Finite Group Databasebeta

← #13138312

Subgroups — #13138312

Order 1728.

428 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 87 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#11720992C211yesmaximal, characteristic
3#11720924C211yesmaximal, characteristic, complemented
4C2×#88426C211yesmaximal, characteristic
5C9×#816C211yesmaximal, characteristic, complemented
6C3×Dic9⋊C8C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590760—31nomaximal
9#10588498—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C2×C3×C8×C9C2211yescharacteristic
15#88524—22no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic9⋊Dic3—21no—
21Dic9⋊Dic3—21no—
22#88426—22no—
23C2×#3744—31no—
24#63894—31no—
25#63866—31no—
26C2×#3717—31no—
27C3×#816D311yescharacteristic
28C3×Dic3⋊C8—31no—
29#63567—31no—
30#63506—31no—
31C2×#3675—31no—
32C2×#3664—31no—
33C2×#3664—61no—
34#63455—31no—
35C9×#103—31no—
36Dic9⋊C8D311yescharacteristic, complemented
37#63368—31no—
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×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D411yescharacteristic
48C2×C3×Dic9D411yescharacteristic
49C9×Dic6—21no—
50C9×Dic6—21no—
51(C3⋊D9)⋊C4—42no—
52C3×C8×C9—21no—
53C3×C9⋊C8—21no—
54#4841—91no—
55#4818—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×#634—31no—
58C2×C3×Dic6C2×D311yescharacteristic
59C2×#624—61no—
60C2×C32×C8C2×D311yescharacteristic
61#3744—62no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C3×C4×Dic3—31no—
64C2×C3×C3⋊C8—31no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67#3722—61no—
68#3717—62no—
69C2×Q8×C9—31no—
70C2×D36—31no—
71C2×D36—61no—
72C2×C4×D9—61no—
73C2×Dic18—31no—
74C2×Dic18—61no—
75C2×C8×C9—21no—
76C2×C8×C9C2×D311yescharacteristic
77C9×C4⋊C4—31no—
78#3675—62no—
79(C2×D9)⋊C4—61no—
80Dic9⋊C4—61no—
81Dic9⋊C4—61no—
82C4×Dic9C2×D311yescharacteristic
83C2×C9⋊C8C2×D311yescharacteristic
84#3664—62no—
85#3664—64no—
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—41no—
94C9×Dic3—41no—
95C3×Dic9—21no—
96C3×Dic9SD1612yescomplemented
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—91no—
107C3×#103—31no—
108#816—91no—
109#816D911yescharacteristic, complemented
110Dic3⋊C8—31no—
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—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—31no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125#624—122no—
126C22×D9—61no—
127C22×D9—121no—
128C32×C8—21no—
129C3×C3⋊C8—61no—
130Q8×C9—61no—
131Q8×C9—61no—
132C2×C4×C9—21no—
133C2×C4×C9—61no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136C2×Dic9—61no—
137C2×Dic9—121no—
138C2×Dic9D1211yescharacteristic
139C2×Dic9D1211yescharacteristic
140D36—32no—
141D36—62no—
142D36—61no—
143D36—121no—
144C4×D9—122no—
145Dic18—62no—
146Dic18—61no—
147Dic18—61no—
148Dic18—121no—
149C8×C9—22no—
150C8×C9—21no—
151C9⋊C8—21no—
152#461—271noSylow
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9C2×SD1611yescharacteristic
155C2×C3×C9#13711yescharacteristic
156C3⋊D9—42no—
157C2×C3×Q8—31no—
158C2×D12—91no—
159C2×D12—91no—
160C2×D12—181no—
161C2×C4×D3—181no—
162C2×C4×D3—181no—
163C2×Dic6—91no—
164C2×Dic6—91no—
165C2×Dic6—181no—
166C2×Dic6C2×D911yescharacteristic
167C2×C3×C8—21no—
168C2×C3×C8—91no—
169C2×C3×C8C2×D911yescharacteristic
170C2×C3×C8D3211yescharacteristic
171C3×C4⋊C4—31no—
172C3×C42—91no—
173#215—182no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—181no—
177C4⋊Dic3—181no—
178C4⋊Dic3C2×D911yescharacteristic
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181C4×Dic3—31no—
182C2×C3⋊C8—31no—
183#204—182no—
184#204—182no—
185#204—184no—
186#204—182no—
187C22×C32D3×D411yescharacteristic
