Finite Group Databasebeta

← #13148826

Subgroups — #13148826

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C3⋊D9)⋊Q8C212yesmaximal
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic
3C9×#888C211yesmaximal, characteristic, complemented
4C3×#63450C211yesmaximal, characteristic, complemented
5#11721000C211yesmaximal, characteristic
6#11720980C212yesmaximal
7C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic
8#11720970C211yesmaximal, characteristic, complemented
9#11720969C211yesmaximal, characteristic, complemented
10#11720904C212yesmaximal, complemented
11#11720903C212yesmaximal, complemented
12#10594374—31nomaximal
13D9×C4⋊Q8—31nomaximal
14C2×C4×C3⋊D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C9×Dic6C2212yes—
17C2×C3×Dic18C2212yes—
18C2×(C3⋊D9)⋊C4C2212yes—
19C2×(C3⋊D9)⋊C4C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×#6189C2212yes—
22C2×#6189C2212yes—
23(C3⋊D9)⋊Q8—28no—
24C3×C9×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26#88534C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C9×Dic3⋊C4C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2211yescharacteristic
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3797—32no—
39C4⋊C4×C3⋊D3—31no—
40C3×#888—31no—
41C3×#888D311yescharacteristic
42#63918—31no—
43#63918—31no—
44#63900—31no—
45#63883—32no—
46C4×#624—31no—
47#63839—32no—
48#63839—32no—
49C2×Q8×D9—32no—
50C9×C4⋊Q8—31no—
51#63506—32no—
52C4⋊C4×D9—31no—
53C4⋊C4×D9—31no—
54C4⋊C4×D9—32no—
55C4⋊C4×D9—61no—
56#63450—32no—
57#63450—31no—
58#63450D311yescharacteristic, complemented
59C42×D9—31no—
60#63431—31no—
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9—24no—
67C4×C3⋊D9Q814yescomplemented
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C9×Dic6—24no—
75C3×Dic18—24no—
76(C3⋊D9)⋊C4—24no—
77(C3⋊D9)⋊C4D414yes—
78(C3⋊D9)⋊C4Q814yescomplemented
79#6189—24no—
80#6189—24no—
81D3×C4⋊Q8—91no—
82D3×C4⋊Q8—91no—
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—32no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6C2×D312yes—
87C2×#627—32no—
88C2×#627—32no—
89C2×#624—31no—
90C2×#624—31no—
91C2×#624—32no—
92#3797—68no—
93C32×C4⋊C4C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)—31no—
95#3750—32no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4C2×D312yes—
100C3×C4×Dic3—31no—
101C3×C4×Dic3C2×D311yescharacteristic
102#3724—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic32—31no—
106Dic32—31no—
107C2×Q8×C9—32no—
108Q8×D9—68no—
109C2×C4×D9—31no—
110C2×C4×D9—31no—
111C2×C4×D9—31no—
112C2×C4×D9—32no—
113C2×C4×D9—32no—
114C2×C4×D9—61no—
115C2×C4×D9—62no—
116C2×Dic18—32no—
117C2×Dic18—32no—
118C2×Dic18C2×D312yes—
119C9×C4⋊C4—21no—
120C9×C4⋊C4—32no—
121C9×C4⋊C4—31no—
122C9×C4⋊C4C2×D311yescharacteristic
123C42×C9—31no—
124C4⋊Dic9—31no—
125C4⋊Dic9—32no—
126C4⋊Dic9—61no—
127C4⋊Dic9C2×D311yescharacteristic
128Dic9⋊C4—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4—32no—
131Dic9⋊C4—62no—
132Dic9⋊C4C2×D312yes—
133C4×Dic9—31no—
134C4×Dic9—31no—
135C4×Dic9C2×D311yescharacteristic
136C22×C3×C9C2411yescharacteristic
137C2×C3⋊D9C2×D412yes—
138C2×C3⋊D9C2×Q812yes—
139C2×C3⋊D9C2×Q812yes—
140C3×C4×C9—22no—
141C3×C4×C9C2×Q812yescomplemented
142C3⋊Dic9—22no—
143C3⋊Dic9C2×Q812yescomplemented
144C9×Dic3—22no—
145C9×Dic3C2×D412yes—
146C9×Dic3C2×Q812yescomplemented
147C3×Dic9—22no—
148C3×Dic9C2×D412yes—
149C3×Dic9C2×Q812yescomplemented
150C2×D3×Q8—92no—
151C2×D3×Q8—92no—
152C3×C4⋊Q8—31no—
153C3×C4⋊Q8—91no—
154#944—92no—
155#944—92no—
156D3×C4⋊C4—92no—
157D3×C4⋊C4—91no—
158D3×C4⋊C4—91no—
159D3×C4⋊C4—91no—
160D3×C4⋊C4—91no—
161D3×C4⋊C4—92no—
162D3×C4⋊C4—181no—
