Finite Group Databasebeta

← #13147501

Subgroups — #13147501

Order 1728.

978 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 213 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×(C3⋊D9)⋊Q8C212yesmaximal
2C42×C3⋊D9C211yesmaximal, characteristic
3C9×#869C211yesmaximal, characteristic, complemented
4C3×#63431C211yesmaximal, characteristic, complemented
5#11721000C214yesmaximal
6#11720971C214yesmaximal, complemented
7#11720930C212yesmaximal, complemented
8#10593909—31nomaximal
9D9×C4⋊Q8—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C4×C3⋊D9C2212yes—
12C2×C9×Dic6C2212yes—
13C2×C3×Dic18C2212yes—
14C2×(C3⋊D9)⋊C4C2214yes—
15C2×#6189C2214yes—
16(C3⋊D9)⋊Q8—28no—
17C3×C42×C9C2211yescharacteristic
18C4×C3⋊Dic9C2212yes—
19C4×C3⋊Dic9C2211yescharacteristic
20C9×C4⋊Dic3C2214yes—
21C3×C4⋊Dic9C2214yes—
22#88442C2218yes—
23C2×#3797—32no—
24C42×C3⋊D3—31no—
25C3×#869—31no—
26C3×#869D311yescharacteristic
27#63919—34no—
28#63900—34no—
29#63873—32no—
30C2×Q8×D9—32no—
31C9×C4⋊Q8—31no—
32#63506—32no—
33C4⋊C4×D9—34no—
34#63450—34no—
35C42×D9—31no—
36C42×D9—61no—
37#63431D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3×C4×C9C2312yes—
41C2×C3⋊Dic9C2311yescharacteristic
42C2×C3⋊Dic9C2312yes—
43C4×C3⋊D9D414yes—
44C4×C3⋊D9Q818yescomplemented
45C2×C9×Dic3C2314yes—
46C2×C3×Dic9C2314yes—
47C9×Dic6—24no—
48C3×Dic18—24no—
49(C3⋊D9)⋊C4—28no—
50#6189—28no—
51D3×C4⋊Q8—91no—
52D3×C4⋊Q8—91no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—32no—
55C2×C3×Dic6—32no—
56C2×C3×Dic6C2×D312yes—
57C2×#627—34no—
58C2×#624—34no—
59#3797—68no—
60C32×C42C2×D311yescharacteristic
61C4×C3⋊Dic3—31no—
62C4×C3⋊Dic3—32no—
63C3×C4⋊Dic3—34no—
64C3×C4⋊Dic3C2×D314yes—
65#3724—38no—
66C2×Q8×C9—32no—
67Q8×D9—68no—
68C2×C4×D9—32no—
69C2×C4×D9—31no—
70C2×C4×D9—34no—
71C2×C4×D9—61no—
72C2×C4×D9—62no—
73C2×Dic18—34no—
74C2×Dic18C2×D312yes—
75C9×C4⋊C4—34no—
76C42×C9—21no—
77C42×C9C2×D311yescharacteristic
78C4⋊Dic9C2×D314yes—
79Dic9⋊C4—38no—
80C4×Dic9—31no—
81C4×Dic9—32no—
82C4×Dic9—62no—
83C4×Dic9—61no—
84C22×C3×C9C2411yescharacteristic
85C2×C3⋊D9C2×D412yes—
86C2×C3⋊D9C2×Q814yes—
87C3×C4×C9C2×D412yes—
88C3×C4×C9C2×Q814yescomplemented
89C3⋊Dic9C2×D412yes—
90C3⋊Dic9C2×Q814yescomplemented
91C9×Dic3—24no—
92C3×Dic9—24no—
93C2×D3×Q8—92no—
94C2×D3×Q8—92no—
95C3×C4⋊Q8—31no—
96C3×C4⋊Q8—91no—
97#944—92no—
98#944—92no—
99D3×C4⋊C4—94no—
100D3×C4⋊C4—94no—
101#888—94no—
102#888—94no—
103C42×D3—91no—
104C42×D3—91no—
105C42×D3—181no—
106#869—31no—
107#869D911yescharacteristic, complemented
