Finite Group Databasebeta

← #13143980

Subgroups — #13143980

Order 1728.

620 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#11721856C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3C2×C8×C3⋊D9C211yesmaximal, characteristic, complemented
4#11721751C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6#11720543C211yesmaximal, characteristic, complemented
7#11720526C211yesmaximal, characteristic, complemented
8#10592506—31nomaximal
9#10588487—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)—21no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88548—22no—
17#88536—21no—
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20#88534—21no—
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22C8×C3⋊D9—22no—
23#64155—31no—
24C2×#3755—31no—
25C2×C8×C3⋊D3—31no—
26#64104—31no—
27C32×#103D311yescharacteristic
28#63648—31no—
29#63635—31no—
30#63508—33no—
31C2×#3675—33no—
32C2×C8×D9—33no—
33#63457—33no—
34C9×#103D313yescomplemented
35#63380—33no—
36#63367—33no—
37C22×C3⋊D9—21no—
38C22×C3⋊D9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D411yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48C4⋊(C3⋊D9)—42no—
49C4×C3⋊D9—21no—
50C4×C3⋊D9—21no—
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)D412yescomplemented
53#4807—93no—
54#4807—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—61no—
58C2×C4×C3⋊D3—31no—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61#3755—62no—
62#3752—61no—
63#3752—61no—
64C4⋊(C3⋊Dic3)—31no—
65#3750—61no—
66C2×C3⋊(C3⋊C8)—31no—
67C8×C3⋊D3—62no—
68C2×Q8×C9C2×D313yes—
69C2×D36—33no—
70C2×D36—63no—
71C2×C4×D9—33no—
72C2×C8×C9C2×D313yes—
73C9×C4⋊C4C2×D313yes—
74#3675—66no—
75(C2×D9)⋊C4—63no—
76(C2×D9)⋊C4—63no—
77C4⋊Dic9—33no—
78Dic9⋊C4—63no—
79C2×C9⋊C8—33no—
80C8×D9—66no—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—21no—
83C2×C3⋊D9—21no—
84C2×C3⋊D9—41no—
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C2×C3⋊D9SD1612yescomplemented
88C3×C4×C9—41no—
89C3×C4×C9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3⋊Dic9—21no—
93C3⋊Dic9—41no—
94#946—93no—
95#946—91no—
96C2×#215—93no—
97C2×#215—91no—
98C2×D3×C8—93no—
99C2×D3×C8—91no—
100#895—93no—
101#895—91no—
102C3×#103D913yescomplemented
103C3×#103C3⋊D311yescharacteristic
104#821—93no—
105#821—91no—
106#808—93no—
107#808—91no—
108C22×C3⋊D3—31no—
109C22×C3⋊D3—61no—
110C32×Q8—21no—
111C32×Q8—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—122no—
120C4×C3⋊D3—61no—
121C4×C3⋊D3—61no—
122C22×D9—33no—
123C22×D9—63no—
124C32×C8—21no—
125C3⋊(C3⋊C8)—32no—
126Q8×C9—23no—
127Q8×C9—23no—
128C2×C4×C9—23no—
129C2×C4×C9—23no—
130C2×C4×C9C22×D313yes—
131C2×Dic9—33no—
132C2×Dic9—63no—
133D36—63no—
134D36—63no—
135D36—126no—
136C4×D9—63no—
137C4×D9—63no—
138C8×C9—23no—
139C9⋊C8—36no—
140#465—271noSylow
141C2×C3×C9#12111yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9#13511yescharacteristic
144C3⋊D9—22no—
145C3⋊D9—81no—
146C2×C3×Q8C2×D913yes—
147C2×C3×Q8C2×C3⋊D311yescharacteristic
148C2×D12—93no—
149C2×D12—91no—
150C2×D12—181no—
151C2×D12—183no—
152C2×C4×D3—93no—
153C2×C4×D3—91no—
154C2×C3×C8C2×D913yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4C2×D913yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158#215—182no—
159#215—186no—
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—183no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—183no—
164C4⋊Dic3—93no—
165C4⋊Dic3—91no—
166Dic3⋊C4—181no—
167Dic3⋊C4—183no—
168C2×C3⋊C8—93no—
169C2×C3⋊C8—91no—
170D3×C8—182no—
171D3×C8—186no—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—32no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—121no—
177C2×C3⋊D3—121no—
178C2×C3⋊D3—121no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4#23511yescharacteristic