163#888—31no—
164#888—91no—
165#888—92no—
166#888—92no—
167#888—91no—
168#888D911yescharacteristic, complemented
169C42×D3—91no—
170C42×D3—91no—
171#869—91no—
172#869—91no—
173C22×C3⋊D3—31no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—32no—
178C4×C3⋊D3—34no—
179C4×C3⋊D3—64no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3C22×D312yes—
184C2×C3×Dic3C22×D311yescharacteristic
185C2×C3×Dic3C22×D311yescharacteristic
186C3×Dic6—24no—
187C3×Dic6—64no—
188#627—64no—
189#627—64no—
190#624—34no—
191#624—34no—
192#624—64no—
193C22×D9—31no—
194C22×D9—61no—
195Q8×C9—64no—
196C2×C4×C9—21no—
197C2×C4×C9—22no—
198C2×C4×C9—31no—
199C2×C4×C9—31no—
200C2×C4×C9—32no—
201C2×C4×C9C22×D312yes—
202C2×C4×C9C22×D311yescharacteristic
203C2×Dic9—31no—
204C2×Dic9—32no—
205C2×Dic9—61no—
206C2×Dic9—62no—
207C2×Dic9C22×D312yes—
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D311yescharacteristic
210C4×D9—34no—
211C4×D9—34no—
212C4×D9—34no—
213C4×D9—64no—
214C4×D9—64no—
215C4×D9—64no—
216C4×D9—124no—
217Dic18—24no—
218Dic18—64no—
219Dic18—64no—
220C2×C4⋊Q8—271noSylow
221C2×C3×C9C22×D411yescharacteristic
222C2×C3×C9C22×Q811yescharacteristic
223C2×C3×C9C22×Q811yescharacteristic
224C3⋊D9C4⋊Q814yescomplemented
225C2×C3×Q8—32no—
226C2×C3×Q8—92no—
227D3×Q8—188no—
228D3×Q8—188no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—91no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3—181no—
240C2×C4×D3—182no—
241C2×Dic6—32no—
242C2×Dic6—92no—
243C2×Dic6—92no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C2×Dic6C2×D912yes—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4C2×D911yescharacteristic
253C3×C4⋊C4D3211yescharacteristic
254C3×C42—31no—
255C3×C42—91no—
256C4⋊Dic3—31no—
257C4⋊Dic3—92no—
258C4⋊Dic3—91no—
259C4⋊Dic3—91no—
260C4⋊Dic3—92no—
261C4⋊Dic3—181no—
262C4⋊Dic3C2×D911yescharacteristic
263Dic3⋊C4—32no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—182no—
271Dic3⋊C4C2×D912yes—
272C4×Dic3—31no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—91no—
276C4×Dic3—91no—
277C4×Dic3C2×D911yescharacteristic
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—32no—
280C2×C3⋊D3—32no—
281C2×C3⋊D3—32no—
282C32×C4—22no—
283C32×C4D3×Q812yes—
284C3⋊Dic3—32no—
285C3⋊Dic3—62no—
286C3×Dic3—22no—
287C3×Dic3—32no—
288C3×Dic3—32no—
289C3×Dic3—62no—
290C3×Dic3D3×D412yes—
291C3×Dic3D3×Q812yes—
292C22×C9—21no—
293C22×C9C23×D311yescharacteristic
294C2×D9—32no—
295C2×D9—32no—
296C2×D9—32no—
297C2×D9—62no—
298C2×D9—62no—
299C2×D9—62no—
300C4×C9—22no—
301C4×C9—22no—
302C4×C9—32no—
303C4×C9—32no—
304C4×C9—42no—
305C4×C9—62no—
306C4×C9D3×Q812yescomplemented
307Dic9—22no—
308Dic9—32no—
309Dic9—62no—
310Dic9—62no—
311Dic9—122no—
312Dic9D3×D412yes—
313Dic9D3×Q812yescomplemented
314C22×Q8—272no—
315C4⋊Q8—31no—
316C4⋊Q8—91no—
317C4⋊Q8—271no—
318C4⋊Q8—271no—
319C4⋊Q8—272no—
320C4⋊Q8—272no—
321C2×C4⋊C4—272no—
322C2×C4⋊C4—271no—
323C2×C4⋊C4—271no—
324C2×C42—271no—
325C3×C9C2×C4⋊Q811yescharacteristic, complemented, Sylow
326C22×D3—91no—
327C22×D3—91no—
328C22×D3—181no—
329C3×Q8—64no—
330C3×Q8—184no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—32no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—92no—
339C2×C3×C4C22×D912yes—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C2×D3212yes—
342C2×C3×C4C2×D3211yescharacteristic
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—32no—
346C2×Dic3—91no—
347C2×Dic3—91no—
348C2×Dic3—92no—
349C2×Dic3—92no—
350C2×Dic3—181no—
351C2×Dic3—182no—
352C2×Dic3C22×D912yes—
353C2×Dic3C22×D911yescharacteristic
354C2×Dic3C22×D911yescharacteristic
355C4×D3—94no—
356C4×D3—94no—
357C4×D3—94no—
