Finite Group Databasebeta

← #13137580

Subgroups — #13137580

Order 1728.

412 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 79 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#11720922C211yesmaximal, characteristic, complemented
3C9×#808C211yesmaximal, characteristic, complemented
4C3×#63366C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590451—31nomaximal
9#10588546—31nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14#88536—21no—
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic3×Dic9—21no—
21#64104—31no—
22#63849—31no—
23C3×#808D311yescharacteristic
24C3×#807—31no—
25#63564—31no—
26#63564—31no—
27#63557—31no—
28#63457—31no—
29#63457—61no—
30#63454—31no—
31C9×#106—31no—
32#63380—31no—
33(C2×D9)⋊C8—31no—
34#63369—31no—
35#63366D311yescharacteristic, 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—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4885—91no—
51#4866—91no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—61no—
54C2×C4×C3⋊D3—31no—
55C32×C4⋊C4C2×D311yescharacteristic
56#3752—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3722—61no—
62Dic32—61no—
63C2×D36—31no—
64C2×D36—61no—
65C2×D36—61no—
66C2×D36—62no—
67C2×C4×D9—31no—
68C2×C4×D9—61no—
69C2×C8×C9—31no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4—31no—
72C9×C4⋊C4C2×D311yescharacteristic
73(C2×D9)⋊C4—61no—
74(C2×D9)⋊C4—61no—
75(C2×D9)⋊C4—62no—
76C4⋊Dic9C2×D311yescharacteristic
77C4×Dic9—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3⋊D9—21no—
81C2×C3⋊D9—21no—
82C2×C3⋊D9—21no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C2×C3⋊D9—41no—
86C3×C4×C9—41no—
87C3×C4×C9#4111yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C9×Dic3—41no—
91C3×Dic9—41no—
92#895—91no—
93#895—91no—
94#895—181no—
95#892—91no—
96#892—91no—
97C3×#107—91no—
98C3×#106—31no—
99#821—91no—
100#821—91no—
101(C2×D3)⋊C8—91no—
102(C2×D3)⋊C8—91no—
103#809—91no—
104#809—91no—
105#808D911yescharacteristic, complemented
106#807—31no—
107C22×C3⋊D3—31no—
108C22×C3⋊D3—61no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C4⋊(C3⋊D3)—61no—
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—122no—
115C4×C3⋊D3—61no—
116C4×C3⋊D3—61no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—61no—
119C22×D9—31no—
120C22×D9—61no—
121C22×D9—62no—
122C22×D9—61no—
123C3×C3⋊C8—21no—
124C3×C3⋊C8—61no—
125C2×C4×C9—22no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—61no—
129C2×C4×C9C22×D311yescharacteristic
130C2×Dic9—21no—
131C2×Dic9—31no—
132C2×Dic9—61no—
133D36—61no—
134D36—61no—
135D36—121no—
136D36—121no—
137D36—124no—
138D36—122no—
139C4×D9—61no—
140C4×D9—61no—
141C4×D9—121no—
142C4×D9—121no—
143C8×C9—61no—
144C9⋊C8—21no—
145#482—271noSylow
146C2×C3×C9#12311yescharacteristic
147C2×C3×C9#13511yescharacteristic
148C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
149C3⋊D9—41no—
150C3⋊D9—81no—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×D12—181no—
155C2×D12—181no—
156C2×D12—182no—
157C2×C4×D3—91no—
158C2×C4×D3—91no—
159C2×C4×D3—181no—
160C2×C3×C8—31no—
161C2×C3×C8—91no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4C2×D911yescharacteristic
166C3×C4⋊C4D3211yescharacteristic
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—182no—
172C4⋊Dic3—31no—
173C4⋊Dic3C2×D911yescharacteristic
174C4×Dic3—181no—
175C4×Dic3—181no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8C2×D911yescharacteristic
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—61no—
180C2×C3⋊D3—61no—
181C2×C3⋊D3—61no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—121no—
185C32×C4—41no—
186C32×C4#23511yescharacteristic
187C32×C4#23911yescharacteristic
188C3⋊Dic3—61no—
189C3×Dic3—41no—
190C3×Dic3—121no—
191C22×C9—21no—
192C22×C9D3×D411yescharacteristic
193C2×D9—62no—
194C2×D9—61no—
195C2×D9—61no—
196C2×D9—61no—
197C2×D9—121no—
198C2×D9—121no—
199C2×D9—122no—
200C2×D9—122no—
201C2×D9—122no—
202C2×D9—121no—
