Finite Group Databasebeta

← #13143981

Subgroups — #13143981

Order 1728.

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

normal subgroups only

This group has 740 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×Q8×C3⋊D9C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic, complemented
3C2×#88521C211yesmaximal, characteristic, complemented
4#11721753C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6#11720542C211yesmaximal, characteristic
7#11720527C211yesmaximal, characteristic, complemented
8#10592508—31nomaximal
9#10588489—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11Q8×C3⋊D9—22no—
12Q8×C3⋊D9—22no—
13C2×C4×C3⋊D9—21no—
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226—21no—
16C2×#6226C2211yescharacteristic
17C2×#6226C2211yescharacteristic
18C2×C3×C8×C9C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20#88547—22no—
21#88536—21no—
22C4⋊(C3⋊Dic9)—21no—
23C2×C3⋊(C9⋊C8)C2211yescharacteristic
24#88521—22no—
25C2×Q8×C3⋊D3—31no—
26C2×#3756—31no—
27C2×#3746—31no—
28#64106—31no—
29C2×Q8×D9—33no—
30C32×#103D311yescharacteristic
31#63647—31no—
32#63637—31no—
33C2×#3674—33no—
34C2×Dic36—33no—
35#63459—33no—
36C9×#103D313yescomplemented
37(C2×D9)⋊C8—33no—
38#63368—33no—
39C22×C3⋊D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9D412yes—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9D411yescharacteristic
48C4×C3⋊D9—24no—
49C4×C3⋊D9—21no—
50C4×C3⋊D9—21no—
51#6226—24no—
52#6226—21no—
53#6226—21no—
54#6226—21no—
55#6226D412yes—
56C3×C8×C9—21no—
57C3⋊(C9⋊C8)—21no—
58#4809—93no—
59#4809—91no—
60C2×C32×Q8C2×D311yescharacteristic
61Q8×C3⋊D3—62no—
62Q8×C3⋊D3—62no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—61no—
65C2×#634—31no—
66C2×#634—31no—
67C2×#634—61no—
68C2×C32×C8C2×D311yescharacteristic
69C32×C4⋊C4C2×D311yescharacteristic
70#3756—62no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—61no—
73C2×C3⋊(C3⋊C8)—31no—
74#3746—62no—
75C2×Q8×C9C2×D313yes—
76Q8×D9—66no—
77Q8×D9—66no—
78C2×C4×D9—33no—
79C2×C4×D9—63no—
80C2×Dic18—33no—
81C2×Dic18—33no—
82C2×Dic18—63no—
83C2×C8×C9C2×D313yes—
84C9×C4⋊C4C2×D313yes—
85#3674—66no—
86(C2×D9)⋊C4—63no—
87C4⋊Dic9—63no—
88C2×C9⋊C8—33no—
89Dic36—66no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—21no—
92C2×C3⋊D9—21no—
93C2×C3⋊D9Q1612yescomplemented
94C3×C4×C9—22no—
95C3×C4×C9—41no—
96C3×C4×C9C2×D411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—21no—
99C3⋊Dic9—22no—
100C3⋊Dic9—41no—
101C2×D3×Q8—93no—
102C2×D3×Q8—91no—
103C2×#216—93no—
104C2×#216—91no—
105C2×Dic12—93no—
106C2×Dic12—91no—
107#897—93no—
108#897—91no—
109C3×#103D913yescomplemented
110C3×#103C3⋊D311yescharacteristic
111(C2×D3)⋊C8—93no—
112(C2×D3)⋊C8—91no—
113#810—93no—
114#810—91no—
115C22×C3⋊D3—31no—
116C32×Q8—21no—
117C32×Q8D1212yes—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124C4×C3⋊D3—64no—
125C4×C3⋊D3—61no—
126C4×C3⋊D3—61no—
127#634—32no—
128#634—64no—
129#634—61no—
130#634—61no—
131#634—61no—
132C22×D9—33no—
133C32×C8—21no—
134C3⋊(C3⋊C8)—61no—
135Q8×C9—23no—
136Q8×C9D1216yes—
137C2×C4×C9—23no—
138C2×C4×C9—23no—
139C2×C4×C9C22×D313yes—
140C2×Dic9—33no—
141C2×Dic9—63no—
142C2×Dic9—63no—
143C4×D9—612no—
144C4×D9—63no—
145C4×D9—63no—
146Dic18—36no—
147Dic18—612no—
148Dic18—63no—
149Dic18—63no—
150Dic18—63no—
151C8×C9—23no—
152C9⋊C8—63no—
153#451—271noSylow
154C2×C3×C9#12011yescharacteristic
155C2×C3×C9C2×Q1611yescharacteristic
156C2×C3×C9#13711yescharacteristic
157C3⋊D9—22no—
158C2×C3×Q8C2×D913yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160D3×Q8—182no—
161D3×Q8—186no—
162D3×Q8—186no—
163D3×Q8—182no—
164C2×C4×D3—93no—
165C2×C4×D3—91no—
166C2×C4×D3—181no—
167C2×C4×D3—183no—
168C2×Dic6—93no—
169C2×Dic6—91no—
170C2×Dic6—93no—
171C2×Dic6—91no—
172C2×Dic6—181no—
173C2×Dic6—183no—
174C2×C3×C8C2×D913yes—
175C2×C3×C8C2×C3⋊D311yescharacteristic
176C3×C4⋊C4C2×D913yes—
177C3×C4⋊C4C2×C3⋊D311yescharacteristic
178#216—186no—
