Finite Group Databasebeta

← #13147546

Subgroups — #13147546

Order 1728.

736 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic
2C4×C9×Dic6C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721000C211yesmaximal, characteristic, complemented
5#11720997C212yesmaximal
6#11720929C211yesmaximal, characteristic, complemented
7#11720929C211yesmaximal, characteristic, complemented
8#11720926C212yesmaximal, complemented
9#11720922C211yesmaximal, characteristic, complemented
10#11720915C212yesmaximal, complemented
11#11720898C212yesmaximal, complemented
12#10593947—31nomaximal
13#10589683—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2212yes—
19C3×C42×C9C2211yescharacteristic
20#88536C2211yescharacteristic
21#88536C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C9×C4⋊Dic3C2211yescharacteristic
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2212yes—
26C3×C4⋊Dic9C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2212yes—
30#88445C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C4×C4⋊(C3⋊D3)—31no—
38C3×#870—31no—
39C3×C4×Dic6D311yescharacteristic
40#63900—31no—
41#63897—32no—
42#63871—31no—
43#63871—31no—
44#63868—32no—
45#63849—31no—
46#63845—32no—
47#63836—32no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63508—31no—
51#63456—32no—
52#63454—31no—
53C4⋊C4×D9—31no—
54#63451—32no—
55#63438—32no—
56C4×D36—31no—
57C4×D36—61no—
58#63434—32no—
59#63432D311yescharacteristic, complemented
60C22×C3⋊D9C2311yescharacteristic
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C4⋊(C3⋊D9)—22no—
68C4×C3⋊D9—22no—
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2312yes—
74C9×Dic6—22no—
75(C3⋊D9)⋊C4—24no—
76#5694—91no—
77#5581—91no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C4×C3⋊D3—31no—
80C2×C4×C3⋊D3—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#624—32no—
83C32×C42C2×D311yescharacteristic
84#3752—31no—
85#3752—31no—
86C4⋊(C3⋊Dic3)—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3C2×D312yes—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97#3722—32no—
98#3722—32no—
99#3722—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C2×D36—31no—
103C2×D36—61no—
104C2×C4×D9—31no—
105C2×C4×D9—31no—
106C2×C4×D9—32no—
107C2×C4×D9—61no—
108C2×C4×D9—61no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—31no—
111C42×C9—21no—
112C42×C9—32no—
113C42×C9C2×D311yescharacteristic
114(C2×D9)⋊C4—31no—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—61no—
120(C2×D9)⋊C4—61no—
121C4⋊Dic9—31no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D311yescharacteristic
124C4⋊Dic9C2×D311yescharacteristic
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4C2×D312yes—
129Dic9⋊C4C2×D312yes—
130C4×Dic9—32no—
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—21no—
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×C3⋊D9—21no—
137C2×C3⋊D9#4112yes—
138C3×C4×C9—21no—
139C3×C4×C9—21no—
140C3×C4×C9#4112yes—
141C3⋊Dic9—21no—
142C3⋊Dic9—21no—
143C9×Dic3—22no—
144C9×Dic3—22no—
145C3×Dic9—22no—
146C3×Dic9—22no—
147C3×#125—91no—
148C3×C4×Q8—31no—
149#946—91no—
150#946—91no—
151#898—92no—
152#897—91no—
153#897—91no—
154#894—92no—
155#894—92no—
156#892—91no—
157#892—91no—
158#892—92no—
159D3×C4⋊C4—91no—
160D3×C4⋊C4—91no—
161#890—92no—
162#889—92no—
