Finite Group Databasebeta

← #13148755

Subgroups — #13148755

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×Dic6C212yesmaximal, complemented
2#11721744C211yesmaximal, characteristic, complemented
3C3×#63455C211yesmaximal, characteristic
4C9×#888C211yesmaximal, characteristic, complemented
5#11720998C211yesmaximal, characteristic
6#11720996C212yesmaximal, complemented
7Dic3×D36C211yesmaximal, characteristic
8#11720984C212yesmaximal, complemented
9#11720969C211yesmaximal, characteristic, complemented
10C4×#6189C211yesmaximal, characteristic, complemented
11#11720939C212yesmaximal, complemented
12#10594376—31nomaximal
13#10589829—31nomaximal
14C2×#6226C2212yes—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C9×Dic6C2212yes—
18C2×Dic3×D9C2212yes—
19C2×Dic3×D9C2212yes—
20C2×#6189C2212yes—
21C2×#6189C2211yescharacteristic
22D9×Dic6—28no—
23C3×C9×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25#88534C2212yes—
26C4×C3⋊Dic9C2211yescharacteristic
27C3×(C2×D9)⋊C4C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C9×Dic3⋊C4C2212yes—
30C4×C9×Dic3C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2211yescharacteristic
35#88442C2211yescharacteristic
36Dic9⋊Dic3C2211yescharacteristic
37Dic9⋊Dic3C2212yes—
38C2×D3×Dic6—32no—
39#64097—31no—
40C3×#893—31no—
41C3×#888D311yescharacteristic
42#63918—31no—
43C4×#627—31no—
44#63898—31no—
45#63896—32no—
46Dic3×D12—31no—
47#63887—32no—
48#63851—32no—
49C2×Q8×D9—32no—
50C9×C4⋊Q8—31no—
51#63508—32no—
52#63458—32no—
53#63458—32no—
54#63455—31no—
55#63455D311yescharacteristic
56#63450—31no—
57#63450—31no—
58#63450—61no—
59C4×D36—31no—
60C4×Dic18—31no—
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67#6226—24no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×D36Q814yescomplemented
74C3×C4×D9—24no—
75C9×Dic6—24no—
76Dic3×D9—24no—
77Dic3×D9—24no—
78#6189—24no—
79#6189D414yescomplemented
80#5727—91no—
81D4×Dic6—91no—
82C2×#634—32no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6C2×D312yes—
86C2×#627—31no—
87C2×#627—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—32no—
90D3×Dic6—68no—
91C32×C4⋊C4C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)—31no—
93#3750—32no—
94C4×C3⋊Dic3—31no—
95C3×(C2×D3)⋊C4—32no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4C2×D312yes—
98C3×C4×Dic3—31no—
99C3×C4×Dic3C2×D311yescharacteristic
100#3724—31no—
101Dic3⋊Dic3—31no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—31no—
104(C2×D3)⋊Dic3—32no—
105(C2×D3)⋊Dic3—32no—
106C2×Q8×C9—32no—
107Q8×D9—68no—
108C2×D36C2×D311yescharacteristic
109C2×C4×D9—32no—
110C2×C4×D9—32no—
111C2×C4×D9C2×D312yes—
112C2×Dic18—31no—
113C2×Dic18—32no—
114C2×Dic18—32no—
115C2×Dic18—62no—
116C9×C4⋊C4—21no—
117C9×C4⋊C4—32no—
118C9×C4⋊C4—31no—
119C9×C4⋊C4C2×D311yescharacteristic
120C42×C9—31no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4C2×D312yes—
124C4⋊Dic9—31no—
125C4⋊Dic9—31no—
126C4⋊Dic9—61no—
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—31no—
130Dic9⋊C4—31no—
131Dic9⋊C4—62no—
132C4×Dic9—31no—
133C4×Dic9—61no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9C2411yescharacteristic
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C2×C3×D9C2×Q814yescomplemented
139C3×C4×C9—22no—
140C3×C4×C9C2×Q812yescomplemented
141C3⋊Dic9—22no—
142C3⋊Dic9—21no—
143C3⋊Dic9C2×D412yes—
144C9×Dic3—22no—
145C9×Dic3—21no—
146C9×Dic3C2×D412yes—
147C3×Dic9—21no—
148C3×Dic9C2×D412yes—
149C2×D3×Q8—92no—
150C22×Dic6—92no—
151C3×C4⋊Q8—31no—
152C3×C4×D4—91no—
153#946—92no—
154D4×Dic3—91no—
155D4×Dic3—91no—
