Finite Group Databasebeta

← #13148992

Subgroups — #13148992

Order 1728.

776 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⋊D9C211yesmaximal, characteristic, complemented
2C3×#63459C211yesmaximal, characteristic, complemented
3C9×#888C211yesmaximal, characteristic, complemented
4C4×#6192C211yesmaximal, characteristic
5D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#11720984C212yesmaximal, complemented
7#11720978C212yesmaximal
8#11720936C211yesmaximal, characteristic, complemented
9#11720929C212yesmaximal, complemented
10#11720916C212yesmaximal, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594393—31nomaximal
13#10589832—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C2×#6192C2211yescharacteristic
20C2×Dic3×D9C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29#88445C2212yes—
30#88445C2211yescharacteristic
31(C2×D9)⋊Dic3C2211yescharacteristic
32Dic3⋊Dic9C2212yes—
33Dic3⋊Dic9C2211yescharacteristic
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C4⋊C4×C3⋊D3—31no—
38C3×#897—31no—
39C3×#888D311yescharacteristic
40C4×#626—31no—
41D3×C4⋊Dic3—31no—
42#63887—32no—
43#63881—32no—
44#63871—32no—
45#63871—31no—
46#63846—32no—
47Dic3×Dic6—31no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50#63508—32no—
51#63459—31no—
52#63459D311yescharacteristic, complemented
53#63455—32no—
54C4⋊C4×D9—31no—
55C4⋊C4×D9—31no—
56C4⋊C4×D9—61no—
57#63452—32no—
58C4×D36—31no—
59C4×Dic18—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9—22no—
67C4×C3⋊D9—42no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—22no—
74C9×Dic6—42no—
75C3×Dic18—22no—
76#6192Q814yescomplemented
77Dic3×D9—42no—
78#5730—91no—
79#5635—91no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3—32no—
82C2×C3×C4×D3—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6C2×D312yes—
85C2×#626—31no—
86C2×D3×Dic3—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4C2×D312yes—
94C3×C4×Dic3C2×D311yescharacteristic
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—31no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99#3722—32no—
100#3722—31no—
101(C2×D3)⋊Dic3—31no—
102Dic32—32no—
103C2×Q8×C9—32no—
104C2×D36—31no—
105C2×C4×D9—31no—
106C2×C4×D9—32no—
107C2×C4×D9—32no—
108C2×C4×D9—61no—
109C2×C4×D9—62no—
110C2×C4×D9C2×D311yescharacteristic
111C2×Dic18C2×D311yescharacteristic
112C9×C4⋊C4—21no—
113C9×C4⋊C4—32no—
114C9×C4⋊C4—31no—
115C9×C4⋊C4C2×D311yescharacteristic
116C42×C9—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4C2×D312yes—
121C4⋊Dic9—31no—
122C4⋊Dic9—32no—
123C4⋊Dic9—31no—
124C4⋊Dic9—61no—
125C4⋊Dic9C2×D312yes—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—62no—
130C4×Dic9—32no—
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—21no—
133C2×C3⋊D9—21no—
134C2×C3⋊D9C2×Q812yes—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9C2×Q812yes—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9—22no—
142C9×Dic3—22no—
143C9×Dic3—21no—
144C9×Dic3C2×Q812yes—
145C3×Dic9—22no—
146C3×Dic9—21no—
147C3×C4⋊Q8—31no—
148C3×C22⋊Q8—91no—
149#946—92no—
150#946—92no—
151#946—91no—
152Q8×Dic3—91no—
153C4×#67—91no—
154C2×C4⋊Dic3—91no—
155#897—31no—
156#897—91no—
157#897—92no—
158#893—92no—
159D3×C4⋊C4—91no—
160D3×C4⋊C4—91no—
161D3×C4⋊C4—91no—
162D3×C4⋊C4—181no—
