Finite Group Databasebeta

← #13140481

Subgroups — #13140481

Order 1728.

800 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D9C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720929C211yesmaximal, characteristic, complemented
4C2×#88426C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10591467—31nomaximal
9#10588849—31nomaximal
10C23×C3⋊D9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6233—21no—
13D4×C3⋊D9—22no—
14D4×C3⋊D9—22no—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic, complemented
18C9×#214—22no—
19C3×C4⋊Dic9C2211yescharacteristic, complemented
20C2×C9×C3⋊C8C2211yescharacteristic, complemented
21C2×C3×C9⋊C8C2211yescharacteristic, complemented
22#88445—21no—
23Dic9⋊Dic3—21no—
24#88426—22no—
25C2×D4×C3⋊D3—31no—
26C2×C3×#214D311yescharacteristic
27#63871—31no—
28C2×#3717—31no—
29C2×D4×D9—31no—
30C2×D4×D9—61no—
31C3×#832—31no—
32#63564—31no—
33#63557—31no—
34C2×C9×SD16—31no—
35#63508—31no—
36C2×#3675—31no—
37#63392D311yescharacteristic, complemented
38#63380—31no—
39(C2×D9)⋊C8—31no—
40C23×C3×C9—21no—
41C22×C3⋊D9—24no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—42no—
44C22×C3⋊D9—42no—
45C22×C3⋊D9D411yescharacteristic
46C3×D4×C9—21no—
47C3×D4×C9D412yes—
48C2×C3×C4×C9C2311yescharacteristic
49#6233—24no—
50C2×C3⋊Dic9D411yescharacteristic
51C4⋊(C3⋊D9)—21no—
52C4⋊(C3⋊D9)D412yes—
53C4×C3⋊D9—21no—
54C4×C3⋊D9—21no—
55C2×C9×Dic3—21no—
56C2×C3×Dic9—21no—
57C9×Dic6—21no—
58C9×Dic6—21no—
59C9×C3⋊C8—21no—
60C3×C9⋊C8—21no—
61#5169—91no—
62#4776—91no—
63C23×C3⋊D3—61no—
64C2×C32×D4C2×D311yescharacteristic
65C2×#638—61no—
66D4×C3⋊D3—62no—
67D4×C3⋊D3—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4×C3⋊D3—31no—
70C2×C3×Dic6C2×D311yescharacteristic
71C23×D9—61no—
72C23×D9—62no—
73C3×#214—22no—
74C3×C4⋊Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8C2×D311yescharacteristic
77Dic3⋊Dic3—61no—
78#3722—61no—
79#3717—62no—
80C2×Q8×C9—31no—
81C2×D4×C9—21no—
82C2×D4×C9C2×D311yescharacteristic
83C2×#611—61no—
84C2×#611—62no—
85D4×D9—62no—
86D4×D9—64no—
87D4×D9—64no—
88D4×D9—62no—
89C2×D36—31no—
90C2×D36—61no—
91C2×C4×D9—31no—
92C2×C4×D9—61no—
93C9×SD16—62no—
94C2×C8×C9—31no—
95#3675—62no—
96(C2×D9)⋊C4—61no—
97C4⋊Dic9C2×D311yescharacteristic, complemented
98Dic9⋊C4—61no—
99C2×C9⋊C8C2×D311yescharacteristic, complemented
100C22×C3×C9—22no—
101C22×C3×C9—41no—
102C22×C3×C9—41no—
103C22×C3×C9C2×D411yescharacteristic
104C2×C3⋊D9—21no—
105C2×C3⋊D9—21no—
106C2×C3⋊D9—22no—
107C2×C3⋊D9—48no—
108C2×C3⋊D9—41no—
109C2×C3⋊D9—41no—
110C2×C3⋊D9SD1612yescomplemented
111C3×C4×C9C2×D411yescharacteristic
112C3×C4×C9C2×D411yescharacteristic
113C3⋊Dic9—21no—
114C9×Dic3—41no—
115C3×Dic9—41no—
116C2×D3×D4—91no—
117C2×D3×D4—91no—
118C2×D3×D4—181no—
119C2×C3×SD16—31no—
120#946—91no—
121C2×#215—91no—
122C2×#214D911yescharacteristic, complemented
123C2×#204—91no—
124#897—91no—
125C3×#102—91no—
126#832—31no—
127#821—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—91no—
130#809—91no—
131C23×C32—21no—
132C22×C3⋊D3—31no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—122no—
136C22×C3⋊D3—122no—
137C32×D4—21no—
138C32×D4#6712yes—
139C2×C32×C4C22×D311yescharacteristic
140#638—64no—
141C2×C3⋊Dic3—31no—
142C4⋊(C3⋊D3)—32no—
143C4⋊(C3⋊D3)—61no—
144C4×C3⋊D3—61no—
145C4×C3⋊D3—61no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—61no—
148C3×Dic6—21no—
149C3×Dic6—21no—
150C23×C9—22no—
151C23×C9—21no—
152C22×D9—31no—
153C22×D9—61no—
154C22×D9—62no—
155C22×D9—68no—
156C22×D9—64no—
157C22×D9—61no—
158C22×D9—122no—
