Finite Group Databasebeta

← #13138670

Subgroups — #13138670

Order 1728.

620 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 109 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×D9×D12C211yesmaximal, characteristic
2C2×C3×#3663C211yesmaximal, characteristic, complemented
3#11720907C211yesmaximal, characteristic, complemented
4#11720543C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590939—31nomaximal
9#10588450—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×C4×D9C2211yescharacteristic
14C2×#6195—21no—
15D9×D12—22no—
16D9×D12C414yescomplemented
17C2×C3×C8×C9C2211yescharacteristic, complemented
18C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
19C9×C4⋊Dic3C2211yescharacteristic, complemented
20C2×C3×C9⋊C8C2211yescharacteristic, complemented
21C3×#3663—22no—
22(C2×D9)⋊Dic3—21no—
23Dic3×Dic9—21no—
24C2×D3×D12—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26#63841—31no—
27C2×D4×D9—31no—
28#63648—31no—
29C3×#821D311yescharacteristic
30#63564—31no—
31#63563—31no—
32C2×#3663D311yescharacteristic, complemented
33#63454—31no—
34C9×#102—31no—
35#63392—31no—
36#63380—31no—
37#63380—61no—
38#63369—31no—
39C22×C3⋊D9—21no—
40C22×D3×C9—21no—
41C22×C3×D9D411yescharacteristic
42C2×D3×D9—24no—
43C2×D3×D9—42no—
44C2×D3×D9—42no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9—21no—
47C4⋊(C3⋊D9)—21no—
48C4⋊(C3⋊D9)C2×C412yes—
49C2×C9×Dic3—21no—
50C2×C3×Dic9D411yescharacteristic
51C9×D12—21no—
52C9×D12C2×C412yes—
53C3×C4×D9C2×C412yes—
54C3×C4×D9D412yes—
55#6195—24no—
56C3×C8×C9—21no—
57C3×C9⋊C8—21no—
58#5131—91no—
59#4770—91no—
60C22×D32—61no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C3×D12C2×D311yescharacteristic
63C2×C3×C4×D3—31no—
64C2×#626—61no—
65D3×D12—34no—
66D3×D12—62no—
67C23×D9—61no—
68C2×C32×C8C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)—31no—
70C3×C4⋊Dic3C2×D311yescharacteristic
71C2×C3×C3⋊C8—31no—
72C3×ΓL(1,𝔽25)—62no—
73(C2×D3)⋊Dic3—61no—
74Dic32—61no—
75C2×D4×C9—31no—
76C2×#611—61no—
77D4×D9—34no—
78D4×D9—62no—
79C2×D36—31no—
80C2×D36—61no—
81C2×C4×D9C2×D311yescharacteristic
82C2×C8×C9—21no—
83C2×C8×C9C2×D311yescharacteristic, complemented
84C9×C4⋊C4—31no—
85(C2×D9)⋊C4—61no—
86C4⋊Dic9—31no—
87C4⋊Dic9—61no—
88C4×Dic9—61no—
89C2×C9⋊C8C2×D311yescharacteristic, complemented
90#3663—22no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—22no—
93C2×C3⋊D9—41no—
94C2×C3⋊D9—41no—
95C2×D3×C9—22no—
96C2×D3×C9—41no—
97C2×D3×C9—41no—
98C2×C3×D9—21no—
99C2×C3×D9—21no—
100C2×C3×D9C22⋊C412yes—
101D3×D9—48no—
102C3×C4×C9C22×C411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—41no—
105C9×Dic3—41no—
106C3×Dic9C22⋊C412yes—
107C2×D3×D4—91no—
108C22×D12—91no—
109C2×C3×M16—91no—
110#926—91no—
111C2×ΓL(1,𝔽25)—31no—
112#892—91no—
113C3×#102—31no—
114#832—91no—
115#821—91no—
116#821—91no—
117#821—91no—
118#821—91no—
119#821—181no—
120#821D911yescharacteristic, complemented
121#809—91no—
122C22×C3⋊D3—61no—
123C22×C3×D3—21no—
124C22×C3×D3—31no—
125C2×D32—64no—
126C2×D32—122no—
127C2×D32—122no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—61no—
130C4⋊(C3⋊D3)—32no—
131C4⋊(C3⋊D3)—61no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—31no—
134C3×D12—21no—
135C3×D12C4×D312yes—
136C3×C4×D3—32no—
137C3×C4×D3—32no—
138#626—64no—
