Finite Group Databasebeta

← #13144327

Subgroups — #13144327

Order 1728.

750 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic
2C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic
3C2×#88534C211yesmaximal, characteristic
4C2×#88534C211yesmaximal, characteristic
5#11720572C211yesmaximal, characteristic
6#11720572C211yesmaximal, characteristic
7#11720572C211yesmaximal, characteristic
8#10592681—31nomaximal
9#10588662—33nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C3×C9×C4⋊C4—22no—
18C4⋊(C3⋊Dic9)—22no—
19#88534—22no—
20#88534—22no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C4⋊(C3⋊Dic3)—31no—
23C2×#3750—31no—
24C2×#3750—31no—
25#63658—31no—
26#63658—31no—
27#63658—31no—
28C2×C9×C4⋊C4D313yes—
29C2×C4⋊Dic9—33no—
30C2×Dic9⋊C4—33no—
31C2×Dic9⋊C4—33no—
32C9⋊#95—33no—
33C9⋊#95—33no—
34C9⋊#95—33no—
35C23×C3×C9C2311yescharacteristic
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9D412yes—
44C2×C3×C4×C9Q812yes—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9D412yes—
56C2×C3⋊Dic9Q812yes—
57#4982—93no—
58#4982—91no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C3⋊Dic3—31no—
63C22×C3⋊Dic3—31no—
64C22×C3⋊Dic3—31no—
65C22×C3⋊Dic3—31no—
66C32×C4⋊C4—22no—
67C4⋊(C3⋊Dic3)—62no—
68#3750—62no—
69#3750—62no—
70C22×C4×C9C2×D313yes—
71C22×C4×C9C2×D313yes—
72C22×C4×C9C2×D313yes—
73C22×Dic9—33no—
74C22×Dic9—33no—
75C22×Dic9—33no—
76C22×Dic9—33no—
77C9×C4⋊C4—26no—
78C4⋊Dic9—66no—
79Dic9⋊C4—66no—
80Dic9⋊C4—66no—
81C22×C3×C9C2×D411yescharacteristic
82C22×C3×C9C2×D411yescharacteristic
83C22×C3×C9C2×Q811yescharacteristic
84C22×C3×C9C2×Q811yescharacteristic
85C22×C3×C9#4111yescharacteristic
86C22×C3×C9#4111yescharacteristic
87C22×C3×C9#4111yescharacteristic
88C3×C4×C9—22no—
89C3×C4×C9—22no—
90C3×C4×C9—41no—
91C3⋊Dic9—22no—
92C3⋊Dic9—22no—
93C3⋊Dic9—41no—
94C3⋊Dic9—41no—
95C2×C3×C4⋊C4D913yes—
96C2×C3×C4⋊C4C3⋊D311yescharacteristic
97C2×C4⋊Dic3—93no—
98C2×C4⋊Dic3—91no—
99C2×Dic3⋊C4—93no—
100C2×Dic3⋊C4—93no—
101C2×Dic3⋊C4—91no—
102C2×Dic3⋊C4—91no—
103C3⋊#95—93no—
104C3⋊#95—93no—
105C3⋊#95—93no—
106C3⋊#95—91no—
107C3⋊#95—91no—
108C3⋊#95—91no—
109C23×C32C22×D311yescharacteristic
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4Dic612yes—
118C2×C32×C4#6712yes—
119C2×C3⋊Dic3—32no—
120C2×C3⋊Dic3—32no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—61no—
131C23×C9C22×D313yes—
132C2×C4×C9—23no—
133C2×C4×C9—23no—
134C2×C4×C9—23no—
135C2×C4×C9—23no—
136C2×C4×C9—23no—
137C2×C4×C9—23no—
138C2×C4×C9—23no—
139C2×C4×C9Dic616yes—
140C2×C4×C9#6716yes—
141C2×Dic9—36no—
142C2×Dic9—36no—
143C2×Dic9—63no—
144C2×Dic9—63no—
145C2×Dic9—63no—
146C2×Dic9—63no—
147C2×Dic9—63no—
148C2×Dic9—63no—
149C2×Dic9—63no—