163#890—92no—
164#889—92no—
165#888D911yescharacteristic, complemented
166#67⋊C4—92no—
167#878—92no—
168C4×D12—91no—
169C4×Dic6—91no—
170C22×C3⋊D3—31no—
171C22×C3×D3—31no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—31no—
176C4×C3⋊D3—62no—
177C4×C3⋊D3—122no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D311yescharacteristic
183C3×C4×D3—62no—
184C3×Dic6—42no—
185C3×Dic6—62no—
186#626—34no—
187D3×Dic3—122no—
188C22×D9—31no—
189C22×D9—61no—
190C22×D9C22×D311yescharacteristic
191Q8×C9—122no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9—32no—
197C2×C4×C9C22×D312yes—
198C2×C4×C9C22×D311yescharacteristic
199C2×Dic9—32no—
200C2×Dic9—31no—
201C2×Dic9—61no—
202C2×Dic9—62no—
203C2×Dic9C22×D312yes—
204C2×Dic9C22×D311yescharacteristic
205D36—34no—
206C4×D9—22no—
207C4×D9—64no—
208C4×D9—62no—
209C4×D9—122no—
210C4×D9—122no—
211C4×D9—124no—
212Dic18—22no—
213#554—271noSylow
214C2×C3×C9C22×Q811yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9H(5,𝔽2)11yescharacteristic
217C3⋊D9—22no—
218C3×D9—22no—
219C2×C3×Q8—32no—
220C2×C3×Q8—91no—
221C22×C3×C4—91no—
222C2×#67—91no—
223C22×Dic3—92no—
224C2×D12—91no—
225C2×C4×D3—31no—
226C2×C4×D3—91no—
227C2×C4×D3—91no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—181no—
232C2×C4×D3—182no—
233C2×Dic6—31no—
234C2×Dic6C2×D912yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4C2×D911yescharacteristic
240C3×C4⋊C4D3211yescharacteristic
241C3×C22⋊C4—92no—
242C3×C42—31no—
243#217—91no—
244(C2×D3)⋊C4—32no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—91no—
250C4⋊Dic3—32no—
251C4⋊Dic3—91no—
252C4⋊Dic3—92no—
253C4⋊Dic3—91no—
254C4⋊Dic3—91no—
255C4⋊Dic3—92no—
256C4⋊Dic3—92no—
257C4⋊Dic3—181no—
258C4⋊Dic3C2×D911yescharacteristic
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—91no—
265Dic3⋊C4—182no—
266Dic3⋊C4C2×D912yes—
267C4×Dic3—92no—
268C4×Dic3—92no—
269C4×Dic3C2×D911yescharacteristic
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—32no—
272C2×C3⋊D3—61no—
273C2×C3⋊D3—61no—
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C32×C4—22no—
278C32×C4#23912yes—
279C3⋊Dic3—61no—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—21no—
283C3×Dic3—62no—
284C3×Dic3—61no—
285C3×Dic3D3×Q812yes—
286C22×C9—21no—
287C22×C9C23×D311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—32no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—61no—
295C2×D9—61no—
296C2×D9D3×Q812yes—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—32no—
300C4×C9—42no—
301C4×C9—62no—
302C4×C9—61no—
303C4×C9#23712yes—
304Dic9—22no—
305Dic9—21no—
306Dic9—62no—
307Dic9—61no—
308Dic9—62no—
309Dic9—122no—
310C4⋊Q8—31no—
311#125—272no—
312C22⋊Q8—91no—
313C22⋊Q8—272no—
314C22⋊Q8—271no—
315C22⋊Q8—272no—
316C4×Q8—271no—
317C4×D4—271no—
318C4×D4—272no—
319C2×C4⋊C4—271no—
320C2×C4⋊C4—271no—
321C3×C9#55411yescharacteristic, complemented, Sylow
322C23×C3—91no—
323C22×D3—31no—
324C22×D3—91no—
325C22×D3—91no—
326C22×D3—181no—
327C3×Q8—122no—
328C3×Q8—182no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—32no—
334C2×C3×C4—92no—
335C2×C3×C4—91no—
336C2×C3×C4—182no—
337C2×C3×C4C22×D912yes—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C2×D3212yes—
340C2×C3×C4C2×D3211yescharacteristic
341#67—94no—
342C2×Dic3—32no—
343C2×Dic3—31no—