108C22×C3⋊D3—31no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D312yes—
111C2×C3⋊Dic3—32no—
112C2×C3⋊Dic3—31no—
113C4×C3⋊D3—34no—
114C4×C3⋊D3—38no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3C22×D314yes—
117C3×Dic6—24no—
118C3×Dic6—64no—
119#627—68no—
120#624—68no—
121C22×D9—31no—
122C22×D9—61no—
123Q8×C9—64no—
124C2×C4×C9—22no—
125C2×C4×C9—21no—
126C2×C4×C9—34no—
127C2×C4×C9C22×D311yescharacteristic
128C2×C4×C9C22×D312yes—
129C2×Dic9—32no—
130C2×Dic9—31no—
131C2×Dic9—62no—
132C2×Dic9—61no—
133C2×Dic9C22×D314yes—
134C4×D9—34no—
135C4×D9—38no—
136C4×D9—64no—
137C4×D9—68no—
138C4×D9—68no—
139Dic18—24no—
140Dic18—68no—
141C2×C4⋊Q8—271noSylow
142C2×C3×C9C22×D411yescharacteristic
143C2×C3×C9C22×Q812yes—
144C3⋊D9C4⋊Q814yescomplemented
145C2×C3×Q8—32no—
146C2×C3×Q8—92no—
147D3×Q8—188no—
148D3×Q8—188no—
149C2×C4×D3—92no—
150C2×C4×D3—91no—
151C2×C4×D3—92no—
152C2×C4×D3—91no—
153C2×C4×D3—94no—
154C2×C4×D3—94no—
155C2×C4×D3—182no—
156C2×C4×D3—181no—
157C2×Dic6—32no—
158C2×Dic6—94no—
159C2×Dic6—94no—
160C2×Dic6C2×D912yes—
161C3×C4⋊C4—34no—
162C3×C4⋊C4—94no—
163C3×C42—21no—
164C3×C42C2×D911yescharacteristic
165C3×C42D3211yescharacteristic
166C4⋊Dic3—34no—
167C4⋊Dic3C2×D914yes—
168Dic3⋊C4—98no—
169Dic3⋊C4—98no—
170C4×Dic3—92no—
171C4×Dic3—91no—
172C4×Dic3—92no—
173C4×Dic3—91no—
174C4×Dic3—181no—
175C4×Dic3—182no—
176C22×C32C23×D311yescharacteristic
177C2×C3⋊D3—32no—
178C2×C3⋊D3—34no—
179C32×C4D3×D412yes—
180C32×C4D3×Q814yes—
181C3⋊Dic3—32no—
182C3⋊Dic3—34no—
183C3×Dic3—24no—
184C3×Dic3—64no—
185C22×C9—21no—
186C22×C9C23×D311yescharacteristic
187C2×D9—32no—
188C2×D9—34no—
189C2×D9—62no—
190C2×D9—64no—
191C4×C9—24no—
192C4×C9—22no—
193C4×C9—64no—
194C4×C9D3×D412yes—
195C4×C9D3×Q814yescomplemented
196Dic9—24no—
197Dic9—32no—
198Dic9—34no—
199Dic9—62no—
200Dic9—64no—
201C22×Q8—272no—
202C4⋊Q8—31no—
203C4⋊Q8—91no—
204C4⋊Q8—274no—
205C4⋊Q8—272no—
206C2×C4⋊C4—274no—
207C2×C42—271no—
208C3×C9C2×C4⋊Q811yescharacteristic, complemented, Sylow
209C22×D3—91no—
210C22×D3—91no—
211C22×D3—181no—
212C3×Q8—64no—
213C3×Q8—184no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—34no—
217C2×C3×C4—94no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C22×D912yes—
220C2×C3×C4C2×D3212yes—
221C2×C3×C4C2×D3211yescharacteristic
222C2×Dic3—34no—
223C2×Dic3—92no—
224C2×Dic3—91no—
225C2×Dic3—92no—
226C2×Dic3—91no—
227C2×Dic3—182no—
228C2×Dic3—181no—
229C2×Dic3C22×D914yes—
230C4×D3—94no—
231C4×D3—98no—
232C4×D3—98no—
