Finite Group Databasebeta

← #13140846

Subgroups — #13140846

Order 1728.

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

normal subgroups only

This group has 372 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
1C3×#63508C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic, complemented
3#11720944C211yesmaximal, characteristic
4C2×#88413C211yesmaximal, characteristic
5#11720582C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720278C211yesmaximal, characteristic, complemented
8#10591638—31nomaximal
9#10588869—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×#216—22no—
16C3×(C2×D9)⋊C4—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21#88442—21no—
22#88413—22no—
23C3×#946—31no—
24C2×C3×#216D311yescharacteristic
25#63856—31no—
26C2×#3711—31no—
27#63662—31no—
28#63573—31no—
29#63565—31no—
30C2×C9×Q16—31no—
31#63508—31no—
32#63508D311yescharacteristic
33C2×#3663—31no—
34#63395—31no—
35#63395—31no—
36#63395—61no—
37#63377—31no—
38C22×C3×D9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C3×C4×D9—21no—
48C3×C4×D9—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8D412yes—
53#5189—91no—
54#5052—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×C4×D3—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C2×C3⋊(C3⋊C8)—31no—
60C3×#216—22no—
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—31no—
63C3×Dic3⋊C4—61no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65#3724—61no—
66(C2×D3)⋊Dic3—61no—
67#3711—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9—31no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×C4×D9C2×D311yescharacteristic
72C9×Q16—62no—
73C2×C8×C9—31no—
74(C2×D9)⋊C4—21no—
75(C2×D9)⋊C4—61no—
76C4⋊Dic9—31no—
77C4⋊Dic9—61no—
78C4⋊Dic9C2×D311yescharacteristic
79Dic9⋊C4—21no—
80Dic9⋊C4—61no—
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83#3663—62no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3×D9—21no—
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C3×C4×C9—41no—
89C3×C4×C9C2×D411yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C3⋊Dic9—41no—
92C9×Dic3—41no—
93C3×Dic9—21no—
94C3×Dic9—41no—
95C2×C3×Q16—31no—
96C3×C22⋊Q8—91no—
97#946—31no—
98#946—91no—
99C2×#216D911yescharacteristic, complemented
100#924—91no—
101C2×#208—91no—
102C2×ΓL(1,𝔽25)—91no—
103#835—91no—
104#835—91no—
105#835—91no—
106#835—181no—
107#817—91no—
108#810—91no—
109#808—91no—
110C22×C3×D3—31no—
111C32×Q8—21no—
112C32×Q8—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—61no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C3×C4×D3—61no—
120C3×C4×D3—61no—
121C3×Dic6—21no—
122C3×Dic6—21no—
123C22×D9#6711yescharacteristic
124C3⋊(C3⋊C8)—61no—
125C3×C3⋊C8#6712yes—
126Q8×C9—22no—
127Q8×C9—21no—
128Q8×C9—21no—
129Q8×C9—41no—
130Q8×C9—61no—
131Q8×C9—61no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—41no—
135C2×C4×C9—61no—
136C2×C4×C9C22×D311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9—61no—
139C2×Dic9—121no—
140C2×Dic9#6711yescharacteristic
141C4×D9—21no—
142C4×D9—21no—
143C8×C9—32no—
144C9⋊C8—61no—
145C9⋊C8—121no—
146#470—271noSylow
147C2×C3×C9#12111yescharacteristic
148C2×C3×C9#13711yescharacteristic
149C2×C3×C9#13711yescharacteristic
150C3×D9—41no—
151C2×C3×Q8—21no—
152C2×C3×Q8—31no—
153C2×C3×Q8C2×D911yescharacteristic
154C2×C3×Q8D3211yescharacteristic
155C22×C3×C4—91no—
156C2×C4×D3—31no—
157C2×Dic6C2×D911yescharacteristic
158C3×Q16—62no—
159C2×C3×C8—31no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C22⋊C4—181no—
163#217—181no—
164#216—22no—
165(C2×D3)⋊C4—61no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—31no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4⋊Dic3—181no—