182C32×C4D3×D411yescharacteristic
183C3⋊Dic3—61no—
184C3⋊Dic3—121no—
185C22×C9D3×D413yes—
186C2×D9—36no—
187C2×D9—63no—
188C2×D9—63no—
189C2×D9—123no—
190C2×D9—123no—
191C2×D9—123no—
192C4×C9—43no—
193C4×C9—43no—
194C4×C9#23513yes—
195C4×C9D3×D413yes—
196Dic9—63no—
197Dic9—123no—
198C2×SD16—271no—
199C22×C8—271no—
200C22⋊Q8—271no—
201#121—271no—
202#106—271no—
203#103C3⋊D911yescharacteristic, complemented
204#102—271no—
205C3×C9#46511yescharacteristic, complemented, Sylow
206C22×D3—93no—
207C22×D3—91no—
208C22×D3—183no—
209C22×D3—181no—
210C3×Q8—23no—
211C3×Q8—23no—
212C3×Q8—21no—
213C3×Q8—21no—
214C2×C3×C4—23no—
215C2×C3×C4—23no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4C22×D913yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—93no—
221C2×Dic3—91no—
222C2×Dic3—183no—
223C2×Dic3—181no—
224D12—183no—
225D12—183no—
226D12—181no—
227D12—181no—
228D12—362no—
229D12—366no—
230C4×D3—183no—
231C4×D3—183no—
232C4×D3—181no—
233C4×D3—181no—
234C3×C8—23no—
235C3×C8—21no—
236C3⋊C8—96no—
237C3⋊C8—92no—
238C2×C32#88411yescharacteristic
239C2×C32D3×SD1611yescharacteristic
240C2×C32#91911yescharacteristic
241C3⋊D3—62no—
242C3⋊D3—241no—
243C2×C9#88413yes—
244C2×C9D3×SD1613yes—
245C2×C9#91913yes—
246D9—66no—
247D9—243no—
248C2×Q8C2×C3⋊D911yescharacteristic
249C2×D4—271no—
250C2×D4—541no—
251C22×C4—271no—
252SD16—542no—
253C2×C8—271no—
254C2×C8—542no—
255C2×C8C2×C3⋊D911yescharacteristic
256C4⋊C4—271no—
257C4⋊C4—541no—
258C4⋊C4C2×C3⋊D911yescharacteristic
259C22⋊C4—541no—
260C22⋊C4—541no—
261C22×C3D4×D913yes—
262C22×C3D4×C3⋊D311yescharacteristic
263C2×D3—92no—
264C2×D3—96no—
265C2×D3—181no—
266C2×D3—181no—
267C2×D3—183no—
268C2×D3—183no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—361no—
272C2×D3—363no—
273C2×D3—363no—
274C2×D3—363no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—43no—
278C3×C4—43no—
279C3×C4#369713yes—
280C3×C4D4×D913yes—
281C3×C4#382811yescharacteristic
282C3×C4D4×C3⋊D311yescharacteristic
283Dic3—181no—
284Dic3—183no—
285Dic3—361no—
286Dic3—363no—
287C32#480711yescharacteristic
288C9#480713yescomplemented
289C23—271no—
290C23—541no—
291Q8—21no—
292Q8—21no—
293D4—541no—
294D4—541no—
295D4—1082no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—271no—
299C2×C4—541no—
300C2×C4—541no—
301C2×C4—541no—
302C2×C4C22×C3⋊D911yescharacteristic
303C8—21no—
304C8—272no—
305C2×C3#6344613yes—
306C2×C3SD16×D913yes—
307C2×C3#6348113yes—
308C2×C3#6409311yescharacteristic
309C2×C3SD16×C3⋊D311yescharacteristic
310C2×C3#6412811yescharacteristic
311D3—186no—
312D3—182no—
313D3—723no—
314D3—721no—
315C22—272no—
316C22—541no—
317C22—541no—
318C22—1081no—
319C22—1081no—
320C22—1081no—
321C22D4×C3⋊D911yescharacteristic
322C4—41no—
323C4—41no—
324C4—541no—
325C4—1081no—
326C4#8873611yescharacteristic
327C4D4×C3⋊D911yescharacteristic
328C3#1058848713yescomplemented
329C3#1059250611yescharacteristic
330C2—542no—
331C2—2161no—
332C2#1172173011yescharacteristic
333C2SD16×C3⋊D911yescharacteristic
334C2#1172180511yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radical#11721751C2
hypercenter#103C3⋊D9
hypocenterC3×C9#465
LCSG ▷ C2×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
1. #11721856C2semidirect
2. C2×#88548C2semidirect
3. C2×C8×C3⋊D9C2semidirect
5. C3×C9×#103C2semidirect
6. #11720543C2semidirect
7. #11720526C2semidirect
34. C9×#103D3semidirect
52. C3⋊(C9⋊C8)D4semidirect
87. C2×C3⋊D9SD16semidirect
102. C3×#103D9semidirect
203. #103C3⋊D9semidirect
205. C3×C9#465semidirect
288. C9#4807semidirect
328. C3#10588487semidirect

Maximal subgroups

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

subgroupindex
#117218562
C2×#885482
C2×C8×C3⋊D92
#117217512
C3×C9×#1032
#117205432
#117205262
#105925063
#105884873