Finite Group Databasebeta

← #13147498

Subgroups — #13147498

Order 1728.

1038 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C2×#88634C212yesmaximal, complemented
2C42×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×#63436C211yesmaximal, characteristic, complemented
4C9×#869C211yesmaximal, characteristic, complemented
5#11721007C214yesmaximal, complemented
6#11720998C214yesmaximal, complemented
7#11720951C212yesmaximal, complemented
8#10593907—31nomaximal
9#10589827—31nomaximal
10C2×C4×C3⋊D9C2212yes—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C3×D36C2214yes—
13C2×C3×D36C2212yes—
14C2×C9×Dic6C2212yes—
15C2×#6192C2214yes—
16C2×Dic3×D9C2214yes—
17#88634—28no—
18C3×C42×C9C2211yescharacteristic
19C4×C3⋊Dic9C2212yes—
20C4×C3⋊Dic9C2211yescharacteristic
21C9×C4⋊Dic3C2214yes—
22(C2×D9)⋊Dic3C2218yes—
23C2×#3796—32no—
24C42×C3⋊D3—31no—
25C3×#874—31no—
26C3×#869D311yescharacteristic
27#63916—34no—
28#63898—34no—
29#63863—32no—
30C2×#3701—32no—
31C9×C4⋊Q8—31no—
32#63509—32no—
33#63457—34no—
34#63455—34no—
35#63436D311yescharacteristic, complemented
36C42×D9—31no—
37C42×D9—61no—
38C22×C3⋊D9C2311yescharacteristic
39C22×C3×D9C2314yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3×C4×C9C2312yes—
42C2×C3⋊Dic9C2312yes—
43C2×C3⋊Dic9C2311yescharacteristic
44C4×C3⋊D9—24no—
45C4×C3⋊D9D414yescomplemented
46C2×C9×Dic3C2314yes—
47C3×D36—24no—
48C3×D36—28no—
49C9×Dic6—24no—
50#6192—28no—
51Dic3×D9—28no—
52#5725—91no—
53#5716—91no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—32no—
56C2×C3×D12—32no—
57C2×C3×D12—34no—
58C2×C3×Dic6C2×D312yes—
59C2×#626—34no—
60C2×D3×Dic3—34no—
61#3796—68no—
62C32×C42C2×D311yescharacteristic
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—32no—
65C3×C4⋊Dic3C2×D314yes—
66(C2×D3)⋊Dic3—38no—
67C2×Q8×C9—32no—
68#3701—68no—
69C2×D36—34no—
70C2×D36C2×D312yes—
71C2×D36C2×D314yes—
72C2×C4×D9—31no—
73C2×C4×D9—34no—
74C2×C4×D9—32no—
75C2×C4×D9—61no—
76C2×C4×D9—62no—
77C9×C4⋊C4—34no—
78C42×C9—21no—
79C42×C9C2×D311yescharacteristic
80(C2×D9)⋊C4—38no—
81C4×Dic9—31no—
82C4×Dic9—32no—
83C4×Dic9—62no—
84C4×Dic9—61no—
85C22×C3×C9C2411yescharacteristic
86C2×C3⋊D9—22no—
87C2×C3⋊D9C2×D412yes—
88C2×C3×D9—24no—
89C2×C3×D9—28no—
90C3×C4×C9C2×D412yes—
91C3×C4×C9#4114yescomplemented
92C3⋊Dic9—22no—
93C3⋊Dic9C2×D412yes—
94C9×Dic3—24no—
95C2×#239—92no—
96C2×#237—92no—
97C3×C4⋊Q8—31no—
98C3×#127—91no—
99#947—92no—
100#938—94no—
101#936—92no—
102D4×Dic3—94no—
103#895—94no—
104#893—94no—
105#874—31no—
106C42×D3—91no—
107C42×D3—91no—
108C42×D3—181no—
