Finite Group Databasebeta

← #13147626

Subgroups — #13147626

Order 1728.

754 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721025C211yesmaximal, characteristic
5#11721025C211yesmaximal, characteristic
6#11721024C212yesmaximal, complemented
7#11721023C212yesmaximal, complemented
8D3×Dic9⋊C4C212yesmaximal, complemented
9#11720971C211yesmaximal, characteristic, complemented
10#11720955C212yesmaximal, complemented
11#11720937C211yesmaximal, characteristic, complemented
12#10593941—31nomaximal
13#10589641—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×D3×Dic9C2212yes—
18C2×D3×Dic9C2212yes—
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2211yescharacteristic
23#88534C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C9×(C2×D3)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37C4×#634—31no—
38C3×C4×D12D311yescharacteristic
39C3×#870—31no—
40#63919—31no—
41#63898—31no—
42#63898—31no—
43#63896—32no—
44#63893—32no—
45D3×Dic3⋊C4—32no—
46#63872—31no—
47#63851—32no—
48C4×D4×C9—31no—
49D4×Dic9—31no—
50D4×Dic9—31no—
51#63486—32no—
52C2×Dic9⋊C4—32no—
53#63452—31no—
54#63450—31no—
55#63440—32no—
56#63440—32no—
57#63432D311yescharacteristic, complemented
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×D3×C9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68#6226—22no—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12Q814yescomplemented
73C4×D3×C9—42no—
74D3×Dic9—24no—
75D3×Dic9—42no—
76#6189—42no—
77#5690—91no—
78#5539—91no—
79C2×#634—31no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×C4×D3C2×D312yes—
82C2×#627—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87#3750—31no—
88#3750—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D4×C9—31no—
103C22×C4×C9—32no—
104C22×Dic9—32no—
105C22×Dic9—32no—
106C2×Dic18—31no—
107C2×Dic18—32no—
108C2×Dic18—61no—
109C9×C4⋊C4—31no—
110C9×C22⋊C4—32no—
111C42×C9—21no—
112C42×C9C2×D311yescharacteristic
113#3676—32no—
114#3676—32no—
115C4⋊Dic9—31no—
116C4⋊Dic9—32no—
117C4⋊Dic9—61no—
118C4⋊Dic9C2×D311yescharacteristic
119C4⋊Dic9C2×D311yescharacteristic
120Dic9⋊C4—31no—
121Dic9⋊C4—31no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—61no—
125Dic9⋊C4—61no—
126Dic9⋊C4C2×D312yes—
127Dic9⋊C4C2×D312yes—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C4×Dic9—61no—
132C22×C3×C9C2411yescharacteristic
133C2×D3×C9—22no—
134C2×D3×C9—22no—
135C2×D3×C9C2×Q814yes—
136C3×C4×C9—21no—
137C3×C4×C9—21no—
138C3×C4×C9C2×Q812yes—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C3⋊Dic9—21no—
142C3⋊Dic9#4112yes—
143C9×Dic3—22no—
144C3×Dic9—22no—
145C3×Dic9—22no—
146C3×#125—91no—
147C3×C4×D4—31no—
148D4×Dic3—91no—
149D4×Dic3—91no—
150#924—92no—
151C2×Dic3⋊C4—92no—
152#897—92no—
153#896—92no—
154#896—92no—
155#893—91no—
156#893—91no—
157D3×C4⋊C4—92no—
158#890—91no—
159#890—91no—
160#888—91no—
161#888—91no—
162#878—92no—
163#878—92no—
164C4×D12D911yescharacteristic
165#870—31no—
166C4×Dic6—91no—
167C4×Dic6—91no—
168C4×Dic6—181no—
169C22×C3×D3C22×D312yes—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177#634—62no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3C22×D312yes—
181C3×D12Dic614yes—
182C3×C4×D3—42no—
183#627—122no—
184D3×Dic3—64no—
185D3×Dic3—122no—
186C23×C9—32no—
187D4×C9—34no—
188C2×C4×C9—21no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—32no—
192C2×C4×C9—122no—
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196C2×Dic9—31no—
197C2×Dic9—31no—
198C2×Dic9—31no—
199C2×Dic9—31no—
200C2×Dic9—61no—
201C2×Dic9—61no—
202C2×Dic9—61no—
203C2×Dic9—61no—
204C2×Dic9—64no—
205C2×Dic9—122no—
206C2×Dic9C22×D312yes—
207C2×Dic9C22×D312yes—
208Dic18—62no—
209Dic18—64no—
210Dic18—122no—
211#554—271noSylow
212C2×C3×C9C22×Q811yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9H(5,𝔽2)11yescharacteristic
215D3×C9—24no—
216C2×C3×D4—31no—
217C22×C3×C4—32no—
218C22×Dic3—92no—
219C22×Dic3—92no—
220C2×D12C2×D911yescharacteristic
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×C4×D3C2×D912yes—
224C2×Dic6—91no—
225C2×Dic6—91no—
226C2×Dic6—92no—
227C2×Dic6—92no—
228C2×Dic6—181no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—92no—
234C3×C22⋊C4—32no—
235C3×C42—21no—
236C3×C42C2×D911yescharacteristic
237C3×C42D3211yescharacteristic
238#217—92no—
239#217—92no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4C2×D912yes—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—92no—
246C4⋊Dic3—91no—
247C4⋊Dic3—91no—
248C4⋊Dic3—92no—
249C4⋊Dic3—181no—
250C4⋊Dic3C2×D911yescharacteristic
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—91no—
256Dic3⋊C4—91no—
