Finite Group Databasebeta

← #13144321

Subgroups — #13144321

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic
2C2×#88534C211yesmaximal, characteristic
3C2×#88534C211yesmaximal, characteristic
4C2×#88534C211yesmaximal, characteristic
5C2×C4×C3⋊Dic9C211yesmaximal, characteristic
6#11720572C211yesmaximal, characteristic
7#11720572C211yesmaximal, characteristic
8#10592675—31nomaximal
9#10588656—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—
18#88534—22no—
19#88534—22no—
20#88534C414yescomplemented
21C4×C3⋊Dic9—22no—
22C2×C32×C4⋊C4D311yescharacteristic
23C2×#3750—31no—
24C2×#3750—31no—
25C2×#3750—31no—
26C2×C4×C3⋊Dic3—31no—
27#63658—31no—
28#63658—31no—
29C2×C9×C4⋊C4D313yes—
30C2×Dic9⋊C4—33no—
31C2×Dic9⋊C4—33no—
32C2×Dic9⋊C4—33no—
33C2×C4×Dic9—33no—
34C9⋊#95—33no—
35C9⋊#95—33no—
36C23×C3×C9C2311yescharacteristic
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×C9—21no—
44C2×C3×C4×C9C2×C412yes—
45C2×C3×C4×C9D412yes—
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⋊Dic9C2×C412yes—
54C2×C3⋊Dic9C2×C412yes—
55C2×C3⋊Dic9D412yes—
56C2×C3⋊Dic9Q812yes—
57C2×C3⋊Dic9Q812yes—
58#4976—93no—
59#4976—91no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C3⋊Dic3—31no—
64C22×C3⋊Dic3—31no—
65C22×C3⋊Dic3—31no—
66C22×C3⋊Dic3—31no—
67C32×C4⋊C4—22no—
68#3750—34no—
69#3750—62no—
70#3750—62no—
71C4×C3⋊Dic3—62no—
72C22×C4×C9C2×D313yes—
73C22×C4×C9C2×D313yes—
74C22×C4×C9C2×D313yes—
75C22×Dic9—33no—
76C22×Dic9—33no—
77C22×Dic9—33no—
78C22×Dic9—33no—
79C9×C4⋊C4—26no—
80Dic9⋊C4—312no—
81Dic9⋊C4—66no—
82Dic9⋊C4—66no—
83C4×Dic9—66no—
84C22×C3×C9C22×C411yescharacteristic
85C22×C3×C9C2×D411yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C22×C3×C9C2×Q811yescharacteristic
88C22×C3×C9C2×Q811yescharacteristic
89C22×C3×C9#4111yescharacteristic
90C22×C3×C9#4111yescharacteristic
91C3×C4×C9—22no—
92C3×C4×C9—22no—
93C3×C4×C9—41no—
94C3⋊Dic9—22no—
95C3⋊Dic9—22no—
96C3⋊Dic9—41no—
97C3⋊Dic9C4⋊C414yescomplemented
98C2×C3×C4⋊C4D913yes—
99C2×C3×C4⋊C4C3⋊D311yescharacteristic
100C2×Dic3⋊C4—93no—
101C2×Dic3⋊C4—91no—
102C2×Dic3⋊C4—93no—
103C2×Dic3⋊C4—93no—
104C2×Dic3⋊C4—91no—
105C2×Dic3⋊C4—91no—
106C2×C4×Dic3—93no—
107C2×C4×Dic3—91no—
108C3⋊#95—93no—
109C3⋊#95—93no—
110C3⋊#95—91no—
111C3⋊#95—91no—
112C23×C32C22×D311yescharacteristic
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4C4×D312yes—
121C2×C32×C4#6712yes—
122C2×C3⋊Dic3—32no—
123C2×C3⋊Dic3—32no—
124C2×C3⋊Dic3—32no—
125C2×C3⋊Dic3—32no—
126C2×C3⋊Dic3—32no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—61no—
134C23×C9C22×D313yes—
135C2×C4×C9—23no—
136C2×C4×C9—23no—
137C2×C4×C9—23no—
138C2×C4×C9—23no—
139C2×C4×C9—23no—
140C2×C4×C9—23no—
141C2×C4×C9—23no—
142C2×C4×C9C4×D316yes—
143C2×C4×C9#6716yes—
144C2×Dic9—36no—
145C2×Dic9—36no—
146C2×Dic9—36no—
147C2×Dic9—36no—
148C2×Dic9—36no—
149C2×Dic9—63no—
