Finite Group Databasebeta

← #13138214

Subgroups — #13138214

Order 1728.

428 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 91 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
2#11720993C211yesmaximal, characteristic
3#11720932C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic
5C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C3×#63376C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10590760—31nomaximal
9#10588525—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
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—
20Dic3⋊Dic9—21no—
21Dic3⋊Dic9—21no—
22#88416—22no—
23C2×#3744—31no—
24#63894—31no—
25#63866—31no—
26C2×#3717—31no—
27C3×#816—31no—
28C3×Dic3⋊C8D311yescharacteristic
29#63567—31no—
30C2×#3664—31no—
31C2×#3664—31no—
32C2×#3664—61no—
33C4×D36—31no—
34#63431—31no—
35C9×C4⋊C8—31no—
36#63376—31no—
37#63376D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)D412yes—
43#6226—21no—
44#6226—21no—
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#4841—91no—
55#4818—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×#634—31no—
58C2×C3×Dic6—31no—
59C2×#624—61no—
60C2×C32×C8C2×D311yescharacteristic
61#3744—62no—
62C3×C4⋊Dic3—31no—
63C3×C4×Dic3C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67#3722—61no—
68#3717—62no—
69C2×D36—31no—
70C2×D36—61no—
71C2×C4×D9—61no—
72C2×Dic18—31no—
73C2×Dic18—61no—
74C2×Dic18C2×D311yescharacteristic
75C2×C8×C9—21no—
76C2×C8×C9—31no—
77C2×C8×C9C2×D311yescharacteristic
78C42×C9—31no—
79(C2×D9)⋊C4—61no—
80C4⋊Dic9—61no—
81C4⋊Dic9—61no—
82C4⋊Dic9C2×D311yescharacteristic
83#3664—62no—
84#3664—64no—
85#3664—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—22no—
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3⋊Dic9—41no—
93C9×Dic3—21no—
94C9×Dic3SD1612yescomplemented
95C3×Dic9—41no—
96C3×Dic9—41no—
97#944—91no—
98C2×#215—91no—
99C2×#204—91no—
100C2×#204—91no—
101C2×#204—91no—
102C2×#204—181no—
103#893—91no—
104C4×D12—91no—
105#869—91no—
106C3×C4⋊C8—31no—
107C3×#103—91no—
108#816—31no—
109#816—91no—
110Dic3⋊C8D911yescharacteristic, complemented
111#810—91no—
112C22×C3⋊D3—61no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—61no—
115C4⋊(C3⋊D3)—32no—
116C4⋊(C3⋊D3)—61no—
117#634—61no—
118#634—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3D1211yescharacteristic
122C2×C3×Dic3D1211yescharacteristic
123C3×Dic6—61no—
124C3×Dic6—61no—
125#624—122no—
126C22×D9—61no—
127C22×D9—121no—
128C32×C8—21no—
129C3×C3⋊C8—21no—
130C2×C4×C9—21no—
131C2×C4×C9—31no—
132C2×C4×C9—31no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—21no—
136C2×Dic9—61no—
137C2×Dic9—121no—
138D36—32no—
139D36—62no—
140D36—61no—
141D36—121no—
142C4×D9—122no—
143Dic18—21no—
144Dic18—21no—
145Dic18—62no—
146Dic18—61no—
147Dic18—61no—
148Dic18—121no—
149C8×C9—22no—
150C8×C9—21no—
151C8×C9—61no—
152#461—271noSylow
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9C2×SD1611yescharacteristic
155C2×C3×C9#13711yescharacteristic
156C3⋊D9—42no—
157C2×C3×Q8—91no—
158C2×D12—91no—
159C2×D12—91no—
160C2×D12—181no—
161C2×C4×D3—181no—
162C2×C4×D3—181no—
163C2×Dic6—31no—
164C2×Dic6—91no—
165C2×Dic6—91no—
166C2×Dic6—181no—
167C2×C3×C8—21no—
168C2×C3×C8—31no—
169C2×C3×C8C2×D911yescharacteristic
170C2×C3×C8D3211yescharacteristic
171C3×C4⋊C4—91no—
172C3×C42—31no—
173#215—182no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—181no—
178C4⋊Dic3—181no—
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181C4×Dic3C2×D911yescharacteristic
182C2×C3⋊C8C2×D911yescharacteristic
183#204—182no—
184#204—182no—
185#204—184no—
186#204—182no—
187C22×C32C2×D1211yescharacteristic
188C2×C3⋊D3—62no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—121no—
191C32×C4D3×D411yescharacteristic
192C32×C4#23911yescharacteristic
193C3⋊Dic3—121no—
194C3×Dic3—21no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C3×Dic3#20412yes—