188C2×C3⋊D3—62no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—121no—
191C32×C4#23511yescharacteristic
192C32×C4D3×D411yescharacteristic
193C3⋊Dic3—121no—
194C3×Dic3—32no—
195C3×Dic3—41no—
196C3×Dic3—41no—
197C3×Dic3—61no—
198C22×C9—21no—
199C22×C9C2×D1211yescharacteristic
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—121no—
209C4×C9—121no—
210C4×C9D3×D411yescharacteristic
211C4×C9#23911yescharacteristic
212Dic9—21no—
213Dic9—122no—
214Dic9—121no—
215Dic9#20412yescomplemented
216C2×SD16—271no—
217C2×SD16—271no—
218C4⋊Q8—271no—
219C4×D4—271no—
220C4⋊C8—91no—
221#103—31no—
222#103—271no—
223C3×C9#46111yescharacteristic, complemented, Sylow
224C22×D3—181no—
225C22×D3—181no—
226C22×D3—361no—
227C3×Q8—61no—
228C3×Q8—61no—
229C2×C3×C4—21no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4C22×D911yescharacteristic
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C2×Dic3—361no—
243D12—92no—
244D12—92no—
245D12—182no—
246D12—181no—
247D12—181no—
248D12—361no—
249C4×D3—362no—
250C4×D3—362no—
251Dic6—21no—
252Dic6—21no—
253Dic6—182no—
254Dic6—181no—
255Dic6—181no—
256Dic6—181no—
257Dic6—181no—
258Dic6—361no—
259C3×C8—22no—
260C3×C8—21no—
261C3×C8—21no—
262C3×C8—181no—
263C3⋊C8—61no—
264C2×C32#88411yescharacteristic
265C2×C32D3×SD1611yescharacteristic
266C2×C32#91811yescharacteristic
267C3⋊D3—122no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9#89511yescharacteristic
272C2×C9C2×#20411yescharacteristic
273C2×C9#90911yescharacteristic
274D9—122no—
275D9—242no—
276C2×Q8—31no—
277C2×Q8—271no—
278C2×D4—271no—
279C22×C4—541no—
280SD16—542no—
281SD16—542no—
282C2×C8—91no—
283C2×C8D3×D911yescharacteristic
284C4⋊C4—31no—
285C4⋊C4—541no—
286C4⋊C4—541no—
287C22⋊C4—541no—
288C42—91no—
289C22×C3—21no—
290C22×C3D4×D911yescharacteristic
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—92no—
304C3×C4—121no—
305C3×C4—121no—
306C3×C4—181no—
307C3×C4#369711yescharacteristic
308C3×C4D4×D911yescharacteristic
309C3×C4#379911yescharacteristic
310C3×C4#381111yescharacteristic
311Dic3—32no—
312Dic3—41no—
313Dic3—41no—
314Dic3—61no—
315Dic3—362no—
316Dic3—361no—
317Dic3—361no—
318C32#481811yescharacteristic
319C9—21no—
320C9#484111yescharacteristic, complemented
321C23—541no—
322Q8—61no—
323Q8—61no—
324Q8—541no—
325Q8—541no—
326D4—272no—
327D4—541no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4—541no—
333C2×C4—1082no—
334C2×C4C2×D3×D911yescharacteristic
335C8—21no—
336C8—181no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3#6344611yescharacteristic
341C2×C3SD16×D911yescharacteristic
342C2×C3#6348011yescharacteristic
343C2×C3D3×#20411yescharacteristic
344C2×C3#6380111yescharacteristic
345C2×C3#6391011yescharacteristic
346D3—362no—
347D3—362no—
348D3—722no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D9×D1211yescharacteristic
353C4—92no—
354C4—121no—
355C4—121no—
356C4—181no—
357C4—1081no—
358C4#8864211yescharacteristic
359C4#8866811yescharacteristic
360C3—21no—
361C3#1058849811yescharacteristic, complemented
362C3#1059076011yescharacteristic
363C2—1082no—
364C2D9×#20411yescharacteristic
365C2#1172082811yescharacteristic
366C2#1172103911yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
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. #11720924C2semidirect
5. C9×#816C2semidirect
6. C3×Dic9⋊C8C2semidirect
7. #11720269C2semidirect
36. Dic9⋊C8D3semidirect
96. C3×Dic9SD16semidirect
109. #816D9semidirect
215. Dic9#204semidirect
223. C3×C9#461semidirect
320. C9#4841semidirect
361. C3#10588498semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
#117209922
#117209242
C2×#884262
C9×#8162
C3×Dic9⋊C82
#117202692
#105907603
#105884983