Finite Group Databasebeta

← #13150155

Subgroups — #13150155

Order 1728.

774 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊Dic9C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic
4#11721267C212yesmaximal
5#11721248C212yesmaximal
6#11720971C211yesmaximal, characteristic, complemented
7#11720930C211yesmaximal, characteristic, complemented
8#11720928C212yesmaximal, complemented
9#11720911C211yesmaximal, characteristic, complemented
10#11720910C211yesmaximal, characteristic, complemented
11#11720902C212yesmaximal, complemented
12#10594810—31nomaximal
13#10589724—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6189C2212yes—
20C4×C3⋊Dic9C2211yescharacteristic
21C4×C3⋊Dic9C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C4×C3⋊Dic9C2211yescharacteristic
24C9×#217C2212yes—
25C3×#3676C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37C2×C4×C3⋊Dic3—31no—
38C3×#924—31no—
39C3×#924D311yescharacteristic
40#63982—32no—
41#63957—32no—
42#63919—31no—
43#63873—31no—
44#63870—32no—
45#63843—31no—
46#63843—31no—
47#63838—32no—
48C9×C22⋊Q8—31no—
49Q8×Dic9—31no—
50#63506—31no—
51#63486D311yescharacteristic
52C2×C4×Dic9—31no—
53C2×C4×Dic9—61no—
54#63451—32no—
55#63450—31no—
56#63449—31no—
57#63449—32no—
58#63440—32no—
59#63439—32no—
60C23×C3×C9C2311yescharacteristic
61C2×C3×C4×C9—22no—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9—22no—
65C2×C3⋊Dic9Q814yescomplemented
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2312yes—
74C9×Dic6—22no—
75C3×Dic18—22no—
76#6189—24no—
77#5622—91no—
78#5622—91no—
79C22×C32×C4C2×D311yescharacteristic
80C22×C3⋊Dic3—31no—
81C22×C3⋊Dic3—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6C2×D311yescharacteristic
84C2×#627—32no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—31no—
89C3×#217—32no—
90C3×#217C2×D312yes—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4C2×D312yes—
95#3724—32no—
96#3724—32no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100Dic32—32no—
101Dic32—32no—
102C2×Q8×C9—31no—
103C22×C4×C9—21no—
104C22×C4×C9C2×D311yescharacteristic
105C22×Dic9—31no—
106C22×Dic9—31no—
107C22×Dic9—61no—
108C22×Dic9—61no—
109C2×Dic18—32no—
110C2×Dic18C2×D311yescharacteristic
111C9×C4⋊C4—32no—
112C9×C4⋊C4—31no—
113C9×C22⋊C4—32no—
114#3676C2×D312yes—
115C4⋊Dic9—32no—
116C4⋊Dic9C2×D311yescharacteristic
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4C2×D312yes—
122C4×Dic9—31no—
123C4×Dic9—31no—
124C4×Dic9—31no—
125C4×Dic9—31no—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9—61no—
129C4×Dic9—61no—
130C4×Dic9—61no—
131C4×Dic9—61no—
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9C2×Q812yes—
135C22×C3×C9C2411yescharacteristic
136C3×C4×C9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—21no—
139C3⋊Dic9C2×Q812yes—
140C3⋊Dic9C2×Q812yes—
141C3⋊Dic9#4112yes—
142C9×Dic3—22no—
143C9×Dic3—22no—
144C3×Dic9—22no—
145C3×Dic9—22no—
146C3×C22⋊Q8—31no—
147C3×C22⋊Q8—91no—
148Q8×Dic3—91no—
149Q8×Dic3—91no—
150#944—91no—
151#944—91no—
152#924—31no—
153#924D911yescharacteristic
154C2×C4×Dic3—91no—
155C2×C4×Dic3—91no—
156C2×C4×Dic3—181no—
157#889—92no—
158#889—92no—
159#888—91no—
160#888—91no—
161#887—92no—
162#887—91no—
163#887—91no—
164#887—92no—
165#878—92no—
166#878—92no—
167#877—92no—
168#877—92no—
169C23×C32C22×D311yescharacteristic
170C2×C32×C4—22no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—34no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—62no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D312yes—
183C3×Dic6—22no—
184C3×Dic6—62no—
185#627—64no—
186C23×C9—21no—
187C23×C9C22×D311yescharacteristic
188Q8×C9—62no—
189C2×C4×C9—24no—
190C2×C4×C9—22no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—32no—
194C2×C4×C9—32no—
195C2×C4×C9C22×D311yescharacteristic
196C2×C4×C9C22×D311yescharacteristic
197C2×Dic9—34no—
198C2×Dic9—31no—
199C2×Dic9—31no—
200C2×Dic9—31no—
201C2×Dic9—31no—
202C2×Dic9—61no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9—61no—
206C2×Dic9—64no—
207C2×Dic9—62no—
208C2×Dic9—64no—
209C2×Dic9C22×D312yes—
210C2×Dic9C22×D312yes—
211Dic18—22no—
212Dic18—64no—
213#533—271noSylow
214C2×C3×C9—22no—
215C2×C3×C9C22×Q811yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×Q8—31no—
219C2×C3×Q8—91no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D911yescharacteristic
222C22×C3×C4D3211yescharacteristic
223C22×Dic3—91no—
224C22×Dic3—91no—
225C22×Dic3—91no—
226C22×Dic3—91no—
227C22×Dic3—181no—
228C22×Dic3—181no—
229C2×Dic6—31no—
230C2×Dic6—92no—
231C2×Dic6—92no—
232C2×Dic6C2×D911yescharacteristic
233C3×C4⋊C4—31no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—92no—
