Finite Group Databasebeta

← #13150091

Subgroups — #13150091

Order 1728.

734 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
1C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
2C3×C4×#611C211yesmaximal, characteristic
3C9×#924C211yesmaximal, characteristic, complemented
4#11721260C212yesmaximal
5#11721248C212yesmaximal, complemented
6#11720949C212yesmaximal, complemented
7#11720944C211yesmaximal, characteristic, complemented
8#11720935C211yesmaximal, characteristic, complemented
9#11720914C212yesmaximal, complemented
10Dic6×Dic9C211yesmaximal, characteristic, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10594853—31nomaximal
13#10589744—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C3×#611C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C9×#217C2212yes—
23C3×#3676C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C3×Dic9⋊C4C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29(C2×D9)⋊Dic3C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34#88442C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37C2×C4⋊(C3⋊Dic3)—31no—
38C3×C4×#67—31no—
39C3×#924D311yescharacteristic
40#63962—32no—
41#63957—32no—
42#63869—31no—
43#63861—32no—
44#63856—31no—
45#63845—32no—
46Dic3×Dic6—31no—
47#63841—31no—
48C9×C22⋊Q8—31no—
49#63508—31no—
50Q8×Dic9—31no—
51C4×#611D311yescharacteristic
52C2×C4⋊Dic9—31no—
53C2×C4⋊Dic9—61no—
54#63460—32no—
55#63454—31no—
56#63452—32no—
57#63452—31no—
58#63448—32no—
59#63439—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9—24no—
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68C3×#611—22no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75C9×Dic6—22no—
76#5642—91no—
77#5541—91no—
78C22×C32×C4C2×D311yescharacteristic
79C22×C3⋊Dic3—32no—
80C2×C3×#67—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6C2×D311yescharacteristic
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—32no—
85C4⋊(C3⋊Dic3)—31no—
86C3×#217—31no—
87C3×#217C2×D312yes—
88C3×(C2×D3)⋊C4—31no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—31no—
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—31no—
93#3724—32no—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99Dic32—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C22×C4×C9—21no—
103C22×C4×C9C2×D311yescharacteristic
104C2×#611C2×D311yescharacteristic
105C22×Dic9—32no—
106C22×Dic9—62no—
107C2×C4×D9C2×D311yescharacteristic
108C9×C4⋊C4—32no—
109C9×C4⋊C4—31no—
110C9×C22⋊C4—32no—
111#3676C2×D311yescharacteristic
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4C2×D311yescharacteristic
115C4⋊Dic9—31no—
116C4⋊Dic9—31no—
117C4⋊Dic9—32no—
118C4⋊Dic9—32no—
119C4⋊Dic9—32no—
120C4⋊Dic9—62no—
121C4⋊Dic9—61no—
122C4⋊Dic9—61no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4C2×D311yescharacteristic
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9#4112yes—
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—21no—
145C3×C22⋊Q8—31no—
146C3×C4×D4—91no—
147#946—91no—
148Q8×Dic3—91no—
149#935—92no—
150C4×#67—31no—
151#926—91no—
152C2×C4⋊Dic3—91no—
153C2×C4⋊Dic3—91no—
154C2×C4⋊Dic3—181no—
155#924—91no—
156#924D911yescharacteristic, complemented
157#898—92no—
158#892—91no—
159#890—91no—
160#890—92no—
161#889—92no—
162#886—92no—
163#879—92no—
164#877—92no—
165#877—92no—
166#870—91no—
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—
175C2×C3⋊Dic3—64no—
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—
183C3×Dic6—22no—
184C23×C9—21no—
185C23×C9C22×D311yescharacteristic
186C22×D9C22×D311yescharacteristic
187Q8×C9—62no—
188C2×C4×C9—24no—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—32no—
193C2×C4×C9—32no—
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196#611—22no—
197C2×Dic9—32no—
198C2×Dic9—32no—
199C2×Dic9—62no—
200C2×Dic9—62no—
201C2×Dic9—64no—
202C2×Dic9—124no—
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D311yescharacteristic
205C2×Dic9C22×D311yescharacteristic
206C4×D9—22no—
207#552—271noSylow
208C2×C3×C9—22no—
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C92−1+411yescharacteristic
212C3×D9—41no—
213C2×C3×Q8—31no—
214C2×C3×D4—91no—
215C22×C3×C4—21no—
216C22×C3×C4—91no—
217C22×C3×C4C2×D911yescharacteristic
218C22×C3×C4D3211yescharacteristic
219C2×#67—31no—
220C22×Dic3—92no—
221C22×Dic3—92no—
222C22×Dic3—182no—
223C2×C4×D3—31no—
224C2×Dic6C2×D911yescharacteristic
225C3×C4⋊C4—31no—
226C3×C4⋊C4—32no—
227C3×C4⋊C4—91no—
228C3×C22⋊C4—32no—
229C3×C22⋊C4—91no—
230C3×C22⋊C4—91no—
231C3×C42—91no—
232#217—31no—
233#217—92no—
234#217—92no—
235#217C2×D912yes—
236(C2×D3)⋊C4—31no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239C4⋊Dic3—92no—
240C4⋊Dic3—92no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—92no—
