Finite Group Databasebeta

← #13147572

Subgroups — #13147572

Order 1728.

796 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
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C3×C42×D9C211yesmaximal, characteristic, complemented
3C4×C9×Dic6C211yesmaximal, characteristic, complemented
4#11721004C211yesmaximal, characteristic
5C4×#6192C212yesmaximal, complemented
6#11720953C211yesmaximal, characteristic, complemented
7#11720944C211yesmaximal, characteristic, complemented
8#11720938C212yesmaximal, complemented
9#11720935C211yesmaximal, characteristic, complemented
10#11720934C212yesmaximal, complemented
11#11720931C212yesmaximal, complemented
12#10593870—31nomaximal
13#10589682—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×#6192C2212yes—
21C3×C42×C9C2211yescharacteristic
22#88536C2212yes—
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C9×C4⋊Dic3C2211yescharacteristic
25C9×Dic3⋊C4C2212yes—
26C4×C9×Dic3C2212yes—
27C3×C4×Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30#88445C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36#88442C2212yes—
37C4×C4⋊(C3⋊D3)—31no—
38C3×C42×D3—31no—
39C3×C4×Dic6D311yescharacteristic
40#63909—31no—
41C4×#626—32no—
42#63874—32no—
43#63869—31no—
44#63868—32no—
45#63864—31no—
46#63856—31no—
47#63850—32no—
48C4×Q8×C9—31no—
49#63509—31no—
50#63508—31no—
51#63460—32no—
52#63457—31no—
53#63456—32no—
54#63452—31no—
55C4×D36—32no—
56C4×D36—31no—
57C4×D36—61no—
58#63434—32no—
59C42×D9D311yescharacteristic, complemented
60C22×C3⋊D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4⋊(C3⋊D9)—22no—
67C4×C3⋊D9—24no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—22no—
74C3×C4×D9—22no—
75C3×C4×D9—22no—
76C9×Dic6—22no—
77#6192—24no—
78#5580—91no—
79#5479—91no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4×C3⋊D3—32no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#626—32no—
87C32×C42C2×D311yescharacteristic
88#3752—32no—
89C4⋊(C3⋊Dic3)—31no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D312yes—
96#3724—32no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99#3722—32no—
100#3722—32no—
101#3722—32no—
102(C2×D3)⋊Dic3—32no—
103C2×Q8×C9—31no—
104C2×D36—32no—
105C2×D36—31no—
106C2×D36—61no—
107C2×C4×D9—32no—
108C2×C4×D9—62no—
109C2×C4×D9C2×D311yescharacteristic
110C2×C4×D9C2×D311yescharacteristic
111C2×C4×D9C2×D311yescharacteristic
112C9×C4⋊C4—32no—
113C9×C4⋊C4—31no—
114C42×C9—21no—
115C42×C9—32no—
116C42×C9C2×D311yescharacteristic
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—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—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—22no—
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9#4112yes—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3×C4×C9#4112yes—
141C3⋊Dic9—22no—
142C9×Dic3—22no—
143C9×Dic3—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9—21no—
147C3×C4×Q8—31no—
148C2×C3×C42—91no—
149#947—91no—
150#946—91no—
151#930—91no—
152#929—92no—
153C4×#67—92no—
154#924—91no—
155#898—92no—
156#895—91no—
157#894—92no—
158#890—91no—
159#876—92no—
160#875—91no—
161C4×D12—92no—
162C4×D12—91no—
163C4×D12—91no—
164C4×D12—181no—
165#872—92no—
166#872—92no—
167C42×D3—31no—
168#870—91no—
169C4×Dic6D911yescharacteristic, complemented
170C22×C3⋊D3—32no—
171C22×C3×D3—31no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C4⋊(C3⋊D3)—62no—
177C4×C3⋊D3—64no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D312yes—
183C3×C4×D3—62no—
184C3×C4×D3—62no—
185C3×C4×D3—62no—
186C3×Dic6—22no—
187#626—64no—
188C22×D9—32no—
189C22×D9—62no—
190C22×D9C22×D311yescharacteristic
191Q8×C9—62no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—21no—
195C2×C4×C9—32no—
196C2×C4×C9—32no—
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—32no—
201C2×Dic9—62no—
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D311yescharacteristic
205D36—64no—
206D36—64no—
207D36—62no—
208C4×D9—22no—
209C4×D9—22no—
210C4×D9—22no—
211C4×D9—64no—
212C4×D9—124no—
213#529—271noSylow
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C3⋊D9—42no—
218C3×D9—22no—
219C2×C3×Q8—31no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C2×#67—92no—
224C2×D12—92no—
225C2×D12—91no—
226C2×D12—91no—
227C2×D12—181no—
228C2×C4×D3—31no—
229C2×C4×D3—31no—
230C2×C4×D3—31no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—182no—
234C2×Dic6C2×D911yescharacteristic
235C3×C4⋊C4—31no—
236C3×C4⋊C4—32no—
237C3×C42—21no—
238C3×C42—32no—
239C3×C42—91no—
240C3×C42—91no—
241C3×C42—91no—
242C3×C42C2×D911yescharacteristic
243C3×C42D3211yescharacteristic
244#217—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—182no—
255C4⋊Dic3—92no—
256C4⋊Dic3—91no—
257C4⋊Dic3—91no—
