Finite Group Databasebeta

← #13150147

Subgroups — #13150147

Order 1728.

740 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721813C211yesmaximal, characteristic
2C2×C4×C9×Dic3C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic
4#11721249C212yesmaximal
5#11721244C212yesmaximal
6#11720970C211yesmaximal, characteristic, complemented
7#11720932C211yesmaximal, characteristic, complemented
8#11720925C212yesmaximal, complemented
9#11720912C211yesmaximal, characteristic, complemented
10Dic3×Dic18C211yesmaximal, characteristic, complemented
11#11720900C212yesmaximal, complemented
12#10594808—31nomaximal
13#10589572—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6189C2212yes—
20#88551C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C3×#3676C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C4×C9×Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30Dic3⋊Dic9C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×#924—31no—
39C2×C3×C4×Dic3D311yescharacteristic
40#63964—32no—
41#63955—32no—
42#63918—31no—
43#63867—32no—
44#63866—31no—
45#63844—31no—
46Dic3×Dic6—31no—
47#63837—32no—
48C2×C42×C9—31no—
49#63488—32no—
50#63486—31no—
51#63486—32no—
52#63486—61no—
53#63486D311yescharacteristic
54#63432—32no—
55#63431—31no—
56#63431—31no—
57C4×Dic18—31no—
58C4×Dic18—31no—
59C4×Dic18—32no—
60C23×C3×C9C2311yescharacteristic
61C2×C3×C4×C9—22no—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3—22no—
68C2×C9×Dic3Q814yescomplemented
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C3×Dic18—22no—
76#6189—24no—
77#5622—91no—
78#5470—91no—
79C22×C32×C4C2×D311yescharacteristic
80C2×#634—31no—
81C22×C3×Dic3C2×D311yescharacteristic
82C22×C3×Dic3C2×D311yescharacteristic
83C2×C3×Dic6—31no—
84C2×#627—32no—
85#3757—32no—
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C3×#217—32no—
89C3×C4⋊Dic3—31no—
90C3×Dic3⋊C4—32no—
91C3×C4×Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100Dic32—32no—
101Dic32—32no—
102C22×C4×C9—21no—
103C22×C4×C9—31no—
104C22×C4×C9—31no—
105C22×C4×C9C2×D311yescharacteristic
106C2×Dic18—31no—
107C2×Dic18—32no—
108C2×Dic18—61no—
109C2×Dic18C2×D311yescharacteristic
110C42×C9—31no—
111C42×C9—31no—
112C42×C9—31no—
113C42×C9—31no—
114#3676—32no—
115#3676—62no—
116#3676C2×D312yes—
117C4⋊Dic9—31no—
118C4⋊Dic9—32no—
119C4⋊Dic9—32no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—62no—
127Dic9⋊C4C2×D312yes—
128C4×Dic9—32no—
129C4×Dic9—32no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2×Q812yes—
133C22×C3×C9C2411yescharacteristic
134C3×C4×C9—21no—
135C3×C4×C9#4112yes—
136C3⋊Dic9—22no—
137C3⋊Dic9—22no—
138C9×Dic3—21no—
139C9×Dic3C2×Q812yes—
140C9×Dic3C2×Q812yes—
141C9×Dic3#4112yes—
142C3×Dic9—22no—
143C3×Dic9—22no—
144C3×C22⋊Q8—91no—
145C2×C3×C42—31no—
146Q8×Dic3—91no—
147#944—91no—
148#926—92no—
149#924—31no—
150#924—92no—
151#924—91no—
152#924—91no—
153#924—181no—
154C2×C4×Dic3D911yescharacteristic
155#889—92no—
156#888—91no—
157#887—92no—
158#887—91no—
159#878—92no—
160#877—92no—
161#870—92no—
162#869—91no—
163#869—91no—
164C4×Dic6—92no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C23×C32C22×D311yescharacteristic
168C2×C32×C4—22no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173#634—62no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3Dic614yes—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C3×Dic6—62no—
183#627—64no—
184C23×C9—21no—
185C23×C9C22×D311yescharacteristic
186C2×C4×C9—22no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—34no—
190C2×C4×C9—31no—
191C2×C4×C9—31no—
192C2×C4×C9—31no—
193C2×C4×C9—31no—
194C2×C4×C9—42no—
195C2×C4×C9—62no—
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—62no—
202C2×Dic9C22×D312yes—
203C2×Dic9C22×D312yes—
204Dic18—22no—
205Dic18—64no—
206Dic18—62no—
207Dic18—122no—
208#533—271noSylow
209C2×C3×C9—22no—
210C2×C3×C9C22×Q811yescharacteristic
211C2×C3×C9C2×#4111yescharacteristic
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×Q8—91no—
214C22×C3×C4—21no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4C2×D911yescharacteristic
218C22×C3×C4D3211yescharacteristic
219C22×Dic3C2×D911yescharacteristic
220C22×Dic3C2×D911yescharacteristic
221C2×Dic6—31no—
222C2×Dic6—92no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—92no—
226C2×Dic6—181no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C22⋊C4—92no—
230C3×C42—31no—
231C3×C42—31no—
232C3×C42—31no—
233C3×C42—31no—
234#217—32no—
235#217—92no—
236#217—92no—
237#217—182no—
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240C4⋊Dic3—92no—
241C4⋊Dic3—92no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—181no—
246Dic3⋊C4—32no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—182no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3C2×D911yescharacteristic