198C22×C9—21no—
199C22×C9D3×D411yescharacteristic
200C2×D9—62no—
201C2×D9—122no—
202C2×D9—121no—
203C2×D9—121no—
204C2×D9—241no—
205C2×D9—241no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—32no—
209C4×C9—61no—
210C4×C9#23511yescharacteristic
211C4×C9D3×D411yescharacteristic
212Dic9—41no—
213Dic9—41no—
214Dic9—122no—
215Dic9—121no—
216C2×SD16—271no—
217C2×SD16—271no—
218C4⋊Q8—271no—
219C4×D4—271no—
220C4⋊C8—31no—
221#103—91no—
222#103—271no—
223C3×C9#46111yescharacteristic, complemented, Sylow
224C22×D3—181no—
225C22×D3—181no—
226C22×D3—361no—
227C3×Q8—181no—
228C3×Q8—181no—
229C2×C3×C4—21no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4C22×D911yescharacteristic
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240C2×Dic3—361no—
241C2×Dic3D3611yescharacteristic
242C2×Dic3D3611yescharacteristic
243D12—92no—
244D12—92no—
245D12—182no—
246D12—181no—
247D12—181no—
248D12—361no—
249C4×D3—362no—
250C4×D3—362no—
251Dic6—61no—
252Dic6—61no—
253Dic6—182no—
254Dic6—181no—
255Dic6—181no—
256Dic6—181no—
257Dic6—181no—
258Dic6—361no—
259C3×C8—22no—
260C3×C8—21no—
261C3×C8—21no—
262C3×C8—61no—
263C3⋊C8—21no—
264C2×C32#89511yescharacteristic
265C2×C32C2×#20411yescharacteristic
266C2×C32#90911yescharacteristic
267C3⋊D3—122no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9#88411yescharacteristic
272C2×C9D3×SD1611yescharacteristic
273C2×C9#91811yescharacteristic
274D9—122no—
275D9—242no—
276C2×Q8—91no—
277C2×Q8—271no—
278C2×D4—271no—
279C22×C4—541no—
280SD16—542no—
281SD16—542no—
282C2×C8—31no—
283C2×C8D3×D911yescharacteristic
284C4⋊C4—91no—
285C4⋊C4—541no—
286C4⋊C4—541no—
287C22⋊C4—541no—
288C42—31no—
289C22×C3—21no—
290C22×C3C2×D3611yescharacteristic
291C22×C3D3×D1211yescharacteristic
292C2×D3—182no—
293C2×D3—182no—
294C2×D3—362no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—361no—
298C2×D3—361no—
299C2×D3—721no—
300C2×D3—721no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—32no—
304C3×C4—61no—
305C3×C4—361no—
306C3×C4—361no—
307C3×C4D4×D911yescharacteristic
308C3×C4#370111yescharacteristic
309C3×C4#379911yescharacteristic
310C3×C4#381111yescharacteristic
311Dic3—21no—
312Dic3—121no—
313Dic3—121no—
314Dic3—362no—
315Dic3—361no—
316Dic3—361no—
317Dic3#366412yescomplemented
318C32#484111yescharacteristic
319C9—21no—
320C9#481811yescharacteristic, complemented
321C23—541no—
322Q8—181no—
323Q8—181no—
324Q8—541no—
325Q8—541no—
326D4—272no—
327D4—541no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—541no—
333C2×C4—1082no—
334C2×C4C2×D3×D911yescharacteristic
335C8—21no—
336C8—61no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3#6345711yescharacteristic
341C2×C3C2×#366411yescharacteristic
342C2×C3#6347111yescharacteristic
343C2×C3D3×#20411yescharacteristic
344C2×C3#6380111yescharacteristic
345C2×C3#6391011yescharacteristic
346D3—362no—
347D3—362no—
348D3—722no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D3×D3611yescharacteristic
353C4—32no—
354C4—61no—
355C4—361no—
356C4—361no—
357C4—1081no—
358C4#8863511yescharacteristic
359C4#8866811yescharacteristic
360C3—21no—
361C3#1058852511yescharacteristic, complemented
362C3#1059076011yescharacteristic
363C2—1082no—
364C2#1172079711yescharacteristic
365C2D3×#366411yescharacteristic
366C2#1172100511yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×C8D3×D9
hypocenterC3×C9#461
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
3. #11720932C2semidirect
5. C9×Dic3⋊C8C2semidirect
6. C3×#63376C2semidirect
7. #11720277C2semidirect
37. #63376D3semidirect
94. C9×Dic3SD16semidirect
110. Dic3⋊C8D9semidirect
223. C3×C9#461semidirect
317. Dic3#3664semidirect
320. C9#4818semidirect
361. C3#10588525semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
#117209932
#117209322
C2×#884162
C9×Dic3⋊C82
C3×#633762
#117202772
#105907603
#105885253