203C2×D9—121no—
204C2×D9—121no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—42no—
208C4×C9—41no—
209C4×C9—121no—
210C4×C9#23511yescharacteristic
211C4×C9#23911yescharacteristic
212Dic9—41no—
213Dic9—61no—
214Dic9—121no—
215#121—271no—
216#117—271no—
217#107—91no—
218#106—31no—
219#102—271no—
220#102—271no—
221C22⋊C8—271no—
222C3×C9#48211yescharacteristic, complemented, Sylow
223C22×D3—91no—
224C22×D3—91no—
225C22×D3—181no—
226C22×D3—182no—
227C22×D3—181no—
228C22×D3—181no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—61no—
234C2×C3×C4—181no—
235C2×C3×C4C22×D911yescharacteristic
236C2×C3×C4C2×D3211yescharacteristic
237C2×Dic3—21no—
238C2×Dic3—61no—
239C2×Dic3—91no—
240C2×Dic3—91no—
241C2×Dic3—181no—
242D12—181no—
243D12—181no—
244D12—181no—
245D12—181no—
246D12—361no—
247D12—361no—
248D12—362no—
249D12—364no—
250D12—362no—
251C4×D3—181no—
252C4×D3—181no—
253C4×D3—181no—
254C4×D3—181no—
255C4×D3—361no—
256C4×D3—361no—
257C3×C8—61no—
258C3×C8—181no—
259C3⋊C8—21no—
260C3⋊C8—61no—
261C2×C32#89411yescharacteristic
262C2×C32#91411yescharacteristic
263C2×C32#91611yescharacteristic
264C3⋊D3—121no—
265C3⋊D3—241no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9#89411yescharacteristic
270C2×C9#91111yescharacteristic
271C2×C9#91911yescharacteristic
272D9—122no—
273D9—121no—
274D9—242no—
275D9—241no—
276C2×D4—271no—
277C2×D4—541no—
278C22×C4—271no—
279C2×C8—31no—
280C2×C8—91no—
281C4⋊C4—31no—
282C4⋊C4—91no—
283C4⋊C4D3×D911yescharacteristic
284C22⋊C4—541no—
285C22⋊C4—541no—
286C42—541no—
287C22×C3—21no—
288C22×C3D4×D911yescharacteristic
289C22×C3#381111yescharacteristic
290C2×D3—182no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—181no—
294C2×D3—181no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—361no—
298C2×D3—361no—
299C2×D3—362no—
300C2×D3—362no—
301C2×D3—362no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—361no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—42no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—121no—
314C3×C4—361no—
315C3×C4#369711yescharacteristic
316C3×C4#370111yescharacteristic
317C3×C4#379911yescharacteristic
318C3×C4#379911yescharacteristic
319Dic3—41no—
320Dic3—121no—
321Dic3—181no—
322Dic3—181no—
323Dic3—361no—
324C32#486611yescharacteristic
325C9—21no—
326C9#488511yescharacteristic, complemented
327C23—271no—
328C23—541no—
329D4—541no—
330D4—541no—
331D4—1082no—
332C2×C4—21no—
333C2×C4—61no—
334C2×C4—181no—
335C2×C4—271no—
336C2×C4—541no—
337C2×C4—541no—
338C2×C4C2×D3×D911yescharacteristic
339C8—61no—
340C8—181no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3#6345611yescharacteristic
345C2×C3#6347611yescharacteristic
346C2×C3#6347811yescharacteristic
347C2×C3#6389711yescharacteristic
348C2×C3#6393711yescharacteristic
349C2×C3#6395011yescharacteristic
350D3—362no—
351D3—361no—
352D3—361no—
353D3—722no—
354D3—721no—
355D3—721no—
356C22—541no—
357C22—541no—
358C22—541no—
359C22—1081no—
360C22—1081no—
361C22—1081no—
362C22#8866811yescharacteristic
363C4—41no—
364C4—121no—
365C4—361no—
366C4—541no—
367C4#8863511yescharacteristic
368C4#8864211yescharacteristic
369C3—21no—
370C3#1058854611yescharacteristic, complemented
371C3#1059045111yescharacteristic
372C2—1081no—
373C2—2161no—
374C2#1172099711yescharacteristic
375C2#1172118711yescharacteristic
376C2#1172121611yescharacteristic
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#482
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. #11720922C2semidirect
3. C9×#808C2semidirect
4. C3×#63366C2semidirect
5. #11720284C2semidirect
6. #11720282C2semidirect
7. #11720267C2semidirect
35. #63366D3semidirect
105. #808D9semidirect
222. C3×C9#482semidirect
326. C9#4885semidirect
370. C3#10588546semidirect

Maximal subgroups

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

subgroupindex
#117217512
#117209222
C9×#8082
C3×#633662
#117202842
#117202822
#117202672
#105904513
#105885463