Finite Group Databasebeta

← #13143917

Subgroups — #13143917

Order 1728.

940 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 157 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×#88543C211yesmaximal, characteristic, complemented
3C2×#88524C211yesmaximal, characteristic, complemented
4#11721753C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6#11720542C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592475—31nomaximal
9#10588456—33nomaximal
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×#6226C2211yescharacteristic, complemented
18C2×C3×C8×C9C2211yescharacteristic, complemented
19C3×C9×C4⋊C4C2211yescharacteristic, complemented
20#88543—22no—
21#88536—21no—
22C4⋊(C3⋊Dic9)—21no—
23C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
24#88524—22no—
25C2×D4×C3⋊D3—31no—
26C2×#3754—31no—
27C2×#3744—31no—
28#64106—31no—
29C2×D4×D9—33no—
30C32×#102D311yescharacteristic
31#63647—31no—
32#63636—31no—
33C2×#3672—33no—
34C2×#3664—33no—
35#63459—33no—
36C9×#102D313yescomplemented
37(C2×D9)⋊C8—33no—
38#63369—33no—
39C23×C3×C9—21no—
40C22×C3⋊D9—24no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—42no—
43C22×C3⋊D9—42no—
44C22×C3⋊D9D411yescharacteristic
45C3×D4×C9—21no—
46C3×D4×C9D412yes—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9C2311yescharacteristic
49#6233—24no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9D411yescharacteristic
52C4⋊(C3⋊D9)—21no—
53C4⋊(C3⋊D9)D412yes—
54C4×C3⋊D9—21no—
55C4×C3⋊D9—21no—
56#6226—21no—
57#6226—21no—
58C3×C8×C9—21no—
59C3⋊(C9⋊C8)—21no—
60#4776—93no—
61#4776—91no—
62C23×C3⋊D3—61no—
63C2×C32×D4C2×D311yescharacteristic
64C2×#638—61no—
65D4×C3⋊D3—62no—
66D4×C3⋊D3—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4×C3⋊D3—31no—
69C2×#634—31no—
70C23×D9—63no—
71C2×C32×C8C2×D311yescharacteristic
72C32×C4⋊C4C2×D311yescharacteristic
73#3754—62no—
74#3752—61no—
75C4⋊(C3⋊Dic3)—61no—
76C2×C3⋊(C3⋊C8)—31no—
77#3744—62no—
78C2×D4×C9C2×D313yes—
79C2×#611—63no—
80D4×D9—66no—
81D4×D9—66no—
82C2×D36—33no—
83C2×C4×D9—33no—
84C2×Dic18—33no—
85C2×C8×C9C2×D313yescomplemented
86C9×C4⋊C4C2×D313yescomplemented
87#3672—66no—
88(C2×D9)⋊C4—63no—
89C4⋊Dic9—63no—
90C2×C9⋊C8—33no—
91#3664—66no—
92C22×C3×C9—22no—
93C22×C3×C9—41no—
94C22×C3×C9—41no—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3⋊D9—21no—
97C2×C3⋊D9—21no—
98C2×C3⋊D9—22no—
99C2×C3⋊D9—48no—
100C2×C3⋊D9—41no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9SD1612yescomplemented
103C3×C4×C9—41no—
104C3×C4×C9C2×D411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C3⋊Dic9—21no—
107C3⋊Dic9—41no—
108C2×D3×D4—93no—
109C2×D3×D4—91no—
110C2×#214—93no—
111C2×#214—91no—
112C2×#204—93no—
113C2×#204—91no—
114#897—93no—
115#897—91no—
116C3×#102D913yescomplemented
117C3×#102C3⋊D311yescharacteristic
118(C2×D3)⋊C8—93no—
119(C2×D3)⋊C8—91no—
120#809—93no—
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×D4D1212yes—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132#638—64no—
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—61no—
135C4⋊(C3⋊D3)—32no—
