Finite Group Databasebeta

← #13140758

Subgroups — #13140758

Order 1728.

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

normal subgroups only

This group has 392 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
1#11721854C211yesmaximal, characteristic
2C2×C3×#3674C211yesmaximal, characteristic
3#11720910C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic
7#11720269C211yesmaximal, characteristic
8#10591580—31nomaximal
9#10588475—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13#88534—21no—
14#88534—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C3×#3674—22no—
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21Dic3×Dic9—21no—
22#88418—22no—
23#64153—31no—
24C2×C3×#216—31no—
25#63843—31no—
26C2×#3719—31no—
27C3×#835D311yescharacteristic
28#63572—31no—
29#63567—31no—
30#63506—31no—
31#63506—61no—
32C2×#3674D311yescharacteristic
33C2×Dic36—31no—
34#63449—31no—
35C9×#103—31no—
36Dic9⋊C8—31no—
37#63368—31no—
38C3×Q8×C9—21no—
39C3×Q8×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45#6226—21no—
46#6226—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18D412yes—
51C9×C3⋊C8—21no—
52C3×C9⋊C8—21no—
53#5182—91no—
54#4795—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×#634—31no—
57C2×C3×Dic6—31no—
58#3750—61no—
59#3750—61no—
60C4×C3⋊Dic3—31no—
61C3×#216—62no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66Dic32—61no—
67#3719—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C2×D311yescharacteristic
70C2×Dic18—31no—
71C2×Dic18—61no—
72C2×Dic18C2×D311yescharacteristic
73C2×C8×C9—31no—
74C9×C4⋊C4—31no—
75#3674—22no—
76Dic9⋊C4—61no—
77Dic9⋊C4—61no—
78Dic9⋊C4—61no—
79Dic9⋊C4—62no—
80Dic9⋊C4—62no—
81C4×Dic9—31no—
82C4×Dic9—61no—
83C4×Dic9—61no—
84C2×C9⋊C8C2×D311yescharacteristic
85Dic36—62no—
86C22×C3×C9C2×D411yescharacteristic
87C3×C4×C9—41no—
88C3×C4×C9C2×D411yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—21no—
91C3⋊Dic9—41no—
92C3⋊Dic9Q1612yescomplemented
93C9×Dic3—41no—
94C3×Dic9—22no—
95C2×C3×Q16—91no—
96Q8×Dic3—91no—
97#944—91no—
98#944—91no—
99#944—181no—
100C2×#216—31no—
101C2×#216—91no—
102C2×Dic12—91no—
103#887—91no—
104C3×#103—31no—
105#835D911yescharacteristic
106#816—91no—
107Dic3⋊C8—91no—
108Dic3⋊C8—91no—
109#810—91no—
110C32×Q8—21no—
111C32×Q8—21no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117#634—61no—
118#634—61no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—61no—
121C3×Dic6—32no—
122C3×Dic6—61no—
123C3×C3⋊C8—21no—
124C3×C3⋊C8—61no—
125Q8×C9—21no—
126Q8×C9—21no—
127Q8×C9—41no—
128Q8×C9—41no—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—31no—
136C2×Dic9—31no—
137C2×Dic9—61no—
138C2×Dic9—61no—
139C2×Dic9—62no—
140C2×Dic9—61no—
141Dic18—21no—
142Dic18—61no—
143Dic18—61no—
144Dic18—121no—
145Dic18—121no—
146Dic18#6712yes—
147C8×C9—61no—
148C9⋊C8—21no—
149#462—271noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9C2×Q1611yescharacteristic
152C2×C3×C9#13711yescharacteristic
153C2×C3×Q8—21no—
154C2×C3×Q8—91no—
155C2×C3×Q8C2×D911yescharacteristic
156C2×C3×Q8D3211yescharacteristic
157C2×Dic6—31no—
158C2×Dic6—91no—
159C2×Dic6—91no—
160C2×Dic6—181no—
161C3×Q16—182no—
162C2×C3×C8—31no—
163C2×C3×C8—91no—
164C3×C4⋊C4—31no—
165#216—62no—
166#216—182no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×D911yescharacteristic
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—182no—
175Dic3⋊C4—182no—
176C4×Dic3—91no—
177C4×Dic3—91no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8C2×D911yescharacteristic
183Dic12—182no—
184C22×C32D3×D411yescharacteristic
