Finite Group Databasebeta

← #13137572

Subgroups — #13137572

Order 1728.

436 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721751C211yesmaximal, characteristic
2#11721000C211yesmaximal, characteristic, complemented
3C9×#808C211yesmaximal, characteristic, complemented
4C3×#63367C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590445—31nomaximal
9#10588543—31nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64104—31no—
22#63900—31no—
23C3×#808—31no—
24C3×#808D311yescharacteristic
25#63564—31no—
26#63564—31no—
27#63557—31no—
28#63457—31no—
29#63457—61no—
30C4⋊C4×D9—31no—
31C9×#106—31no—
32#63380—31no—
33(C2×D9)⋊C8—31no—
34#63369—31no—
35#63367D311yescharacteristic, complemented
36C22×C3⋊D9—21no—
37C22×C3⋊D9D411yescharacteristic
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—42no—
44C4×C3⋊D9—21no—
45C4×C3⋊D9—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48(C3⋊D9)⋊C4—42no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8—21no—
51#4863—91no—
52#4863—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—61no—
55C2×C4×C3⋊D3—31no—
56C2×#624—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58#3752—61no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63#3724—61no—
64C2×D36—31no—
65C2×D36—61no—
66C2×D36—61no—
67C2×D36—62no—
68C2×C4×D9—31no—
69C2×C4×D9—61no—
70C2×C4×D9—61no—
71C2×C8×C9—31no—
72C9×C4⋊C4—21no—
73C9×C4⋊C4—31no—
74C9×C4⋊C4C2×D311yescharacteristic
75(C2×D9)⋊C4—61no—
76(C2×D9)⋊C4—62no—
77C4⋊Dic9C2×D311yescharacteristic
78Dic9⋊C4—61no—
79C2×C9⋊C8C2×D311yescharacteristic
80C22×C3×C9C2×D411yescharacteristic
81C2×C3⋊D9—21no—
82C2×C3⋊D9—21no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C2×C3⋊D9—41no—
86C2×C3⋊D9SD1612yes—
87C3×C4×C9—41no—
88C3×C4×C9#4111yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—21no—
91C9×Dic3—41no—
92C3×Dic9—41no—
93#895—91no—
94#895—91no—
95#895—181no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—91no—
98C3×#106—31no—
99C3×#106—91no—
100#821—91no—
101#821—91no—
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#809—91no—
105#809—91no—
106#808—31no—
107#808D911yescharacteristic, complemented
108C22×C3⋊D3—31no—
109C22×C3⋊D3—61no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—61no—
115C4⋊(C3⋊D3)—122no—
116C4×C3⋊D3—61no—
117C4×C3⋊D3—61no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—61no—
120#624—122no—
121C22×D9—31no—
122C22×D9—61no—
123C22×D9—62no—
124C22×D9—61no—
125C3×C3⋊C8—21no—
126C3×C3⋊C8—61no—
127C2×C4×C9—22no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—61no—
131C2×C4×C9C22×D311yescharacteristic
132C2×Dic9—21no—
133C2×Dic9—31no—
134C2×Dic9—61no—
135D36—61no—
136D36—61no—
137D36—121no—
138D36—121no—
139D36—122no—
140D36—124no—
141C4×D9—61no—
142C4×D9—61no—
143C4×D9—121no—
144C4×D9—121no—
145C4×D9—122no—
146C8×C9—61no—
147C9⋊C8—21no—
148#481—271noSylow
149C2×C3×C9#12311yescharacteristic
150C2×C3×C9C2×SD1611yescharacteristic
151C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
152C3⋊D9—22no—
153C3⋊D9—81no—
154C2×D12—91no—
155C2×D12—91no—
156C2×D12—181no—
157C2×D12—181no—
158C2×D12—181no—
159C2×D12—182no—
160C2×C4×D3—91no—
161C2×C4×D3—91no—
162C2×C4×D3—181no—
163C2×C4×D3—181no—
164C2×C4×D3—181no—
165C2×C3×C8—31no—
166C2×C3×C8—91no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4C2×D911yescharacteristic
171C3×C4⋊C4D3211yescharacteristic
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—182no—
175C4⋊Dic3—31no—
176C4⋊Dic3C2×D911yescharacteristic
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8C2×D911yescharacteristic
181C22×C32D3×D411yescharacteristic
182C2×C3⋊D3—32no—
183C2×C3⋊D3—61no—
184C2×C3⋊D3—61no—
185C2×C3⋊D3—121no—
186C2×C3⋊D3—121no—
187C2×C3⋊D3—121no—
188C32×C4—41no—
189C32×C4#23511yescharacteristic
190C32×C4#23911yescharacteristic
191C3⋊Dic3—61no—
192C3×Dic3—41no—
193C3×Dic3—121no—
194C22×C9—21no—
195C22×C9D3×D411yescharacteristic
196C2×D9—32no—
197C2×D9—62no—
198C2×D9—61no—
199C2×D9—61no—
200C2×D9—121no—
201C2×D9—121no—
202C2×D9—122no—
