Finite Group Databasebeta

← #13143912

Subgroups — #13143912

Order 1728.

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

normal subgroups only

This group has 810 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×#88742C211yesmaximal, characteristic
2C2×#88523C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6#11720539C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592470—31nomaximal
9#10588451—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C22×C3⋊Dic9—21no—
13#88742—22no—
14#88742C414yescomplemented
15C2×C4×C3⋊D9—21no—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×#6226C2211yescharacteristic
18C2×C3×C8×C9C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534—21no—
22C2×C3⋊(C9⋊C8)C2211yescharacteristic
23#88523—22no—
24C2×#3830—31no—
25C2×#3743—31no—
26C4⋊C4×C3⋊D3—31no—
27C2×#3699—33no—
28C32×#102D311yescharacteristic
29#63659—31no—
30#63643—31no—
31#63637—31no—
32C2×#3663—33no—
33C4⋊C4×D9—33no—
34C9×#102D313yescomplemented
35#63392—33no—
36#63376—33no—
37#63368—33no—
38C23×C3×C9—21no—
39C22×C3⋊D9D411yescharacteristic
40C3×D4×C9—21no—
41C3×D4×C9C2×C412yescomplemented
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44#6233—24no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9—24no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9D411yescharacteristic
49C4×C3⋊D9—42no—
50C4×C3⋊D9C2×C412yes—
51C4×C3⋊D9D412yes—
52#6226—21no—
53#6226C2×C412yescomplemented
54C3×C8×C9—21no—
55C3⋊(C9⋊C8)—21no—
56#4771—93no—
57#4771—91no—
58C2×C32×D4C2×D311yescharacteristic
59C2×#638—61no—
60C22×C3⋊Dic3—61no—
61#3830—34no—
62#3830—62no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—61no—
65C2×#634—31no—
66C2×C32×C8C2×D311yescharacteristic
67C32×C4⋊C4C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)—31no—
69#3750—61no—
70C2×C3⋊(C3⋊C8)—31no—
71#3743—62no—
72C2×D4×C9C2×D313yes—
73C2×#611—63no—
74C22×Dic9—63no—
75#3699—312no—
76#3699—66no—
77C2×C4×D9—33no—
78C2×C4×D9—63no—
79C2×Dic18—33no—
80C2×C8×C9C2×D313yes—
81C9×C4⋊C4C2×D313yes—
82C4⋊Dic9—33no—
83Dic9⋊C4—63no—
84C2×C9⋊C8—33no—
85#3663—66no—
86C22×C3×C9—22no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3⋊D9—21no—
91C2×C3⋊D9—21no—
92C2×C3⋊D9C22⋊C412yes—
93C3×C4×C9—41no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—22no—
97C3⋊Dic9—41no—
98C3⋊Dic9C22⋊C412yes—
99C2×#237—93no—
100C2×#237—91no—
101C2×ΓL(1,𝔽25)—93no—
102C2×ΓL(1,𝔽25)—91no—
103D3×C4⋊C4—93no—
104D3×C4⋊C4—91no—
105C3×#102D913yescomplemented
106C3×#102C3⋊D311yescharacteristic
107#832—93no—
108#832—91no—
109#816—93no—
110#816—91no—
111#810—93no—
112#810—91no—
113C23×C32—21no—
114C22×C3⋊D3—31no—
115C32×D4—21no—
116C32×D4C4×D312yes—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119#638—64no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—32no—
126C4×C3⋊D3—122no—
127#634—32no—
128#634—61no—
129C23×C9—23no—
130C22×D9—33no—
131C32×C8—21no—
132C3⋊(C3⋊C8)—61no—
133D4×C9—23no—
134D4×C9C4×D316yescomplemented
135C2×C4×C9—23no—
136C2×C4×C9C22×D313yes—
137#611—612no—
138C2×Dic9—33no—
139C2×Dic9—612no—
140C2×Dic9—63no—
141C2×Dic9—63no—
142C4×D9—36no—
143C4×D9—36no—
144C4×D9—126no—
145Dic18—36no—
146Dic18—63no—
147C8×C9—23no—
148C9⋊C8—63no—
149#417—271noSylow
150C2×C3×C9—42no—
151C2×C3×C9C2×C22⋊C411yescharacteristic
152C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
153C2×C3×C9#13711yescharacteristic
154C3⋊D9—22no—
155C2×C3×D4C2×D913yes—
156C2×C3×D4C2×C3⋊D311yescharacteristic
157C2×#67—181no—
158C2×#67—183no—
159C22×Dic3—181no—
160C22×Dic3—183no—
161#237—912no—
162#237—94no—
163#237—182no—
164#237—186no—
165C2×C4×D3—93no—
166C2×C4×D3—91no—
167C2×C4×D3—181no—
168C2×C4×D3—183no—
169C2×Dic6—93no—
170C2×Dic6—91no—
171C2×C3×C8C2×D913yes—
172C2×C3×C8C2×C3⋊D311yescharacteristic
173C3×C4⋊C4C2×D913yes—
174C3×C4⋊C4C2×C3⋊D311yescharacteristic
175C4⋊Dic3—93no—
176C4⋊Dic3—91no—
177Dic3⋊C4—181no—
178Dic3⋊C4—183no—
179C2×C3⋊C8—93no—
180C2×C3⋊C8—91no—
181ΓL(1,𝔽25)—182no—
182ΓL(1,𝔽25)—186no—
183C22×C32—22no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—32no—
