Finite Group Databasebeta

← #13147675

Subgroups — #13147675

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic, complemented
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11721040C211yesmaximal, characteristic
5#11721035C212yesmaximal, complemented
6D12×Dic9C211yesmaximal, characteristic, complemented
7#11721020C212yesmaximal, complemented
8#11721014C212yesmaximal, complemented
9#11720957C212yesmaximal, complemented
10#11720953C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10593939—31nomaximal
13#10589783—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×D12C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6195C2212yes—
20C2×#6195C2212yes—
21C2×D3×Dic9C2212yes—
22C3×C42×C9C2211yescharacteristic
23#88536C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C9×(C2×D3)⋊C4C2212yes—
26C9×(C2×D3)⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33#88445C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic3×Dic9C2212yes—
37C4×C4⋊(C3⋊D3)—31no—
38C3×#875D311yescharacteristic
39C3×C4×Dic6—31no—
40#63911—31no—
41#63907—32no—
42Dic3×D12—31no—
43#63890—32no—
44#63881—32no—
45#63864—31no—
46#63853—32no—
47Dic3×Dic6—31no—
48C9×#124—31no—
49#63509—31no—
50Q8×Dic9—31no—
51#63502—31no—
52D4×Dic9—31no—
53#63448—32no—
54#63447—32no—
55#63445—32no—
56#611⋊C4—32no—
57C4×D36—31no—
58C4×D36—61no—
59C4×Dic18D311yescharacteristic, complemented
60C22×C3⋊D9C2312yes—
61C22×D3×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4⋊(C3⋊D9)—22no—
67C4×C3⋊D9—42no—
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×D12—22no—
74C9×Dic6—22no—
75C3×Dic18—22no—
76#6195—24no—
77#6195—42no—
78D3×Dic9—42no—
79#5681—91no—
80#5585—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4×C3⋊D3—32no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×Dic6—31no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#626—32no—
87C2×#626—32no—
88C2×D3×Dic3—32no—
89C32×C42C2×D311yescharacteristic
90#3752—32no—
91C4⋊(C3⋊Dic3)—31no—
92C3×(C2×D3)⋊C4C2×D312yes—
93C3×(C2×D3)⋊C4C2×D312yes—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—31no—
96C3×Dic3⋊C4—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99Dic3⋊Dic3—32no—
100#3722—32no—
101#3722—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×Q8×C9—31no—
105C2×D4×C9—31no—
106C2×#611—32no—
107C2×#611—32no—
108C22×Dic9—32no—
109C2×D36—31no—
110C2×D36—61no—
111C2×C4×D9—32no—
112C2×C4×D9—62no—
113C2×Dic18C2×D311yescharacteristic
114C9×C22⋊C4—32no—
115C9×C22⋊C4—32no—
116C42×C9—21no—
117C42×C9C2×D311yescharacteristic
118#3676—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—62no—
123C4⋊Dic9—31no—
124C4⋊Dic9—32no—
125C4⋊Dic9—61no—
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4C2×D311yescharacteristic
128Dic9⋊C4C2×D311yescharacteristic
129C4×Dic9—32no—
130C4×Dic9C2×D311yescharacteristic
131C4×Dic9C2×D311yescharacteristic
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3⋊Dic9—22no—
143C9×Dic3—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9—21no—
147C3×Dic9#4112yes—
148C3×#124—31no—
149C3×C4×Q8—91no—
150#947—91no—
151#947—92no—
152Q8×Dic3—91no—
153#940—91no—
154D4×Dic3—91no—
155#894—92no—
156#894—92no—
157#893—92no—
158#886—92no—
159#885—92no—
160#883—92no—
161#67⋊C4—92no—
162#875—91no—
163#875D911yescharacteristic, complemented
164#874—91no—
165C4×D12—91no—
