Finite Group Databasebeta

← #13139798

Subgroups — #13139798

Order 1728.

800 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C2×#88426C212yesmaximal, complemented
4C3×#63389C211yesmaximal, characteristic, complemented
5#11720267C212yesmaximal, complemented
6#10591183—31nomaximal
7#10588729—31nomaximal
8C23×C3⋊D9—21no—
9C22×C3×C4×C9C2211yescharacteristic
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)—24no—
12C2×C4⋊(C3⋊D9)C2212yes—
13C2×C9×Dic6C2211yescharacteristic, complemented
14C9×#217—21no—
15C9×C4⋊Dic3C2211yescharacteristic, complemented
16C9×Dic3⋊C4—21no—
17C2×C3×C9⋊C8C2212yescomplemented
18#88426—24no—
19C22×C4⋊(C3⋊D3)—31no—
20C3×#924D311yescharacteristic
21C2×#3717—32no—
22C22×D36—31no—
23C22×D36—61no—
24C3×#830—31no—
25#63564—32no—
26C9×C22⋊Q8—31no—
27C2×#3675—32no—
28#63389D311yescharacteristic, complemented
29#63369—32no—
30C23×C3×C9D411yescharacteristic
31C22×C3⋊D9—24no—
32C22×C3⋊D9—22no—
33C22×C3⋊D9—42no—
34C22×C3⋊D9—42no—
35C2×C3×C4×C9—22no—
36C2×C3×C4×C9D411yescharacteristic
37C2×C3×C4×C9C2311yescharacteristic
38C4⋊(C3⋊D9)—24no—
39C4⋊(C3⋊D9)—22no—
40C4⋊(C3⋊D9)D414yes—
41C2×C9×Dic3—21no—
42C2×C9×Dic3—21no—
43C9×Dic6—22no—
44C3×C9⋊C8—22no—
45#5046—91no—
46#4731—91no—
47C23×C3⋊D3—61no—
48C22×C32×C4C2×D311yescharacteristic
49C2×C4⋊(C3⋊D3)—32no—
50C2×C4⋊(C3⋊D3)—64no—
51C2×C4⋊(C3⋊D3)—61no—
52C2×C3×Dic6C2×D311yescharacteristic
53C23×D9—62no—
54C23×D9—61no—
55C3×#217—21no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×Dic3⋊C4—21no—
58C2×C3×C3⋊C8—32no—
59#3717—64no—
60C2×Q8×C9—31no—
61C22×C4×C9—21no—
62C22×C4×C9C2×D311yescharacteristic
63C2×D36—32no—
64C2×D36—64no—
65C2×D36—62no—
66C2×D36—68no—
67C2×D36—62no—
68C2×D36—61no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4—61no—
71C9×C22⋊C4—61no—
72#3675—64no—
73C2×C9⋊C8C2×D312yescomplemented
74C22×C3×C9—21no—
75C22×C3×C9—21no—
76C22×C3×C9SD1612yescomplemented
77C22×C3×C9C2×D411yescharacteristic
78C2×C3⋊D9—24no—
79C2×C3⋊D9—48no—
80C2×C3⋊D9—42no—
81C2×C3⋊D9—42no—
82C3×C4×C9—21no—
83C3×C4×C9C2×D412yes—
84C9×Dic3—41no—
85C9×Dic3—41no—
86C22×D12—91no—
87C22×D12—91no—
88C22×D12—181no—
89C3×C22⋊Q8—31no—
90C2×#215—92no—
91#924D911yescharacteristic, complemented
92C2×#204—92no—
93C3×C22⋊C8—91no—
94#830—31no—
95#821—92no—
96#809—92no—
97C23×C32#6711yescharacteristic
98C22×C3⋊D3—64no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—122no—
101C22×C3⋊D3—122no—
102C2×C32×C4—22no—
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C4⋊(C3⋊D3)—34no—
106C4⋊(C3⋊D3)—64no—
107C4⋊(C3⋊D3)—62no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C3×Dic6—22no—
111C23×C9—21no—
112C23×C9D1211yescharacteristic
113C22×D9—68no—
114C22×D9—64no—
