Finite Group Databasebeta

← #13140372

Subgroups — #13140372

Order 1728.

620 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 121 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×C3×D4×D9C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic, complemented
3C2×#88413C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10591421—31nomaximal
9#10588450—31nomaximal
10C23×C3×D9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×C3×#611—21no—
13C3×D4×D9—22no—
14C3×D4×D9C414yescomplemented
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
18C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
19C9×C4⋊Dic3C2211yescharacteristic, complemented
20C2×C9×C3⋊C8C2211yescharacteristic, complemented
21(C2×D9)⋊Dic3—21no—
22Dic3×Dic9—21no—
23#88413—22no—
24C2×C3×D3×D4—31no—
25#63841—31no—
26C2×#3711—31no—
27C2×D4×D9D311yescharacteristic
28#63659—31no—
29C3×#832D311yescharacteristic
30#63563—31no—
31#63562—31no—
32C2×#3663—31no—
33#63454—31no—
34C9×#102—31no—
35#63392—31no—
36#63392—61no—
37#63380—31no—
38#63369—31no—
39C23×C3×C9—21no—
40C22×C3×D9—21no—
41C22×C3×D9—24no—
42C22×C3×D9—42no—
43C22×C3×D9—42no—
44C22×C3×D9D411yescharacteristic
45C3×D4×C9—21no—
46C3×D4×C9C2×C412yes—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9—21no—
49C3×#611—24no—
50C2×C9×Dic3—21no—
51C2×C3×Dic9D411yescharacteristic
52C3×D36—21no—
53C3×D36C2×C412yes—
54C3×C4×D9C2×C412yes—
55C3×C4×D9D412yes—
56C3⋊(C9⋊C8)—21no—
57C9×C3⋊C8—21no—
58#5215—91no—
59#4770—91no—
60C23×C3×D3—61no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C3×#67—61no—
63C3×D3×D4—34no—
64C3×D3×D4—62no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—31no—
67C23×D9—21no—
68C4⋊(C3⋊Dic3)—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C3×C4⋊Dic3C2×D311yescharacteristic
71C2×C3×C3⋊C8C2×D311yescharacteristic
72(C2×D3)⋊Dic3—61no—
73Dic32—61no—
74#3711—62no—
75C2×D4×C9—21no—
76C2×D4×C9C2×D311yescharacteristic
77C2×#611—21no—
78D4×D9—22no—
79D4×D9Dic314yescomplemented
80C2×D36C2×D311yescharacteristic
81C2×C4×D9C2×D311yescharacteristic
82C2×C8×C9—31no—
83C9×C4⋊C4—31no—
84(C2×D9)⋊C4—61no—
85C4⋊Dic9—31no—
86C4⋊Dic9—61no—
87C4×Dic9—61no—
88C2×C9⋊C8—31no—
89C2×C9⋊C8—61no—
90#3663—62no—
91C22×C3×C9—22no—
92C22×C3×C9—41no—
93C22×C3×C9—41no—
94C22×C3×C9C2×D411yescharacteristic
95C2×C3×D9—21no—
96C2×C3×D9—21no—
97C2×C3×D9—22no—
98C2×C3×D9—48no—
99C2×C3×D9—41no—
100C2×C3×D9—41no—
101C2×C3×D9C22⋊C412yes—
102C3×C4×C9C22×C411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—41no—
105C9×Dic3—41no—
106C3×Dic9C22⋊C412yes—
107C22×C3×D4—91no—
108C2×D3×D4—31no—
109#926—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112#892—91no—
113C3×#102—31no—
114#832—91no—
115#832—91no—
116#832—91no—
117#832—91no—
118#832—181no—
119#832D911yescharacteristic, complemented
120#821—91no—
121#809—91no—
122C23×C32—21no—
123C22×C3×D3—31no—
124C22×C3×D3—64no—
125C22×C3×D3—61no—
126C22×C3×D3—122no—
127C22×C3×D3—122no—
128C32×D4—21no—
129C32×D4C4×D312yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—61no—
132C3×#67—64no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—31no—
135C3×D12—32no—
136C3×D12—61no—
137C3×C4×D3—32no—
138C3×C4×D3—32no—
139C23×C9—21no—
