Finite Group Databasebeta

← #13143977

Subgroups — #13143977

Order 1728.

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

normal subgroups only

This group has 790 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×#88746C211yesmaximal, characteristic
2C2×C8×C3⋊D9C211yesmaximal, characteristic, complemented
3#11721748C211yesmaximal, characteristic, complemented
4C3×C9×#103C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720543C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592503—31nomaximal
9#10588484—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11#88746—22no—
12#88746C414yescomplemented
13C2×C4⋊(C3⋊D9)—21no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9—21no—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×C8×C9C2211yescharacteristic
18C3×C9×C4⋊C4C2211yescharacteristic
19#88536—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4×C3⋊Dic9—21no—
22C2×C3⋊(C9⋊C8)C2211yescharacteristic
23C8×C3⋊D9—22no—
24C2×#3832—31no—
25C2×C8×C3⋊D3—31no—
26#64101—31no—
27C2×#3701—33no—
28C32×#103D311yescharacteristic
29#63662—31no—
30#63648—31no—
31#63636—31no—
32C2×C8×D9—33no—
33#63454—33no—
34C9×#103D313yescomplemented
35#63395—33no—
36#63380—33no—
37#63369—33no—
38C22×C3⋊D9—21no—
39C22×C3⋊D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9C2×C412yes—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C4⋊(C3⋊D9)—24no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)C2×C412yes—
50C4×C3⋊D9—24no—
51C4×C3⋊D9C2×C412yes—
52C4×C3⋊D9D412yes—
53C3×C8×C9—21no—
54C3⋊(C9⋊C8)—21no—
55#4804—93no—
56#4804—91no—
57C2×C32×Q8C2×D311yescharacteristic
58#3832—34no—
59#3832—62no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—61no—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)—31no—
70C8×C3⋊D3—62no—
71C2×Q8×C9C2×D313yes—
72#3701—312no—
73#3701—66no—
74C2×D36—33no—
75C2×D36—63no—
76C2×C4×D9—33no—
77C2×C4×D9—63no—
78C2×C8×C9C2×D313yes—
79C9×C4⋊C4C2×D313yes—
80(C2×D9)⋊C4—63no—
81C4⋊Dic9—33no—
82C4×Dic9—63no—
83C2×C9⋊C8—33no—
84C8×D9—66no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3⋊D9—21no—
87C2×C3⋊D9—21no—
88C2×C3⋊D9—22no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—41no—
91C2×C3⋊D9C22⋊C412yes—
92C3×C4×C9—22no—
93C3×C4×C9—41no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—41no—
97C3⋊Dic9C22⋊C412yes—
98C2×#239—93no—
99C2×#239—91no—
100C2×D3×C8—93no—
101C2×D3×C8—91no—
102#892—93no—
103#892—91no—
104C3×#103D913yescomplemented
105C3×#103C3⋊D311yescharacteristic
106#835—93no—
107#835—91no—
108#821—93no—
109#821—91no—
110#809—93no—
111#809—91no—
112C22×C3⋊D3—31no—
113C22×C3⋊D3—61no—
114C32×Q8—21no—
115C32×Q8C4×D312yes—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—61no—
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—64no—
123C4⋊(C3⋊D3)—61no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—32no—
126C4×C3⋊D3—64no—
127C22×D9—33no—
128C22×D9—63no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—61no—
131Q8×C9—23no—
132Q8×C9C4×D316yes—
133C2×C4×C9—23no—
134C2×C4×C9—23no—
135C2×C4×C9C22×D313yes—
136C2×Dic9—33no—
137C2×Dic9—63no—
138D36—36no—
139D36—612no—
140D36—63no—
141C4×D9—36no—
142C4×D9—36no—
143C4×D9—612no—
144C8×C9—23no—
145C9⋊C8—63no—
146#416—271noSylow
147C2×C3×C9C2×C22⋊C411yescharacteristic
148C2×C3×C9#13511yescharacteristic
149C2×C3×C9#13511yescharacteristic
150C3⋊D9—22no—
151C3⋊D9—42no—
152C2×C3×Q8C2×D913yes—
153C2×C3×Q8C2×C3⋊D311yescharacteristic
154#239—912no—
155#239—94no—
156#239—182no—
157#239—186no—
158C2×D12—93no—
159C2×D12—91no—
160C2×D12—181no—
161C2×D12—183no—
162C2×C4×D3—93no—
163C2×C4×D3—91no—
164C2×C4×D3—181no—
165C2×C4×D3—183no—
166C2×C3×C8C2×D913yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168C3×C4⋊C4C2×D913yes—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—183no—
172C4⋊Dic3—93no—
173C4⋊Dic3—91no—
174C4×Dic3—181no—
175C4×Dic3—183no—
176C2×C3⋊C8—93no—
177C2×C3⋊C8—91no—
178D3×C8—182no—
179D3×C8—186no—
180C22×C32D3×D411yescharacteristic
181C2×C3⋊D3—32no—