109#869D911yescharacteristic, complemented
110C22×C3⋊D3—31no—
111C22×C3×D3—34no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D312yes—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—32no—
116C4×C3⋊D3—34no—
117C4×C3⋊D3—64no—
118C2×C3×Dic3C22×D314yes—
119C3×D12—64no—
120C3×D12—68no—
121C3×Dic6—24no—
122#626—68no—
123D3×Dic3—68no—
124C22×D9—31no—
125C22×D9—61no—
126C22×D9C22×D314yes—
127Q8×C9—64no—
128C2×C4×C9—22no—
129C2×C4×C9—21no—
130C2×C4×C9—34no—
131C2×C4×C9C22×D311yescharacteristic
132C2×C4×C9C22×D312yes—
133C2×Dic9—31no—
134C2×Dic9—32no—
135C2×Dic9—61no—
136C2×Dic9—62no—
137D36—24no—
138D36—28no—
139D36—68no—
140C4×D9—34no—
141C4×D9—64no—
142C4×D9—68no—
143C4×D9—68no—
144C4×D9—64no—
145#532—271noSylow
146C2×C3×C9C22×D411yescharacteristic
147C2×C3×C9C2×#4112yes—
148C3⋊D9—22no—
149C3×D9—44no—
150C2×C3×Q8—32no—
151C2×C3×D4—94no—
152C2×C3×D4—92no—
153C2×#67—94no—
154C22×Dic3—94no—
155#239—188no—
156#237—188no—
157C2×D12—34no—
158C2×D12—32no—
159C2×D12—94no—
160C2×C4×D3—91no—
161C2×C4×D3—91no—
162C2×C4×D3—94no—
163C2×C4×D3—92no—
164C2×C4×D3—92no—
165C2×C4×D3—181no—
166C2×C4×D3—182no—
167C2×Dic6C2×D912yes—
168C3×C4⋊C4—34no—
169C3×C42—21no—
170C3×C42C2×D911yescharacteristic
171C3×C42D3211yescharacteristic
172#217—98no—
173(C2×D3)⋊C4—98no—
174C4⋊Dic3C2×D914yes—
175C4×Dic3—92no—
176C4×Dic3—91no—
177C4×Dic3—92no—
178C4×Dic3—91no—
179C4×Dic3—182no—
180C4×Dic3—181no—
181C22×C32C23×D311yescharacteristic
182C2×C3⋊D3—32no—
183C2×C3⋊D3—62no—
184C2×C3×D3—64no—
185C2×C3×D3—68no—
186C32×C4D3×D412yes—
187C32×C4#23914yes—
188C3⋊Dic3—32no—
189C3⋊Dic3—62no—
190C3×Dic3—24no—
191C22×C9—21no—
192C22×C9C23×D311yescharacteristic
193C2×D9—24no—
194C2×D9—28no—
195C2×D9—32no—
196C2×D9—62no—
197C2×D9—64no—
198C2×D9—62no—
199C4×C9—24no—
200C4×C9—22no—
201C4×C9—64no—
202C4×C9D3×D412yes—
203C4×C9#23714yescomplemented
204Dic9—32no—
205Dic9—62no—
206Dic9—64no—
207Dic9—62no—
208C2×#41—272no—
209C4⋊Q8—31no—
210#127—91no—
211#124—272no—
212#121—274no—
213C4×D4—274no—
214C2×C42—271no—
215C3×C9#53211yescharacteristic, complemented, Sylow
216C23×C3—94no—
217C22×D3—34no—
218C22×D3—91no—
219C22×D3—91no—
220C22×D3—181no—
221C3×Q8—64no—
222C3×D4—184no—
223C3×D4—188no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—34no—
227C2×C3×C4C22×D911yescharacteristic
228C2×C3×C4C22×D912yes—
229C2×C3×C4C2×D3212yes—
230C2×C3×C4C2×D3211yescharacteristic
231#67—188no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—92no—
235C2×Dic3—92no—
236C2×Dic3—181no—
237C2×Dic3—182no—
238C2×Dic3—188no—
239C2×Dic3C22×D914yes—
240D12—68no—
241D12—64no—