140C23×C9—41no—
141C22×D9—21no—
142C22×D9—24no—
143C22×D9—42no—
144C22×D9—42no—
145C22×D9#6711yescharacteristic
146C3⋊(C3⋊C8)—61no—
147C3×C3⋊C8—21no—
148D4×C9—21no—
149D4×C9—22no—
150D4×C9—41no—
151D4×C9C2×Dic312yes—
152C2×C4×C9—21no—
153C2×C4×C9—61no—
154C2×C4×C9C22×D311yescharacteristic
155#611—24no—
156C2×Dic9—61no—
157C2×Dic9—121no—
158C2×Dic9#6711yescharacteristic
159D36—21no—
160D36C2×Dic312yes—
161C4×D9C2×Dic312yes—
162C4×D9#6712yes—
163C8×C9—61no—
164C9⋊C8—61no—
165C9⋊C8—121no—
166#418—271noSylow
167C2×C3×C9—42no—
168C2×C3×C9C2×C22⋊C411yescharacteristic
169C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
170C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
171C3×D9—22no—
172C3×D9—42no—
173C24×C3—181no—
174C23×D3—61no—
175C2×C3×D4—21no—
176C2×C3×D4—94no—
177C2×C3×D4—91no—
178C2×C3×D4—182no—
179C2×C3×D4—181no—
180C2×C3×D4C2×D911yescharacteristic
181C2×C3×D4D3211yescharacteristic
182C22×C3×C4—91no—
183C2×#67—61no—
184D3×D4—34no—
185D3×D4—62no—
186C2×D12—31no—
187C2×C4×D3—31no—
188C2×C3×C8—31no—
189C3×C4⋊C4—31no—
190#217—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—91no—
193C4⋊Dic3—91no—
194C4⋊Dic3—181no—
195C4⋊Dic3C2×D911yescharacteristic, complemented
196C4×Dic3—181no—
197C4×Dic3—181no—
198#208—182no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—181no—
202C2×C3⋊C8C2×D911yescharacteristic, complemented
203ΓL(1,𝔽25)—182no—
204C22×C32—22no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32D3×D411yescharacteristic
208C2×C3×D3—32no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—62no—
212C2×C3×D3—128no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C32×C4C2×C4×D311yescharacteristic
216C32×C4D3×D411yescharacteristic
217C3⋊Dic3—121no—
218C3×Dic3—32no—
219C3×Dic3—41no—
220C22×C9—22no—
221C22×C9—21no—
222C22×C9—42no—
223C22×C9—41no—
224C22×C9—41no—
225C22×C9—81no—
226C22×C9—81no—
227C22×C9C2×#6711yescharacteristic
228C2×D9—21no—
229C2×D9—21no—
230C2×D9—22no—
231C2×D9—48no—
232C2×D9—41no—
233C2×D9—41no—
234C2×D9#21712yes—
235C4×C9—21no—
236C4×C9—21no—
237C4×C9—121no—
238C4×C9C22×Dic311yescharacteristic
239C4×C9C2×#6711yescharacteristic
240Dic9—121no—
241Dic9—241no—
242Dic9#21712yes—
243C22×D4—91no—
244C2×M16—271no—
245#117—271no—
246#102—31no—
247#102—271no—
248#102—271no—
249#102—271no—
250C3×C9#41811yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—41no—
254C23×C3—91no—
255C23×C3—184no—
256C23×C3—181no—
257C23×C3—362no—
258C23×C3—362no—
259C22×D3—31no—
260C22×D3—64no—
261C22×D3—61no—
262C22×D3—122no—
263C22×D3—122no—
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—22no—
267C3×D4—41no—
268C3×D4—92no—
269C3×D4—184no—
270C3×D4—181no—
271C3×D4C4×D912yes—
272C3×D4D3×Dic312yes—
273C2×C3×C4—21no—
274C2×C3×C4—61no—
275C2×C3×C4—91no—
276C2×C3×C4—92no—
277C2×C3×C4—92no—
278C2×C3×C4C22×D911yescharacteristic
279C2×C3×C4C2×D3211yescharacteristic
280#67—64no—
281C2×Dic3—21no—
282C2×Dic3—31no—
283C2×Dic3—181no—
284C2×Dic3—181no—
285C2×Dic3—361no—
286D12—32no—
287D12—61no—
288C4×D3—32no—
289C4×D3—32no—
290C3×C8—61no—
291C3⋊C8—21no—
292C3⋊C8—181no—
293C3⋊C8—181no—
294C3⋊C8—361no—
295C2×C32—42no—
296C2×C32D3×C22⋊C411yescharacteristic