188C2×C3⋊D3—61no—
189C2×C3⋊D3—61no—
190C32×C4—41no—
191C32×C4C2×C4×D311yescharacteristic
192C32×C4D3×D411yescharacteristic
193C3⋊Dic3—32no—
194C3⋊Dic3—62no—
195C3⋊Dic3—121no—
196C22×C9—26no—
197C22×C9—43no—
198C22×C9—43no—
199C22×C9D3×D413yes—
200C2×D9—36no—
201C2×D9—63no—
202C2×D9—63no—
203C4×C9—43no—
204C4×C9C2×C4×D313yes—
205C4×C9D3×D413yes—
206Dic9—36no—
207Dic9—66no—
208Dic9—123no—
209C2×#41—271no—
210C2×M16—271no—
211C2×C4⋊C4—271no—
212#103—271no—
213#103—271no—
214#102—271no—
215#102C3⋊D911yescharacteristic, complemented
216C3×C9#41711yescharacteristic, complemented, Sylow
217C23×C3—23no—
218C23×C3—21no—
219C22×D3—93no—
220C22×D3—91no—
221C3×D4—23no—
222C3×D4—21no—
223C3×D4C4×D916yescomplemented
224C3×D4C4×C3⋊D312yes—
225C2×C3×C4—23no—
226C2×C3×C4—21no—
227C2×C3×C4C22×D913yes—
228C2×C3×C4C22×C3⋊D311yescharacteristic
229#67—184no—
230#67—1812no—
231C2×Dic3—93no—
232C2×Dic3—91no—
233C2×Dic3—184no—
234C2×Dic3—1812no—
235C2×Dic3—183no—
236C2×Dic3—181no—
237C2×Dic3—183no—
238C2×Dic3—181no—
239C4×D3—96no—
240C4×D3—96no—
241C4×D3—92no—
242C4×D3—92no—
243C4×D3—362no—
244C4×D3—366no—
245Dic6—96no—
246Dic6—92no—
247Dic6—183no—
248Dic6—181no—
249C3×C8—23no—
250C3×C8—21no—
251C3⋊C8—183no—
252C3⋊C8—181no—
253C2×C32—42no—
254C2×C32D3×C22⋊C411yescharacteristic
255C2×C32#91111yescharacteristic
256C2×C32#91511yescharacteristic
257C3⋊D3—62no—
258C2×C9—46no—
259C2×C9D3×C22⋊C413yes—
260C2×C9#91113yes—
261C2×C9#91513yes—
262D9—66no—
263#41—274no—
264#41—542no—
265C2×Q8—271no—
266C2×D4—541no—
267C2×D4C2×C3⋊D911yescharacteristic
268C22×C4—271no—
269C22×C4—541no—
270C22×C4—541no—
271M16—542no—
272C2×C8—271no—
273C2×C8C2×C3⋊D911yescharacteristic
274C4⋊C4—271no—
275C4⋊C4—541no—
276C4⋊C4C2×C3⋊D911yescharacteristic
277C22×C3—22no—
278C22×C3—26no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—43no—
282C22×C3—43no—
283C22×C3D4×D913yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C2×D3—92no—
286C2×D3—96no—
287C2×D3—181no—
288C2×D3—181no—
289C2×D3—183no—
290C2×D3—183no—
291C3×C4—41no—
292C3×C4—43no—
293C3×C4C2×C4×D913yes—
294C3×C4D4×D913yes—
295C3×C4C2×C4×C3⋊D311yescharacteristic
296C3×C4D4×C3⋊D311yescharacteristic
297Dic3—92no—
298Dic3—96no—
299Dic3—182no—
300Dic3—186no—
301Dic3—361no—
302Dic3—363no—
303C32#477111yescharacteristic
304C9#477113yescomplemented
305C23—21no—
306C23—271no—
307Q8—272no—
308Q8—541no—
309D4—21no—
310D4—544no—
311D4C4×C3⋊D912yescomplemented
312C2×C4—21no—
313C2×C4—271no—
314C2×C4—272no—
315C2×C4—272no—
316C2×C4—544no—
317C2×C4—541no—
318C2×C4—541no—
319C2×C4—1082no—
320C2×C4C22×C3⋊D911yescharacteristic
321C8—21no—
322C8—541no—
323C2×C3—46no—
324C2×C3—42no—
325C2×C3C22⋊C4×D913yes—
326C2×C3#6347313yes—
327C2×C3#6347713yes—
328C2×C3C22⋊C4×C3⋊D311yescharacteristic
329C2×C3#6412011yescharacteristic
330C2×C3#6412411yescharacteristic
331D3—186no—
332D3—182no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—272no—
337C22—541no—
338C22—541no—
339C22D4×C3⋊D911yescharacteristic
340C4—41no—
341C4—272no—
342C4—542no—
343C4—1081no—
344C4C2×C4×C3⋊D911yescharacteristic
345C4D4×C3⋊D911yescharacteristic
346C3#1058845113yescomplemented
347C3#1059247011yescharacteristic
348C2—42no—
349C2—542no—
350C2C22⋊C4×C3⋊D911yescharacteristic
351C2#1172179011yescharacteristic
352C2#1172179711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×#88742C2
cosocleC2×C3×C4×C9C23
hypercenter#102C3⋊D9
hypocenterC3×C9#417
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. C4⋊C4×C3⋊D9C2semidirect
4. C3×C9×#102C2semidirect
5. #11720576C2semidirect
6. #11720539C2semidirect
7. #11720527C2semidirect
14. #88742C4semidirect
34. C9×#102D3semidirect
41. C3×D4×C9C2×C4semidirect
53. #6226C2×C4semidirect
105. C3×#102D9semidirect
134. D4×C9C4×D3semidirect
215. #102C3⋊D9semidirect
216. C3×C9#417semidirect
223. C3×D4C4×D9semidirect
304. C9#4771semidirect
311. D4C4×C3⋊D9semidirect
346. C3#10588451semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C2×#885232
C4⋊C4×C3⋊D92
C3×C9×#1022
#117205762
#117205392
#117205272
#105924703
#105884513