159C22×D9—124no—
160C22×D9—124no—
161C22×D9—122no—
162C3×C3⋊C8—21no—
163C3×C3⋊C8—61no—
164Q8×C9—61no—
165Q8×C9—61no—
166D4×C9—22no—
167D4×C9—21no—
168D4×C9—22no—
169D4×C9D1212yes—
170C2×C4×C9—21no—
171C2×C4×C9—61no—
172C2×C4×C9C22×D311yescharacteristic
173#611—68no—
174#611—64no—
175C2×Dic9—21no—
176C2×Dic9—31no—
177C2×Dic9—61no—
178D36—32no—
179D36—62no—
180D36—62no—
181D36—61no—
182C4×D9—62no—
183C4×D9—62no—
184C4×D9—61no—
185C4×D9—61no—
186C8×C9—61no—
187C9⋊C8—21no—
188#450—271noSylow
189C2×C3×C9—42no—
190C2×C3×C9#12011yescharacteristic
191C2×C3×C9C2×SD1611yescharacteristic
192C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
193C3⋊D9—22no—
194C3⋊D9—42no—
195C23×D3—181no—
196C23×D3—181no—
197C23×D3—182no—
198C2×C3×Q8—31no—
199C2×C3×D4—21no—
200C2×C3×D4C2×D911yescharacteristic
201C2×C3×D4D3211yescharacteristic
202C2×#67—181no—
203C2×#67—181no—
204C2×#67—182no—
205D3×D4—182no—
206D3×D4—182no—
207D3×D4—184no—
208D3×D4—184no—
209D3×D4—182no—
210D3×D4—182no—
211C2×D12—91no—
212C2×D12—91no—
213C2×D12—181no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×C4×D3—181no—
217C2×Dic6C2×D911yescharacteristic, complemented
218C3×SD16—62no—
219C2×C3×C8—31no—
220C2×C3×C8—91no—
221C3×C4⋊C4—91no—
222#215—182no—
223#214—22no—
224(C2×D3)⋊C4—181no—
225(C2×D3)⋊C4—181no—
226C4⋊Dic3—31no—
227C4⋊Dic3—181no—
228Dic3⋊C4—181no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8C2×D911yescharacteristic, complemented
231#204—182no—
232C22×C32—22no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32D3×D411yescharacteristic
236C2×C3⋊D3—32no—
237C2×C3⋊D3—61no—
238C2×C3⋊D3—61no—
239C2×C3⋊D3—62no—
240C2×C3⋊D3—128no—
241C2×C3⋊D3—121no—
242C2×C3⋊D3—121no—
243C32×C4D3×D411yescharacteristic
244C32×C4C2×#6711yescharacteristic
245C3⋊Dic3—61no—
246C3×Dic3—41no—
247C3×Dic3—121no—
248C22×C9—24no—
249C22×C9—22no—
250C22×C9—21no—
251C22×C9—42no—
252C22×C9—42no—
253C22×C9—41no—
254C22×C9—41no—
255C22×C9D3×D411yescharacteristic
256C2×D9—32no—
257C2×D9—62no—
258C2×D9—62no—
259C2×D9—62no—
260C2×D9—61no—
261C2×D9—61no—
262C2×D9—64no—
263C2×D9—62no—
264C2×D9—1216no—
265C2×D9—128no—
266C2×D9—122no—
267C2×D9—122no—
268C2×D9—121no—
269C2×D9—121no—
270C4×C9—21no—
271C4×C9—21no—
272C4×C9—121no—
273C4×C9C2×D1211yescharacteristic
274C4×C9D3×D411yescharacteristic
275Dic9—41no—
276Dic9—62no—
277Dic9—61no—
278C22×D4—271no—
279C2×SD16—31no—
280C2×SD16—271no—
281C22⋊Q8—271no—
282#102—91no—
283#102—271no—
284C22⋊C8—271no—
285C3×C9#45011yescharacteristic, complemented, Sylow
286C23×C3—22no—
287C23×C3—21no—
288C23×C3—21no—
289C22×D3—91no—
290C22×D3—91no—
291C22×D3—181no—
292C22×D3—184no—
293C22×D3—182no—
294C22×D3—188no—
295C22×D3—184no—
296C22×D3—181no—
297C22×D3—181no—
298C22×D3—362no—
299C22×D3—362no—
300C22×D3—362no—
301C22×D3—364no—
302C22×D3—364no—
303C22×D3—362no—
304C3×Q8—61no—
305C3×Q8—61no—
306C3×D4—22no—
307C3×D4—21no—
308C3×D4—21no—
309C3×D4—22no—
310C3×D4#61112yes—
311C3×D4#62612yes—
312C2×C3×C4—21no—
313C2×C3×C4—61no—
314C2×C3×C4—181no—
315C2×C3×C4C22×D911yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317#67—184no—
318#67—188no—
319#67—184no—
320C2×Dic3—21no—
321C2×Dic3—61no—
322C2×Dic3—91no—
323C2×Dic3—91no—
324C2×Dic3—181no—
325D12—92no—
326D12—92no—
327D12—182no—
328D12—182no—
329D12—181no—
330D12—181no—
331C4×D3—182no—
332C4×D3—182no—
333C4×D3—181no—
334C4×D3—181no—
335C4×D3—181no—
336C4×D3—181no—
337Dic6—21no—
338Dic6—21no—
339C3×C8—61no—
340C3×C8—181no—
341C3⋊C8—21no—
342C3⋊C8—61no—
343C2×C32—42no—
344C2×C32D3×SD1611yescharacteristic
