Finite Group Databasebeta

← #13138229

Subgroups — #13138229

Order 1728.

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

normal subgroups only

This group has 440 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×#88524C211yesmaximal, characteristic, complemented
2#11720993C211yesmaximal, characteristic, complemented
3#11720948C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic
5C3×#63380C211yesmaximal, characteristic, complemented
6C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10590860—31nomaximal
9#10588528—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C2×C3×C8×C9C2211yescharacteristic
15#88524—22no—
16C4×C9×Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20(C2×D9)⋊Dic3—21no—
21(C2×D9)⋊Dic3—21no—
22#88417—22no—
23C2×#3744—31no—
24#63894—31no—
25#63860—31no—
26C2×#3714—31no—
27C3×#821—31no—
28C3×Dic3⋊C8D311yescharacteristic
29#63567—31no—
30C2×D72—31no—
31C2×#3664—31no—
32C2×#3664—61no—
33#63437—31no—
34C4×D36—31no—
35C9×C4⋊C8—31no—
36#63380D311yescharacteristic, complemented
37#63376—31no—
38C22×C3⋊D9—21no—
39C22×C3×D9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)D412yes—
44#6226—21no—
45#6226—21no—
46C2×C9×Dic3D411yescharacteristic
47C2×C9×Dic3D411yescharacteristic
48C2×C3×Dic9—21no—
49C3×D36—21no—
50C3×D36—21no—
51(C3⋊D9)⋊C4—42no—
52C3×C8×C9—21no—
53C9×C3⋊C8—21no—
54#4844—91no—
55#4787—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×#634—31no—
58C2×C3×D12—31no—
59C2×#624—61no—
60C2×C32×C8C2×D311yescharacteristic
61#3744—62no—
62C3×C4⋊Dic3—31no—
63C3×C4×Dic3C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65#3722—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—61no—
68#3714—62no—
69C2×D36—31no—
70C2×D36—61no—
71C2×D36C2×D311yescharacteristic
72C2×C4×D9—61no—
73C2×Dic18—31no—
74C2×Dic18—61no—
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—
84#3664—62no—
85#3664—64no—
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
95C3⋊Dic9—41no—
96C9×Dic3—21no—
97C9×Dic3—21no—
98C3×Dic9—41no—
99#936—91no—
100C2×#213—91no—
101C2×D24—91no—
102C2×#204—91no—
103C2×#204—91no—
104C2×#204—181no—
105#893—91no—
106#875—91no—
107C4×D12—91no—
108C3×C4⋊C8—31no—
109C3×#102—91no—
110#821—31no—
111#816—91no—
112Dic3⋊C8D911yescharacteristic, complemented
113#810—91no—
114C22×C3⋊D3—61no—
115C22×C3×D3—61no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C4⋊(C3⋊D3)—32no—
119C4⋊(C3⋊D3)—61no—
120#634—61no—
121#634—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3D1211yescharacteristic
124C2×C3×Dic3D1211yescharacteristic
125C3×D12—61no—
126C3×D12—61no—
127#624—122no—
128C22×D9—21no—
129C22×D9—61no—
130C22×D9—121no—
131C32×C8—21no—
132C3×C3⋊C8—21no—
133C2×C4×C9—21no—
134C2×C4×C9—31no—
135C2×C4×C9—31no—
136C2×C4×C9C22×D311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9—61no—
139C2×Dic9—121no—
140D36—21no—
141D36—21no—
142D36—32no—
143D36—62no—
144D36—61no—
145D36—121no—
146C4×D9—122no—
147Dic18—62no—
148Dic18—61no—
149Dic18—61no—
150Dic18—121no—
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×D4—91no—
161C2×D12—31no—
162C2×D12—91no—
163C2×D12—91no—
164C2×D12—181no—
165C2×C4×D3—181no—
166C2×C4×D3—181no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×Dic6—181no—
170C2×C3×C8—21no—
171C2×C3×C8—31no—
172C2×C3×C8C2×D911yescharacteristic
173C2×C3×C8D3211yescharacteristic
174C3×C4⋊C4—91no—
175C3×C42—31no—
176#217—181no—
177#217—181no—
178#213—182no—
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—
187#204—182no—
188#204—182no—
189#204—184no—
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—121no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9D3×D411yescharacteristic
205C2×D9—41no—
206C2×D9—41no—
207C2×D9—41no—
208C2×D9—62no—
209C2×D9—122no—
210C2×D9—121no—
211C2×D9—121no—
212C2×D9—241no—
213C2×D9—241no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—61no—