257Dic3⋊C4—91no—
258Dic3⋊C4—91no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—181no—
262Dic3⋊C4—181no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—22no—
271C2×C3×D3—22no—
272C2×C3×D3C2×Dic614yes—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4C2×Dic612yes—
276C3⋊Dic3—32no—
277C3⋊Dic3—61no—
278C3⋊Dic3—61no—
279C3⋊Dic3—61no—
280C3×Dic3—22no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C22×C9—21no—
284C22×C9—34no—
285C22×C9—62no—
286C22×C9—62no—
287C22×C9C23×D311yescharacteristic
288C4×C9—22no—
289C4×C9—21no—
290C4×C9—21no—
291C4×C9—22no—
292C4×C9—41no—
293C4×C9—62no—
294C4×C9D3×Q812yes—
295Dic9—22no—
296Dic9—22no—
297Dic9—32no—
298Dic9—62no—
299Dic9—62no—
300Dic9—61no—
301Dic9—61no—
302Dic9—61no—
303Dic9—121no—
304Dic9—121no—
305C4⋊Q8—271no—
306#125—91no—
307#125—271no—
308C22⋊Q8—272no—
309C22⋊Q8—272no—
310C22⋊Q8—272no—
311C4×Q8—271no—
312C4×D4—31no—
313C4×D4—271no—
314C4×D4—271no—
315C2×C4⋊C4—272no—
316C3×C9#55411yescharacteristic, complemented, Sylow
317C23×C3—32no—
318C22×D3C22×D912yes—
319C3×D4—34no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—32no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—122no—
327C2×C3×C4C22×D911yescharacteristic
328C2×C3×C4C22×D911yescharacteristic
329C2×C3×C4C22×D911yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341C2×Dic3—91no—
342C2×Dic3—91no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—184no—
348C2×Dic3—362no—
349C2×Dic3C22×D912yes—
350D12Dic1814yescomplemented
351C4×D3—42no—
352C4×D3—184no—
353C4×D3—362no—
354Dic6—182no—
355Dic6—182no—
356Dic6—184no—
357Dic6—362no—
358Dic6—362no—
359C2×C32C22×Dic611yescharacteristic
360C2×C32#100411yescharacteristic
361C2×C32D3×#4111yescharacteristic
362C3×D3—24no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—64no—
367C2×C9C2×D3×Q811yescharacteristic
368C2×C9D3×#4111yescharacteristic
369C2×C9#100911yescharacteristic
370C2×Q8—272no—
371C2×Q8—271no—
372C2×D4—31no—
373C22×C4—32no—
374C22×C4—272no—
375C22×C4—272no—
376C4⋊C4—31no—
377C4⋊C4—92no—
378C4⋊C4—91no—
379C4⋊C4—91no—
380C4⋊C4—92no—
381C4⋊C4—272no—
382C4⋊C4—272no—
383C4⋊C4—272no—
384C4⋊C4—271no—
385C4⋊C4—271no—
386C4⋊C4—271no—
387C22⋊C4—32no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C42—271no—
391C42—271no—
392C42D3×D911yescharacteristic
393C22×C3—21no—
394C22×C3—34no—
395C22×C3—62no—
396C22×C3—62no—
397C22×C3C23×D911yescharacteristic
398C22×C3C22×D3211yescharacteristic
399C2×D3—22no—
400C2×D3—22no—
401C2×D3C2×Dic1814yes—
402C3×C4—22no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—41no—
409C3×C4—62no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4C2×Dic1812yes—
413C3×C4D3×Dic612yes—
414Dic3—22no—
415Dic3—62no—
416Dic3—62no—
417Dic3—92no—
418Dic3—92no—
419Dic3—182no—
420Dic3—182no—
421Dic3—181no—
422Dic3—181no—
423Dic3—181no—
424Dic3—181no—
425Dic3—181no—
426Dic3—181no—
427Dic3—361no—
428Dic3—361no—
429C32#553911yescharacteristic
430C9—21no—
431C9#569011yescharacteristic, complemented
432C23—32no—
433Q8—542no—
434Q8—1082no—
435D4—34no—
436C2×C4—32no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—122no—
440C2×C4—271no—
441C2×C4—271no—
442C2×C4—271no—
443C2×C4—271no—
444C2×C4—544no—
445C2×C4—1082no—
446C2×C4C2×D3×D911yescharacteristic
447C2×C4C2×D3×D911yescharacteristic
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—64no—
453C2×C3C22×Dic1811yescharacteristic
454C2×C3#6371011yescharacteristic
455C2×C3#41×D911yescharacteristic
456C2×C3C2×D3×Dic611yescharacteristic
457C2×C3#6430611yescharacteristic
458C2×C3#6430711yescharacteristic
459D3—24no—
460C22—34no—
461C22—62no—
462C22—62no—
463C22C22×D3×D911yescharacteristic
464C4—21no—
465C4—21no—
466C4—62no—
467C4—182no—
468C4—182no—
469C4—272no—
470C4—541no—
471C4—541no—
472C4—541no—
473C4D3×Dic1812yes—
474C3—21no—
475C3#1058964111yescharacteristic, complemented
476C3#1059394111yescharacteristic
477C2—64no—
478C2C2×D3×Dic1811yescharacteristic
479C2#1172247511yescharacteristic
480C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D12C2×D3
hypercenterC42D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
3. C3×#63432C2semidirect
6. #11721024C2semidirect
7. #11721023C2semidirect
8. D3×Dic9⋊C4C2semidirect
9. #11720971C2semidirect
10. #11720955C2semidirect
11. #11720937C2semidirect
57. #63432D3semidirect
72. C9×D12Q8semidirect
316. C3×C9#554semidirect
350. D12Dic18semidirect
431. C9#5690semidirect
475. C3#10589641semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C4×C9×D122
C3×#634322
#117210252
#117210242
#117210232
D3×Dic9⋊C42
#117209712
#117209552
#117209372
#105939413
#105896413