139C23×C9—61no—
140C22×D9—61no—
141C22×D9—64no—
142C22×D9—121no—
143C22×D9—122no—
144C22×D9—122no—
145C22×D9D1211yescharacteristic
146C32×C8—21no—
147C3×C3⋊C8—61no—
148D4×C9—32no—
149D4×C9—61no—
150C2×C4×C9—21no—
151C2×C4×C9—61no—
152C2×C4×C9C22×D311yescharacteristic
153#611—64no—
154C2×Dic9—61no—
155C2×Dic9—121no—
156C2×Dic9D1211yescharacteristic
157D36—32no—
158D36—62no—
159D36—61no—
160D36—121no—
161C4×D9C4×D312yes—
162C4×D9#6712yes—
163C8×C9—21no—
164C8×C9—41no—
165C9⋊C8—21no—
166#418—271noSylow
167C2×C3×C9C2×C22⋊C411yescharacteristic
168C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
169C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
170C3⋊D9—42no—
171D3×C9—42no—
172C3×D9—22no—
173C23×D3—181no—
174C23×D3—181no—
175C2×C3×D4—31no—
176C22×C3×C4—91no—
177C2×#67—181no—
178D3×D4—94no—
179D3×D4—182no—
180C2×D12—91no—
181C2×D12—91no—
182C2×D12—94no—
183C2×D12—181no—
184C2×D12—182no—
185C2×D12—181no—
186C2×D12C2×D911yescharacteristic
187C2×C4×D3—31no—
188C3×M16—182no—
189C2×C3×C8—21no—
190C2×C3×C8—91no—
191C2×C3×C8C2×D911yescharacteristic, complemented
192C2×C3×C8D3211yescharacteristic
193C3×C4⋊C4—31no—
194#217—181no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—91no—
197C4⋊Dic3—91no—
198C4⋊Dic3—181no—
199C4⋊Dic3C2×D911yescharacteristic, complemented
200C4×Dic3—181no—
201C4×Dic3—181no—
202C2×C3⋊C8—31no—
203ΓL(1,𝔽25)—62no—
204C22×C32D3×D411yescharacteristic
205C2×C3⋊D3—62no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—121no—
208C2×C3×D3—22no—
209C2×C3×D3—32no—
210C2×C3×D3—41no—
211C2×C3×D3—41no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214D32—128no—
215C32×C4C2×C4×D311yescharacteristic
216C32×C4D3×D411yescharacteristic
217C3⋊Dic3—121no—
218C3×Dic3—32no—
219C3×Dic3—41no—
220C22×C9—21no—
221C22×C9—62no—
222C22×C9—121no—
223C22×C9—121no—
224C22×C9C2×D1211yescharacteristic
225C2×D9—21no—
226C2×D9—21no—
227C2×D9—62no—
228C2×D9—122no—
229C2×D9—128no—
230C2×D9—121no—
231C2×D9—121no—
232C2×D9—241no—
233C2×D9—241no—
234C2×D9(C2×D3)⋊C412yes—
235C4×C9—21no—
236C4×C9—21no—
237C4×C9—121no—
238C4×C9C2×C4×D311yescharacteristic
239C4×C9C2×#6711yescharacteristic
240Dic9—121no—
241Dic9—241no—
242Dic9(C2×D3)⋊C412yes—
243C22×D4—271no—
244C2×M16—91no—
245#117—271no—
246#102—31no—
247#102—271no—
248#102—271no—
249#102—271no—
250C3×C9#41811yescharacteristic, complemented, Sylow
251C23×C3—61no—
252C23×C3—91no—
253C22×D3—21no—
254C22×D3—31no—
255C22×D3—184no—
256C22×D3—184no—
257C22×D3—181no—
258C22×D3—181no—
259C22×D3—361no—
260C22×D3—362no—
261C22×D3—362no—
262C22×D3—362no—
263C22×D3—362no—
264C3×D4—32no—
265C3×D4—61no—
266C2×C3×C4—21no—
267C2×C3×C4—61no—
268C2×C3×C4—91no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4C22×D911yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273#67—184no—
274C2×Dic3—21no—
275C2×Dic3—31no—
276C2×Dic3—181no—
277C2×Dic3—181no—
278C2×Dic3—361no—
279D12—21no—
280D12—92no—
281D12—92no—
282D12—182no—
283D12—184no—
284D12—181no—
285D12—181no—
286D12—361no—
287D12C4×D912yes—
288C4×D3—32no—
289C4×D3—32no—
290C3×C8—21no—
291C3×C8—21no—
292C3×C8—41no—
293C3×C8—181no—
294C3⋊C8—61no—
295C2×C32D3×C22⋊C411yescharacteristic
296C2×C32#91111yescharacteristic
297C2×C32#91411yescharacteristic
298C3⋊D3—122no—
299C3×D3—42no—
