Finite Group Databasebeta

← #13140836

Subgroups — #13140836

Order 1728.

394 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 93 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic
3C2×C3×#3674C211yesmaximal, characteristic
4#11720930C211yesmaximal, characteristic
5#11720298C211yesmaximal, characteristic
6#11720276C211yesmaximal, characteristic
7#11720273C211yesmaximal, characteristic
8#10591633—31nomaximal
9#10588862—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C4⋊(C3⋊Dic9)—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C4×C3⋊Dic9—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C9×#216—22no—
18C3×#3674—22no—
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88442—21no—
22#88442—21no—
23Q8×C3⋊Dic3—31no—
24C2×C3×#216—31no—
25C2×C3×#216D311yescharacteristic
26#63873—31no—
27#63572—31no—
28#63566—31no—
29#63566—31no—
30C2×C9×Q16—31no—
31Q8×Dic9—31no—
32Q8×Dic9—61no—
33#63506—31no—
34C2×#3674D311yescharacteristic
35#63395—31no—
36#63376—31no—
37Dic9⋊C8—31no—
38C3×Q8×C9—21no—
39C3×Q8×C9D412yes—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C9×Dic6—21no—
48C9×Dic6—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C9×C3⋊C8—21no—
52C3×C9⋊C8—21no—
53#5182—91no—
54#5182—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—61no—
60C4×C3⋊Dic3—31no—
61C4×C3⋊Dic3—61no—
62C3×#216—22no—
63C3×#216—62no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66#3724—61no—
67#3724—61no—
68C2×Q8×C9—21no—
69C2×Q8×C9—31no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×Dic18C2×D311yescharacteristic
72C9×Q16—62no—
73C2×C8×C9—31no—
74#3674—22no—
75C4⋊Dic9—31no—
76C4⋊Dic9—61no—
77C4⋊Dic9—61no—
78C4⋊Dic9—62no—
79Dic9⋊C4—61no—
80Dic9⋊C4—61no—
81C4×Dic9—31no—
82C4×Dic9—61no—
83C4×Dic9—61no—
84C4×Dic9—62no—
85C2×C9⋊C8C2×D311yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C3×C4×C9—22no—
88C3×C4×C9C2×D411yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—21no—
91C3⋊Dic9—41no—
92C3⋊Dic9Q1612yescomplemented
93C9×Dic3—41no—
94C3×Dic9—41no—
95C2×C3×Q16—31no—
96C2×C3×Q16—91no—
97Q8×Dic3—91no—
98Q8×Dic3—91no—
99Q8×Dic3—181no—
100#944—91no—
101#944—91no—
102C2×#216—31no—
103C2×#216D911yescharacteristic
104#835—91no—
105#835—91no—
106#816—91no—
107#816—91no—
108Dic3⋊C8—91no—
109Dic3⋊C8—91no—
110C32×Q8—21no—
111C32×Q8#6712yes—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—61no—
119C3×Dic6—21no—
120C3×Dic6—21no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C3×C3⋊C8—21no—
124C3×C3⋊C8—61no—
125Q8×C9—22no—
126Q8×C9—21no—
127Q8×C9—22no—
128Q8×C9—61no—
129Q8×C9—61no—
130Q8×C9#6712yes—
131C2×C4×C9—22no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136C2×Dic9—21no—
137C2×Dic9—31no—
138C2×Dic9—31no—
139C2×Dic9—61no—
140C2×Dic9—61no—
141C2×Dic9—62no—
142C2×Dic9—61no—
143Dic18—21no—
144Dic18—21no—
145C8×C9—61no—
146C9⋊C8—21no—
147#462—271noSylow
148C2×C3×C9#12111yescharacteristic
149C2×C3×C9C2×Q1611yescharacteristic
150C2×C3×C9#13711yescharacteristic
151C2×C3×Q8—21no—
152C2×C3×Q8—31no—
153C2×C3×Q8—91no—
154C2×C3×Q8C2×D911yescharacteristic
155C2×C3×Q8D3211yescharacteristic
156C2×Dic6—31no—
157C2×Dic6C2×D911yescharacteristic
158C3×Q16—62no—
159C3×Q16—182no—
160C2×C3×C8—31no—
161C2×C3×C8—91no—
162#216—22no—
163#216—62no—
164C4⋊Dic3—91no—
165C4⋊Dic3—91no—
166C4⋊Dic3—181no—
167C4⋊Dic3—181no—
168C4⋊Dic3—181no—
169C4⋊Dic3—182no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174C4×Dic3—91no—
