Finite Group Databasebeta

← #13147605

Subgroups — #13147605

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×#88639C212yesmaximal, complemented
2C42×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#874C211yesmaximal, characteristic, complemented
4C3×#63431C211yesmaximal, characteristic, complemented
5#11721045C214yesmaximal, complemented
6#11721025C214yesmaximal, complemented
7#11720967C212yesmaximal, complemented
8#10593907—31nomaximal
9#10589818—31nomaximal
10C2×C4×C3⋊D9C2212yes—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2212yes—
13C2×C9×D12C2214yes—
14C2×C3×Dic18C2212yes—
15C2×#6195C2214yes—
16C2×D3×Dic9C2214yes—
17#88639—28no—
18C3×C42×C9C2211yescharacteristic
19C4×C3⋊Dic9C2211yescharacteristic
20C4×C3⋊Dic9C2212yes—
21C3×C4⋊Dic9C2214yes—
22(C2×D3)⋊Dic9C2218yes—
23C2×#3796—32no—
24C42×C3⋊D3—31no—
25C3×#874D311yescharacteristic
26C3×#869—31no—
27#63916—34no—
28#63898—34no—
29#63863—32no—
30C2×#3699—32no—
31C9×#127—31no—
32#63500—34no—
33#63498—32no—
34D4×Dic9—34no—
35C42×D9—31no—
36C42×D9—61no—
37#63431D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C22×D3×C9C2314yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3×C4×C9C2312yes—
42C2×C3⋊Dic9C2312yes—
43C2×C3⋊Dic9C2311yescharacteristic
44C4×C3⋊D9—24no—
45C4×C3⋊D9D414yescomplemented
46C2×C3×Dic9C2314yes—
47C9×D12—24no—
48C9×D12—28no—
49C3×Dic18—24no—
50#6195—28no—
51D3×Dic9—28no—
52#5725—91no—
53#5716—91no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—32no—
56C2×C3×D12C2×D312yes—
57C2×C3×D12C2×D314yes—
58C2×C3×Dic6—32no—
59C2×#626—34no—
60C2×D3×Dic3—34no—
61#3796—68no—
62C32×C42C2×D311yescharacteristic
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—32no—
65C3×C4⋊Dic3—34no—
66(C2×D3)⋊Dic3—38no—
67C2×D4×C9—34no—
68C2×D4×C9—32no—
69C2×#611—34no—
70C22×Dic9—34no—
71#3699—68no—
72C2×C4×D9—31no—
73C2×C4×D9—32no—
74C2×C4×D9—61no—
75C2×C4×D9—62no—
76C2×Dic18C2×D312yes—
77C42×C9—21no—
78C42×C9C2×D311yescharacteristic
79#3676—38no—
80C4⋊Dic9C2×D314yes—
81C4×Dic9—32no—
82C4×Dic9—31no—
83C4×Dic9—62no—
84C4×Dic9—61no—
85C22×C3×C9C2411yescharacteristic
86C2×C3⋊D9—22no—
87C2×C3⋊D9C2×D412yes—
88C2×D3×C9—24no—
89C2×D3×C9—28no—
90C3×C4×C9C2×D412yes—
91C3×C4×C9#4114yescomplemented
92C3⋊Dic9—22no—
93C3⋊Dic9C2×D412yes—
94C3×Dic9—24no—
95C2×#239—92no—
96C2×#237—92no—
97C3×C4⋊Q8—91no—
98C3×#127—31no—
99#947—92no—
100#938—94no—
101#936—92no—
102D4×Dic3—94no—
103#895—94no—
104#893—94no—
105#874D911yescharacteristic, complemented
106C42×D3—91no—
107C42×D3—91no—
108C42×D3—181no—
109#869—31no—
110C22×C3⋊D3—31no—
111C22×C3×D3C22×D314yes—
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×Dic3—34no—
119C3×D12—24no—
120C3×D12—28no—
121C3×Dic6—64no—
122#626—68no—
123D3×Dic3—68no—
124C23×C9—34no—
125C22×D9—31no—
126C22×D9—61no—
127D4×C9—64no—
128D4×C9—68no—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9C22×D311yescharacteristic
132C2×C4×C9C22×D312yes—
133#611—68no—
134C2×Dic9—31no—
135C2×Dic9—32no—
136C2×Dic9—61no—
137C2×Dic9—62no—
138C2×Dic9—68no—
139C2×Dic9C22×D314yes—
140C4×D9—34no—
141C4×D9—64no—
142C4×D9—68no—
143C4×D9—64no—
144Dic18—24no—
145#532—271noSylow
146C2×C3×C9C22×D411yescharacteristic
147C2×C3×C9C2×#4112yes—
148C3⋊D9—22no—
149D3×C9—44no—
150C2×C3×Q8—92no—
151C2×C3×D4—34no—
152C2×C3×D4—32no—
153C2×#67—94no—
154C22×Dic3—94no—
155#239—188no—
156#237—188no—
157C2×D12—94no—
158C2×D12C2×D914yes—
159C2×D12C2×D912yes—
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×Dic6—32no—
168C3×C4⋊C4—94no—
169C3×C42—21no—
170C3×C42C2×D911yescharacteristic