156#924—92no—
157#896—92no—
158#896—92no—
159#893—31no—
160#893—91no—
161#888—91no—
162#888—91no—
163#888—91no—
164#888—181no—
165#888D911yescharacteristic, complemented
166#878—92no—
167#878—92no—
168C4×D12—91no—
169#869—91no—
170C4×Dic6—91no—
171C4×Dic6—91no—
172C22×C3×D3—32no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—32no—
178#634—64no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D311yescharacteristic
184C3×D12—34no—
185C3×C4×D3—64no—
186C3×Dic6—24no—
187#627—34no—
188#627—64no—
189D3×Dic3—64no—
190D3×Dic3—64no—
191C22×D9C22×D312yes—
192Q8×C9—64no—
193C2×C4×C9—21no—
194C2×C4×C9—22no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—32no—
198C2×C4×C9C22×D312yes—
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—31no—
201C2×Dic9—31no—
202C2×Dic9—32no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9—62no—
206C2×Dic9C22×D311yescharacteristic
207C2×Dic9C22×D311yescharacteristic
208D36Dic614yescomplemented
209C4×D9—24no—
210C4×D9—64no—
211C4×D9—64no—
212Dic18—34no—
213Dic18—64no—
214Dic18—64no—
215Dic18—124no—
216D4×Q8—271noSylow
217C2×C3×C9C22×D411yescharacteristic
218C2×C3×C9C22×Q811yescharacteristic
219C2×C3×C92−1+411yescharacteristic
220C3×D9—24no—
221C2×C3×Q8—32no—
222C2×C3×D4—91no—
223C22×C3×C4—92no—
224C22×Dic3—92no—
225C22×Dic3—92no—
226D3×Q8—188no—
227C2×D12—31no—
228C2×C4×D3—32no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×Dic6—91no—
232C2×Dic6—91no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C2×Dic6—92no—
236C2×Dic6—92no—
237C2×Dic6—182no—
238C2×Dic6—188no—
239C2×Dic6C2×D912yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4C2×D911yescharacteristic
244C3×C4⋊C4D3211yescharacteristic
245C3×C22⋊C4—92no—
246C3×C42—31no—
247C3×C42—91no—
248#217—92no—
249#217—92no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253C4⋊Dic3—91no—
254C4⋊Dic3—91no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—181no—
259C4⋊Dic3C2×D911yescharacteristic
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—91no—
266Dic3⋊C4—91no—
267Dic3⋊C4—182no—
268Dic3⋊C4C2×D912yes—
269C4×Dic3—31no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—181no—
273C4×Dic3C2×D911yescharacteristic
274C22×C32C23×D311yescharacteristic
275C2×C3×D3—34no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C32×C4—22no—
279C32×C4D3×Q812yes—
280C3⋊Dic3—32no—
281C3⋊Dic3—61no—
282C3⋊Dic3—62no—
283C3×Dic3—22no—
284C3×Dic3—21no—
285C3×Dic3—32no—
286C3×Dic3—61no—
287C3×Dic3D3×D412yes—
288C22×C9—21no—
289C22×C9C23×D311yescharacteristic
290C2×D9—22no—
291C2×D9—22no—
292C2×D9C2×Dic614yescomplemented
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—32no—
296C4×C9—42no—
297C4×C9—62no—
298C4×C9—61no—
299C4×C9C2×Dic612yescomplemented
300Dic9—21no—
301Dic9—32no—
302Dic9—62no—
303Dic9—62no—
304Dic9—62no—
305Dic9—61no—
306Dic9—122no—
307Dic9D3×D412yes—
308C22×Q8—272no—
309C4⋊Q8—31no—
310C4⋊Q8—271no—
311C4⋊Q8—271no—
312C22⋊Q8—272no—
313C22⋊Q8—272no—
314C22⋊Q8—272no—
315C4×Q8—271no—
316C4×D4—91no—
317C4×D4—271no—
318C4×D4—271no—
319C3×C9D4×Q811yescharacteristic, complemented, Sylow
320C23×C3—92no—
321C22×D3—32no—
322C3×Q8—64no—
323C3×D4—94no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328C2×C3×C4—32no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—184no—
332C2×C3×C4C22×D912yes—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C2×D3212yes—
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341C2×Dic3—92no—
342C2×Dic3—91no—
343C2×Dic3—92no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—182no—
347C2×Dic3—184no—
