122no—
452C2×C4—182no—
453C2×C4—271no—
454C2×C4—272no—
455C2×C4—271no—
456C2×C4C2×D3×D911yescharacteristic
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—42no—
461C2×C3—42no—
462C2×C3—82no—
463C2×C3—361no—
464C2×C3C2×#369711yescharacteristic
465C2×C3#6371011yescharacteristic
466C2×C3#6371611yescharacteristic
467C2×C3#6430911yescharacteristic
468C2×C3#6431311yescharacteristic
469C2×C3C2×#380411yescharacteristic
470D3—121no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—181no—
475C22—181no—
476C22—181no—
477C22C22×D3×D911yescharacteristic
478C4—61no—
479C4—62no—
480C4—61no—
481C4—182no—
482C4—181no—
483C4—541no—
484C4—542no—
485C4—541no—
486C4#8865812yes—
487C3—21no—
488C3#1058966311yescharacteristic, complemented
489C3#1059512411yescharacteristic
490C2—42no—
491C2—361no—
492C2#1172249411yescharacteristic
493C2#1172249911yescharacteristic
494C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC3×#63500C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721836C2semidirect
3. D4×C9×Dic3C2semidirect
4. #11721259C2semidirect
5. #11721257C2semidirect
6. #11721249C2semidirect
7. #11721246C2semidirect
8. #11720971C2semidirect
9. #11720945C2semidirect
10. #11720943C2semidirect
11. #11720912C2semidirect
154. D4×Dic3D9semidirect
318. C3×C9#542semidirect
439. C9#5603semidirect
488. C3#10589663semidirect

Maximal subgroups

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

subgroupindex
#117218362
C3×#635002
D4×C9×Dic32
#117212592
#117212572
#117212492
#117212462
#117209712
#117209452
#117209432
#117209122
#105951243
#105896633