182C2×C3⋊D3—61no—
183C2×C3⋊D3—61no—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—121no—
186C2×C3⋊D3—121no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4C2×C4×D311yescharacteristic
190C32×C4D3×D411yescharacteristic
191C3⋊Dic3—32no—
192C3⋊Dic3—121no—
193C22×C9D3×D413yes—
194C2×D9—36no—
195C2×D9—63no—
196C2×D9—63no—
197C2×D9—66no—
198C2×D9—123no—
199C2×D9—123no—
200C4×C9—26no—
201C4×C9—43no—
202C4×C9C2×C4×D313yes—
203C4×C9D3×D413yes—
204Dic9—36no—
205Dic9—123no—
206C2×#41—271no—
207C22×C8—271no—
208#117—271no—
209#103—271no—
210#103C3⋊D911yescharacteristic, complemented
211#102—271no—
212#102—271no—
213C3×C9#41611yescharacteristic, complemented, Sylow
214C22×D3—93no—
215C22×D3—91no—
216C22×D3—183no—
217C22×D3—181no—
218C3×Q8—23no—
219C3×Q8—21no—
220C3×Q8C4×D916yes—
221C3×Q8C4×C3⋊D312yes—
222C2×C3×C4—23no—
223C2×C3×C4—21no—
224C2×C3×C4—23no—
225C2×C3×C4—21no—
226C2×C3×C4C22×D913yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228C2×Dic3—93no—
229C2×Dic3—91no—
230C2×Dic3—183no—
231C2×Dic3—181no—
232D12—96no—
233D12—92no—
234D12—184no—
235D12—1812no—
236D12—183no—
237D12—181no—
238C4×D3—96no—
239C4×D3—96no—
240C4×D3—92no—
241C4×D3—92no—
242C4×D3—184no—
243C4×D3—1812no—
244C3×C8—23no—
245C3×C8—21no—
246C3⋊C8—183no—
247C3⋊C8—181no—
248C2×C32D3×C22⋊C411yescharacteristic
249C2×C32#91611yescharacteristic
250C2×C32#91911yescharacteristic
251C3⋊D3—62no—
252C3⋊D3—122no—
253C2×C9D3×C22⋊C413yes—
254C2×C9#91613yes—
255C2×C9#91913yes—
256D9—66no—
257D9—126no—
258#41—274no—
259#41—542no—
260C2×Q8C2×C3⋊D911yescharacteristic
261C2×D4—271no—
262C2×D4—541no—
263C22×C4—271no—
264C22×C4—541no—
265C2×C8—271no—
266C2×C8—542no—
267C2×C8C2×C3⋊D911yescharacteristic
268C4⋊C4—271no—
269C4⋊C4C2×C3⋊D911yescharacteristic
270C22⋊C4—541no—
271C42—541no—
272C22×C3D4×D913yes—
273C22×C3D4×C3⋊D311yescharacteristic
274C2×D3—92no—
275C2×D3—96no—
276C2×D3—181no—
277C2×D3—181no—
278C2×D3—183no—
279C2×D3—183no—
280C2×D3—182no—
281C2×D3—186no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—363no—
285C2×D3—363no—
286C3×C4—22no—
287C3×C4—26no—
288C3×C4—41no—
289C3×C4—43no—
290C3×C4C2×C4×D913yes—
291C3×C4D4×D913yes—
292C3×C4C2×C4×C3⋊D311yescharacteristic
293C3×C4D4×C3⋊D311yescharacteristic
294Dic3—92no—
295Dic3—96no—
296Dic3—361no—
297Dic3—363no—
298C32#480411yescharacteristic
299C9#480413yescomplemented
300C23—271no—
301C23—541no—
302Q8—21no—
303Q8C4×C3⋊D912yes—
304D4—272no—
305D4—544no—
306D4—541no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—271no—
310C2×C4—272no—
311C2×C4—272no—
312C2×C4—544no—
313C2×C4—541no—
314C2×C4C22×C3⋊D911yescharacteristic
315C8—21no—
316C8—541no—
317C2×C3C22⋊C4×D913yes—
318C2×C3#6347813yes—
319C2×C3#6348113yes—
320C2×C3C22⋊C4×C3⋊D311yescharacteristic
321C2×C3#6412511yescharacteristic
322C2×C3#6412811yescharacteristic
323D3—186no—
324D3—182no—
325D3—366no—
326D3—362no—
327C22—272no—
328C22—541no—
329C22—541no—
330C22—542no—
331C22—1081no—
332C22—1081no—
333C22D4×C3⋊D911yescharacteristic
334C4—22no—
335C4—41no—
336C4—272no—
337C4—1081no—
338C4C2×C4×C3⋊D911yescharacteristic
339C4D4×C3⋊D911yescharacteristic
340C3#1058848413yescomplemented
341C3#1059250311yescharacteristic
342C2—542no—
343C2—1082no—
344C2C22⋊C4×C3⋊D911yescharacteristic
345C2#1172179811yescharacteristic
346C2#1172180511yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC2×#88746C2
cosocleC2×C3×C4×C9C23
hypercenter#103C3⋊D9
hypocenterC3×C9#416
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×C8×C3⋊D9C2semidirect
3. #11721748C2semidirect
4. C3×C9×#103C2semidirect
5. #11720582C2semidirect
6. #11720543C2semidirect
7. #11720528C2semidirect
12. #88746C4semidirect
34. C9×#103D3semidirect
104. C3×#103D9semidirect
210. #103C3⋊D9semidirect
213. C3×C9#416semidirect
299. C9#4804semidirect
340. C3#10588484semidirect

Maximal subgroups

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

subgroupindex
C2×#887462
C2×C8×C3⋊D92
#117217482
C3×C9×#1032
#117205822
#117205432
#117205282
#105925033
#105884843