Finite Group Databasebeta

← #13140724

Subgroups — #13140724

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×#88548C211yesmaximal, characteristic
2C3×#63506C211yesmaximal, characteristic, complemented
3#11720992C211yesmaximal, characteristic
4C2×#88416C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic, complemented
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10591565—31nomaximal
9#10588498—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14#88548—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×Dic9⋊C4—21no—
18C3×Dic9⋊C4—21no—
19C3×C4×Dic9C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21#88445—21no—
22#88416—22no—
23C2×#3755—31no—
24C3×#944—31no—
25#63894—31no—
26C2×#3717—31no—
27C3×#835D311yescharacteristic
28Dic3⋊(C3⋊C8)—31no—
29#63565—31no—
30#63506D311yescharacteristic, complemented
31C2×#3675—31no—
32C2×#3675—61no—
33C2×#3664—31no—
34#63455—31no—
35C9×#103—31no—
36Dic9⋊C8—31no—
37#63368—31no—
38C22×C3⋊D9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)D412yes—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C2×C3×Dic9D411yescharacteristic
49C3×Dic18—21no—
50C3×Dic18—21no—
51(C3⋊D9)⋊C4—42no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8—21no—
54#5083—91no—
55#4818—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)—31no—
58C2×C3×Dic6—31no—
59C2×#624—61no—
60#3755—62no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C3×Dic3⋊C4—61no—
64C3×Dic3⋊C4—61no—
65C3×C4×Dic3—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67#3722—61no—
68#3717—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D36—31no—
72C2×D36—61no—
73C2×C4×D9—61no—
74C2×Dic18C2×D311yescharacteristic
75C2×C8×C9—31no—
76C9×C4⋊C4—31no—
77#3675—62no—
78#3675—64no—
79(C2×D9)⋊C4—61no—
80Dic9⋊C4—21no—
81Dic9⋊C4—21no—
82C4×Dic9C2×D311yescharacteristic
83C2×C9⋊C8—31no—
84C2×C9⋊C8—61no—
85#3664—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—22no—
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C3×C4×C9—41no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C9×Dic3—41no—
94C3×Dic9—21no—
95C3×Dic9—41no—
96C3×Dic9SD1612yescomplemented
97C3×C4⋊Q8—91no—
98#944—31no—
99C2×#215—91no—
100C2×#215—91no—
101C2×#215—91no—
102C2×#215—181no—
103C2×#204—91no—
104#893—91no—
105C4×D12—91no—
106C3×#103—31no—
107#835—91no—
108#835D911yescharacteristic, complemented
109Dic3⋊C8—91no—
110#810—91no—
111C4⋊(C3⋊C8)—91no—
112C22×C3⋊D3—61no—
113C32×Q8—21no—
114C32×Q8—21no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C4⋊(C3⋊D3)—32no—
118C4⋊(C3⋊D3)—61no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—61no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125#624—122no—
126C22×D9—61no—
127C22×D9—121no—
128C3⋊(C3⋊C8)—61no—
129C3×C3⋊C8—21no—
130Q8×C9—22no—
131Q8×C9—21no—
132Q8×C9—21no—
133Q8×C9—41no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—41no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9—21no—
140C2×Dic9#6711yescharacteristic
141C2×Dic9#6711yescharacteristic
142D36—32no—
143D36—62no—
144D36—61no—
145D36—121no—
146C4×D9—122no—
147Dic18—21no—
148Dic18—21no—
149C8×C9—61no—
150C9⋊C8—62no—
151C9⋊C8—61no—
152#461—271noSylow
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9C2×SD1611yescharacteristic
155C2×C3×C9#13711yescharacteristic
156C3⋊D9—42no—
157C2×C3×Q8—21no—
158C2×C3×Q8—91no—
159C2×C3×Q8C2×D911yescharacteristic
160C2×C3×Q8D3211yescharacteristic
161C2×D12—91no—
162C2×D12—91no—
163C2×D12—181no—
164C2×C4×D3—181no—
165C2×C4×D3—181no—
166C2×Dic6—31no—
167C2×C3×C8—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4—181no—
171C3×C42—91no—
172#215—182no—
173#215—182no—
174#215—182no—
175#215—184no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3C2×D911yescharacteristic
179Dic3⋊C4—61no—
180Dic3⋊C4—61no—