163#876—92no—
164C4×D12—91no—
165C4×D12—91no—
166C4×D12—181no—
167#872—92no—
168#870—31no—
169C4×Dic6D911yescharacteristic, complemented
170C22×C3⋊D3—31no—
171C22×C3⋊D3—31no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C4⋊(C3⋊D3)—62no—
178C4×C3⋊D3—62no—
179C4×C3⋊D3—62no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D312yes—
184C3×Dic6—22no—
185#624—64no—
186C22×D9—31no—
187C22×D9—31no—
188C22×D9—61no—
189C22×D9—61no—
190Q8×C9—62no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—32no—
195C2×C4×C9—32no—
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199C2×Dic9—31no—
200C2×Dic9—31no—
201C2×Dic9—61no—
202C2×Dic9—61no—
203C2×Dic9C22×D312yes—
204C2×Dic9C22×D312yes—
205D36—62no—
206D36—64no—
207C4×D9—64no—
208C4×D9—62no—
209C4×D9—62no—
210C4×D9—122no—
211C4×D9—122no—
212#552—271noSylow
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C92−1+411yescharacteristic
216C3⋊D9—22no—
217C3⋊D9—41no—
218C2×C3×Q8—31no—
219C2×D12—91no—
220C2×D12—91no—
221C2×D12—181no—
222C2×C4×D3—92no—
223C2×C4×D3—91no—
224C2×C4×D3—91no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—92no—
228C2×C4×D3—181no—
229C2×C4×D3—181no—
230C2×Dic6C2×D911yescharacteristic
231C3×C4⋊C4—32no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C42—21no—
238C3×C42—32no—
239C3×C42C2×D911yescharacteristic
240C3×C42D3211yescharacteristic
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—91no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—91no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—181no—
252(C2×D3)⋊C4—181no—
253C4⋊Dic3—31no—
254C4⋊Dic3—31no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—92no—
259C4⋊Dic3—181no—
260C4⋊Dic3C2×D911yescharacteristic
261Dic3⋊C4—32no—
262Dic3⋊C4—32no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4C2×D912yes—
268C4×Dic3—92no—
269C4×Dic3—92no—
270C4×Dic3C2×D912yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—32no—
273C2×C3⋊D3—61no—
274C2×C3⋊D3—61no—
275C2×C3⋊D3—61no—
276C2×C3⋊D3—61no—
277C2×C3⋊D3—61no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4#23512yes—
281C3⋊Dic3—61no—
282C3⋊Dic3—61no—
283C3×Dic3—22no—
284C3×Dic3—22no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C22×C9—21no—
288C22×C9C23×D311yescharacteristic
289C2×D9—32no—
290C2×D9—62no—
291C2×D9—61no—
292C2×D9—61no—
293C2×D9—62no—
294C2×D9—61no—
295C2×D9—61no—
296C2×D9—61no—
297C2×D9—121no—
298C2×D9—121no—
299C2×D9—121no—
300C2×D9—121no—
301C4×C9—22no—
302C4×C9—21no—
303C4×C9—21no—
304C4×C9—22no—
305C4×C9—41no—
306C4×C9—62no—
307C4×C9—62no—
308C4×C9#23912yes—
309Dic9—22no—
310Dic9—22no—
311Dic9—61no—
312Dic9—61no—
313Dic9—121no—
314Dic9—121no—
315#126—272no—
316#125—91no—
317#125—272no—
318#123—272no—
319C22⋊Q8—271no—
320C22⋊Q8—271no—
321C4×Q8—31no—
322C4×D4—271no—
323#117—272no—
324#117—271no—
325C2×C4⋊C4—271no—
326C3×C9#55211yescharacteristic, complemented, Sylow
327C22×D3—91no—
328C22×D3—91no—
329C22×D3—91no—
330C22×D3—91no—
331C22×D3—181no—
332C22×D3—181no—
333C3×Q8—62no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347C2×Dic3—32no—
348C2×Dic3—32no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3C22×D912yes—
356C2×Dic3C22×D912yes—
357D12—182no—
358D12—182no—
359D12—184no—
360C4×D3—184no—
361C4×D3—184no—
362C4×D3—182no—
363C4×D3—182no—
364C4×D3—182no—