185C32×C4—41no—
186C32×C4#23511yescharacteristic
187C32×C4D3×D411yescharacteristic
188C3⋊Dic3—32no—
189C3⋊Dic3—61no—
190C3⋊Dic3—121no—
191C3×Dic3—41no—
192C3×Dic3—62no—
193C22×C9—21no—
194C22×C9D3×D411yescharacteristic
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—42no—
198C4×C9—41no—
199C4×C9—121no—
200C4×C9#23711yescharacteristic
201C4×C9C2×#6711yescharacteristic
202Dic9—22no—
203Dic9—32no—
204Dic9—62no—
205Dic9—62no—
206Dic9—61no—
207Dic9—122no—
208Dic9—121no—
209C2×Q16—91no—
210C2×Q16—271no—
211C4⋊Q8—271no—
212C4×Q8—271no—
213C4⋊C8—271no—
214#103—31no—
215#103—271no—
216C3×C9#46211yescharacteristic, complemented, Sylow
217C3×Q8—21no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—21no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—92no—
224C3×Q8—181no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—61no—
230C2×C3×C4—181no—
231C2×C3×C4C22×D911yescharacteristic
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—61no—
235C2×Dic3—91no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241C2×Dic3—182no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244Dic6—32no—
245Dic6—61no—
246Dic6—181no—
247Dic6—181no—
248Dic6—181no—
249Dic6—181no—
250Dic6—361no—
251Dic6—361no—
252C3×C8—61no—
253C3×C8—181no—
254C3⋊C8—21no—
255C3⋊C8—61no—
256C2×C32#88411yescharacteristic
257C2×C32#91511yescharacteristic
258C2×C32D3×Q1611yescharacteristic
259C2×C9—21no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9D3×Q1611yescharacteristic
263C2×C9#91811yescharacteristic
264C2×C9#93911yescharacteristic
265C2×Q8—91no—
266C2×Q8—271no—
267C2×Q8D3×D911yescharacteristic
268Q16—182no—
269Q16—542no—
270C2×C8—31no—
271C2×C8—91no—
272C4⋊C4—31no—
273C4⋊C4—541no—
274C4⋊C4—541no—
275C4⋊C4—541no—
276C42—271no—
277C42—541no—
278C22×C3—21no—
279C22×C3D4×D911yescharacteristic
280C22×C3#381111yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—42no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—121no—
287C3×C4—182no—
288C3×C4#369711yescharacteristic
289C3×C4D4×D911yescharacteristic
290C3×C4#380411yescharacteristic
291C3×C4D3×#6711yescharacteristic
292Dic3—41no—
293Dic3—62no—
294Dic3—92no—
295Dic3—92no—
296Dic3—182no—
297Dic3—182no—
298Dic3—181no—
299Dic3—181no—
300Dic3—362no—
301Dic3—361no—
302Dic3—361no—
303C32#479511yescharacteristic
304C9—21no—
305C9#518211yescharacteristic, complemented
306Q8—21no—
307Q8—21no—
308Q8—92no—
309Q8—181no—
310Q8—541no—
311Q8—541no—
312C2×C4—21no—
313C2×C4—61no—
314C2×C4—181no—
315C2×C4—271no—
316C2×C4—271no—
317C2×C4—541no—
318C2×C4C2×D3×D911yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3#6344611yescharacteristic
325C2×C3#6347711yescharacteristic
326C2×C3Q16×D911yescharacteristic
327C2×C3#6394411yescharacteristic
328C2×C3#6395111yescharacteristic
329C2×C3#6397111yescharacteristic
330C22#8866811yescharacteristic
331C4—41no—
332C4—121no—
333C4—182no—
334C4—272no—
335C4—541no—
336C4—1081no—
337C4#8865811yescharacteristic
338C4D9×#6711yescharacteristic
339C3—21no—
340C3#1058847511yescharacteristic, complemented
341C3#1059158011yescharacteristic
342C2#1172121711yescharacteristic
343C2#1172122011yescharacteristic
344C2#1172127111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#462
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
92. C3⋊Dic9Q16semidirect
216. C3×C9#462semidirect
305. C9#5182semidirect
340. C3#10588475semidirect

Maximal subgroups

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

subgroupindex
#117218542
C2×C3×#36742
#117209102
C2×#884182
C9×#8352
#117202732
#117202692
#105915803
#105884753