Finite Group Databasebeta

← #13140838

Subgroups — #13140838

Order 1728.

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

normal subgroups only

This group has 366 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
1C3×#63506C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic
3Dic6×Dic9C211yesmaximal, characteristic
4C2×#88427C211yesmaximal, characteristic
5#11720582C211yesmaximal, characteristic
6#11720276C211yesmaximal, characteristic
7Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591574—31nomaximal
9#10588862—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×#216—22no—
16C3×Dic9⋊C4—21no—
17C3×Dic9⋊C4—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic9—21no—
21Dic3×Dic9—21no—
22#88427—22no—
23C3×#944—31no—
24C2×C3×#216D311yescharacteristic
25Dic3×Dic6—31no—
26C2×#3718—31no—
27#63662—31no—
28Dic3⋊(C3⋊C8)—31no—
29#63566—31no—
30C2×C9×Q16—31no—
31Q8×Dic9—31no—
32#63506D311yescharacteristic
33C2×#3674—31no—
34#63395—31no—
35#63395—61no—
36#63376—31no—
37Dic9⋊C8—31no—
38C3×Q8×C9—21no—
39C3×Q8×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C9×Dic3—21no—
44C2×C3×Dic9—21no—
45C2×C3×Dic9D411yescharacteristic
46C2×C3×Dic9D411yescharacteristic
47C9×Dic6—21no—
48C9×Dic6D412yes—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8—21no—
53#5182—91no—
54#5090—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C2×C3⋊(C3⋊C8)—31no—
60C3×#216—22no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×C4×Dic3—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66Dic32—61no—
67#3718—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9—31no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×Dic18C2×D311yescharacteristic
72C9×Q16—62no—
73C2×C8×C9—31no—
74#3674—62no—
75C4⋊Dic9—31no—
76C4⋊Dic9—61no—
77C4⋊Dic9—61no—
78Dic9⋊C4—21no—
79Dic9⋊C4—21no—
80C4×Dic9—61no—
81C4×Dic9C2×D311yescharacteristic
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84C22×C3×C9C2×D411yescharacteristic
85C3×C4×C9—41no—
86C3×C4×C9C2×D411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—41no—
89C9×Dic3—22no—
90C3×Dic9—21no—
91C3×Dic9—41no—
92C3×Dic9Q1612yescomplemented
93C2×C3×Q16—31no—
94C3×C4⋊Q8—91no—
95Q8×Dic3—91no—
96#944—31no—
97C2×#216—91no—
98C2×#216—91no—
99C2×#216D911yescharacteristic
100C4×Dic6—91no—
101#835—91no—
102#835—91no—
103#835—91no—
104#835—181no—
105#816—91no—
106Dic3⋊C8—91no—
107C4⋊(C3⋊C8)—91no—
108C32×Q8—21no—
109C32×Q8—21no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—61no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—61no—
117C3×Dic6—21no—
118C3×Dic6—61no—
119C3×Dic6—61no—
120C3×Dic6#6712yes—
121C3⋊(C3⋊C8)—61no—
122C3×C3⋊C8—21no—
123Q8×C9—22no—
124Q8×C9—21no—
125Q8×C9—21no—
126Q8×C9—32no—
127Q8×C9—41no—
128Q8×C9—61no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—41no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—61no—
136C2×Dic9—121no—
137C2×Dic9#6711yescharacteristic
138C2×Dic9#6711yescharacteristic
139Dic18—21no—
140Dic18—21no—
141C8×C9—61no—
142C9⋊C8—61no—
143C9⋊C8—121no—
144#462—271noSylow
145C2×C3×C9#12111yescharacteristic
146C2×C3×C9C2×Q1611yescharacteristic
147C2×C3×C9#13711yescharacteristic
148C2×C3×Q8—21no—
149C2×C3×Q8—31no—
150C2×C3×Q8—91no—
151C2×C3×Q8C2×D911yescharacteristic
152C2×C3×Q8D3211yescharacteristic
153C2×Dic6—31no—
154C2×Dic6C2×D911yescharacteristic
155C3×Q16—62no—
156C2×C3×C8—31no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4—181no—
159C3×C42—91no—
160#216—22no—
161#216—182no—
162#216—182no—
163C4⋊Dic3—91no—
164C4⋊Dic3—91no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167Dic3⋊C4—61no—