242D12—188no—
243C4×D3—94no—
244C4×D3—94no—
245C4×D3—184no—
246C4×D3—188no—
247C4×D3—188no—
248C4×D3—184no—
249C4×D3—184no—
250Dic6—24no—
251C2×C32C2×D3×D411yescharacteristic
252C2×C32C2×#23912yes—
253C3⋊D3—62no—
254C3×D3—124no—
255C2×C9—22no—
256C2×C9—21no—
257C2×C9C2×D3×D411yescharacteristic
258C2×C9C2×#23712yes—
259D9—44no—
260D9—64no—
261D9—62no—
262#41—548no—
263C2×Q8—32no—
264C2×D4—94no—
265C2×D4—92no—
266C2×D4—274no—
267C22×C4—274no—
268C22×C4—271no—
269C22×C4—272no—
270C4⋊C4—34no—
271C22⋊C4—278no—
272C42—272no—
273C42—271no—
274C42D3×D911yescharacteristic
275C22×C3—21no—
276C22×C3—184no—
277C22×C3—188no—
278C22×C3C23×D911yescharacteristic
279C22×C3C22×D3211yescharacteristic
280C2×D3—68no—
281C2×D3—64no—
282C2×D3—92no—
283C2×D3—92no—
284C2×D3—182no—
285C2×D3—184no—
286C2×D3—182no—
287C2×D3—182no—
288C3×C4—24no—
289C3×C4—22no—
290C3×C4—64no—
291C3×C4D4×D912yes—
292C3×C4#370114yescomplemented
293C3×C4#379614yes—
294C3×C4#380212yes—
295Dic3—24no—
296Dic3—92no—
297Dic3—92no—
298Dic3—182no—
299Dic3—184no—
300Dic3—182no—
301Dic3—182no—
302C32#572511yescharacteristic
303C9—21no—
304C9#571611yescharacteristic, complemented
305C23—94no—
306C23—271no—
307Q8—64no—
308D4—188no—
309D4—184no—
310D4—548no—
311C2×C4—34no—
312C2×C4—271no—
313C2×C4—274no—
314C2×C4—272no—
315C2×C4—548no—
316C2×C4—544no—
317C2×C4C2×D3×D912yes—
318C2×C4C2×D3×D911yescharacteristic
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—364no—
322C2×C3C2×D4×D911yescharacteristic
323C2×C3C2×#370112yes—
324C2×C3C2×#379612yes—
325C2×C3C2×#380211yescharacteristic
326D3—124no—
327D3—184no—
328D3—182no—
329D3—182no—
330C22—188no—
331C22—184no—
332C22—272no—
333C22—542no—
334C22C22×D3×D911yescharacteristic
335C4—64no—
336C4—272no—
337C4—542no—
338C4#8863414yescomplemented
339C4#8864612yes—
340C3—21no—
341C3#1058982711yescharacteristic, complemented
342C3#1059390711yescharacteristic
343C2—364no—
344C2—542no—
345C2C2×#8863412yes—
346C2C2×#8864611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88634C2semidirect
2. C42×C3⋊D9C2semidirect
3. C3×#63436C2semidirect
4. C9×#869C2semidirect
5. #11721007C2semidirect
6. #11720998C2semidirect
7. #11720951C2semidirect
35. #63436D3semidirect
45. C4×C3⋊D9D4semidirect
91. C3×C4×C9#41semidirect
109. #869D9semidirect
203. C4×C9#237semidirect
215. C3×C9#532semidirect
292. C3×C4#3701semidirect
304. C9#5716semidirect
338. C4#88634semidirect
341. C3#10589827semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C42×C3⋊D92
C3×#634362
C9×#8692
#117210072
#117209982
#117209512
#105939073
#105898273