203C2×D9—122no—
204C2×D9—122no—
205C2×D9—121no—
206C2×D9—121no—
207C2×D9—121no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—42no—
211C4×C9—41no—
212C4×C9—121no—
213C4×C9#23511yescharacteristic
214C4×C9#23911yescharacteristic
215Dic9—41no—
216Dic9—61no—
217Dic9—121no—
218#121—271no—
219C2×C4⋊C4—271no—
220#106—31no—
221#106—91no—
222#102—271no—
223#102—271no—
224C22⋊C8—271no—
225C3×C9#48111yescharacteristic, complemented, Sylow
226C22×D3—91no—
227C22×D3—91no—
228C22×D3—181no—
229C22×D3—182no—
230C22×D3—181no—
231C22×D3—181no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—61no—
237C2×C3×C4—181no—
238C2×C3×C4C22×D911yescharacteristic
239C2×C3×C4C2×D3211yescharacteristic
240C2×Dic3—21no—
241C2×Dic3—61no—
242C2×Dic3—91no—
243C2×Dic3—91no—
244C2×Dic3—181no—
245D12—181no—
246D12—181no—
247D12—181no—
248D12—181no—
249D12—361no—
250D12—361no—
251D12—362no—
252D12—362no—
253D12—364no—
254C4×D3—181no—
255C4×D3—181no—
256C4×D3—181no—
257C4×D3—181no—
258C4×D3—361no—
259C4×D3—361no—
260C4×D3—362no—
261C4×D3—362no—
262C3×C8—61no—
263C3×C8—181no—
264C3⋊C8—21no—
265C3⋊C8—61no—
266C2×C32#89411yescharacteristic
267C2×C32D3×SD1611yescharacteristic
268C2×C32#91411yescharacteristic
269C3⋊D3—62no—
270C3⋊D3—241no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9#89411yescharacteristic
275C2×C9D3×SD1611yescharacteristic
276C2×C9#91411yescharacteristic
277D9—62no—
278D9—122no—
279D9—242no—
280D9—241no—
281C2×D4—271no—
282C2×D4—541no—
283C22×C4—271no—
284C22×C4—541no—
285C2×C8—31no—
286C2×C8—91no—
287C4⋊C4—31no—
288C4⋊C4—91no—
289C4⋊C4—541no—
290C4⋊C4D3×D911yescharacteristic
291C22⋊C4—541no—
292C22×C3—21no—
293C22×C3D4×D911yescharacteristic
294C22×C3#381111yescharacteristic
295C2×D3—92no—
296C2×D3—92no—
297C2×D3—182no—
298C2×D3—181no—
299C2×D3—181no—
300C2×D3—181no—
301C2×D3—181no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—362no—
305C2×D3—362no—
306C2×D3—362no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—361no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—121no—
319C3×C4—361no—
320C3×C4#369711yescharacteristic
321C3×C4#370111yescharacteristic
322C3×C4#379911yescharacteristic
323C3×C4#379911yescharacteristic
324Dic3—41no—
325Dic3—121no—
326Dic3—181no—
327Dic3—181no—
328Dic3—361no—
329C32#486311yescharacteristic
330C9—21no—
331C9#486311yescharacteristic, complemented
332C23—271no—
333C23—541no—
334D4—541no—
335D4—541no—
336D4—1082no—
337C2×C4—21no—
338C2×C4—61no—
339C2×C4—181no—
340C2×C4—271no—
341C2×C4—541no—
342C2×C4—541no—
343C2×C4—1082no—
344C2×C4C2×D3×D911yescharacteristic
345C8—61no—
346C8—181no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#6345611yescharacteristic
351C2×C3SD16×D911yescharacteristic
352C2×C3#6347611yescharacteristic
353C2×C3#6389711yescharacteristic
354C2×C3#6393011yescharacteristic
355C2×C3#6394111yescharacteristic
356D3—182no—
357D3—182no—
358D3—362no—
359D3—722no—
360D3—721no—
361D3—721no—
362C22—272no—
363C22—541no—
364C22—541no—
365C22—1081no—
366C22—1081no—
367C22—1081no—
368C22#8866811yescharacteristic
369C4—41no—
370C4—121no—
371C4—361no—
372C4—541no—
373C4#8863511yescharacteristic
374C4#8864211yescharacteristic
375C3—21no—
376C3#1058854311yescharacteristic, complemented
377C3#1059044511yescharacteristic
378C2—542no—
379C2—2161no—
380C2#1172099711yescharacteristic
381C2#1172118211yescharacteristic
382C2#1172121311yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721751C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#481
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721000C2semidirect
3. C9×#808C2semidirect
4. C3×#63367C2semidirect
5. #11720284C2semidirect
6. #11720282C2semidirect
7. #11720267C2semidirect
35. #63367D3semidirect
107. #808D9semidirect
225. C3×C9#481semidirect
331. C9#4863semidirect
376. C3#10588543semidirect

Maximal subgroups

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

subgroupindex
#117217512
#117210002
C9×#8082
C3×#633672
#117202842
#117202822
#117202672
#105904453
#105885433