Finite Group Databasebeta

← #13144854

Subgroups — #13144854

Order 1728.

760 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×#41C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic, complemented
3C2×#88543C211yesmaximal, characteristic, complemented
4#11721813C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720576C211yesmaximal, characteristic, complemented
7#11720571C211yesmaximal, characteristic, complemented
8#10592938—31nomaximal
9#10588918—33nomaximal
10C3×C9×#41—22no—
11C3×C9×#41—22no—
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×D4×C9—21no—
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9—21no—
16C22×C3×C4×C9C2211yescharacteristic
17C2×#6226C2211yescharacteristic
18#88551—21no—
19#88547—22no—
20#88543—22no—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534—21no—
23C2×C3⋊(C9⋊C8)C2211yescharacteristic
24C2×C3⋊(C9⋊C8)C2211yescharacteristic
25C2×C32×#41D311yescharacteristic
26C2×#3756—31no—
27C2×#3754—31no—
28#64133—31no—
29C2×C9×#41D313yes—
30#63662—31no—
31#63659—31no—
32#63657—31no—
33C2×#3674—33no—
34C2×#3672—33no—
35#63486—33no—
36#63395—33no—
37#63392—33no—
38#63389—33no—
39C23×C3×C9—21no—
40C23×C3×C9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9D412yes—
43C3×D4×C9—24no—
44C3×D4×C9—21no—
45C3×D4×C9D412yes—
46C2×C3×C4×C9—24no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9D411yescharacteristic
51C2×C3×C4×C9C2311yescharacteristic
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9—21no—
54#6226—21no—
55#6226—21no—
56C3⋊(C9⋊C8)—21no—
57C3⋊(C9⋊C8)—21no—
58#5239—93no—
59#5239—91no—
60C32×#41—22no—
61C32×#41—22no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×D4—21no—
64C2×C32×D4C2×D311yescharacteristic
65C22×C32×C4—21no—
66C22×C32×C4C2×D311yescharacteristic
67C2×#634—31no—
68#3757—61no—
69#3756—62no—
70#3754—62no—
71C4⋊(C3⋊Dic3)—31no—
72#3750—61no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C9×#41—26no—
76C9×#41—26no—
77C2×Q8×C9C2×D313yes—
78C2×D4×C9—23no—
79C2×D4×C9C2×D313yes—
80C22×C4×C9—23no—
81C22×C4×C9C2×D313yes—
82C2×Dic18—33no—
83#3676—63no—
84#3674—66no—
85#3672—66no—
86C4⋊Dic9—33no—
87Dic9⋊C4—63no—
88C2×C9⋊C8—33no—
89C2×C9⋊C8—33no—
90C22×C3×C9—21no—
91C22×C3×C9—22no—
92C22×C3×C9—21no—
93C22×C3×C9—21no—
94C22×C3×C9—41no—
95C22×C3×C9—41no—
96C22×C3×C9C2×D411yescharacteristic
97C3×C4×C9—21no—
98C3×C4×C9—22no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—41no—
102C3⋊Dic9—41no—
103C2×C3×#41D913yes—
104C2×C3×#41C3⋊D311yescharacteristic
105C2×#216—93no—
106C2×#216—91no—
107C2×#214—93no—
108C2×#214—91no—
109#924—93no—
110#924—91no—
111#835—93no—
112#835—91no—
113#832—93no—
114#832—91no—
115#830—93no—
116#830—91no—
117C23×C32—21no—
118C23×C32#6711yescharacteristic
119C32×Q8—21no—
120C32×Q8#6712yes—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4#6712yes—
124C2×C32×C4—24no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—61no—
132#634—61no—
133#634—61no—
134C23×C9—23no—
135C23×C9#6713yes—
136C3⋊(C3⋊C8)—61no—
137C3⋊(C3⋊C8)—61no—
138Q8×C9—23no—
139Q8×C9#6716yes—
140D4×C9—212no—
141D4×C9—23no—
142D4×C9#6716yes—
143C2×C4×C9—212no—
144C2×C4×C9—23no—
145C2×C4×C9—23no—
146C2×C4×C9—23no—
147C2×C4×C9#6713yes—
148C2×C4×C9C22×D313yes—
149C2×Dic9—63no—
150C2×Dic9—63no—
151Dic18—63no—
152Dic18—63no—
153C9⋊C8—63no—
154C9⋊C8—63no—
155#452—271noSylow
156C2×C3×C9—41no—
157C2×C3×C9—42no—
158C2×C3×C9#12011yescharacteristic
159C2×C3×C9#13511yescharacteristic
160C2×C3×C9#13711yescharacteristic
161C3×#41—26no—
162C3×#41—22no—
163C3×#41—26no—
164C3×#41—22no—
165C2×C3×Q8C2×D913yes—
166C2×C3×Q8C2×C3⋊D311yescharacteristic
167C2×C3×D4—23no—
168C2×C3×D4—21no—
169C2×C3×D4C2×D913yes—
170C2×C3×D4C2×C3⋊D311yescharacteristic
171C22×C3×C4—23no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D913yes—
174C22×C3×C4C2×C3⋊D311yescharacteristic
175C2×Dic6—93no—
176C2×Dic6—91no—
177#217—183no—
178#217—181no—
179#216—186no—
180#216—182no—
181#214—186no—
182#214—182no—
183C4⋊Dic3—93no—
184C4⋊Dic3—91no—
185Dic3⋊C4—183no—
186Dic3⋊C4—181no—
187C2×C3⋊C8—93no—