171Dic3⋊C4—61no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174#208—182no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—181no—
178C2×C3⋊C8C2×D911yescharacteristic
179ΓL(1,𝔽25)—182no—
180C22×C32D3×D411yescharacteristic
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C32×C4—41no—
185C32×C4#23711yescharacteristic
186C32×C4C2×#6711yescharacteristic
187C3⋊Dic3—121no—
188C3×Dic3—41no—
189C3×Dic3—61no—
190C3×Dic3—121no—
191C22×C9—21no—
192C22×C9C2×#6711yescharacteristic
193C2×D9—21no—
194C2×D9—21no—
195C2×D9—21no—
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—42no—
199C4×C9—41no—
200C4×C9—121no—
201C4×C9D3×D411yescharacteristic
202C4×C9#23711yescharacteristic
203Dic9—21no—
204Dic9—41no—
205Dic9—121no—
206Dic9—241no—
207C2×Q16—31no—
208C2×M16—271no—
209C22⋊Q8—91no—
210C22⋊Q8—271no—
211#106—271no—
212#103—271no—
213#103—271no—
214C3×C9#47011yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—31no—
217C3×Q8—22no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—41no—
223C3×Q8—61no—
224C3×Q8—61no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4C22×D911yescharacteristic
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—31no—
238C2×Dic3—61no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241C2×Dic3—361no—
242C4×D3—61no—
243C4×D3—61no—
244Dic6—21no—
245Dic6—21no—
246C3×C8—32no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C3⋊C8#61112yes—
251C2×C32#91811yescharacteristic
252C2×C32#91811yescharacteristic
253C2×C32#93811yescharacteristic
254C3×D3—121no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9#93911yescharacteristic
259C2×C9#94211yescharacteristic
260C2×C9#95111yescharacteristic
261D9—41no—
262C2×Q8—31no—
263C2×Q8D3×D911yescharacteristic
264C22×C4—91no—
265Q16—62no—
266M16—542no—
267C2×C8—31no—
268C2×C8—271no—
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—271no—
272C4⋊C4—541no—
273C22⋊C4—181no—
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D4×D911yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—61no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—121no—
290C3×C4—181no—
291C3×C4—361no—
292C3×C4#369911yescharacteristic
293C3×C4C2×#61111yescharacteristic
294C3×C4#380511yescharacteristic
295C3×C4D3×#6711yescharacteristic
296Dic3—41no—
297Dic3—61no—
298Dic3—121no—
299Dic3—361no—
300Dic3—361no—
301Dic3—721no—
302C32#518911yescharacteristic
303C9—21no—
304C9#505211yescharacteristic, complemented
305C23—91no—
306Q8—21no—
307Q8—21no—
308Q8—61no—
309Q8—61no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×D911yescharacteristic
318C8—32no—
319C8—541no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—361no—
324C2×C3#6348011yescharacteristic
325C2×C3#6348011yescharacteristic
326C2×C3#6350011yescharacteristic
327C2×C3#6394311yescharacteristic
328C2×C3#6394811yescharacteristic
329C2×C3#6397011yescharacteristic
330D3—121no—
331C22—181no—
332C22—181no—
333C22—181no—
334C22D9×#6711yescharacteristic
335C4—41no—
336C4—121no—
337C4—181no—
338C4—361no—
339C4—1081no—
340C4#8866311yescharacteristic
341C4D3×#61111yescharacteristic
342C3—21no—
343C3#1058886911yescharacteristic, complemented
344C3#1059163811yescharacteristic
345C2—361no—
346C2#1172121911yescharacteristic
347C2#1172122211yescharacteristic
348C2#1172127011yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#470
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. C2×C9×#216C2semidirect
5. #11720582C2semidirect
6. #11720297C2semidirect
7. #11720278C2semidirect
99. C2×#216D9semidirect
214. C3×C9#470semidirect
304. C9#5052semidirect
343. C3#10588869semidirect

Maximal subgroups

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

subgroupindex
C3×#635082
C2×C9×#2162
#117209442
C2×#884132
#117205822
#117202972
#117202782
#105916383
#105888693