344C2×Dic3—92no—
345C2×Dic3—91no—
346C2×Dic3—92no—
347C2×Dic3—91no—
348C2×Dic3—181no—
349C2×Dic3—182no—
350C2×Dic3—362no—
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D912yes—
353C2×Dic3C22×D911yescharacteristic
354D12—94no—
355C4×D3—62no—
356C4×D3—184no—
357C4×D3—182no—
358C4×D3—182no—
359C4×D3—362no—
360C4×D3—362no—
361C4×D3—362no—
362C4×D3—364no—
363Dic6—42no—
364Dic6—62no—
365C2×C32#100411yescharacteristic
366C2×C32C2×D3×Q811yescharacteristic
367C2×C32C2×#23911yescharacteristic
368C3⋊D3—62no—
369C3×D3—62no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9C2×#23711yescharacteristic
374C2×C9C2×D3×Q811yescharacteristic
375C2×C9#100911yescharacteristic
376D9—22no—
377D9—64no—
378D9—62no—
379C2×Q8—32no—
380C2×Q8—91no—
381C2×D4—271no—
382C22×C4—91no—
383C22×C4—272no—
384C22×C4—271no—
385C22×C4—272no—
386C4⋊C4—31no—
387C4⋊C4—32no—
388C4⋊C4—92no—
389C4⋊C4—272no—
390C4⋊C4—271no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—271no—
394C4⋊C4—272no—
395C4⋊C4D3×D911yescharacteristic
396C22⋊C4—92no—
397C22⋊C4—271no—
398C22⋊C4—272no—
399C22⋊C4—271no—
400C42—31no—
401C42—272no—
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—92no—
412C2×D3—92no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—32no—
424C3×C4—42no—
425C3×C4—62no—
426C3×C4—61no—
427C3×C4—181no—
428C3×C4—182no—
429C3×C4#370112yes—
430C3×C4#379612yes—
431Dic3—21no—
432Dic3—22no—
433Dic3—62no—
434Dic3—61no—
435Dic3—182no—
436Dic3—181no—
437Dic3—182no—
438Dic3—182no—
439Dic3—181no—
440Dic3—362no—
441Dic3Q8×D912yes—
442C32#573011yescharacteristic
443C9—21no—
444C9#563511yescharacteristic, complemented
445C23—91no—
446C23—271no—
447Q8—122no—
448Q8—182no—
449D4—274no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—32no—
453C2×C4—92no—
454C2×C4—91no—
455C2×C4—182no—
456C2×C4—272no—
457C2×C4—271no—
458C2×C4—542no—
459C2×C4—1082no—
460C2×C4—1082no—
461C2×C4C2×D3×D912yes—
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—182no—
467C2×C3#6371011yescharacteristic
468C2×C3C2×Q8×D911yescharacteristic
469C2×C3C2×#370111yescharacteristic
470C2×C3C2×#379611yescharacteristic
471C2×C3#6431411yescharacteristic
472C2×C3D32×Q811yescharacteristic
473D3—62no—
474D3—184no—
475D3—182no—
476D3—182no—
477C22—92no—
478C22—181no—
479C22—181no—
480C22—272no—
481C22—541no—
482C22—541no—
483C22C22×D3×D911yescharacteristic
484C4—22no—
485C4—32no—
486C4—62no—
487C4—61no—
488C4—181no—
489C4—182no—
490C4—542no—
491C4—541no—
492C4#8863412yes—
493C3—21no—
494C3#1058983211yescharacteristic, complemented
495C3#1059439311yescharacteristic
496C2—182no—
497C2—542no—
498C2C2×#8863411yescharacteristic
499C2#1172250011yescharacteristic
500C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×C3⋊D9C2semidirect
2. C3×#63459C2semidirect
3. C9×#888C2semidirect
5. D9×C4⋊Dic3C2semidirect
6. #11720984C2semidirect
8. #11720936C2semidirect
9. #11720929C2semidirect
10. #11720916C2semidirect
11. Dic3×Dic18C2semidirect
52. #63459D3semidirect
76. #6192Q8semidirect
165. #888D9semidirect
321. C3×C9#554semidirect
444. C9#5635semidirect
494. C3#10589832semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D92
C3×#634592
C9×#8882
C4×#61922
D9×C4⋊Dic32
#117209842
#117209782
#117209362
#117209292
#117209162
Dic3×Dic182
#105943933
#105898323