Finite Group Databasebeta

← #13143911

Subgroups — #13143911

Order 1728.

950 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 167 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×#88523C211yesmaximal, characteristic, complemented
3#11721748C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6#11720543C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592469—31nomaximal
9#10588450—33nomaximal
10C23×C3⋊D9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6233—21no—
13D4×C3⋊D9—22no—
14D4×C3⋊D9C414yescomplemented
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×C8×C9C2211yescharacteristic, complemented
18C3×C9×C4⋊C4C2211yescharacteristic, complemented
19#88536—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
21C4×C3⋊Dic9—21no—
22C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
23#88523—22no—
24C2×D4×C3⋊D3—31no—
25C2×#3743—31no—
26#64101—31no—
27C2×D4×D9—33no—
28C32×#102D311yescharacteristic
29#63659—31no—
30#63648—31no—
31#63636—31no—
32C2×#3663—33no—
33#63454—33no—
34C9×#102D313yescomplemented
35#63392—33no—
36#63380—33no—
37#63369—33no—
38C23×C3×C9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—24no—
41C22×C3⋊D9—42no—
42C22×C3⋊D9—42no—
43C22×C3⋊D9D411yescharacteristic
44C3×D4×C9—21no—
45C3×D4×C9C2×C412yes—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48#6233—24no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9D411yescharacteristic
51C4⋊(C3⋊D9)—21no—
52C4⋊(C3⋊D9)C2×C412yes—
53C4×C3⋊D9C2×C412yes—
54C4×C3⋊D9D412yes—
55C3×C8×C9—21no—
56C3⋊(C9⋊C8)—21no—
57#4770—93no—
58#4770—91no—
59C23×C3⋊D3—61no—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—61no—
62D4×C3⋊D3—34no—
63D4×C3⋊D3—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4×C3⋊D3—31no—
66C23×D9—63no—
67C2×C32×C8C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—61no—
70C4⋊(C3⋊Dic3)—31no—
71C4×C3⋊Dic3—61no—
72C2×C3⋊(C3⋊C8)—31no—
73#3743—62no—
74C2×D4×C9C2×D313yes—
75C2×#611—63no—
76D4×D9—312no—
77D4×D9—66no—
78C2×D36—33no—
79C2×C4×D9—33no—
80C2×C8×C9C2×D313yescomplemented
81C9×C4⋊C4C2×D313yescomplemented
82(C2×D9)⋊C4—63no—
83C4⋊Dic9—33no—
84C4×Dic9—63no—
85C2×C9⋊C8—33no—
86#3663—66no—
87C22×C3×C9—22no—
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—21no—
92C2×C3⋊D9—21no—
93C2×C3⋊D9—22no—
94C2×C3⋊D9—48no—
95C2×C3⋊D9—41no—
96C2×C3⋊D9—41no—
97C2×C3⋊D9C22⋊C412yes—
98C3×C4×C9—41no—
99C3×C4×C9C22×C411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—41no—
102C3⋊Dic9C22⋊C412yes—
103C2×D3×D4—93no—
104C2×D3×D4—91no—
105C2×ΓL(1,𝔽25)—93no—
106C2×ΓL(1,𝔽25)—91no—
107#892—93no—
108#892—91no—
109C3×#102D913yescomplemented
110C3×#102C3⋊D311yescharacteristic
111#832—93no—
112#832—91no—
113#821—93no—
114#821—91no—
115#809—93no—
116#809—91no—
117C23×C32—21no—
118C22×C3⋊D3—31no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—61no—
121C22×C3⋊D3—122no—
122C22×C3⋊D3—122no—
123C32×D4—21no—
124C32×D4C4×D312yes—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127#638—64no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—61no—