171C3×C42D3211yescharacteristic
172#217—98no—
173(C2×D3)⋊C4—98no—
174C4⋊Dic3—34no—
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—24no—
185C2×C3×D3—28no—
186C32×C4D3×D412yes—
187C32×C4#23714yes—
188C3⋊Dic3—32no—
189C3⋊Dic3—62no—
190C3×Dic3—64no—
191C22×C9—21no—
192C22×C9—64no—
193C22×C9—68no—
194C22×C9C23×D311yescharacteristic
195C2×D9—32no—
196C2×D9—62no—
197C2×D9—64no—
198C2×D9—62no—
199C4×C9—24no—
200C4×C9—22no—
201C4×C9D3×D412yes—
202C4×C9#23914yescomplemented
203Dic9—24no—
204Dic9—32no—
205Dic9—62no—
206Dic9—64no—
207Dic9—62no—
208C2×#41—272no—
209C4⋊Q8—91no—
210#127—31no—
211#124—272no—
212#121—274no—
213C4×D4—274no—
214C2×C42—271no—
215C3×C9#53211yescharacteristic, complemented, Sylow
216C23×C3—34no—
217C22×D3—91no—
218C22×D3—91no—
219C22×D3—181no—
220C22×D3C22×D914yes—
221C3×Q8—184no—
222C3×D4—68no—
223C3×D4—64no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—94no—
227C2×C3×C4C22×D911yescharacteristic
228C2×C3×C4C22×D912yes—
229C2×C3×C4C2×D3212yes—
230C2×C3×C4C2×D3211yescharacteristic
231#67—188no—
232C2×Dic3—34no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—92no—
236C2×Dic3—92no—
237C2×Dic3—181no—
238C2×Dic3—182no—
239C2×Dic3—188no—
240D12—24no—
241D12—28no—
242D12—188no—
243C4×D3—94no—
244C4×D3—94no—
245C4×D3—184no—
246C4×D3—188no—
247C4×D3—188no—
248C4×D3—184no—
249C4×D3—184no—
250Dic6—64no—
251C2×C32C2×D3×D411yescharacteristic
252C2×C32C2×#23712yes—
253C3⋊D3—62no—
254C3×D3—44no—
255C2×C9—22no—
256C2×C9—21no—
257C2×C9—124no—
258C2×C9C2×D3×D411yescharacteristic
259C2×C9C2×#23912yes—
260D9—64no—
261D9—62no—
262#41—548no—
263C2×Q8—92no—
264C2×D4—34no—
265C2×D4—32no—
266C2×D4—274no—
267C22×C4—274no—
268C22×C4—271no—
269C22×C4—272no—
270C4⋊C4—94no—
271C22⋊C4—278no—
272C42—272no—
273C42—271no—
274C42D3×D911yescharacteristic
275C22×C3—21no—
276C22×C3—68no—
277C22×C3—64no—
278C22×C3C23×D911yescharacteristic
279C22×C3C22×D3211yescharacteristic
280C2×D3—24no—
281C2×D3—28no—
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—184no—
291C3×C4D4×D912yes—
292C3×C4#369914yescomplemented
293C3×C4#379614yes—
294C3×C4#380212yes—
295Dic3—64no—
296Dic3—92no—
297Dic3—92no—
298Dic3—182no—
299Dic3—184no—
300Dic3—182no—
301Dic3—182no—
302C32#571611yescharacteristic
303C9—21no—
304C9#572511yescharacteristic, complemented
305C23—34no—
306C23—271no—
307Q8—184no—
308D4—68no—
309D4—64no—
310D4—548no—
311C2×C4—94no—
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—124no—
322C2×C3C2×D4×D911yescharacteristic
323C2×C3C2×#369912yes—
324C2×C3C2×#379612yes—
325C2×C3C2×#380211yescharacteristic
326D3—44no—
327D3—184no—
328D3—182no—
329D3—182no—
330C22—68no—
331C22—64no—
332C22—272no—
333C22—542no—
334C22C22×D3×D911yescharacteristic
335C4—184no—
336C4—272no—
337C4—542no—
338C4#8863914yescomplemented
339C4#8864612yes—
340C3—21no—
341C3#1058981811yescharacteristic, complemented
342C3#1059390711yescharacteristic
343C2—124no—
344C2—542no—
345C2C2×#8863912yes—
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×#88639C2semidirect
2. C42×C3⋊D9C2semidirect
3. C9×#874C2semidirect
4. C3×#63431C2semidirect
5. #11721045C2semidirect
6. #11721025C2semidirect
7. #11720967C2semidirect
37. #63431D3semidirect
45. C4×C3⋊D9D4semidirect
91. C3×C4×C9#41semidirect
105. #874D9semidirect
202. C4×C9#239semidirect
215. C3×C9#532semidirect
292. C3×C4#3699semidirect
304. C9#5725semidirect
338. C4#88639semidirect
341. C3#10589818semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
C42×C3⋊D92
C9×#8742
C3×#634312
#117210452
#117210252
#117209672
#105939073
#105898183