348C2×Dic3—184no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D912yes—
351C2×Dic3C22×D911yescharacteristic
352D12—34no—
353C4×D3—64no—
354C4×D3—184no—
355C4×D3—184no—
356Dic6—24no—
357Dic6—94no—
358Dic6—94no—
359Dic6—184no—
360Dic6—184no—
361Dic6—184no—
362Dic6—184no—
363Dic6—364no—
364C2×C32C2×D3×D411yescharacteristic
365C2×C32C2×D3×Q811yescharacteristic
366C2×C32#100711yescharacteristic
367C3×D3—64no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9C22×Dic611yescharacteristic
372C2×C9C2×D3×D411yescharacteristic
373C2×C9#101011yescharacteristic
374D9—24no—
375C2×Q8—32no—
376C2×Q8—271no—
377C2×Q8—272no—
378C2×Q8—272no—
379C2×Q8—548no—
380C2×D4—91no—
381C22×C4—92no—
382C22×C4—272no—
383C22×C4—272no—
384C4⋊C4—31no—
385C4⋊C4—32no—
386C4⋊C4—271no—
387C4⋊C4—271no—
388C4⋊C4—271no—
389C4⋊C4—272no—
390C4⋊C4—271no—
391C4⋊C4—272no—
392C4⋊C4D3×D911yescharacteristic
393C22⋊C4—92no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C42—31no—
397C42—91no—
398C42—271no—
399C22×C3—21no—
400C22×C3—94no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3C23×D911yescharacteristic
404C22×C3C22×D3211yescharacteristic
405C2×D3—34no—
406C2×D3—62no—
407C2×D3—62no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—32no—
412C3×C4—42no—
413C3×C4—62no—
414C3×C4—61no—
415C3×C4—92no—
416C3×C4—181no—
417C3×C4Q8×D912yescomplemented
418C3×C4D3×Dic612yes—
419Dic3—22no—
420Dic3—21no—
421Dic3—32no—
422Dic3—61no—
423Dic3—92no—
424Dic3—92no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—181no—
429Dic3—182no—
430Dic3—181no—
431Dic3—362no—
432Dic3D4×D912yes—
433C32#572711yescharacteristic
434C9—21no—
435C9D4×Dic611yescharacteristic, complemented
436C23—92no—
437Q8—64no—
438Q8—274no—
439Q8—544no—
440Q8—544no—
441D4—94no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—32no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—184no—
448C2×C4—271no—
449C2×C4—271no—
450C2×C4—272no—
451C2×C4—544no—
452C2×C4—544no—
453C2×C4C2×D3×D912yes—
454C2×C4C2×D3×D911yescharacteristic
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—184no—
459C2×C3C2×D4×D911yescharacteristic
460C2×C3C2×Q8×D911yescharacteristic
461C2×C3#6371311yescharacteristic
462C2×C3C2×D3×Dic611yescharacteristic
463C2×C3D32×D411yescharacteristic
464C2×C3#6431511yescharacteristic
465D3—64no—
466C22—94no—
467C22—182no—
468C22—182no—
469C22C22×D3×D911yescharacteristic
470C4—22no—
471C4—32no—
472C4—61no—
473C4—62no—
474C4—92no—
475C4—181no—
476C4—272no—
477C4—541no—
478C4—542no—
479C4D9×Dic612yescomplemented
480C3—21no—
481C3#1058982911yescharacteristic, complemented
482C3#1059437611yescharacteristic
483C2—184no—
484C2C2×D9×Dic611yescharacteristic
485C2D3×D4×D911yescharacteristic
486C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9D4×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
1. C2×D9×Dic6C2semidirect
2. #11721744C2semidirect
4. C9×#888C2semidirect
6. #11720996C2semidirect
8. #11720984C2semidirect
9. #11720969C2semidirect
10. C4×#6189C2semidirect
11. #11720939C2semidirect
73. C3×D36Q8semidirect
79. #6189D4semidirect
138. C2×C3×D9C2×Q8semidirect
140. C3×C4×C9C2×Q8semidirect
165. #888D9semidirect
208. D36Dic6semidirect
292. C2×D9C2×Dic6semidirect
299. C4×C9C2×Dic6semidirect
319. C3×C9D4×Q8semidirect
417. C3×C4Q8×D9semidirect
435. C9D4×Dic6semidirect
479. C4D9×Dic6semidirect
481. C3#10589829semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
#117217442
C3×#634552
C9×#8882
#117209982
#117209962
Dic3×D362
#117209842
#117209692
C4×#61892
#117209392
#105943763
#105898293