136C4⋊(C3⋊D3)—61no—
137C4×C3⋊D3—61no—
138C4×C3⋊D3—61no—
139#634—61no—
140#634—61no—
141C23×C9—23no—
142C22×D9—33no—
143C22×D9—612no—
144C22×D9—63no—
145C22×D9—126no—
146C22×D9—126no—
147C32×C8—21no—
148C3⋊(C3⋊C8)—61no—
149D4×C9—23no—
150D4×C9D1216yes—
151C2×C4×C9—23no—
152C2×C4×C9C22×D313yes—
153#611—612no—
154C2×Dic9—33no—
155C2×Dic9—63no—
156D36—36no—
157D36—63no—
158C4×D9—63no—
159C4×D9—63no—
160Dic18—63no—
161Dic18—63no—
162C8×C9—23no—
163C9⋊C8—63no—
164#450—271noSylow
165C2×C3×C9—42no—
166C2×C3×C9#12011yescharacteristic
167C2×C3×C9C2×SD1611yescharacteristic
168C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
169C3⋊D9—22no—
170C3⋊D9—42no—
171C23×D3—181no—
172C23×D3—183no—
173C2×C3×D4C2×D913yes—
174C2×C3×D4C2×C3⋊D311yescharacteristic
175C2×#67—181no—
176C2×#67—183no—
177D3×D4—182no—
178D3×D4—186no—
179D3×D4—186no—
180D3×D4—182no—
181C2×D12—93no—
182C2×D12—91no—
183C2×C4×D3—93no—
184C2×C4×D3—91no—
185C2×Dic6—93no—
186C2×Dic6—91no—
187C2×C3×C8C2×D913yescomplemented
188C2×C3×C8C2×C3⋊D311yescharacteristic
189C3×C4⋊C4C2×D913yescomplemented
190C3×C4⋊C4C2×C3⋊D311yescharacteristic
191#214—186no—
192#214—182no—
193(C2×D3)⋊C4—181no—
194(C2×D3)⋊C4—183no—
195C4⋊Dic3—181no—
196C4⋊Dic3—183no—
197C2×C3⋊C8—93no—
198C2×C3⋊C8—91no—
199#204—182no—
200#204—186no—
201C22×C32—22no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32D3×D411yescharacteristic
205C2×C3⋊D3—32no—
206C2×C3⋊D3—61no—
207C2×C3⋊D3—61no—
208C2×C3⋊D3—62no—
209C2×C3⋊D3—128no—
210C2×C3⋊D3—121no—
211C2×C3⋊D3—121no—
212C32×C4—41no—
213C32×C4C2×D1211yescharacteristic
214C32×C4D3×D411yescharacteristic
215C3⋊Dic3—61no—
216C3⋊Dic3—121no—
217C22×C9—26no—
218C22×C9—43no—
219C22×C9—43no—
220C22×C9D3×D413yes—
221C2×D9—36no—
222C2×D9—63no—
223C2×D9—63no—
224C2×D9—66no—
225C2×D9—1224no—
226C2×D9—123no—
227C2×D9—123no—
228C4×C9—43no—
229C4×C9C2×D1213yes—
230C4×C9D3×D413yes—
231Dic9—63no—
232Dic9—123no—
233C22×D4—271no—
234C2×SD16—271no—
235C2×SD16—271no—
236C22⋊Q8—271no—
237#102—271no—
238#102C3⋊D911yescharacteristic, complemented
239C22⋊C8—271no—
240C3×C9#45011yescharacteristic, complemented, Sylow
241C23×C3—23no—
242C23×C3—21no—
243C22×D3—93no—
244C22×D3—91no—
245C22×D3—184no—
246C22×D3—1812no—
247C22×D3—183no—
248C22×D3—181no—
249C22×D3—362no—
250C22×D3—366no—
251C22×D3—362no—
252C22×D3—366no—
253C3×D4—23no—
254C3×D4—21no—
255C3×D4D3616yes—
256C3×D4C4⋊(C3⋊D3)12yes—
257C2×C3×C4—23no—
258C2×C3×C4—21no—
259C2×C3×C4C22×D913yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261#67—184no—
262#67—1812no—
263C2×Dic3—93no—
264C2×Dic3—91no—
265C2×Dic3—183no—
266C2×Dic3—181no—
267D12—96no—
268D12—92no—
269D12—183no—
270D12—181no—
271C4×D3—183no—
272C4×D3—183no—
273C4×D3—181no—
274C4×D3—181no—
275Dic6—183no—
276Dic6—183no—
277Dic6—181no—
278Dic6—181no—
279C3×C8—23no—
280C3×C8—21no—
281C3⋊C8—183no—
282C3⋊C8—181no—
283C2×C32—42no—
284C2×C32#88211yescharacteristic
285C2×C32#91111yescharacteristic
286C2×C32D3×SD1611yescharacteristic
287C3⋊D3—62no—
288C3⋊D3—122no—
289C2×C9—46no—
290C2×C9#88213yes—
291C2×C9#91113yes—