179#216—182no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—183no—
182C4⋊Dic3—181no—
183C4⋊Dic3—183no—
184C2×C3⋊C8—93no—
185C2×C3⋊C8—91no—
186Dic12—182no—
187Dic12—186no—
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—32no—
190C2×C3⋊D3—61no—
191C2×C3⋊D3—61no—
192C32×C4—22no—
193C32×C4—41no—
194C32×C4C2×D1211yescharacteristic
195C32×C4D3×D411yescharacteristic
196C3⋊Dic3—62no—
197C3⋊Dic3—61no—
198C3⋊Dic3—121no—
199C22×C9D3×D413yes—
200C2×D9—36no—
201C2×D9—63no—
202C2×D9—63no—
203C4×C9—26no—
204C4×C9—43no—
205C4×C9C2×D1213yes—
206C4×C9D3×D413yes—
207Dic9—66no—
208Dic9—63no—
209Dic9—123no—
210C22×Q8—271no—
211C2×Q16—271no—
212C2×Q16—271no—
213C22⋊Q8—271no—
214#103—271no—
215#103C3⋊D911yescharacteristic, complemented
216C22⋊C8—271no—
217C3×C9#45111yescharacteristic, complemented, Sylow
218C22×D3—93no—
219C22×D3—91no—
220C3×Q8—23no—
221C3×Q8—21no—
222C3×Q8D3616yes—
223C3×Q8C4⋊(C3⋊D3)12yes—
224C2×C3×C4—23no—
225C2×C3×C4—21no—
226C2×C3×C4—23no—
227C2×C3×C4—21no—
228C2×C3×C4C22×D913yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—93no—
231C2×Dic3—91no—
232C2×Dic3—183no—
233C2×Dic3—181no—
234C2×Dic3—183no—
235C2×Dic3—181no—
236C4×D3—184no—
237C4×D3—1812no—
238C4×D3—183no—
239C4×D3—183no—
240C4×D3—181no—
241C4×D3—181no—
242Dic6—96no—
243Dic6—92no—
244Dic6—184no—
245Dic6—1812no—
246Dic6—183no—
247Dic6—181no—
248Dic6—183no—
249Dic6—183no—
250Dic6—181no—
251Dic6—181no—
252C3×C8—23no—
253C3×C8—21no—
254C3⋊C8—183no—
255C3⋊C8—181no—
256C2×C32#88211yescharacteristic
257C2×C32#91511yescharacteristic
258C2×C32D3×Q1611yescharacteristic
259C3⋊D3—62no—
260C2×C9#88213yes—
261C2×C9#91513yes—
262C2×C9D3×Q1613yes—
263D9—66no—
264C2×Q8—271no—
265C2×Q8—271no—
266C2×Q8—541no—
267C2×Q8—542no—
268C2×Q8—542no—
269C2×Q8C2×C3⋊D911yescharacteristic
270C22×C4—271no—
271C22×C4—541no—
272Q16—542no—
273Q16—542no—
274C2×C8—271no—
275C2×C8C2×C3⋊D911yescharacteristic
276C4⋊C4—541no—
277C4⋊C4C2×C3⋊D911yescharacteristic
278C22⋊C4—541no—
279C22×C3D4×D913yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—92no—
282C2×D3—96no—
283C2×D3—181no—
284C2×D3—181no—
285C2×D3—183no—
286C2×D3—183no—
287C3×C4—22no—
288C3×C4—26no—
289C3×C4—41no—
290C3×C4—43no—
291C3×C4C2×D3613yes—
292C3×C4D4×D913yes—
293C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
294C3×C4D4×C3⋊D311yescharacteristic
295Dic3—182no—
296Dic3—181no—
297Dic3—186no—
298Dic3—183no—
299Dic3—361no—
300Dic3—363no—
301C32#480911yescharacteristic
302C9#480913yescomplemented
303C23—271no—
304Q8—21no—
305Q8—272no—
306Q8—544no—
307Q8—541no—
308Q8—541no—
309Q8—541no—
310Q8C4⋊(C3⋊D9)12yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—271no—
314C2×C4—544no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4—541no—
318C2×C4—541no—
319C2×C4C22×C3⋊D911yescharacteristic
320C8—21no—
321C8—541no—
322C2×C3#6344413yes—
323C2×C3#6347713yes—
324C2×C3Q16×D913yes—
325C2×C3#6409111yescharacteristic
326C2×C3#6412411yescharacteristic
327C2×C3Q16×C3⋊D311yescharacteristic
328D3—186no—
329D3—182no—
330C22—272no—
331C22—541no—
332C22—541no—
333C22D4×C3⋊D911yescharacteristic
334C4—22no—
335C4—41no—
336C4—541no—
337C4—542no—
338C4—1081no—
339C4C2×C4⋊(C3⋊D9)11yescharacteristic
340C4D4×C3⋊D911yescharacteristic
341C3#1058848913yescomplemented
342C3#1059250811yescharacteristic
343C2—542no—
344C2#1172172811yescharacteristic
345C2#1172179711yescharacteristic
346C2Q16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenter#103C3⋊D9
hypocenterC3×C9#451
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
2. C2×#88547C2semidirect
3. C2×#88521C2semidirect
5. C3×C9×#103C2semidirect
7. #11720527C2semidirect
36. C9×#103D3semidirect
93. C2×C3⋊D9Q16semidirect
109. C3×#103D9semidirect
215. #103C3⋊D9semidirect
217. C3×C9#451semidirect
302. C9#4809semidirect
341. C3#10588489semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
C2×#885472
C2×#885212
#117217532
C3×C9×#1032
#117205422
#117205272
#105925083
#105884893