Finite Group Databasebeta

← #13138224

Subgroups — #13138224

Order 1728.

448 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 85 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×#88525C211yesmaximal, characteristic
2#11720993C211yesmaximal, characteristic, complemented
3#11720952C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic, complemented
5C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C3×#63376C211yesmaximal, characteristic, complemented
7#11720284C211yesmaximal, characteristic, complemented
8#10590770—31nomaximal
9#10588528—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C2×C3×C8×C9C2211yescharacteristic
15#88525—22no—
16C4×C9×Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20#88445—21no—
21#88445—21no—
22#88416—22no—
23C2×#3745—31no—
24#63894—31no—
25#63864—31no—
26C2×#3717—31no—
27C3×#816—31no—
28C3×Dic3⋊C8D311yescharacteristic
29#63564—31no—
30C2×D72—31no—
31C2×D72—61no—
32C2×#3664—31no—
33#63437—31no—
34C4×D36—31no—
35C9×C4⋊C8—31no—
36#63380—31no—
37#63376D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)D412yes—
45C2×C9×Dic3D411yescharacteristic
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51(C3⋊D9)⋊C4—42no—
52C3×C8×C9—21no—
53C9×C3⋊C8—21no—
54#4844—91no—
55#4819—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C3×Dic6—31no—
59C2×#624—61no—
60C2×C32×C8C2×D311yescharacteristic
61#3745—62no—
62C3×C4⋊Dic3—31no—
63C3×C4×Dic3C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65#3722—61no—
66#3722—61no—
67#3722—61no—
68#3717—62no—
69C2×D36—31no—
70C2×D36—31no—
71C2×D36—61no—
72C2×D36—61no—
73C2×C4×D9—61no—
74C2×Dic18C2×D311yescharacteristic
75C2×C8×C9—21no—
76C2×C8×C9—31no—
77C2×C8×C9C2×D311yescharacteristic
78C42×C9—31no—
79(C2×D9)⋊C4—61no—
80(C2×D9)⋊C4—61no—
81(C2×D9)⋊C4—61no—
82C4⋊Dic9C2×D311yescharacteristic
83D72—62no—
84D72—64no—
85#3664—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—22no—
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—41no—
91C2×C3⋊D9—41no—
92C2×C3⋊D9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C9×Dic3—21no—
96C9×Dic3—21no—
97C3×Dic9—41no—
98C3×Dic9—41no—
99#947—91no—
100C2×#215—91no—
101C2×D24—91no—
102C2×D24—91no—
103C2×D24—181no—
104C2×#204—91no—
105#893—91no—
106#875—91no—
107C4×D12—91no—
108C3×C4⋊C8—31no—
109C3×#103—91no—
110#821—91no—
111#816—31no—
112Dic3⋊C8D911yescharacteristic, complemented
113#809—91no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—61no—
116C2×C32×C4C22×D311yescharacteristic
117C4⋊(C3⋊D3)—32no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—61no—
120C4⋊(C3⋊D3)—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3D1211yescharacteristic
124C2×C3×Dic3D1211yescharacteristic
125C3×Dic6—61no—
126C3×Dic6—61no—
127#624—122no—
128C22×D9—61no—
129C22×D9—61no—
130C22×D9—121no—
131C22×D9—121no—
132C32×C8—21no—
133C3×C3⋊C8—21no—
134C2×C4×C9—21no—
135C2×C4×C9—31no—
136C2×C4×C9—31no—
137C2×C4×C9C22×D311yescharacteristic
138C2×Dic9—21no—
139C2×Dic9—21no—
140D36—32no—
141D36—62no—
142D36—62no—
143D36—61no—
144D36—61no—
145D36—61no—
146D36—121no—
147D36—121no—
148C4×D9—122no—
149Dic18—21no—
150Dic18—21no—
151C8×C9—22no—
152C8×C9—21no—
153C8×C9—61no—
154#463—271noSylow
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9#13511yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
158C3⋊D9—42no—
159C3⋊D9—81no—
160C2×C3×Q8—91no—
161C2×D12—91no—
162C2×D12—91no—
163C2×D12—91no—
164C2×D12—91no—
165C2×D12—181no—
166C2×D12—181no—
167C2×C4×D3—181no—
168C2×C4×D3—181no—
169C2×Dic6—31no—
170C2×C3×C8—21no—
171C2×C3×C8—31no—
172C2×C3×C8C2×D911yescharacteristic
173C2×C3×C8D3211yescharacteristic
174C3×C4⋊C4—91no—
175C3×C42—31no—
176#215—182no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—31no—
184C4×Dic3C2×D911yescharacteristic
185C2×C3⋊C8C2×D911yescharacteristic
186D24—182no—
187D24—182no—
188D24—184no—
189#204—182no—
190C22×C32C2×D1211yescharacteristic
191C2×C3⋊D3—62no—
192C2×C3⋊D3—121no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—121no—
197C32×C4D3×D411yescharacteristic
198C32×C4#23911yescharacteristic
