Finite Group Databasebeta

← #13137832

Subgroups — #13137832

Order 1728.

646 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 101 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×#88646C211yesmaximal, characteristic
2#11721748C211yesmaximal, characteristic, complemented
3C2×#88414C211yesmaximal, characteristic, complemented
4C9×#809C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590594—31nomaximal
9#10588450—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6194—21no—
15#88646—22no—
16#88646C414yescomplemented
17C3×C9×C4⋊C4C2211yescharacteristic, complemented
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
20C4×C3⋊Dic9—21no—
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22C2×C3×C9⋊C8C2211yescharacteristic, complemented
23#88414—22no—
24C2×#3802—31no—
25#64101—31no—
26C2×#3712—31no—
27C2×D4×D9—31no—
28C3×#809—31no—
29C3×#809D311yescharacteristic
30#63563—31no—
31#63563—31no—
32C2×#3663—31no—
33#63454—31no—
34#63454—61no—
35C9×#102—31no—
36#63392—31no—
37#63380—31no—
38#63369D311yescharacteristic, complemented
39C22×C3⋊D9D411yescharacteristic
40C22×D3×C9—21no—
41C22×C3×D9—21no—
42C2×D3×D9—24no—
43C2×D3×D9—42no—
44C2×D3×D9—42no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9D411yescharacteristic
49C4×C3⋊D9C2×C412yes—
50C4×C3⋊D9D412yes—
51C9×D12—21no—
52C9×D12C2×C412yes—
53C3×D36—21no—
54C3×D36C2×C412yes—
55#6194—24no—
56C9×C3⋊C8—21no—
57C3×C9⋊C8—21no—
58#4770—91no—
59#4770—91no—
60C22×D32—61no—
61C2×C4×C3⋊D3—31no—
62C2×C3×D12—31no—
63C2×C3×D12C2×D311yescharacteristic
64C2×#625—61no—
65#3802—34no—
66#3802—62no—
67C23×D9—61no—
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—61no—
70C4⋊(C3⋊Dic3)—31no—
71C4×C3⋊Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74#3712—62no—
75C2×D4×C9—31no—
76C2×#611—61no—
77D4×D9—34no—
78D4×D9—62no—
79C2×D36C2×D311yescharacteristic
80C2×C4×D9—31no—
81C2×C4×D9—61no—
82C2×C8×C9—31no—
83C9×C4⋊C4—21no—
84C9×C4⋊C4C2×D311yescharacteristic, complemented
85(C2×D9)⋊C4—61no—
86(C2×D9)⋊C4—62no—
87C4⋊Dic9—31no—
88C4⋊Dic9—61no—
89C4×Dic9—61no—
90C4×Dic9—62no—
91C2×C9⋊C8C2×D311yescharacteristic, complemented
92#3663—62no—
93C22×C3×C9C2×D411yescharacteristic
94C2×C3⋊D9—21no—
95C2×C3⋊D9—21no—
96C2×C3⋊D9C22⋊C412yes—
97C2×D3×C9—22no—
98C2×D3×C9—41no—
99C2×D3×C9—41no—
100C2×C3×D9—22no—
101C2×C3×D9—41no—
102C2×C3×D9—41no—
103D3×D9—48no—
104C3×C4×C9—41no—
105C3×C4×C9C22×C411yescharacteristic
106C3×C4×C9C2×D411yescharacteristic
107C3⋊Dic9—41no—
108C3⋊Dic9C22⋊C412yes—
109C2×D3×D4—91no—
110C2×D3×D4—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113#892—91no—
114#892—91no—
115#892—181no—
116C3×#102—31no—
117C3×#102—91no—
118#832—91no—
119#832—91no—
120#821—91no—
121#821—91no—
122#809—31no—
123#809D911yescharacteristic, complemented
124C22×C3⋊D3—31no—
125C22×C3×D3—21no—
126C22×C3×D3—61no—
127C2×D32—64no—
128C2×D32—122no—
129C2×D32—122no—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—61no—
134C4×C3⋊D3—32no—
135C4×C3⋊D3—32no—
136C3×D12—21no—
137C3×D12—32no—
138C3×D12—61no—
139C3×D12C4×D312yes—
140#625—64no—
141C23×C9—61no—
142C22×D9—21no—
143C22×D9—31no—
144C22×D9—61no—
145C22×D9—64no—
146C22×D9—122no—
147C22×D9—122no—
148C3×C3⋊C8—21no—
149C3×C3⋊C8—61no—
150D4×C9—32no—
151D4×C9—61no—