217C4×C9—61no—
218C4×C9#23511yescharacteristic
219C4×C9D3×D411yescharacteristic
220Dic9—41no—
221Dic9—122no—
222Dic9—121no—
223C2×SD16—271no—
224C2×D8—271no—
225#124—271no—
226C4×D4—271no—
227C4⋊C8—31no—
228#103—271no—
229#102—91no—
230C3×C9#46311yescharacteristic, complemented, Sylow
231C23×C3—181no—
232C22×D3—61no—
233C22×D3—181no—
234C22×D3—181no—
235C22×D3—361no—
236C3×D4—181no—
237C3×D4—181no—
238C2×C3×C4—21no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—181no—
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—61no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—361no—
248C2×Dic3D3611yescharacteristic
249C2×Dic3D3611yescharacteristic
250D12—61no—
251D12—61no—
252D12—92no—
253D12—92no—
254D12—182no—
255D12—181no—
256D12—181no—
257D12—361no—
258C4×D3—362no—
259C4×D3—362no—
260Dic6—182no—
261Dic6—181no—
262Dic6—181no—
263Dic6—181no—
264Dic6—181no—
265Dic6—361no—
266C3×C8—22no—
267C3×C8—21no—
268C3×C8—21no—
269C3×C8—61no—
270C3⋊C8—21no—
271C2×C32#89511yescharacteristic
272C2×C32#90411yescharacteristic
273C2×C32#90811yescharacteristic
274C3⋊D3—122no—
275C3×D3—241no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9#88411yescharacteristic
280C2×C9#91111yescharacteristic
281C2×C9#91611yescharacteristic
282D9—81no—
283D9—122no—
284D9—242no—
285C2×Q8—271no—
286C2×D4—91no—
287C2×D4—271no—
288C22×C4—541no—
289SD16—542no—
290D8—542no—
291C2×C8—31no—
292C2×C8D3×D911yescharacteristic
293C4⋊C4—91no—
294C22⋊C4—541no—
295C22⋊C4—541no—
296C22⋊C4—541no—
297C42—31no—
298C22×C3—21no—
299C22×C3—361no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3C2×D3611yescharacteristic
303C22×C3D3×D1211yescharacteristic
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—121no—
307C2×D3—182no—
308C2×D3—182no—
309C2×D3—362no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—721no—
315C2×D3—721no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—361no—
321C3×C4D4×D911yescharacteristic
322C3×C4#370111yescharacteristic
323C3×C4#379911yescharacteristic
324C3×C4#381111yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—121no—
328Dic3—362no—
329Dic3—361no—
330Dic3—361no—
331C32#484411yescharacteristic
332C9—21no—
333C9#478711yescharacteristic, complemented
334C23—181no—
335C23—541no—
336Q8—541no—
337Q8—541no—
338D4—181no—
339D4—181no—
340D4—272no—
341D4—541no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—181no—
345C2×C4—541no—
346C2×C4—1082no—
347C2×C4C2×D3×D911yescharacteristic
348C8—21no—
349C8—61no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—721no—
354C2×C3#6345711yescharacteristic
355C2×C3#6346711yescharacteristic
356C2×C3#6347011yescharacteristic
357C2×C3#6379511yescharacteristic
358C2×C3#6380611yescharacteristic
359C2×C3#6391011yescharacteristic
360D3—241no—
361D3—362no—
362D3—362no—
363D3—722no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22—542no—
368C22—1081no—
369C22—1081no—
370C22D3×D3611yescharacteristic
371C4—61no—
372C4—61no—
373C4—361no—
374C4—1081no—
375C4#8863511yescharacteristic
376C4#8866811yescharacteristic
377C3—21no—
378C3#1058852811yescharacteristic, complemented
379C3#1059086011yescharacteristic
380C2—721no—
381C2—1082no—
382C2#1172080511yescharacteristic
383C2#1172081011yescharacteristic
384C2#1172100511yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×#88417C2
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
1. C2×#88524C2semidirect
2. #11720993C2semidirect
3. #11720948C2semidirect
5. C3×#63380C2semidirect
6. C9×Dic3⋊C8C2semidirect
7. #11720277C2semidirect
36. #63380D3semidirect
112. Dic3⋊C8D9semidirect
230. C3×C9#463semidirect
333. C9#4787semidirect
378. C3#10588528semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
#117209932
#117209482
C2×#884172
C3×#633802
C9×Dic3⋊C82
#117202772
#105908603
#105885283