150C2×Dic9—63no—
151C2×Dic9—63no—
152C2×Dic9—63no—
153C2×Dic9—63no—
154C2×Dic9—63no—
155C2×Dic9—63no—
156#389—271noSylow
157C2×C3×C9C2×C4⋊C411yescharacteristic
158C2×C3×C9C4×D411yescharacteristic
159C2×C3×C9C4×Q811yescharacteristic
160C2×C3×C9#12111yescharacteristic
161C2×C3×C9C22⋊Q811yescharacteristic
162C2×C3×C9#12511yescharacteristic
163C2×C3×C9C4⋊Q811yescharacteristic
164C22×C3×C4C2×D913yes—
165C22×C3×C4C2×D913yes—
166C22×C3×C4C2×D913yes—
167C22×C3×C4C2×C3⋊D311yescharacteristic
168C22×C3×C4C2×C3⋊D311yescharacteristic
169C22×C3×C4C2×C3⋊D311yescharacteristic
170C22×Dic3—93no—
171C22×Dic3—91no—
172C22×Dic3—93no—
173C22×Dic3—93no—
174C22×Dic3—91no—
175C22×Dic3—91no—
176C22×Dic3—93no—
177C22×Dic3—91no—
178C3×C4⋊C4—26no—
179C3×C4⋊C4—22no—
180Dic3⋊C4—912no—
181Dic3⋊C4—94no—
182Dic3⋊C4—182no—
183Dic3⋊C4—186no—
184Dic3⋊C4—186no—
185Dic3⋊C4—182no—
186C4×Dic3—182no—
187C4×Dic3—186no—
188C22×C32C2×C4×D311yescharacteristic
189C22×C32#23511yescharacteristic
190C22×C32D3×D411yescharacteristic
191C22×C32#23711yescharacteristic
192C22×C32D3×Q811yescharacteristic
193C22×C32D3×Q811yescharacteristic
194C22×C32C2×#6711yescharacteristic
195C32×C4—22no—
196C32×C4—22no—
197C32×C4—41no—
198C3⋊Dic3—34no—
199C3⋊Dic3—62no—
200C3⋊Dic3—62no—
201C3⋊Dic3—121no—
202C22×C9C2×C4×D313yes—
203C22×C9#23513yes—
204C22×C9D3×D413yes—
205C22×C9#23713yes—
206C22×C9D3×Q813yes—
207C22×C9D3×Q813yes—
208C22×C9C2×#6713yes—
209C4×C9—26no—
210C4×C9—26no—
211C4×C9—43no—
212Dic9—312no—
213Dic9—66no—
214Dic9—66no—
215Dic9—123no—
216C2×C4⋊C4—271no—
217C2×C4⋊C4—271no—
218C2×C4⋊C4—271no—
219C2×C4⋊C4C3⋊D911yescharacteristic
220C2×C42—271no—
221#95—271no—
222#95—271no—
223C3×C9#38911yescharacteristic, complemented, Sylow
224C23×C3C22×D913yes—
225C23×C3C22×C3⋊D311yescharacteristic
226C2×C3×C4—23no—
227C2×C3×C4—23no—
228C2×C3×C4—23no—
229C2×C3×C4—23no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—23no—
235C2×C3×C4—23no—
236C2×C3×C4—23no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4C4×D916yes—
241C2×C3×C4#61116yes—
242C2×C3×C4C4×C3⋊D312yes—
243C2×C3×C4#63812yes—
244C2×Dic3—96no—
245C2×Dic3—96no—
246C2×Dic3—96no—
247C2×Dic3—92no—
248C2×Dic3—92no—
249C2×Dic3—92no—
250C2×Dic3—96no—
251C2×Dic3—96no—
252C2×Dic3—92no—
253C2×Dic3—92no—
254C2×Dic3—183no—
255C2×Dic3—183no—
256C2×Dic3—183no—
257C2×Dic3—183no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261C2×Dic3—181no—
262C2×Dic3—183no—
263C2×Dic3—183no—
264C2×Dic3—183no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268C2×C32#88711yescharacteristic
269C2×C32#88911yescharacteristic
270C2×C32D3×C4⋊C411yescharacteristic
271C2×C32#89611yescharacteristic
272C2×C32C4×#6711yescharacteristic
273C2×C32#93911yescharacteristic
274C2×C32#94411yescharacteristic
275C2×C9#88713yes—
276C2×C9#88913yes—
277C2×C9D3×C4⋊C413yes—
278C2×C9#89613yes—