300C3×D3—62no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—122no—
305C2×C9#90811yescharacteristic
306C2×C9#90811yescharacteristic
307C2×C9C2×(C2×D3)⋊C411yescharacteristic
308D9—22no—
309D9—122no—
310D9—242no—
311C24—541no—
312C2×D4—31no—
313C2×D4—274no—
314C2×D4—271no—
315C2×D4—542no—
316C2×D4—541no—
317C22×C4—91no—
318M16—182no—
319C2×C8—91no—
320C2×C8D3×D911yescharacteristic, complemented
321C4⋊C4—31no—
322C4⋊C4—271no—
323C22⋊C4—541no—
324C42—541no—
325C22×C3—21no—
326C22×C3—62no—
327C22×C3—92no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3D4×D911yescharacteristic
333C22×C3D3×D1211yescharacteristic
334C2×D3—22no—
335C2×D3—32no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—182no—
341C2×D3—182no—
342C2×D3—362no—
343C2×D3—368no—
344C2×D3—368no—
345C2×D3—361no—
346C2×D3—361no—
347C2×D3—361no—
348C2×D3—361no—
349C2×D3—721no—
350C2×D3—721no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—92no—
354C3×C4—121no—
355C3×C4C2×C4×D911yescharacteristic
356C3×C4D4×D911yescharacteristic
357C3×C4C4×D3211yescharacteristic
358C3×C4D3×#6711yescharacteristic
359Dic3—32no—
360Dic3—41no—
361Dic3—361no—
362Dic3—361no—
363Dic3—721no—
364C32#477011yescharacteristic
365C9—21no—
366C9#513111yescharacteristic, complemented
367C23—61no—
368C23—91no—
369C23—544no—
370C23—541no—
371C23—1082no—
372C23—1082no—
373D4—32no—
374D4—61no—
375D4—272no—
376D4—544no—
377D4—541no—
378C2×C4—61no—
379C2×C4—91no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—541no—
383C2×C4C2×D3×D911yescharacteristic
384C8—21no—
385C8—181no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—122no—
390C2×C3—182no—
391C2×C3C22⋊C4×D911yescharacteristic
392C2×C3#6347311yescharacteristic
393C2×C3#6347611yescharacteristic
394C2×C3#6379411yescharacteristic
395C2×C3#6379511yescharacteristic
396C2×C3D3×(C2×D3)⋊C411yescharacteristic
397D3—42no—
398D3—62no—
399D3—362no—
400D3—362no—
401D3—722no—
402C22—62no—
403C22—92no—
404C22—121no—
405C22—121no—
406C22—181no—
407C22—181no—
408C22—542no—
409C22—1088no—
410C22—1081no—
411C22—1081no—
412C22D9×D1211yescharacteristic
413C4—92no—
414C4—121no—
415C4—1081no—
416C4C4×D3×D911yescharacteristic
417C4D9×#6711yescharacteristic
418C3—21no—
419C3#1058845011yescharacteristic, complemented
420C3#1059093911yescharacteristic
421C2—122no—
422C2—182no—
423C2—1082no—
424C2#1172081511yescharacteristic
425C2#1172081611yescharacteristic
426C2D9×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×D9×D12C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#418
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3663C2semidirect
3. #11720907C2semidirect
4. #11720543C2semidirect
5. C9×#821C2semidirect
6. #11720295C2semidirect
7. #11720267C2semidirect
16. D9×D12C4semidirect
17. C2×C3×C8×C9C22semidirect
18. C4⋊(C3⋊Dic9)C22semidirect
19. C9×C4⋊Dic3C22semidirect
20. C2×C3×C9⋊C8C22semidirect
32. C2×#3663D3semidirect
83. C2×C8×C9C2×D3semidirect
89. C2×C9⋊C8C2×D3semidirect
120. #821D9semidirect
191. C2×C3×C8C2×D9semidirect
199. C4⋊Dic3C2×D9semidirect
250. C3×C9#418semidirect
320. C2×C8D3×D9semidirect
366. C9#5131semidirect
419. C3#10588450semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C2×C3×#36632
#117209072
#117205432
C9×#8212
#117202952
#117202672
#105909393
#105884503