292C2×C9D3×SD1613yes—
293D9—66no—
294D9—126no—
295C24—541no—
296C2×Q8—271no—
297C2×D4—271no—
298C2×D4—541no—
299C2×D4—542no—
300C2×D4—542no—
301C2×D4C2×C3⋊D911yescharacteristic
302C22×C4—271no—
303SD16—542no—
304SD16—542no—
305C2×C8—271no—
306C2×C8C2×C3⋊D911yescharacteristic, complemented
307C4⋊C4—541no—
308C4⋊C4C2×C3⋊D911yescharacteristic, complemented
309C22⋊C4—541no—
310C22×C3—22no—
311C22×C3—26no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—43no—
315C22×C3—43no—
316C22×C3D4×D913yes—
317C22×C3D4×C3⋊D311yescharacteristic
318C2×D3—92no—
319C2×D3—96no—
320C2×D3—181no—
321C2×D3—181no—
322C2×D3—183no—
323C2×D3—183no—
324C2×D3—182no—
325C2×D3—186no—
326C2×D3—368no—
327C2×D3—3624no—
328C2×D3—361no—
329C2×D3—361no—
330C2×D3—363no—
331C2×D3—363no—
332C3×C4—41no—
333C3×C4—43no—
334C3×C4C2×D3613yes—
335C3×C4D4×D913yes—
336C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
337C3×C4D4×C3⋊D311yescharacteristic
338Dic3—181no—
339Dic3—183no—
340Dic3—361no—
341Dic3—363no—
342C32#477611yescharacteristic
343C9#477613yescomplemented
344C23—21no—
345C23—271no—
346C23—544no—
347C23—541no—
348C23—1082no—
349C23—1082no—
350Q8—541no—
351Q8—541no—
352D4—21no—
353D4—272no—
354D4—544no—
355D4—541no—
356D4C4⋊(C3⋊D9)12yes—
357C2×C4—21no—
358C2×C4—271no—
359C2×C4—541no—
360C2×C4—541no—
361C2×C4—541no—
362C2×C4C22×C3⋊D911yescharacteristic
363C8—21no—
364C8—541no—
365C2×C3—46no—
366C2×C3—42no—
367C2×C3#6344413yes—
368C2×C3#6347313yes—
369C2×C3SD16×D913yes—
370C2×C3#6409111yescharacteristic
371C2×C3#6412011yescharacteristic
372C2×C3SD16×C3⋊D311yescharacteristic
373D3—186no—
374D3—182no—
375D3—366no—
376D3—362no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—272no—
381C22—541no—
382C22—541no—
383C22—542no—
384C22—1088no—
385C22—1081no—
386C22—1081no—
387C22D4×C3⋊D911yescharacteristic
388C4—41no—
389C4—541no—
390C4—1081no—
391C4C2×C4⋊(C3⋊D9)11yescharacteristic
392C4D4×C3⋊D911yescharacteristic
393C3#1058845613yescomplemented
394C3#1059247511yescharacteristic
395C2—42no—
396C2—542no—
397C2—1082no—
398C2#1172172811yescharacteristic
399C2#1172179011yescharacteristic
400C2SD16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×D4×C3⋊D9C2
hypercenter#102C3⋊D9
hypocenterC3×C9#450
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#88543C2semidirect
3. C2×#88524C2semidirect
4. #11721753C2semidirect
5. C3×C9×#102C2semidirect
6. #11720542C2semidirect
7. #11720528C2semidirect
17. C2×#6226C22semidirect
18. C2×C3×C8×C9C22semidirect
19. C3×C9×C4⋊C4C22semidirect
23. C2×C3⋊(C9⋊C8)C22semidirect
36. C9×#102D3semidirect
85. C2×C8×C9C2×D3semidirect
86. C9×C4⋊C4C2×D3semidirect
102. C2×C3⋊D9SD16semidirect
116. C3×#102D9semidirect
187. C2×C3×C8C2×D9semidirect
189. C3×C4⋊C4C2×D9semidirect
238. #102C3⋊D9semidirect
240. C3×C9#450semidirect
306. C2×C8C2×C3⋊D9semidirect
308. C4⋊C4C2×C3⋊D9semidirect
343. C9#4776semidirect
393. C3#10588456semidirect

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×#885432
C2×#885242
#117217532
C3×C9×#1022
#117205422
#117205282
#105924753
#105884563