279C2×C9C4×#6713yes—
280C2×C9#93913yes—
281C2×C9#94413yes—
282C22×C4—271no—
283C22×C4—271no—
284C22×C4—271no—
285C22×C4—271no—
286C22×C4C2×C3⋊D911yescharacteristic
287C22×C4C2×C3⋊D911yescharacteristic
288C22×C4C2×C3⋊D911yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—274no—
291C4⋊C4—542no—
292C4⋊C4—542no—
293C42—542no—
294C22×C3C2×C4×D913yes—
295C22×C3#369713yes—
296C22×C3D4×D913yes—
297C22×C3#369913yes—
298C22×C3Q8×D913yes—
299C22×C3Q8×D913yes—
300C22×C3C2×#61113yes—
301C22×C3C2×C4×C3⋊D311yescharacteristic
302C22×C3#382811yescharacteristic
303C22×C3D4×C3⋊D311yescharacteristic
304C22×C3#383011yescharacteristic
305C22×C3Q8×C3⋊D311yescharacteristic
306C22×C3Q8×C3⋊D311yescharacteristic
307C22×C3C2×#63811yescharacteristic
308C3×C4—26no—
309C3×C4—26no—
310C3×C4—22no—
311C3×C4—22no—
312C3×C4—43no—
313C3×C4—41no—
314Dic3—94no—
315Dic3—912no—
316Dic3—186no—
317Dic3—182no—
318Dic3—182no—
319Dic3—186no—
320Dic3—363no—
321Dic3—361no—
322C32#497611yescharacteristic
323C9#497613yescomplemented
324C23C22×C3⋊D911yescharacteristic
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—272no—
333C2×C4—272no—
334C2×C4—272no—
335C2×C4—272no—
336C2×C4—272no—
337C2×C4—541no—
338C2×C4—541no—
339C2×C4—541no—
340C2×C4—541no—
341C2×C4—541no—
342C2×C4—541no—
343C2×C4—541no—
344C2×C4C4×C3⋊D912yes—
345C2×C4#623312yes—
346C2×C3#6344913yes—
347C2×C3#6345113yes—
348C2×C3C4⋊C4×D913yes—
349C2×C3#6345813yes—
350C2×C3C4×#61113yes—
351C2×C3#6350113yes—
352C2×C3#6350613yes—
353C2×C3#6409611yescharacteristic
354C2×C3#6409811yescharacteristic
355C2×C3C4⋊C4×C3⋊D311yescharacteristic
356C2×C3#6410511yescharacteristic
357C2×C3C4×#63811yescharacteristic
358C2×C3#6414811yescharacteristic
359C2×C3#6415311yescharacteristic
360C22C2×C4×C3⋊D911yescharacteristic
361C22#8873611yescharacteristic
362C22D4×C3⋊D911yescharacteristic
363C22#8874211yescharacteristic
364C22Q8×C3⋊D911yescharacteristic
365C22Q8×C3⋊D911yescharacteristic
366C22C2×#623311yescharacteristic
367C4—22no—
368C4—22no—
369C4—41no—
370C4—274no—
371C4—542no—
372C4—542no—
373C4—1081no—
374C3#1058865613yescomplemented
375C3#1059267511yescharacteristic
376C2#1172174311yescharacteristic
377C2#1172174511yescharacteristic
378C2C4⋊C4×C3⋊D911yescharacteristic
379C2#1172175211yescharacteristic
380C2C4×#623311yescharacteristic
381C2#1172183911yescharacteristic
382C2#1172185411yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC2×C3×C9×C4⋊C4C2
socle, prime radicalC23×C32C22×D3
cosocleC23×C3×C9C23
hypercenterC2×C4⋊C4C3⋊D9
hypocenterC3×C9#389
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
20. #88534C4semidirect
97. C3⋊Dic9C4⋊C4semidirect
223. C3×C9#389semidirect
323. C9#4976semidirect
374. C3#10588656semidirect

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×#885342
C2×C4×C3⋊Dic92
#117205722
#105926753
#105886563