150C2×Dic9—63no—
151C2×Dic9—63no—
152C2×Dic9—63no—
153#398—271noSylow
154C2×C3×C9#12111yescharacteristic
155C2×C3×C9C22⋊Q811yescharacteristic
156C2×C3×C9C22⋊Q811yescharacteristic
157C2×C3×C9#12311yescharacteristic
158C2×C3×C9#12511yescharacteristic
159C2×C3×C9#12511yescharacteristic
160C2×C3×C9C4⋊Q811yescharacteristic
161C22×C3×C4C2×D913yes—
162C22×C3×C4C2×D913yes—
163C22×C3×C4C2×D913yes—
164C22×C3×C4C2×C3⋊D311yescharacteristic
165C22×C3×C4C2×C3⋊D311yescharacteristic
166C22×C3×C4C2×C3⋊D311yescharacteristic
167C22×Dic3—93no—
168C22×Dic3—93no—
169C22×Dic3—91no—
170C22×Dic3—91no—
171C22×Dic3—93no—
172C22×Dic3—93no—
173C22×Dic3—91no—
174C22×Dic3—91no—
175C3×C4⋊C4—26no—
176C3×C4⋊C4—22no—
177C4⋊Dic3—182no—
178C4⋊Dic3—186no—
179Dic3⋊C4—182no—
180Dic3⋊C4—186no—
181Dic3⋊C4—186no—
182Dic3⋊C4—182no—
183C22×C32C2×Dic611yescharacteristic
184C22×C32#23511yescharacteristic
185C22×C32D3×D411yescharacteristic
186C22×C32#23711yescharacteristic
187C22×C32D3×Q811yescharacteristic
188C22×C32#23911yescharacteristic
189C22×C32C2×#6711yescharacteristic
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—41no—
193C3⋊Dic3—62no—
194C3⋊Dic3—62no—
195C3⋊Dic3—121no—
196C3⋊Dic3—121no—
197C22×C9C2×Dic613yes—
198C22×C9#23513yes—
199C22×C9D3×D413yes—
200C22×C9#23713yes—
201C22×C9D3×Q813yes—
202C22×C9#23913yes—
203C22×C9C2×#6713yes—
204C4×C9—26no—
205C4×C9—26no—
206C4×C9—43no—
207Dic9—66no—
208Dic9—66no—
209Dic9—123no—
210Dic9—123no—
211C2×C4⋊C4—271no—
212C2×C4⋊C4—271no—
213C2×C4⋊C4—271no—
214C2×C4⋊C4C3⋊D911yescharacteristic
215#95—271no—
216#95—271no—
217#95—271no—
218C3×C9#39811yescharacteristic, complemented, Sylow
219C23×C3C22×D913yes—
220C23×C3C22×C3⋊D311yescharacteristic
221C2×C3×C4—23no—
222C2×C3×C4—23no—
223C2×C3×C4—23no—
224C2×C3×C4—23no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—23no—
230C2×C3×C4—23no—
231C2×C3×C4—23no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4Dic1816yes—
236C2×C3×C4#61116yes—
237C2×C3×C4#63412yes—
238C2×C3×C4#63812yes—
239C2×Dic3—96no—
240C2×Dic3—96no—
241C2×Dic3—92no—
242C2×Dic3—92no—
243C2×Dic3—183no—
244C2×Dic3—183no—
245C2×Dic3—183no—
246C2×Dic3—183no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—183no—
252C2×Dic3—183no—
253C2×Dic3—183no—
254C2×Dic3—183no—
255C2×Dic3—183no—
256C2×Dic3—183no—
257C2×Dic3—181no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263C2×C32#88811yescharacteristic
264C2×C32#88911yescharacteristic
265C2×C32#89011yescharacteristic
266C2×C32#89411yescharacteristic
267C2×C32#92411yescharacteristic
268C2×C32#93911yescharacteristic
269C2×C32#94611yescharacteristic
270C2×C9#88813yes—
271C2×C9#88913yes—
272C2×C9#89013yes—
273C2×C9#89413yes—
274C2×C9#92413yes—