345C2×C32#91411yescharacteristic
346C2×C32#93711yescharacteristic
347C3⋊D3—62no—
348C3⋊D3—122no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—21no—
352C2×C9—44no—
353C2×C9—42no—
354C2×C9#88211yescharacteristic
355C2×C9#91111yescharacteristic
356C2×C9D3×SD1611yescharacteristic
357D9—64no—
358D9—62no—
359D9—124no—
360D9—122no—
361C24—541no—
362C2×Q8—31no—
363C2×D4—271no—
364C2×D4—541no—
365C2×D4—542no—
366C2×D4—542no—
367C2×D4D3×D911yescharacteristic
368C22×C4—271no—
369SD16—62no—
370SD16—542no—
371C2×C8—31no—
372C2×C8—91no—
373C4⋊C4—91no—
374C4⋊C4—541no—
375C22⋊C4—541no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—21no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3D4×D911yescharacteristic
387C22×C3#381111yescharacteristic
388C2×D3—92no—
389C2×D3—92no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—181no—
394C2×D3—181no—
395C2×D3—181no—
396C2×D3—181no—
397C2×D3—184no—
398C2×D3—182no—
399C2×D3—182no—
400C2×D3—368no—
401C2×D3—3616no—
402C2×D3—368no—
403C2×D3—362no—
404C2×D3—362no—
405C2×D3—361no—
406C2×D3—361no—
407C2×D3—361no—
408C2×D3—361no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—121no—
412C3×C4—361no—
413C3×C4D4×D911yescharacteristic
414C3×C4C2×#61111yescharacteristic
415C3×C4C2×#62611yescharacteristic
416C3×C4#381111yescharacteristic
417Dic3—41no—
418Dic3—121no—
419Dic3—182no—
420Dic3—181no—
421Dic3—181no—
422C32#516911yescharacteristic
423C9—21no—
424C9#477611yescharacteristic, complemented
425C23—21no—
426C23—271no—
427C23—544no—
428C23—541no—
429C23—1082no—
430C23—1082no—
431Q8—61no—
432Q8—61no—
433D4—21no—
434D4—272no—
435D4—544no—
436D4—541no—
437D4#619512yes—
438C2×C4—61no—
439C2×C4—181no—
440C2×C4—271no—
441C2×C4—541no—
442C2×C4—541no—
443C2×C4C2×D3×D911yescharacteristic
444C8—61no—
445C8—181no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—44no—
450C2×C3—42no—
451C2×C3—42no—
452C2×C3SD16×D911yescharacteristic
453C2×C3#6347611yescharacteristic
454C2×C3#6349911yescharacteristic
455C2×C3#6393011yescharacteristic
456C2×C3#6393711yescharacteristic
457C2×C3#6398111yescharacteristic
458D3—184no—
459D3—182no—
460D3—182no—
461D3—364no—
462D3—362no—
463D3—362no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—272no—
468C22—541no—
469C22—541no—
470C22—542no—
471C22—1088no—
472C22—1081no—
473C22—1081no—
474C22#8866811yescharacteristic
475C4—121no—
476C4—361no—
477C4—541no—
478C4C2×#619511yescharacteristic
479C4#8866811yescharacteristic
480C3—21no—
481C3#1058884911yescharacteristic, complemented
482C3#1059146711yescharacteristic
483C2—42no—
484C2—542no—
485C2—1082no—
486C2#1172118211yescharacteristic
487C2#1172118711yescharacteristic
488C2#1172129111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×D4×C3⋊D9C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#450
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C9×#214C2semidirect
3. #11720929C2semidirect
4. C2×#88426C2semidirect
5. C3×#63392C2semidirect
6. #11720284C2semidirect
7. #11720282C2semidirect
17. C2×C9×Dic6C22semidirect
19. C3×C4⋊Dic9C22semidirect
20. C2×C9×C3⋊C8C22semidirect
21. C2×C3×C9⋊C8C22semidirect
37. #63392D3semidirect
97. C4⋊Dic9C2×D3semidirect
99. C2×C9⋊C8C2×D3semidirect
110. C2×C3⋊D9SD16semidirect
122. C2×#214D9semidirect
217. C2×Dic6C2×D9semidirect
230. C2×C3⋊C8C2×D9semidirect
285. C3×C9#450semidirect
424. C9#4776semidirect
481. C3#10588849semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C2×C9×#2142
#117209292
C2×#884262
C3×#633922
#117202842
#117202822
#105914673
#105888493