175C4×Dic3—91no—
176C4×Dic3—181no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C4×Dic3—182no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8C2×D911yescharacteristic
182C22×C32D3×D411yescharacteristic
183C32×C4—22no—
184C32×C4#23711yescharacteristic
185C32×C4C2×#6711yescharacteristic
186C3⋊Dic3—32no—
187C3⋊Dic3—61no—
188C3⋊Dic3—121no—
189C3×Dic3—41no—
190C3×Dic3—121no—
191C22×C9—21no—
192C22×C9D3×D411yescharacteristic
193C4×C9—24no—
194C4×C9—22no—
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—121no—
198C4×C9#23711yescharacteristic
199C4×C9C2×#6711yescharacteristic
200Dic9—32no—
201Dic9—41no—
202Dic9—62no—
203Dic9—61no—
204Dic9—121no—
205Dic9—122no—
206Dic9—121no—
207C2×Q16—31no—
208C2×Q16—91no—
209C4⋊Q8—271no—
210C4×Q8—271no—
211C4⋊C8—271no—
212#103—271no—
213#103—271no—
214C3×C9#46211yescharacteristic, complemented, Sylow
215C3×Q8—22no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—22no—
219C3×Q8—61no—
220C3×Q8—61no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×Q8#61112yes—
224C3×Q8#62512yes—
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—21no—
245Dic6—21no—
246Dic6—61no—
247Dic6—61no—
248C3×C8—61no—
249C3×C8—181no—
250C3⋊C8—21no—
251C3⋊C8—61no—
252C2×C32D3×Q1611yescharacteristic
253C2×C32#91811yescharacteristic
254C2×C32#93911yescharacteristic
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9D3×Q1611yescharacteristic
259C2×C9#91811yescharacteristic
260C2×C9#93911yescharacteristic
261C2×Q8—31no—
262C2×Q8—91no—
263C2×Q8D3×D911yescharacteristic
264Q16—62no—
265Q16—182no—
266C2×C8—31no—
267C2×C8—91no—
268C4⋊C4—271no—
269C4⋊C4—541no—
270C4⋊C4—541no—
271C4⋊C4—541no—
272C42—271no—
273C42—541no—
274C22×C3—21no—
275C22×C3D4×D911yescharacteristic
276C22×C3#381111yescharacteristic
277C3×C4—24no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—121no—
283C3×C4—361no—
284C3×C4#369911yescharacteristic
285C3×C4C2×#61111yescharacteristic
286C3×C4#380511yescharacteristic
287C3×C4C2×#62511yescharacteristic
288Dic3—41no—
289Dic3—92no—
290Dic3—92no—
291Dic3—121no—
292Dic3—182no—
293Dic3—181no—
294Dic3—181no—
295Dic3—361no—
296Dic3—362no—
297Dic3—361no—
298Dic3—361no—
299C32#518211yescharacteristic
300C9—21no—
301C9#518211yescharacteristic, complemented
302Q8—21no—
303Q8—61no—
304Q8—61no—
305Q8—181no—
306Q8—181no—
307Q8#619412yes—
308C2×C4—21no—
309C2×C4—61no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4C2×D3×D911yescharacteristic
315C8—61no—
316C8—181no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3Q16×D911yescharacteristic
321C2×C3#6348011yescharacteristic
322C2×C3#6350111yescharacteristic
323C2×C3#6394411yescharacteristic
324C2×C3#6394511yescharacteristic
325C2×C3#6398011yescharacteristic
326C22#8866811yescharacteristic
327C4—22no—
328C4—121no—
329C4—272no—
330C4—361no—
331C4—541no—
332C4—1081no—
333C4#8866311yescharacteristic
334C4C2×#619411yescharacteristic
335C3—21no—
336C3#1058886211yescharacteristic, complemented
337C3#1059163311yescharacteristic
338C2#1172122011yescharacteristic
339C2#1172122111yescharacteristic
340C2#1172129011yescharacteristic
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
214. C3×C9#462semidirect
301. C9#5182semidirect
336. C3#10588862semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
C2×C9×#2162
C2×C3×#36742
#117209302
#117202982
#117202762
#117202732
#105916333
#105888623