358C4×D3—94no—
359C4×D3—94no—
360C4×D3—94no—
361C4×D3—184no—
362C4×D3—184no—
363C4×D3—184no—
364C4×D3—184no—
365C4×D3—184no—
366C4×D3—364no—
367Dic6—24no—
368Dic6—64no—
369Dic6—184no—
370Dic6—184no—
371Dic6—184no—
372Dic6—184no—
373C2×C32C2×D3×D411yescharacteristic
374C2×C32C2×D3×Q811yescharacteristic
375C2×C32C2×D3×Q811yescharacteristic
376C3⋊D3—34no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9C2×D3×D411yescharacteristic
381C2×C9C2×D3×Q811yescharacteristic
382C2×C9C2×D3×Q811yescharacteristic
383D9—34no—
384D9—64no—
385C2×Q8—32no—
386C2×Q8—92no—
387C2×Q8—272no—
388C2×Q8—272no—
389C2×Q8—548no—
390C22×C4—271no—
391C22×C4—271no—
392C22×C4—272no—
393C22×C4—271no—
394C22×C4—272no—
395C4⋊C4—32no—
396C4⋊C4—31no—
397C4⋊C4—92no—
398C4⋊C4—91no—
399C4⋊C4—272no—
400C4⋊C4—272no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C4⋊C4—271no—
404C4⋊C4D3×D911yescharacteristic
405C42—31no—
406C42—91no—
407C42—271no—
408C42—271no—
409C22×C3—21no—
410C22×C3C23×D911yescharacteristic
411C22×C3C22×D3211yescharacteristic
412C2×D3—92no—
413C2×D3—92no—
414C2×D3—92no—
415C2×D3—92no—
416C2×D3—92no—
417C2×D3—92no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—182no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—32no—
425C3×C4—32no—
426C3×C4—42no—
427C3×C4—62no—
428C3×C4—92no—
429C3×C4—92no—
430C3×C4—182no—
431C3×C4Q8×D912yescomplemented
432C3×C4#379712yes—
433Dic3—22no—
434Dic3—32no—
435Dic3—32no—
436Dic3—62no—
437Dic3—92no—
438Dic3—92no—
439Dic3—182no—
440Dic3—182no—
441Dic3—182no—
442Dic3—362no—
443Dic3D4×D912yes—
444Dic3Q8×D912yescomplemented
445C32D3×C4⋊Q811yescharacteristic
446C9—21no—
447C9D3×C4⋊Q811yescharacteristic, complemented
448C23—271no—
449Q8—64no—
450Q8—184no—
451Q8—544no—
452Q8—544no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—32no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—92no—
459C2×C4—274no—
460C2×C4—274no—
461C2×C4—271no—
462C2×C4—274no—
463C2×C4—272no—
464C2×C4—544no—
465C2×C4—544no—
466C2×C4C2×D3×D912yes—
467C2×C4C2×D3×D911yescharacteristic
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3C2×D4×D911yescharacteristic
472C2×C3C2×Q8×D911yescharacteristic
473C2×C3C2×Q8×D911yescharacteristic
474C2×C3C2×#379711yescharacteristic
475C2×C3D32×D411yescharacteristic
476C2×C3D32×Q811yescharacteristic
477D3—94no—
478D3—94no—
479D3—184no—
480C22—272no—
481C22—272no—
482C22—272no—
483C22C22×D3×D911yescharacteristic
484C4—22no—
485C4—32no—
486C4—32no—
487C4—62no—
488C4—92no—
489C4—92no—
490C4—182no—
491C4—272no—
492C4—542no—
493C4(C3⋊D9)⋊Q812yescomplemented
494C3—21no—
495C3D9×C4⋊Q811yescharacteristic, complemented
496C3#1059437411yescharacteristic
497C2—274no—
498C2C2×(C3⋊D9)⋊Q811yescharacteristic
499C2D3×D4×D911yescharacteristic
500C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×C4⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C9×#888C2semidirect
4. C3×#63450C2semidirect
8. #11720970C2semidirect
9. #11720969C2semidirect
10. #11720904C2semidirect
11. #11720903C2semidirect
58. #63450D3semidirect
67. C4×C3⋊D9Q8semidirect
78. (C3⋊D9)⋊C4Q8semidirect
141. C3×C4×C9C2×Q8semidirect
143. C3⋊Dic9C2×Q8semidirect
146. C9×Dic3C2×Q8semidirect
149. C3×Dic9C2×Q8semidirect
168. #888D9semidirect
224. C3⋊D9C4⋊Q8semidirect
306. C4×C9D3×Q8semidirect
313. Dic9D3×Q8semidirect
325. C3×C9C2×C4⋊Q8semidirect
431. C3×C4Q8×D9semidirect
444. Dic3Q8×D9semidirect
447. C9D3×C4⋊Q8semidirect
493. C4(C3⋊D9)⋊Q8semidirect
495. C3D9×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C4⋊C4×C3⋊D92
C9×#8882
C3×#634502
#117210002
#117209802
C4×(C3⋊D9)⋊C42
#117209702
#117209692
#117209042
#117209032
#105943743
D9×C4⋊Q83