152C2×C4×C9—22no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9C22×D311yescharacteristic
156#611—64no—
157C2×Dic9—31no—
158C2×Dic9—61no—
159C2×Dic9—62no—
160C2×Dic9—61no—
161D36—21no—
162D36C4×D312yes—
163C4×D9—32no—
164C4×D9—32no—
165C4×D9—62no—
166C4×D9—62no—
167C8×C9—61no—
168C9⋊C8—21no—
169#418—271noSylow
170C2×C3×C9C2×C22⋊C411yescharacteristic
171C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
172C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
173C3⋊D9—22no—
174D3×C9—42no—
175C3×D9—42no—
176C23×D3—181no—
177C23×D3—181no—
178C2×C3×D4—31no—
179C2×C3×D4—91no—
180C2×#67—181no—
181C2×#67—181no—
182D3×D4—94no—
183D3×D4—94no—
184D3×D4—182no—
185D3×D4—182no—
186C2×D12—31no—
187C2×D12C2×D911yescharacteristic
188C2×C4×D3—91no—
189C2×C4×D3—91no—
190C2×C4×D3—181no—
191C2×C3×C8—31no—
192C2×C3×C8—91no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4C2×D911yescharacteristic, complemented
195C3×C4⋊C4D3211yescharacteristic
196(C2×D3)⋊C4—181no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—182no—
199C4⋊Dic3—91no—
200C4⋊Dic3—91no—
201C4⋊Dic3—181no—
202C4×Dic3—181no—
203C4×Dic3—181no—
204C4×Dic3—182no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8C2×D911yescharacteristic, complemented
207ΓL(1,𝔽25)—182no—
208ΓL(1,𝔽25)—182no—
209C22×C32D3×D411yescharacteristic
210C2×C3⋊D3—32no—
211C2×C3⋊D3—61no—
212C2×C3⋊D3—61no—
213C2×C3×D3—22no—
214C2×C3×D3—41no—
215C2×C3×D3—41no—
216C2×C3×D3—62no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219D32—128no—
220C32×C4—41no—
221C32×C4C2×C4×D311yescharacteristic
222C32×C4D3×D411yescharacteristic
223C3⋊Dic3—32no—
224C3⋊Dic3—121no—
225C22×C9—21no—
226C22×C9—62no—
227C22×C9—121no—
228C22×C9—121no—
229C22×C9D3×D411yescharacteristic
230C2×D9—22no—
231C2×D9—32no—
232C2×D9—41no—
233C2×D9—41no—
234C2×D9—62no—
235C2×D9—61no—
236C2×D9—61no—
237C2×D9—121no—
238C2×D9—121no—
239C2×D9—128no—
240C4×C9—21no—
241C4×C9—21no—
242C4×C9—42no—
243C4×C9—41no—
244C4×C9C2×C4×D311yescharacteristic
245C4×C9D3×D411yescharacteristic
246Dic9—32no—
247Dic9—62no—
248Dic9—122no—
249Dic9—121no—
250C22×D4—271no—
251C2×M16—271no—
252#117—271no—
253#102—31no—
254#102—91no—
255#102—271no—
256#102—271no—
257C3×C9#41811yescharacteristic, complemented, Sylow
258C23×C3—61no—
259C23×C3—181no—
260C22×D3—21no—
261C22×D3—61no—
262C22×D3—91no—
263C22×D3—91no—
264C22×D3—181no—
265C22×D3—184no—
266C22×D3—184no—
267C22×D3—362no—
268C22×D3—362no—
269C22×D3—362no—
270C22×D3—362no—
271C3×D4—32no—
272C3×D4—61no—
273C3×D4—92no—
274C3×D4—181no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4C22×D911yescharacteristic
280C2×C3×C4C2×D3211yescharacteristic
281#67—184no—
282#67—184no—
283C2×Dic3—91no—
284C2×Dic3—91no—
285C2×Dic3—181no—
286C2×Dic3—182no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289D12—21no—
290D12—32no—
291D12—61no—
292D12C4×D912yes—
293C4×D3—92no—
294C4×D3—92no—
295C4×D3—92no—
296C4×D3—92no—
297C4×D3—182no—
298C4×D3—182no—
299C3×C8—61no—
300C3×C8—181no—
301C3⋊C8—21no—
302C3⋊C8—61no—
303C2×C32D3×C22⋊C411yescharacteristic
304C2×C32#91111yescharacteristic
305C2×C32#91411yescharacteristic
306C3⋊D3—62no—
307C3×D3—42no—
308C3×D3—122no—
309C2×C9—21no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9—122no—