261C4×Dic3C2×D911yescharacteristic
262C4×Dic3C2×D911yescharacteristic
263C4×Dic3C2×D911yescharacteristic
264C22×C32—21no—
265C22×C32—21no—
266C22×C32C2×Dic612yes—
267C22×C32C23×D311yescharacteristic
268C32×C4—21no—
269C32×C4#23512yes—
270C3⋊Dic3—62no—
271C3⋊Dic3—62no—
272C3×Dic3—21no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3C2×Dic612yes—
276C3×Dic3C2×Dic612yes—
277C3×Dic3#23512yes—
278C22×C9—21no—
279C22×C9—21no—
280C22×C9—21no—
281C22×C9—22no—
282C22×C9—41no—
283C22×C9—41no—
284C22×C9D3×Q812yes—
285C22×C9C23×D311yescharacteristic
286C4×C9—21no—
287C4×C9—22no—
288C4×C9—32no—
289C4×C9—32no—
290C4×C9—32no—
291C4×C9—41no—
292C4×C9—61no—
293C4×C9#23712yes—
294Dic9—22no—
295Dic9—22no—
296Dic9—62no—
297Dic9—62no—
298Dic9—122no—
299Dic9—122no—
300C4⋊Q8—271no—
301C4⋊Q8—271no—
302#125—272no—
303C22⋊Q8—91no—
304C22⋊Q8—272no—
305C22⋊Q8—271no—
306C4×Q8—272no—
307C4×Q8—271no—
308C4×Q8—271no—
309#117—272no—
310C2×C42—31no—
311C3×C9#53311yescharacteristic, complemented, Sylow
312C23×C3—21no—
313C23×C3C22×D911yescharacteristic
314C23×C3C2×D3211yescharacteristic
315C3×Q8—182no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—34no—
324C2×C3×C4—31no—
325C2×C3×C4—42no—
326C2×C3×C4—62no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4C22×D911yescharacteristic
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×Dic3—22no—
334C2×Dic3—32no—
335C2×Dic3—32no—
336C2×Dic3—92no—
337C2×Dic3—92no—
338C2×Dic3—92no—
339C2×Dic3—92no—
340C2×Dic3—182no—
341C2×Dic3—182no—
342C2×Dic3Dic1814yescomplemented
343C2×Dic3C22×D911yescharacteristic
344C2×Dic3C22×D911yescharacteristic
345C2×Dic3C22×D911yescharacteristic
346C2×Dic3C22×D911yescharacteristic
347Dic6—62no—
348Dic6—184no—
349Dic6—182no—
350Dic6—182no—
351Dic6—184no—
352Dic6—362no—
353C2×C32—22no—
354C2×C32C22×Dic611yescharacteristic
355C2×C32C2×#23511yescharacteristic
356C2×C32C2×#23511yescharacteristic
357C2×C9—22no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—42no—
362C2×C9C2×#23711yescharacteristic
363C2×C9C2×D3×Q811yescharacteristic
364C2×C9D3×#4111yescharacteristic
365C2×Q8—91no—
366C2×Q8—272no—
367C2×Q8—271no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4D3×D911yescharacteristic
371C4⋊C4—92no—
372C4⋊C4—91no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C22⋊C4—92no—
381C22⋊C4—272no—
382C42—31no—
383C42—31no—
384C42—31no—
385C42—31no—
386C42—272no—
387C42—272no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3C2×Dic1812yes—
397C22×C3C23×D911yescharacteristic
398C22×C3D3×Dic612yes—
399C22×C3C22×D3211yescharacteristic
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—22no—
403C3×C4—32no—
404C3×C4—32no—
405C3×C4—32no—
406C3×C4—41no—
407C3×C4—61no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4#369712yes—
411C3×C4#380412yes—
412Dic3—21no—
413Dic3—62no—
414Dic3—62no—
415Dic3—182no—
416Dic3—182no—
417Dic3—182no—
418Dic3—182no—
419Dic3—362no—
420Dic3—362no—
421Dic3C2×Dic1812yes—
422Dic3C2×Dic1812yes—
423Dic3#369712yes—
424C32#547011yescharacteristic
425C9—21no—
426C9#562211yescharacteristic, complemented
427C23C2×D3×D911yescharacteristic
428Q8—182no—
429Q8—544no—
430Q8—542no—
431C2×C4—22no—
432C2×C4—34no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—62no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—272no—
441C2×C4—272no—
442C2×C4C2×D3×D911yescharacteristic
443C2×C4C2×D3×D911yescharacteristic
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—42no—
450C2×C3C22×Dic1811yescharacteristic
451C2×C3C2×#369711yescharacteristic
452C2×C3C2×#369711yescharacteristic
453C2×C3C2×D3×Dic611yescharacteristic
454C2×C3D3×#23511yescharacteristic
455C2×C3C2×#380411yescharacteristic
456C22—21no—
457C22—21no—
458C22D3×Dic1812yes—
459C22C22×D3×D911yescharacteristic
460C4—21no—
461C4—32no—
462C4—32no—
463C4—32no—
464C4—61no—
465C4—182no—
466C4—182no—
467C4—542no—
468C4—542no—
469C4#8865812yes—
470C3—21no—
471C3#1058957211yescharacteristic, complemented
472C3#1059480811yescharacteristic
473C2—22no—
474C2C2×D3×Dic1811yescharacteristic
475C2D3×#369711yescharacteristic
476C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#533
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11720970C2semidirect
7. #11720932C2semidirect
8. #11720925C2semidirect
9. #11720912C2semidirect
10. Dic3×Dic18C2semidirect
11. #11720900C2semidirect
68. C2×C9×Dic3Q8semidirect
311. C3×C9#533semidirect
342. C2×Dic3Dic18semidirect
426. C9#5622semidirect
471. C3#10589572semidirect

Maximal subgroups

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

subgroupindex
#117218132
C2×C4×C9×Dic32
C3×#634862
#117212492
#117212442
#117209702
#117209322
#117209252
#117209122
Dic3×Dic182
#117209002
#105948083
#105895723