168Dic3⋊C4—61no—
169Dic3⋊C4—181no—
170C4×Dic3—31no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C2×C3⋊C8C2×D911yescharacteristic
177C22×C32D3×D411yescharacteristic
178C32×C4—41no—
179C32×C4#23711yescharacteristic
180C32×C4C2×#6711yescharacteristic
181C3⋊Dic3—121no—
182C3×Dic3—22no—
183C3×Dic3—32no—
184C3×Dic3—61no—
185C3×Dic3—121no—
186C22×C9—21no—
187C22×C9C2×#6711yescharacteristic
188C4×C9—21no—
189C4×C9—21no—
190C4×C9—42no—
191C4×C9—41no—
192C4×C9—62no—
193C4×C9D3×D411yescharacteristic
194C4×C9#23711yescharacteristic
195Dic9—21no—
196Dic9—41no—
197Dic9—121no—
198Dic9—241no—
199Dic9#21612yescomplemented
200C2×Q16—31no—
201C2×Q16—271no—
202C4⋊Q8—91no—
203C4×Q8—271no—
204C4⋊C8—271no—
205#103—271no—
206#103—271no—
207C3×C9#46211yescharacteristic, complemented, Sylow
208C3×Q8—22no—
209C3×Q8—21no—
210C3×Q8—21no—
211C3×Q8—21no—
212C3×Q8—21no—
213C3×Q8—32no—
214C3×Q8—41no—
215C3×Q8—61no—
216C3×Q8—181no—
217C3×Q8—181no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4C22×D911yescharacteristic
227C2×C3×C4C2×D3211yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—31no—
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—181no—
233C2×Dic3—181no—
234C2×Dic3—361no—
235Dic6—21no—
236Dic6—61no—
237Dic6—61no—
238Dic6#61112yes—
239C3×C8—61no—
240C3⋊C8—21no—
241C3⋊C8—181no—
242C3⋊C8—181no—
243C3⋊C8—361no—
244C2×C32D3×Q1611yescharacteristic
245C2×C32#91811yescharacteristic
246C2×C32#93911yescharacteristic
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—21no—
250C2×C9#93811yescharacteristic
251C2×C9#94211yescharacteristic
252C2×C9C2×#21611yescharacteristic
253C2×Q8—31no—
254C2×Q8—91no—
255C2×Q8D3×D911yescharacteristic
256Q16—62no—
257Q16—542no—
258C2×C8—31no—
259C2×C8—271no—
260C4⋊C4—181no—
261C4⋊C4—181no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C42—91no—
265C42—541no—
266C22×C3—21no—
267C22×C3D4×D911yescharacteristic
268C22×C3D3×#6711yescharacteristic
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—62no—
275C3×C4—92no—
276C3×C4—181no—
277C3×C4—361no—
278C3×C4#369911yescharacteristic
279C3×C4C2×#61111yescharacteristic
280C3×C4#380511yescharacteristic
281C3×C4D3×#6711yescharacteristic
282Dic3—22no—
283Dic3—32no—
284Dic3—61no—
285Dic3—121no—
286Dic3—361no—
287Dic3—361no—
288Dic3—721no—
289C32#518211yescharacteristic
290C9—21no—
291C9#509011yescharacteristic, complemented
292Q8—21no—
293Q8—21no—
294Q8—32no—
295Q8—61no—
296Q8—181no—
297Q8—181no—
298C2×C4—21no—
299C2×C4—61no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4C2×D3×D911yescharacteristic
305C8—61no—
306C8—541no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3Q16×D911yescharacteristic
311C2×C3#6348011yescharacteristic
312C2×C3#6350111yescharacteristic
313C2×C3D3×#21611yescharacteristic
314C2×C3#6394811yescharacteristic
315C2×C3#6397011yescharacteristic
316C22D9×#6711yescharacteristic
317C4—41no—
318C4—62no—
319C4—92no—
320C4—181no—
321C4—361no—
322C4—1081no—
323C4#8866311yescharacteristic
324C4D3×#61111yescharacteristic
325C3—21no—
326C3#1058886211yescharacteristic, complemented
327C3#1059157411yescharacteristic
328C2D9×#21611yescharacteristic
329C2#1172122211yescharacteristic
330C2#1172128111yescharacteristic
subgroupG/H
centerC22D9×#67
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
199. Dic9#216semidirect
207. C3×C9#462semidirect
291. C9#5090semidirect
326. C3#10588862semidirect

Maximal subgroups

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

subgroupindex
C3×#635062
C2×C9×#2162
Dic6×Dic92
C2×#884272
#117205822
#117202762
Dic9⋊(C3⋊C8)2
#105915743
#105888623