313C2×C9D3×C22⋊C411yescharacteristic
314C2×C9#91111yescharacteristic
315C2×C9#91411yescharacteristic
316D9—42no—
317D9—62no—
318D9—122no—
319C24—541no—
320C2×D4—31no—
321C2×D4—91no—
322C2×D4—274no—
323C2×D4—542no—
324C2×D4—541no—
325C22×C4—271no—
326M16—542no—
327C2×C8—31no—
328C2×C8—91no—
329C4⋊C4—271no—
330C4⋊C4D3×D911yescharacteristic, complemented
331C22⋊C4—541no—
332C42—541no—
333C22×C3—21no—
334C22×C3—62no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—182no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3D4×D911yescharacteristic
341C22×C3#381111yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—62no—
346C2×D3—92no—
347C2×D3—92no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—182no—
351C2×D3—181no—
352C2×D3—181no—
353C2×D3—181no—
354C2×D3—181no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—368no—
358C2×D3—368no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—42no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4C2×C4×D911yescharacteristic
365C3×C4D4×D911yescharacteristic
366C3×C4C4×D3211yescharacteristic
367C3×C4#380211yescharacteristic
368Dic3—92no—
369Dic3—92no—
370Dic3—182no—
371Dic3—362no—
372Dic3—361no—
373Dic3—361no—
374C32#477011yescharacteristic
375C9—21no—
376C9#477011yescharacteristic, complemented
377C23—61no—
378C23—181no—
379C23—271no—
380C23—544no—
381C23—1082no—
382C23—1082no—
383D4—32no—
384D4—61no—
385D4—92no—
386D4—181no—
387D4—544no—
388C2×C4—21no—
389C2×C4—271no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—541no—
393C2×C4C2×D3×D911yescharacteristic
394C8—61no—
395C8—181no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—122no—
400C2×C3—362no—
401C2×C3C22⋊C4×D911yescharacteristic
402C2×C3#6347311yescharacteristic
403C2×C3#6347611yescharacteristic
404C2×C3#6392111yescharacteristic
405C2×C3#6392711yescharacteristic
406C2×C3#6394111yescharacteristic
407D3—42no—
408D3—122no—
409D3—182no—
410D3—182no—
411D3—362no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—182no—
416C22—272no—
417C22—361no—
418C22—361no—
419C22—541no—
420C22—541no—
421C22—1088no—
422C22#8866811yescharacteristic
423C4—41no—
424C4—272no—
425C4—1081no—
426C4C4×D3×D911yescharacteristic
427C4#8864611yescharacteristic
428C3—21no—
429C3#1058845011yescharacteristic, complemented
430C3#1059059411yescharacteristic
431C2—122no—
432C2—362no—
433C2—542no—
434C2#1172104811yescharacteristic
435C2#1172117511yescharacteristic
436C2#1172121311yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#418
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721748C2semidirect
3. C2×#88414C2semidirect
4. C9×#809C2semidirect
5. C3×#63369C2semidirect
6. #11720295C2semidirect
7. #11720283C2semidirect
16. #88646C4semidirect
17. C3×C9×C4⋊C4C22semidirect
19. C4⋊(C3⋊Dic9)C22semidirect
21. C2×C9×C3⋊C8C22semidirect
22. C2×C3×C9⋊C8C22semidirect
38. #63369D3semidirect
84. C9×C4⋊C4C2×D3semidirect
91. C2×C9⋊C8C2×D3semidirect
123. #809D9semidirect
194. C3×C4⋊C4C2×D9semidirect
206. C2×C3⋊C8C2×D9semidirect
257. C3×C9#418semidirect
330. C4⋊C4D3×D9semidirect
376. C9#4770semidirect
429. C3#10588450semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
#117217482
C2×#884142
C9×#8092
C3×#633692
#117202952
#117202832
#105905943
#105884503