275C2×C9#93913yes—
276C2×C9#94613yes—
277C22×C4—271no—
278C22×C4—271no—
279C22×C4—271no—
280C22×C4—271no—
281C22×C4C2×C3⋊D911yescharacteristic
282C22×C4C2×C3⋊D911yescharacteristic
283C22×C4C2×C3⋊D911yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—542no—
286C4⋊C4—542no—
287C4⋊C4—542no—
288C22×C3C2×Dic1813yes—
289C22×C3#369713yes—
290C22×C3D4×D913yes—
291C22×C3#369913yes—
292C22×C3Q8×D913yes—
293C22×C3#370113yes—
294C22×C3C2×#61113yes—
295C22×C3C2×#63411yescharacteristic
296C22×C3#382811yescharacteristic
297C22×C3D4×C3⋊D311yescharacteristic
298C22×C3#383011yescharacteristic
299C22×C3Q8×C3⋊D311yescharacteristic
300C22×C3#383211yescharacteristic
301C22×C3C2×#63811yescharacteristic
302C3×C4—26no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—26no—
306C3×C4—43no—
307C3×C4—41no—
308Dic3—186no—
309Dic3—182no—
310Dic3—182no—
311Dic3—186no—
312Dic3—363no—
313Dic3—361no—
314Dic3—361no—
315Dic3—363no—
316C32#498211yescharacteristic
317C9#498213yescomplemented
318C23C22×C3⋊D911yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—272no—
327C2×C4—272no—
328C2×C4—541no—
329C2×C4—541no—
330C2×C4—541no—
331C2×C4—541no—
332C2×C4—541no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4—541no—
336C2×C4—541no—
337C2×C4—541no—
338C2×C4#622612yes—
339C2×C4#623312yes—
340C2×C3#6345013yes—
341C2×C3#6345113yes—
342C2×C3#6345213yes—
343C2×C3#6345613yes—
344C2×C3#6348613yes—
345C2×C3#6350113yes—
346C2×C3#6350813yes—
347C2×C3#6409711yescharacteristic
348C2×C3#6409811yescharacteristic
349C2×C3#6409911yescharacteristic
350C2×C3#6410311yescharacteristic
351C2×C3#6413311yescharacteristic
352C2×C3#6414811yescharacteristic
353C2×C3#6415511yescharacteristic
354C22C2×#622611yescharacteristic
355C22#8873611yescharacteristic
356C22D4×C3⋊D911yescharacteristic
357C22#8874211yescharacteristic
358C22Q8×C3⋊D911yescharacteristic
359C22#8874611yescharacteristic
360C22C2×#623311yescharacteristic
361C4—22no—
362C4—22no—
363C4—41no—
364C4—542no—
365C4—542no—
366C4—1081no—
367C4—1081no—
368C3#1058866213yescomplemented
369C3#1059268111yescharacteristic
370C2#1172174411yescharacteristic
371C2#1172174511yescharacteristic
372C2#1172174611yescharacteristic
373C2#1172175011yescharacteristic
374C2#1172181311yescharacteristic
375C2#1172183911yescharacteristic
376C2#1172185611yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC2×C3×C9×C4⋊C4C2
socle, prime radicalC23×C32C22×D3
hypercenterC2×C4⋊C4C3⋊D9
hypocenterC3×C9#398
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
218. C3×C9#398semidirect
317. C9#4982semidirect
368. C3#10588662semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C2×C4⋊(C3⋊Dic9)2
C2×#885342
#117205722
#105926813
#105886623