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9D3×D411yescharacteristic
205C2×D9—62no—
206C2×D9—122no—
207C2×D9—121no—
208C2×D9—121no—
209C2×D9—122no—
210C2×D9—121no—
211C2×D9—121no—
212C2×D9—121no—
213C2×D9—241no—
214C2×D9—241no—
215C2×D9—241no—
216C2×D9—241no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—61no—
220C4×C9—61no—
221C4×C9#23511yescharacteristic
222C4×C9D3×D411yescharacteristic
223Dic9—41no—
224Dic9—41no—
225C2×SD16—271no—
226C2×D8—271no—
227#124—271no—
228C4×D4—271no—
229C4⋊C8—31no—
230#103—91no—
231#102—271no—
232C3×C9#46311yescharacteristic, complemented, Sylow
233C22×D3—181no—
234C22×D3—181no—
235C22×D3—181no—
236C22×D3—181no—
237C22×D3—361no—
238C22×D3—361no—
239C3×Q8—181no—
240C3×Q8—181no—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C22×D911yescharacteristic
247C2×C3×C4C2×D3211yescharacteristic
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3D3611yescharacteristic
251C2×Dic3D3611yescharacteristic
252D12—92no—
253D12—92no—
254D12—182no—
255D12—182no—
256D12—181no—
257D12—181no—
258D12—181no—
259D12—181no—
260D12—181no—
261D12—181no—
262D12—361no—
263D12—361no—
264C4×D3—362no—
265C4×D3—362no—
266Dic6—61no—
267Dic6—61no—
268C3×C8—22no—
269C3×C8—21no—
270C3×C8—21no—
271C3×C8—61no—
272C3⋊C8—21no—
273C2×C32#89511yescharacteristic
274C2×C32#90411yescharacteristic
275C2×C32#90811yescharacteristic
276C3⋊D3—122no—
277C3⋊D3—241no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—21no—
281C2×C9#88411yescharacteristic
282C2×C9#91411yescharacteristic
283C2×C9#91911yescharacteristic
284D9—122no—
285D9—242no—
286D9—242no—
287D9—241no—
288C2×Q8—91no—
289C2×D4—271no—
290C2×D4—271no—
291C22×C4—541no—
292SD16—542no—
293D8—542no—
294C2×C8—31no—
295C2×C8D3×D911yescharacteristic
296C4⋊C4—91no—
297C22⋊C4—541no—
298C22⋊C4—541no—
299C22⋊C4—541no—
300C42—31no—
301C22×C3—21no—
302C22×C3C2×D3611yescharacteristic
303C22×C3D3×D1211yescharacteristic
304C2×D3—182no—
305C2×D3—182no—
306C2×D3—362no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—362no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—721no—
319C2×D3—721no—
320C2×D3—721no—
321C2×D3—721no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4D4×D911yescharacteristic
329C3×C4#370111yescharacteristic
330C3×C4#379911yescharacteristic
331C3×C4#381111yescharacteristic
332Dic3—21no—
333Dic3—21no—
334Dic3—121no—
335Dic3—121no—
336C32#484411yescharacteristic
337C9—21no—
338C9#481911yescharacteristic, complemented
339C23—541no—
340C23—541no—
341Q8—181no—
342Q8—181no—
343D4—272no—
344D4—541no—
345D4—541no—
346D4—541no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—1082no—
352C2×C4C2×D3×D911yescharacteristic
353C8—21no—
354C8—61no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3#6345711yescharacteristic
359C2×C3#6346711yescharacteristic
360C2×C3#6347011yescharacteristic
361C2×C3#6379411yescharacteristic
362C2×C3#6380811yescharacteristic
363C2×C3#6391011yescharacteristic
364D3—362no—
365D3—362no—
366D3—722no—
367D3—722no—
368D3—721no—
369D3—721no—
370C22—542no—
371C22—1081no—
372C22—1081no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22D3×D3611yescharacteristic
377C4—61no—
378C4—61no—
379C4—361no—
380C4—361no—
381C4#8863511yescharacteristic
382C4#8866811yescharacteristic
383C3—21no—
384C3#1058852811yescharacteristic, complemented
385C3#1059077011yescharacteristic
386C2—1082no—
387C2—2161no—
388C2#1172080411yescharacteristic
389C2#1172081211yescharacteristic
390C2#1172100511yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×#88525C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#463
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720993C2semidirect
3. #11720952C2semidirect
4. C2×#88416C2semidirect
5. C9×Dic3⋊C8C2semidirect
6. C3×#63376C2semidirect
7. #11720284C2semidirect
37. #63376D3semidirect
112. Dic3⋊C8D9semidirect
232. C3×C9#463semidirect
338. C9#4819semidirect
384. C3#10588528semidirect

Maximal subgroups

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

subgroupindex
C2×#885252
#117209932
#117209522
C2×#884162
C9×Dic3⋊C82
C3×#633762
#117202842
#105907703
#105885283