297C2×C32#91111yescharacteristic
298C2×C32#91411yescharacteristic
299C3×D3—62no—
300C3×D3—122no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—42no—
305C2×C9—82no—
306C2×C9#93211yescharacteristic
307C2×C9#93211yescharacteristic
308C2×C9C2×#21711yescharacteristic
309D9—22no—
310D9—42no—
311C24—181no—
312C2×D4—94no—
313C2×D4—91no—
314C2×D4—182no—
315C2×D4—181no—
316C2×D4D3×D911yescharacteristic
317C22×C4—91no—
318M16—542no—
319C2×C8—31no—
320C2×C8—271no—
321C4⋊C4—31no—
322C4⋊C4—271no—
323C22⋊C4—541no—
324C42—541no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—92no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3—182no—
339C22×C3—368no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3D4×D911yescharacteristic
343C22×C3D3×#6711yescharacteristic
344C2×D3—32no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—62no—
348C2×D3—128no—
349C2×D3—121no—
350C2×D3—121no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—92no—
354C3×C4—121no—
355C3×C4C2×C4×D911yescharacteristic
356C3×C4D4×D911yescharacteristic
357C3×C4C2×D3×Dic311yescharacteristic
358C3×C4D3×#6711yescharacteristic
359Dic3—32no—
360Dic3—41no—
361Dic3—361no—
362Dic3—361no—
363Dic3—721no—
364C32#477011yescharacteristic
365C9—21no—
366C9#521511yescharacteristic, complemented
367C23—21no—
368C23—91no—
369C23—184no—
370C23—181no—
371C23—362no—
372C23—362no—
373D4—21no—
374D4—92no—
375D4—184no—
376D4—181no—
377D4Dic3×D912yes—
378C2×C4—61no—
379C2×C4—91no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—541no—
383C2×C4C2×D3×D911yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—42no—
390C2×C3—42no—
391C2×C3—82no—
392C2×C3—182no—
393C2×C3—362no—
394C2×C3C22⋊C4×D911yescharacteristic
395C2×C3#6347311yescharacteristic
396C2×C3#6347611yescharacteristic
397C2×C3#6393211yescharacteristic
398C2×C3#6393411yescharacteristic
399C2×C3D3×#21711yescharacteristic
400D3—62no—
401D3—122no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—92no—
406C22—181no—
407C22—181no—
408C22—182no—
409C22—368no—
410C22—361no—
411C22—361no—
412C22D9×#6711yescharacteristic
413C4—92no—
414C4—121no—
415C4—1081no—
416C4C2×Dic3×D911yescharacteristic
417C4D9×#6711yescharacteristic
418C3—21no—
419C3#1058845011yescharacteristic, complemented
420C3#1059142111yescharacteristic
421C2—42no—
422C2—182no—
423C2—362no—
424C2#1172117611yescharacteristic
425C2#1172118411yescharacteristic
426C2D9×#21711yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D4×D9C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#418
LCSG ▷ 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. #11720907C2semidirect
3. C2×#88413C2semidirect
4. #11720576C2semidirect
5. C9×#832C2semidirect
6. #11720283C2semidirect
7. #11720266C2semidirect
14. C3×D4×D9C4semidirect
17. C4⋊(C3⋊Dic9)C22semidirect
18. C2×C3⋊(C9⋊C8)C22semidirect
19. C9×C4⋊Dic3C22semidirect
20. C2×C9×C3⋊C8C22semidirect
79. D4×D9Dic3semidirect
119. #832D9semidirect
195. C4⋊Dic3C2×D9semidirect
202. C2×C3⋊C8C2×D9semidirect
250. C3×C9#418semidirect
366. C9#5215semidirect
419. C3#10588450semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
#117209072
C2×#884132
#117205762
C9×#8322
#117202832
#117202662
#105914213
#105884503