166C4×D12—91no—
167C4×D12—91no—
168C4×D12—181no—
169C4×Dic6—31no—
170C4×Dic6—91no—
171C22×C3⋊D3—32no—
172C22×C3×D3C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—32no—
177C4⋊(C3⋊D3)—62no—
178C4×C3⋊D3—122no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3C22×D312yes—
184C3×D12—22no—
185C3×Dic6—22no—
186C3×Dic6—62no—
187#626—64no—
188#626—122no—
189D3×Dic3—122no—
190C23×C9—32no—
191C22×D9—32no—
192C22×D9—62no—
193Q8×C9—62no—
194D4×C9—62no—
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—32no—
199C2×C4×C9C22×D311yescharacteristic
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202#611—64no—
203#611—122no—
204C2×Dic9—32no—
205C2×Dic9—62no—
206C2×Dic9—122no—
207C2×Dic9C22×D311yescharacteristic
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D311yescharacteristic
210C2×Dic9C22×D311yescharacteristic
211D36—64no—
212D36—62no—
213C4×D9—122no—
214C4×D9—124no—
215Dic18—22no—
216#559—271noSylow
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9H(5,𝔽2)11yescharacteristic
220C3⋊D9—42no—
221D3×C9—42no—
222C2×C3×Q8—31no—
223C2×C3×Q8—91no—
224C2×C3×D4—31no—
225C2×#67—92no—
226C2×#67—92no—
227C22×Dic3—92no—
228C2×D12—92no—
229C2×D12—91no—
230C2×D12—91no—
231C2×D12—92no—
232C2×D12—181no—
233C2×D12C2×D911yescharacteristic
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×C4×D3—182no—
238C2×Dic6—31no—
239C2×Dic6C2×D911yescharacteristic
240C3×C4⋊C4—91no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—91no—
243C3×C22⋊C4—32no—
244C3×C22⋊C4—32no—
245C3×C42—21no—
246C3×C42—91no—
247C3×C42—91no—
248C3×C42C2×D911yescharacteristic
249C3×C42D3211yescharacteristic
250#217—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—182no—
259(C2×D3)⋊C4C2×D912yes—
260(C2×D3)⋊C4C2×D912yes—
261C4⋊Dic3—31no—
262C4⋊Dic3—92no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—181no—
266Dic3⋊C4—31no—
267Dic3⋊C4—31no—
268Dic3⋊C4—92no—
269C4×Dic3—31no—
270C4×Dic3—31no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C22×C32C23×D311yescharacteristic
274C2×C3⋊D3—62no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C2×C3×D3—22no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4#23712yes—
283C3⋊Dic3—62no—
284C3×Dic3—22no—
285C3×Dic3—32no—
286C3×Dic3—61no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C22×C9—21no—
290C22×C9—62no—
291C22×C9—62no—
292C22×C9—62no—
293C22×C9C23×D311yescharacteristic
294C2×D9—64no—
295C2×D9—62no—
296C2×D9—62no—
297C2×D9—62no—
298C2×D9—122no—
299C2×D9—122no—
300C4×C9—22no—
301C4×C9—21no—
302C4×C9—21no—
303C4×C9—22no—
304C4×C9—41no—
305C4×C9—62no—
306C4×C9#23912yes—
307Dic9—21no—
308Dic9—21no—
309Dic9—21no—
310Dic9—62no—
311Dic9—122no—
312Dic9#23512yes—
313#127—271no—
314#124—31no—
315#124—271no—
316#124—272no—
317#123—272no—
318#123—272no—
319C4×Q8—91no—
320C4×Q8—271no—
321C4×D4—271no—
322C4×D4—272no—
323C4×D4—271no—
324C3×C9#55911yescharacteristic, complemented, Sylow
325C23×C3—32no—
326C22×D3—92no—
327C22×D3—92no—
328C22×D3—182no—
329C22×D3C22×D912yes—
330C3×Q8—62no—
331C3×Q8—182no—
332C3×D4—62no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—32no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347#67—184no—
348#67—362no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—31no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—182no—
356C2×Dic3—362no—
357C2×Dic3C22×D912yes—