244C4⋊Dic3—92no—
245C4⋊Dic3—91no—
246C4⋊Dic3—91no—
247C4⋊Dic3—181no—
248C4⋊Dic3—181no—
249C4⋊Dic3—182no—
250C4⋊Dic3C2×D911yescharacteristic
251Dic3⋊C4—31no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4C2×D912yes—
259C4×Dic3—31no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3—92no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32#23912yes—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—21no—
272C32×C4#23712yes—
273C3⋊Dic3—62no—
274C3⋊Dic3—62no—
275C3×Dic3—22no—
276C3×Dic3—22no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C3×Dic3—61no—
280C22×C9—22no—
281C22×C9—22no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—22no—
286C22×C9#23512yes—
287C22×C9C23×D311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—21no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—62no—
295C4×C9—62no—
296C4×C9#23712yes—
297Dic9—21no—
298Dic9—21no—
299Dic9—21no—
300Dic9—62no—
301Dic9—62no—
302Dic9—122no—
303Dic9—122no—
304#126—272no—
305#125—271no—
306#125—272no—
307#123—272no—
308C22⋊Q8—31no—
309C22⋊Q8—271no—
310C4×Q8—271no—
311C4×D4—91no—
312#117—272no—
313#117—271no—
314C2×C4⋊C4—271no—
315C3×C9#55211yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—91no—
318C23×C3C22×D911yescharacteristic
319C23×C3C2×D3211yescharacteristic
320C22×D3—31no—
321C3×Q8—62no—
322C3×D4—182no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—182no—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338#67—62no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—182no—
347C2×Dic3—182no—
348C2×Dic3—184no—
349C2×Dic3—184no—
350C2×Dic3—364no—
351C2×Dic3C22×D912yes—
352C2×Dic3C22×D912yes—
353C4×D3—62no—
354Dic6—22no—
355C2×C32—22no—
356C2×C32C2×#23711yescharacteristic
357C2×C32C2×#23911yescharacteristic
358C2×C32#101011yescharacteristic
359C3×D3—121no—
360C2×C9—24no—
361C2×C9—22no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9C2×#23511yescharacteristic
366C2×C9C2×#23711yescharacteristic
367C2×C9#101011yescharacteristic
368D9—41no—
369C2×Q8—31no—
370C2×D4—91no—
371C22×C4—91no—
372C22×C4—272no—
373C22×C4D3×D911yescharacteristic
374C4⋊C4—31no—
375C4⋊C4—32no—
376C4⋊C4—91no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4—271no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C22⋊C4—32no—
385C22⋊C4—91no—
386C22⋊C4—91no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—91no—
390C42—272no—
391C42—272no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3#370112yes—
404C22×C3C23×D911yescharacteristic
405C22×C3#379912yes—
406C22×C3C22×D3211yescharacteristic
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—181no—
417C3×C4—181no—
418C3×C4—181no—
419C3×C4#369912yes—
420C3×C4#380512yes—
421Dic3—22no—
422Dic3—22no—
423Dic3—61no—
424Dic3—61no—
425Dic3—61no—
426Dic3—182no—
427Dic3—182no—
428Dic3—182no—
429Dic3—182no—
430Dic3—362no—
431Dic3—362no—
432C32#564211yescharacteristic
433C9—21no—
434C9#554111yescharacteristic, complemented
435C23—91no—
436C23C2×D3×D911yescharacteristic
437Q8—62no—
438D4—182no—
439C2×C4—22no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—182no—
446C2×C4—272no—
447C2×C4—272no—
448C2×C4—544no—
449C2×C4C2×D3×D911yescharacteristic
450C2×C4C2×D3×D911yescharacteristic
451C2×C3—24no—
452C2×C3—22no—
453C2×C3—22no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—361no—
458C2×C3C2×#369911yescharacteristic
459C2×C3C2×#370111yescharacteristic
460C2×C3#6371611yescharacteristic
461C2×C3#6429811yescharacteristic
462C2×C3C2×#379911yescharacteristic
463C2×C3C2×#380511yescharacteristic
464D3—121no—
465C22—21no—
466C22—21no—
467C22—181no—
468C22—181no—
469C22—181no—
470C22#8863512yes—
471C22C22×D3×D911yescharacteristic
472C4—21no—
473C4—62no—
474C4—62no—
475C4—181no—
476C4—181no—
477C4—181no—
478C4—542no—
479C4—542no—
480C4#8866312yes—
481C3—21no—
482C3#1058974411yescharacteristic, complemented
483C3#1059485311yescharacteristic
484C2—22no—
485C2—361no—
486C2C2×#8863511yescharacteristic
487C2#1172246511yescharacteristic
488C2C2×#8866311yescharacteristic
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#552
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. C2×C4⋊(C3⋊Dic9)C2semidirect
3. C9×#924C2semidirect
5. #11721248C2semidirect
6. #11720949C2semidirect
7. #11720944C2semidirect
8. #11720935C2semidirect
9. #11720914C2semidirect
10. Dic6×Dic9C2semidirect
11. #11720907C2semidirect
156. #924D9semidirect
315. C3×C9#552semidirect
434. C9#5541semidirect
482. C3#10589744semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊Dic9)2
C3×C4×#6112
C9×#9242
#117212602
#117212482
#117209492
#117209442
#117209352
#117209142
Dic6×Dic92
#117209072
#105948533
#105897443