130C4⋊(C3⋊D3)—32no—
131C4⋊(C3⋊D3)—61no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—32no—
134C23×C9—23no—
135C22×D9—33no—
136C22×D9—612no—
137C22×D9—63no—
138C22×D9—126no—
139C22×D9—126no—
140C32×C8—21no—
141C3⋊(C3⋊C8)—61no—
142D4×C9—23no—
143D4×C9C4×D316yes—
144C2×C4×C9—23no—
145C2×C4×C9C22×D313yes—
146#611—612no—
147C2×Dic9—33no—
148C2×Dic9—63no—
149D36—36no—
150D36—63no—
151C4×D9—36no—
152C4×D9—36no—
153C8×C9—23no—
154C9⋊C8—63no—
155#418—271noSylow
156C2×C3×C9—42no—
157C2×C3×C9C2×C22⋊C411yescharacteristic
158C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
159C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
160C3⋊D9—22no—
161C3⋊D9—42no—
162C23×D3—181no—
163C23×D3—183no—
164C2×C3×D4C2×D913yes—
165C2×C3×D4C2×C3⋊D311yescharacteristic
166C2×#67—181no—
167C2×#67—183no—
168D3×D4—912no—
169D3×D4—94no—
170D3×D4—182no—
171D3×D4—186no—
172C2×D12—93no—
173C2×D12—91no—
174C2×C4×D3—93no—
175C2×C4×D3—91no—
176C2×C3×C8C2×D913yescomplemented
177C2×C3×C8C2×C3⋊D311yescharacteristic
178C3×C4⋊C4C2×D913yescomplemented
179C3×C4⋊C4C2×C3⋊D311yescharacteristic
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—183no—
182C4⋊Dic3—93no—
183C4⋊Dic3—91no—
184C4×Dic3—181no—
185C4×Dic3—183no—
186C2×C3⋊C8—93no—
187C2×C3⋊C8—91no—
188ΓL(1,𝔽25)—182no—
189ΓL(1,𝔽25)—186no—
190C22×C32—22no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—32no—
195C2×C3⋊D3—61no—
196C2×C3⋊D3—61no—
197C2×C3⋊D3—62no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C32×C4—41no—
202C32×C4C2×C4×D311yescharacteristic
203C32×C4D3×D411yescharacteristic
204C3⋊Dic3—32no—
205C3⋊Dic3—121no—
206C22×C9—26no—
207C22×C9—43no—
208C22×C9—43no—
209C22×C9D3×D413yes—
210C2×D9—36no—
211C2×D9—63no—
212C2×D9—63no—
213C2×D9—66no—
214C2×D9—1224no—
215C2×D9—123no—
216C2×D9—123no—
217C4×C9—43no—
218C4×C9C2×C4×D313yes—
219C4×C9D3×D413yes—
220Dic9—36no—
221Dic9—123no—
222C22×D4—271no—
223C2×M16—271no—
224#117—271no—
225#102—271no—
226#102—271no—
227#102—271no—
228#102C3⋊D911yescharacteristic, complemented
229C3×C9#41811yescharacteristic, complemented, Sylow
230C23×C3—23no—
231C23×C3—21no—
232C22×D3—93no—
233C22×D3—91no—
234C22×D3—184no—
235C22×D3—1812no—
236C22×D3—183no—
237C22×D3—181no—
238C22×D3—362no—
239C22×D3—362no—
240C22×D3—366no—
241C22×D3—366no—
242C3×D4—23no—
243C3×D4—21no—
244C3×D4C4×D916yes—
245C3×D4C4×C3⋊D312yes—
246C2×C3×C4—23no—
247C2×C3×C4—21no—
248C2×C3×C4C22×D913yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250#67—184no—
251#67—1812no—
252C2×Dic3—93no—
253C2×Dic3—91no—
254C2×Dic3—183no—
255C2×Dic3—181no—
256D12—96no—
257D12—92no—
258D12—183no—
259D12—181no—
260C4×D3—96no—
261C4×D3—96no—
262C4×D3—92no—
263C4×D3—92no—
264C3×C8—23no—
265C3×C8—21no—
266C3⋊C8—183no—
267C3⋊C8—181no—
268C2×C32—42no—
269C2×C32D3×C22⋊C411yescharacteristic
270C2×C32#91111yescharacteristic
271C2×C32#91411yescharacteristic
272C3⋊D3—62no—
273C3⋊D3—122no—
274C2×C9—46no—
275C2×C9D3×C22⋊C413yes—
276C2×C9#91113yes—
277C2×C9#91413yes—
278D9—66no—