188C2×C3⋊C8—93no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—91no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32C2×#6711yescharacteristic
198C32×C4—21no—
199C32×C4—22no—
200C32×C4C2×#6711yescharacteristic
201C32×C4C2×#6711yescharacteristic
202C3⋊Dic3—121no—
203C3⋊Dic3—121no—
204C22×C9—23no—
205C22×C9—23no—
206C22×C9—23no—
207C22×C9—26no—
208C22×C9—43no—
209C22×C9—43no—
210C22×C9C2×#6713yes—
211C4×C9—23no—
212C4×C9—26no—
213C4×C9C2×#6713yes—
214C4×C9C2×#6713yes—
215Dic9—123no—
216Dic9—123no—
217C2×#41C3⋊D911yescharacteristic
218C2×Q16—271no—
219C2×SD16—271no—
220C22⋊Q8—271no—
221#103—271no—
222#102—271no—
223C22⋊C8—271no—
224C3×C9#45211yescharacteristic, complemented, Sylow
225C23×C3—23no—
226C23×C3—21no—
227C23×C3#61113yes—
228C23×C3#63811yescharacteristic
229C3×Q8—23no—
230C3×Q8—21no—
231C3×Q8#61116yes—
232C3×Q8#63812yes—
233C3×D4—24no—
234C3×D4—212no—
235C3×D4—23no—
236C3×D4—21no—
237C3×D4#61116yes—
238C3×D4#63812yes—
239C2×C3×C4—24no—
240C2×C3×C4—212no—
241C2×C3×C4—23no—
242C2×C3×C4—21no—
243C2×C3×C4—23no—
244C2×C3×C4—23no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4#61113yes—
248C2×C3×C4C22×D913yes—
249C2×C3×C4#63811yescharacteristic
250C2×C3×C4C22×C3⋊D311yescharacteristic
251C2×Dic3—183no—
252C2×Dic3—183no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255Dic6—183no—
256Dic6—183no—
257Dic6—181no—
258Dic6—181no—
259C3⋊C8—183no—
260C3⋊C8—183no—
261C3⋊C8—181no—
262C3⋊C8—181no—
263C2×C32—41no—
264C2×C32—42no—
265C2×C32#95011yescharacteristic
266C2×C32#95111yescharacteristic
267C2×C32#95311yescharacteristic
268C2×C9—43no—
269C2×C9—46no—
270C2×C9#95013yes—
271C2×C9#95113yes—
272C2×C9#95313yes—
273#41—22no—
274#41—22no—
275C2×Q8—271no—
276C2×Q8C2×C3⋊D911yescharacteristic
277C2×D4—21no—
278C2×D4C2×C3⋊D911yescharacteristic
279C22×C4—21no—
280C22×C4C2×C3⋊D911yescharacteristic
281Q16—542no—
282SD16—542no—
283C2×C8—271no—
284C2×C8—271no—
285C4⋊C4—271no—
286C4⋊C4—541no—
287C22⋊C4—541no—
288C22×C3—23no—
289C22×C3—23no—
290C22×C3—23no—
291C22×C3—26no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—43no—
297C22×C3—43no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C2×#61113yes—
301C22×C3C2×#63811yescharacteristic
302C3×C4—23no—
303C3×C4—26no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4C2×#61113yes—
307C3×C4C2×#61113yes—
308C3×C4C2×#63811yescharacteristic
309C3×C4C2×#63811yescharacteristic
310Dic3—363no—
311Dic3—363no—
312Dic3—361no—
313Dic3—361no—
314C32#523911yescharacteristic
315C9#523913yescomplemented
316C23—21no—
317C23#623311yescharacteristic
318Q8—21no—
319Q8—541no—
320Q8—541no—
321Q8#623312yes—
322D4—24no—
323D4—21no—
324D4#623312yes—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—541no—
330C2×C4—541no—
331C2×C4#623311yescharacteristic
332C2×C4C22×C3⋊D911yescharacteristic
333C8—541no—
334C8—541no—
335C2×C3—41no—
336C2×C3—42no—
337C2×C3—43no—
338C2×C3—46no—
339C2×C3#6351113yes—
340C2×C3#6351313yes—
341C2×C3#6351513yes—
342C2×C3#6415911yescharacteristic
343C2×C3#6416011yescharacteristic
344C2×C3#6416211yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22C2×#623311yescharacteristic
352C4—21no—
353C4—22no—
354C4—1081no—
355C4—1081no—
356C4C2×#623311yescharacteristic
357C4C2×#623311yescharacteristic
358C3#1058891813yescomplemented
359C3#1059293811yescharacteristic
360C2—41no—
361C2—42no—
362C2#1172186611yescharacteristic
363C2#1172186811yescharacteristic
364C2#1172187411yescharacteristic
subgroupG/H
centerC22C2×#6233
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C3×C9×#41C2
socleC22×C32C2×#67
prime radicalC2×C32×#41D3
hypercenterC2×#41C3⋊D9
hypocenterC3×C9#452
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×#88547C2semidirect
3. C2×#88543C2semidirect
4. #11721813C2semidirect
5. #11720582C2semidirect
6. #11720576C2semidirect
7. #11720571C2semidirect
224. C3×C9#452semidirect
315. C9#5239semidirect
358. C3#10588918semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×#412
C2×#885472
C2×#885432
#117218132
#117205822
#117205762
#117205712
#105929383
#105889183