181C4×Dic3—31no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—181no—
185C2×C3⋊C8C2×D911yescharacteristic
186#204—182no—
187C22×C32D3×D411yescharacteristic
188C2×C3⋊D3—62no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—121no—
191C32×C4—41no—
192C32×C4#23511yescharacteristic
193C32×C4D3×D411yescharacteristic
194C3×Dic3—32no—
195C3×Dic3—41no—
196C3×Dic3—61no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9C2×#6711yescharacteristic
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—42no—
209C4×C9—41no—
210C4×C9—121no—
211C4×C9D3×D411yescharacteristic
212C4×C9#23711yescharacteristic
213Dic9—21no—
214Dic9—41no—
215Dic9#21512yescomplemented
216C2×SD16—271no—
217C2×SD16—271no—
218C4⋊Q8—91no—
219C4×D4—271no—
220C4⋊C8—271no—
221#103—31no—
222#103—271no—
223C3×C9#46111yescharacteristic, complemented, Sylow
224C22×D3—181no—
225C22×D3—181no—
226C22×D3—361no—
227C3×Q8—22no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—41no—
233C3×Q8—181no—
234C3×Q8—181no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249D12—92no—
250D12—92no—
251D12—182no—
252D12—181no—
253D12—181no—
254D12—361no—
255C4×D3—362no—
256C4×D3—362no—
257Dic6—61no—
258Dic6—61no—
259C3×C8—61no—
260C3⋊C8—21no—
261C3⋊C8—182no—
262C3⋊C8—181no—
263C3⋊C8—181no—
264C2×C32#88411yescharacteristic
265C2×C32D3×SD1611yescharacteristic
266C2×C32#91811yescharacteristic
267C3⋊D3—122no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9#93811yescharacteristic
272C2×C9C2×#21511yescharacteristic
273C2×C9#94211yescharacteristic
274D9—122no—
275D9—242no—
276C2×Q8—91no—
277C2×Q8D3×D911yescharacteristic
278C2×D4—271no—
279C22×C4—541no—
280SD16—542no—
281SD16—542no—
282C2×C8—31no—
283C2×C8—271no—
284C4⋊C4—31no—
285C4⋊C4—181no—
286C4⋊C4—181no—
287C22⋊C4—541no—
288C42—91no—
289C22×C3—21no—
290C22×C3D4×D911yescharacteristic
291C22×C3D3×#6711yescharacteristic
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—42no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—92no—
307C3×C4—121no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4#369711yescharacteristic
311C3×C4D4×D911yescharacteristic
312C3×C4#380411yescharacteristic
313C3×C4#381111yescharacteristic
314Dic3—32no—
315Dic3—41no—
316Dic3—61no—
317Dic3—121no—
318C32#481811yescharacteristic
319C9—21no—
320C9#508311yescharacteristic, complemented
321C23—541no—
322Q8—21no—
323Q8—21no—
324Q8—181no—
325Q8—181no—
326D4—272no—
327D4—541no—
328C2×C4—21no—
329C2×C4—61no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—1082no—
334C2×C4C2×D3×D911yescharacteristic
335C8—61no—
336C8—541no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3#6344611yescharacteristic
341C2×C3SD16×D911yescharacteristic
342C2×C3#6348011yescharacteristic
343C2×C3D3×#21511yescharacteristic
344C2×C3#6394811yescharacteristic
345C2×C3#6395811yescharacteristic
346D3—362no—
347D3—362no—
348D3—722no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D9×#6711yescharacteristic
353C4—41no—
354C4—92no—
355C4—121no—
356C4—181no—
357C4—361no—
358C4#8865811yescharacteristic
359C4#8866811yescharacteristic
360C3—21no—
361C3#1058849811yescharacteristic, complemented
362C3#1059156511yescharacteristic
363C2—1082no—
364C2D9×#21511yescharacteristic
365C2#1172122211yescharacteristic
366C2#1172125611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#461
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#63506C2semidirect
5. C9×#835C2semidirect
6. Dic9⋊(C3⋊C8)C2semidirect
7. #11720268C2semidirect
30. #63506D3semidirect
96. C3×Dic9SD16semidirect
108. #835D9semidirect
215. Dic9#215semidirect
223. C3×C9#461semidirect
320. C9#5083semidirect
361. C3#10588498semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
C3×#635062
#117209922
C2×#884162
C9×#8352
Dic9⋊(C3⋊C8)2
#117202682
#105915653
#105884983