233C4×D3—94no—
234C4×D3—188no—
235C4×D3—184no—
236C4×D3—188no—
237C4×D3—188no—
238Dic6—24no—
239Dic6—64no—
240Dic6—188no—
241Dic6—188no—
242C2×C32C2×D3×D411yescharacteristic
243C2×C32C2×D3×Q812yes—
244C3⋊D3—34no—
245C2×C9—22no—
246C2×C9—21no—
247C2×C9C2×D3×D411yescharacteristic
248C2×C9C2×D3×Q812yes—
249D9—34no—
250D9—64no—
251C2×Q8—32no—
252C2×Q8—92no—
253C2×Q8—274no—
254C2×Q8—548no—
255C22×C4—274no—
256C22×C4—272no—
257C22×C4—271no—
258C4⋊C4—34no—
259C4⋊C4—94no—
260C4⋊C4—278no—
261C42—271no—
262C42—272no—
263C42D3×D911yescharacteristic
264C22×C3—21no—
265C22×C3C23×D911yescharacteristic
266C22×C3C22×D3211yescharacteristic
267C2×D3—94no—
268C2×D3—92no—
269C2×D3—92no—
270C2×D3—94no—
271C2×D3—184no—
272C2×D3—182no—
273C3×C4—24no—
274C3×C4—22no—
275C3×C4—64no—
276C3×C4—184no—
277C3×C4D4×D912yes—
278C3×C4Q8×D914yescomplemented
279C3×C4#379714yes—
280C3×C4#380212yes—
281Dic3—24no—
282Dic3—64no—
283Dic3—94no—
284Dic3—92no—
285Dic3—92no—
286Dic3—94no—
287Dic3—184no—
288Dic3—182no—
289C32D3×C4⋊Q811yescharacteristic
290C9—21no—
291C9D3×C4⋊Q811yescharacteristic, complemented
292C23—271no—
293Q8—64no—
294Q8—184no—
295Q8—548no—
296C2×C4—34no—
297C2×C4—94no—
298C2×C4—272no—
299C2×C4—271no—
300C2×C4—278no—
301C2×C4—274no—
302C2×C4—548no—
303C2×C4C2×D3×D912yes—
304C2×C4C2×D3×D911yescharacteristic
305C2×C3—22no—
306C2×C3—21no—
307C2×C3C2×D4×D911yescharacteristic
308C2×C3C2×Q8×D912yes—
309C2×C3C2×#379712yes—
310C2×C3C2×#380211yescharacteristic
311D3—94no—
312D3—94no—
313D3—184no—
314C22—274no—
315C22—272no—
316C22C22×D3×D911yescharacteristic
317C4—64no—
318C4—184no—
319C4—274no—
320C4—272no—
321C4(C3⋊D9)⋊Q814yescomplemented
322C4#8864612yes—
323C3—21no—
324C3D9×C4⋊Q811yescharacteristic, complemented
325C3#1059390911yescharacteristic
326C2—274no—
327C2C2×(C3⋊D9)⋊Q812yes—
328C2C2×#8864611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC42D3×D9
hypocenterC3×C9C2×C4⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. C9×#869C2semidirect
4. C3×#63431C2semidirect
6. #11720971C2semidirect
7. #11720930C2semidirect
37. #63431D3semidirect
44. C4×C3⋊D9Q8semidirect
88. C3×C4×C9C2×Q8semidirect
90. C3⋊Dic9C2×Q8semidirect
107. #869D9semidirect
144. C3⋊D9C4⋊Q8semidirect
195. C4×C9D3×Q8semidirect
208. C3×C9C2×C4⋊Q8semidirect
278. C3×C4Q8×D9semidirect
291. C9D3×C4⋊Q8semidirect
321. C4(C3⋊D9)⋊Q8semidirect
324. C3D9×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C42×C3⋊D92
C9×#8692
C3×#634312
#117210002
#117209712
#117209302
#105939093
D9×C4⋊Q83