239#217—32no—
240#217C2×D912yes—
241C4⋊Dic3—31no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3C2×D911yescharacteristic
245Dic3⋊C4—32no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×D912yes—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C4×Dic3—91no—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—181no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32D3×Q812yes—
274C22×C32C23×D311yescharacteristic
275C32×C4—21no—
276C32×C4#23712yes—
277C3⋊Dic3—32no—
278C3⋊Dic3—32no—
279C3⋊Dic3—32no—
280C3⋊Dic3—61no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C22×C9—22no—
286C22×C9—22no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—22no—
291C22×C9D3×Q812yes—
292C22×C9C23×D311yescharacteristic
293C4×C9—22no—
294C4×C9—21no—
295C4×C9—22no—
296C4×C9—62no—
297C4×C9—62no—
298C4×C9#23712yes—
299Dic9—22no—
300Dic9—22no—
301Dic9—32no—
302Dic9—32no—
303Dic9—32no—
304Dic9—62no—
305Dic9—62no—
306Dic9—62no—
307Dic9—62no—
308Dic9—61no—
309C4⋊Q8—271no—
310C4⋊Q8—271no—
311#125—272no—
312C22⋊Q8—31no—
313C22⋊Q8—91no—
314C22⋊Q8—272no—
315C4×Q8—272no—
316C4×Q8—271no—
317C4×Q8—271no—
318#117—272no—
319C2×C42—271no—
320C3×C9#53311yescharacteristic, complemented, Sylow
321C23×C3—21no—
322C23×C3C22×D911yescharacteristic
323C23×C3C2×D3211yescharacteristic
324C3×Q8—62no—
325C3×Q8—182no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—92no—
334C2×C3×C4—92no—
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—91no—
342C2×Dic3—91no—
343C2×Dic3—91no—
344C2×Dic3—94no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—91no—
348C2×Dic3—94no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—184no—
356C2×Dic3—182no—
357C2×Dic3—182no—
358C2×Dic3—184no—
359C2×Dic3C22×D912yes—
360C2×Dic3C22×D912yes—
361Dic6—22no—
362Dic6—62no—
363Dic6—184no—
364Dic6—184no—
365C2×C32—22no—
366C2×C32C2×#23711yescharacteristic
367C2×C32C2×D3×Q811yescharacteristic
368C2×C32D3×#4111yescharacteristic
369C2×C9—24no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9C2×#23711yescharacteristic
375C2×C9C2×D3×Q811yescharacteristic
376C2×C9D3×#4111yescharacteristic
377C2×Q8—31no—
378C2×Q8—91no—
379C2×Q8—272no—
380C22×C4—271no—
381C22×C4—271no—
382C22×C4D3×D911yescharacteristic
383C4⋊C4—32no—
384C4⋊C4—31no—
385C4⋊C4—92no—
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—32no—
393C22⋊C4—92no—
394C42—272no—
395C42—272no—
396C42—271no—
397C42—271no—
398C42—271no—
399C42—271no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3Q8×D912yes—
409C22×C3C23×D911yescharacteristic
410C22×C3#379712yes—
411C22×C3C22×D3211yescharacteristic
412C3×C4—22no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—22no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—182no—
419C3×C4—182no—
420C3×C4#369912yes—
421C3×C4#380512yes—
422Dic3—22no—
423Dic3—22no—
424Dic3—62no—
425Dic3—62no—
426Dic3—92no—
427Dic3—92no—
428Dic3—92no—
429Dic3—92no—
430Dic3—92no—
431Dic3—92no—
432Dic3—182no—
433Dic3—182no—
434Dic3—182no—
435Dic3—182no—
436Dic3—181no—
437Dic3—181no—
438C32#562211yescharacteristic
439C9—21no—
440C9#562211yescharacteristic, complemented
441C23C2×D3×D911yescharacteristic
442Q8—62no—
443Q8—182no—
444Q8—544no—
445C2×C4—22no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—274no—
451C2×C4—271no—
452C2×C4—271no—
453C2×C4—271no—
454C2×C4—271no—
455C2×C4—542no—
456C2×C4C2×D3×D911yescharacteristic
457C2×C4C2×D3×D911yescharacteristic
458C2×C3—24no—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3C2×#369911yescharacteristic
465C2×C3C2×Q8×D911yescharacteristic
466C2×C3#41×D911yescharacteristic
467C2×C3C2×#379711yescharacteristic
468C2×C3#6430611yescharacteristic
469C2×C3C2×#380511yescharacteristic
470C22—21no—
471C22—21no—
472C22(C3⋊D9)⋊Q812yes—
473C22C22×D3×D911yescharacteristic
474C4—21no—
475C4—62no—
476C4—62no—
477C4—182no—
478C4—182no—
479C4—272no—
480C4—272no—
481C4—272no—
482C4—541no—
483C4#8866312yes—
484C3—21no—
485C3#1058972411yescharacteristic, complemented
486C3#1059481011yescharacteristic
487C2—22no—
488C2C2×(C3⋊D9)⋊Q811yescharacteristic
489C2#1172247511yescharacteristic
490C2C2×#8866311yescharacteristic
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. #11720971C2semidirect
7. #11720930C2semidirect
8. #11720928C2semidirect
9. #11720911C2semidirect
10. #11720910C2semidirect
11. #11720902C2semidirect
65. C2×C3⋊Dic9Q8semidirect
320. C3×C9#533semidirect
440. C9#5622semidirect
485. C3#10589724semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊Dic92
C9×#9242
C3×#634862
#117212672
#117212482
#117209712
#117209302
#117209282
#117209112
#117209102
#117209022
#105948103
#105897243