279D9—126no—
280C24—541no—
281C2×D4—274no—
282C2×D4—271no—
283C2×D4—542no—
284C2×D4—541no—
285C2×D4C2×C3⋊D911yescharacteristic
286C22×C4—271no—
287M16—542no—
288C2×C8—271no—
289C2×C8C2×C3⋊D911yescharacteristic, complemented
290C4⋊C4—271no—
291C4⋊C4C2×C3⋊D911yescharacteristic, complemented
292C22⋊C4—541no—
293C42—541no—
294C22×C3—22no—
295C22×C3—26no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—43no—
299C22×C3—43no—
300C22×C3D4×D913yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C2×D3—92no—
303C2×D3—96no—
304C2×D3—181no—
305C2×D3—181no—
306C2×D3—183no—
307C2×D3—183no—
308C2×D3—182no—
309C2×D3—186no—
310C2×D3—368no—
311C2×D3—3624no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—363no—
315C2×D3—363no—
316C3×C4—41no—
317C3×C4—43no—
318C3×C4C2×C4×D913yes—
319C3×C4D4×D913yes—
320C3×C4C2×C4×C3⋊D311yescharacteristic
321C3×C4D4×C3⋊D311yescharacteristic
322Dic3—92no—
323Dic3—96no—
324Dic3—361no—
325Dic3—363no—
326C32#477011yescharacteristic
327C9#477013yescomplemented
328C23—21no—
329C23—271no—
330C23—544no—
331C23—541no—
332C23—1082no—
333C23—1082no—
334D4—21no—
335D4—272no—
336D4—544no—
337D4—541no—
338D4C4×C3⋊D912yes—
339C2×C4—21no—
340C2×C4—271no—
341C2×C4—272no—
342C2×C4—272no—
343C2×C4—541no—
344C2×C4C22×C3⋊D911yescharacteristic
345C8—21no—
346C8—541no—
347C2×C3—46no—
348C2×C3—42no—
349C2×C3C22⋊C4×D913yes—
350C2×C3#6347313yes—
351C2×C3#6347613yes—
352C2×C3C22⋊C4×C3⋊D311yescharacteristic
353C2×C3#6412011yescharacteristic
354C2×C3#6412311yescharacteristic
355D3—186no—
356D3—182no—
357D3—366no—
358D3—362no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—272no—
363C22—541no—
364C22—541no—
365C22—542no—
366C22—1088no—
367C22—1081no—
368C22—1081no—
369C22D4×C3⋊D911yescharacteristic
370C4—41no—
371C4—272no—
372C4—1081no—
373C4C2×C4×C3⋊D911yescharacteristic
374C4D4×C3⋊D911yescharacteristic
375C3#1058845013yescomplemented
376C3#1059246911yescharacteristic
377C2—42no—
378C2—542no—
379C2—1082no—
380C2C22⋊C4×C3⋊D911yescharacteristic
381C2#1172179011yescharacteristic
382C2#1172179611yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×D4×C3⋊D9C2
cosocleC2×C3×C4×C9C23
hypercenter#102C3⋊D9
hypocenterC3×C9#418
LCSG ▷ 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×#88523C2semidirect
3. #11721748C2semidirect
4. C3×C9×#102C2semidirect
5. #11720576C2semidirect
6. #11720543C2semidirect
7. #11720528C2semidirect
14. D4×C3⋊D9C4semidirect
17. C2×C3×C8×C9C22semidirect
18. C3×C9×C4⋊C4C22semidirect
20. C4⋊(C3⋊Dic9)C22semidirect
22. C2×C3⋊(C9⋊C8)C22semidirect
34. C9×#102D3semidirect
80. C2×C8×C9C2×D3semidirect
81. C9×C4⋊C4C2×D3semidirect
109. C3×#102D9semidirect
176. C2×C3×C8C2×D9semidirect
178. C3×C4⋊C4C2×D9semidirect
228. #102C3⋊D9semidirect
229. C3×C9#418semidirect
289. C2×C8C2×C3⋊D9semidirect
291. C4⋊C4C2×C3⋊D9semidirect
327. C9#4770semidirect
375. C3#10588450semidirect

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×#885232
#117217482
C3×C9×#1022
#117205762
#117205432
#117205282
#105924693
#105884503