258C4⋊Dic3—181no—
259C4⋊Dic3C2×D911yescharacteristic
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4C2×D912yes—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—31no—
269C4×Dic3C2×D912yes—
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—62no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—62no—
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C32×C4—21no—
278C32×C4#23512yes—
279C32×C4#23912yes—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C22×C9—21no—
287C22×C9C23×D311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—64no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—122no—
295C2×D9—122no—
296C2×D9#23512yes—
297C4×C9—21no—
298C4×C9—22no—
299C4×C9—22no—
300C4×C9—41no—
301C4×C9—62no—
302C4×C9—62no—
303C4×C9#23512yes—
304C4×C9#23512yes—
305Dic9—21no—
306Dic9—21no—
307Dic9—21no—
308Dic9—62no—
309Dic9—122no—
310#126—272no—
311#125—271no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—271no—
316C4×Q8—31no—
317C4×D4—272no—
318C4×D4—271no—
319#117—272no—
320C2×C42—91no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—91no—
323C22×D3—31no—
324C22×D3—92no—
325C22×D3—92no—
326C22×D3—182no—
327C3×Q8—62no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—182no—
337C2×C3×C4—182no—
338C2×C3×C4—182no—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345#67—184no—
346C2×Dic3—31no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—182no—
352C2×Dic3C22×D912yes—
353C2×Dic3C22×D912yes—
354D12—184no—
355D12—184no—
356D12—182no—
357D12—182no—
358C4×D3—62no—
359C4×D3—62no—
360C4×D3—62no—
361C4×D3—184no—
362C4×D3—184no—
363C4×D3—364no—
364Dic6—22no—
365C2×C32C2×#23511yescharacteristic
366C2×C32C2×#23911yescharacteristic
367C2×C32D3×#4111yescharacteristic
368C3⋊D3—122no—
369C3×D3—62no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9C2×#23511yescharacteristic
374C2×C9C2×#23511yescharacteristic
375C2×C9C2×#23511yescharacteristic
376D9—22no—
377D9—124no—
378D9—122no—
379C2×Q8—31no—
380C2×D4—272no—
381C2×D4—271no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—272no—
386C4⋊C4—31no—
387C4⋊C4—32no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—32no—
398C42—91no—
399C42—91no—
400C42—91no—
401C42D3×D911yescharacteristic
402C22×C3—21no—
403C22×C3—92no—
404C22×C3—181no—
405C22×C3—181no—
406C22×C3C23×D911yescharacteristic
407C22×C3C22×D3211yescharacteristic
408C2×D3—32no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—184no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—182no—
418C2×D3—362no—
419C2×D3—362no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—41no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—181no—
428C3×C4—181no—
429C3×C4—181no—
430C3×C4#369712yes—
431C3×C4#370112yes—
432C3×C4#379812yes—
433C3×C4#379912yes—
434Dic3—22no—
435Dic3—22no—
436Dic3—61no—
437Dic3—61no—
438Dic3—61no—
439Dic3—182no—
440Dic3—182no—
441Dic3—362no—
442C32#558011yescharacteristic
443C9—21no—
444C9#547911yescharacteristic, complemented
445C23—91no—
446C23—272no—
447Q8—62no—
448D4—544no—
449D4—542no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—182no—
456C2×C4—182no—
457C2×C4—182no—
458C2×C4—272no—
459C2×C4—544no—
460C2×C4C2×D3×D911yescharacteristic
461C2×C4C2×D3×D911yescharacteristic
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—182no—
467C2×C3C2×#369711yescharacteristic
468C2×C3C2×#370111yescharacteristic
469C2×C3#41×D911yescharacteristic
470C2×C3C2×#379811yescharacteristic
471C2×C3C2×#379911yescharacteristic
472C2×C3D3×#23511yescharacteristic
473D3—62no—
474D3—364no—
475D3—362no—
476D3—362no—
477C22—92no—
478C22—181no—
479C22—181no—
480C22—542no—
481C22—542no—
482C22—542no—
483C22C22×D3×D911yescharacteristic
484C4—21no—
485C4—62no—
486C4—62no—
487C4—181no—
488C4—181no—
489C4—181no—
490C4—542no—
491C4#8863512yes—
492C4#8864012yes—
493C3—21no—
494C3#1058968211yescharacteristic, complemented
495C3#1059387011yescharacteristic
496C2—182no—
497C2—1082no—
498C2C2×#8863511yescharacteristic
499C2C2×#8864011yescharacteristic
500C2D9×#23511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721004C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D9)C2semidirect
2. C3×C42×D9C2semidirect
3. C4×C9×Dic6C2semidirect
5. C4×#6192C2semidirect
6. #11720953C2semidirect
7. #11720944C2semidirect
8. #11720938C2semidirect
9. #11720935C2semidirect
10. #11720934C2semidirect
11. #11720931C2semidirect
59. C42×D9D3semidirect
169. C4×Dic6D9semidirect
321. C3×C9#529semidirect
444. C9#5479semidirect
494. C3#10589682semidirect

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
C3×C42×D92
C4×C9×Dic62
#117210042
C4×#61922
#117209532
#117209442
#117209382
#117209352
#117209342
#117209312
#105938703
#105896823