365C4×D3—182no—
366C4×D3—362no—
367C4×D3—362no—
368Dic6—22no—
369C2×C32C2×#23511yescharacteristic
370C2×C32#100711yescharacteristic
371C2×C32D3×#4111yescharacteristic
372C3⋊D3—62no—
373C3⋊D3—121no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9C2×#23911yescharacteristic
378C2×C9D3×#4111yescharacteristic
379C2×C9#101011yescharacteristic
380D9—62no—
381D9—122no—
382D9—122no—
383D9—121no—
384C2×Q8—31no—
385C2×D4—271no—
386C22×C4—272no—
387C22×C4—271no—
388C22×C4—271no—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—91no—
392C4⋊C4—91no—
393C4⋊C4—92no—
394C4⋊C4—92no—
395C4⋊C4—272no—
396C4⋊C4—272no—
397C4⋊C4—272no—
398C4⋊C4—271no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—271no—
403C22⋊C4—271no—
404C42—32no—
405C42—272no—
406C42D3×D911yescharacteristic
407C22×C3—21no—
408C22×C3C23×D911yescharacteristic
409C22×C3C22×D3211yescharacteristic
410C2×D3—92no—
411C2×D3—92no—
412C2×D3—182no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—182no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—181no—
422C2×D3—181no—
423C2×D3—181no—
424C2×D3—361no—
425C2×D3—361no—
426C2×D3—361no—
427C2×D3—361no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—22no—
434C3×C4—41no—
435C3×C4—62no—
436C3×C4—62no—
437C3×C4—182no—
438C3×C4—182no—
439C3×C4#369712yes—
440C3×C4#379912yes—
441Dic3—22no—
442Dic3—22no—
443Dic3—62no—
444Dic3—62no—
445Dic3—181no—
446Dic3—181no—
447Dic3—181no—
448Dic3—181no—
449Dic3—361no—
450Dic3—361no—
451C32#558111yescharacteristic
452C9—21no—
453C9#569411yescharacteristic, complemented
454C23—271no—
455C23—271no—
456Q8—62no—
457D4—542no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4—271no—
463C2×C4—271no—
464C2×C4—544no—
465C2×C4—542no—
466C2×C4—542no—
467C2×C4C2×D3×D911yescharacteristic
468C2×C4C2×D3×D911yescharacteristic
469C2×C4C2×D3×D911yescharacteristic
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3C2×#369711yescharacteristic
474C2×C3#6371311yescharacteristic
475C2×C3#41×D911yescharacteristic
476C2×C3#6429711yescharacteristic
477C2×C3C2×#379911yescharacteristic
478C2×C3#6430611yescharacteristic
479D3—182no—
480D3—182no—
481D3—362no—
482D3—362no—
483D3—361no—
484D3—361no—
485C22—272no—
486C22—541no—
487C22—541no—
488C22—541no—
489C22—541no—
490C22—541no—
491C22C22×D3×D911yescharacteristic
492C4—21no—
493C4—21no—
494C4—62no—
495C4—62no—
496C4—182no—
497C4—182no—
498C4—541no—
499C4—541no—
500C4#8864212yes—
501C3—21no—
502C3#1058968311yescharacteristic, complemented
503C3#1059394711yescharacteristic
504C2—542no—
505C2—1081no—
506C2#1172245811yescharacteristic
507C2C2×#8864211yescharacteristic
508C2#1172247511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC42D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C9×Dic6C2semidirect
3. C3×#63432C2semidirect
4. #11721000C2semidirect
6. #11720929C2semidirect
7. #11720929C2semidirect
8. #11720926C2semidirect
9. #11720922C2semidirect
10. #11720915C2semidirect
11. #11720898C2semidirect
59. #63432D3semidirect
169. C4×Dic6D9semidirect
326. C3×C9#552semidirect
453. C9#5694semidirect
502. C3#10589683semidirect

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
C4×C9×Dic62
C3×#634322
#117210002
#117209972
#117209292
#117209262
#117209222
#117209152
#117208982
#105939473
#105896833