358D12—22no—
359D12—184no—
360D12—184no—
361D12—182no—
362D12—182no—
363D12—362no—
364C4×D3—362no—
365C4×D3—362no—
366C4×D3—362no—
367C4×D3—364no—
368Dic6—22no—
369Dic6—62no—
370C2×C32C2×#23711yescharacteristic
371C2×C32D3×#4111yescharacteristic
372C2×C32#100911yescharacteristic
373C3⋊D3—122no—
374C3×D3—42no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—122no—
379C2×C9C2×#23511yescharacteristic
380C2×C9C2×#23911yescharacteristic
381C2×C9#100911yescharacteristic
382D9—124no—
383D9—122no—
384C2×Q8—31no—
385C2×Q8—91no—
386C2×D4—31no—
387C2×D4—272no—
388C2×D4—272no—
389C2×D4—271no—
390C22×C4—272no—
391C22×C4—272no—
392C4⋊C4—91no—
393C4⋊C4—91no—
394C4⋊C4—91no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—32no—
398C22⋊C4—32no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—91no—
404C42—91no—
405C42—272no—
406C42D3×D911yescharacteristic
407C22×C3—21no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3C23×D911yescharacteristic
412C22×C3C22×D3211yescharacteristic
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—22no—
416C2×D3—184no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—362no—
424C2×D3—362no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—62no—
433C3×C4—92no—
434C3×C4—181no—
435C3×C4—181no—
436C3×C4—181no—
437C3×C4#369912yes—
438C3×C4#379612yes—
439Dic3—22no—
440Dic3—32no—
441Dic3—61no—
442Dic3—61no—
443Dic3—61no—
444Dic3—182no—
445Dic3—182no—
446Dic3—362no—
447C32#568111yescharacteristic
448C9—21no—
449C9#558511yescharacteristic, complemented
450C23—32no—
451C23—272no—
452Q8—62no—
453Q8—182no—
454D4—62no—
455D4—544no—
456D4—542no—
457D4—1082no—
458C2×C4—32no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—272no—
464C2×C4—1082no—
465C2×C4—1082no—
466C2×C4C2×D3×D911yescharacteristic
467C2×C4C2×D3×D911yescharacteristic
468C2×C4C2×D3×D911yescharacteristic
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—122no—
473C2×C3C2×#369911yescharacteristic
474C2×C3#41×D911yescharacteristic
475C2×C3#6371511yescharacteristic
476C2×C3C2×#379611yescharacteristic
477C2×C3D3×#23511yescharacteristic
478C2×C3#6430811yescharacteristic
479D3—42no—
480D3—364no—
481D3—362no—
482D3—362no—
483C22—62no—
484C22—62no—
485C22—62no—
486C22—542no—
487C22—542no—
488C22—542no—
489C22C22×D3×D911yescharacteristic
490C4—21no—
491C4—21no—
492C4—62no—
493C4—92no—
494C4—181no—
495C4—181no—
496C4—181no—
497C4—542no—
498C4#8863912yes—
499C3—21no—
500C3#1058978311yescharacteristic, complemented
501C3#1059393911yescharacteristic
502C2—122no—
503C2—1082no—
504C2C2×#8863911yescharacteristic
505C2D9×#23511yescharacteristic
506C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721040C2
hypercenterC42D3×D9
hypocenterC3×C9#559
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×C4⋊(C3⋊D9)C2semidirect
2. C9×#875C2semidirect
3. C3×C4×Dic18C2semidirect
5. #11721035C2semidirect
6. D12×Dic9C2semidirect
7. #11721020C2semidirect
8. #11721014C2semidirect
9. #11720957C2semidirect
10. #11720953C2semidirect
11. Dic6×Dic9C2semidirect
59. C4×Dic18D3semidirect
163. #875D9semidirect
324. C3×C9#559semidirect
449. C9#5585semidirect
500. C3#10589783semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D9)2
C9×#8752
C3×C4×Dic182
#117210402
#117210352
D12×Dic92
#117210202
#117210142
#117209572
#117209532
Dic6×Dic92
#105939393
#105897833