115C22×D9—64no—
116C22×D9—62no—
117C22×D9—124no—
118C22×D9—122no—
119C22×D9—124no—
120C22×D9—122no—
121C3×C3⋊C8—62no—
122Q8×C9—62no—
123C2×C4×C9—24no—
124C2×C4×C9—22no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—61no—
128C2×C4×C9—61no—
129C2×C4×C9D1211yescharacteristic
130C2×C4×C9C22×D311yescharacteristic
131D36—34no—
132D36—64no—
133D36—68no—
134D36—64no—
135D36—64no—
136D36—62no—
137C9⋊C8—22no—
138#450—271noSylow
139C2×C3×C9—22no—
140C2×C3×C9#12011yescharacteristic
141C2×C3×C9C2×SD1611yescharacteristic
142C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
143C3⋊D9—44no—
144C23×D3—182no—
145C23×D3—181no—
146C23×D3—181no—
147C2×C3×Q8—31no—
148C22×C3×C4—21no—
149C22×C3×C4C2×D911yescharacteristic
150C22×C3×C4D3211yescharacteristic
151C2×D12—92no—
152C2×D12—92no—
153C2×D12—182no—
154C2×D12—184no—
155C2×D12—184no—
156C2×D12—188no—
157C2×D12—182no—
158C2×D12—181no—
159C2×D12—181no—
160C2×Dic6C2×D911yescharacteristic, complemented
161C2×C3×C8—92no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C22⋊C4—61no—
165#217—21no—
166#215—184no—
167C4⋊Dic3C2×D911yescharacteristic, complemented
168Dic3⋊C4—21no—
169C2×C3⋊C8—32no—
170#204—184no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32#21512yes—
174C22×C32C2×#6711yescharacteristic
175C2×C3⋊D3—64no—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—122no—
178C2×C3⋊D3—122no—
179C32×C4—21no—
180C32×C4D3×D412yes—
181C3×Dic3—41no—
182C3×Dic3—41no—
183C22×C9—22no—
184C22×C9—22no—
185C22×C9—21no—
186C22×C9—21no—
187C22×C9—22no—
188C22×C9—21no—
189C22×C9#20412yescomplemented
190C22×C9C2×D1211yescharacteristic
191C2×D9—68no—
192C2×D9—64no—
193C2×D9—1216no—
194C2×D9—128no—
195C2×D9—124no—
196C2×D9—124no—
197C2×D9—122no—
198C2×D9—122no—
199C4×C9—22no—
200C4×C9—21no—
201C4×C9—22no—
202C4×C9—121no—
203C4×C9—121no—
204C4×C9D3×D412yes—
205C22×D4—271no—
206C2×SD16—272no—
207C22⋊Q8—31no—
208#102—272no—
209C22⋊C8—91no—
210C3×C9#45011yescharacteristic, complemented, Sylow
211C23×C3—21no—
212C23×C3#61111yescharacteristic
213C23×C3#62611yescharacteristic
214C22×D3—184no—
215C22×D3—188no—
216C22×D3—184no—
217C22×D3—184no—
218C22×D3—182no—
219C22×D3—182no—
220C22×D3—362no—
221C22×D3—364no—
222C22×D3—362no—
223C22×D3—364no—
224C22×D3—362no—
225C22×D3—362no—
226C3×Q8—62no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4#61111yescharacteristic
235C2×C3×C4C22×D911yescharacteristic
236C2×C3×C4#62611yescharacteristic
237C2×C3×C4C2×D3211yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—21no—
240D12—94no—
241D12—94no—
242D12—184no—
243D12—184no—
244D12—188no—
245D12—184no—
246D12—184no—
247D12—182no—
248D12—182no—
249Dic6—22no—
250C3×C8—182no—
251C3⋊C8—62no—
252C2×C32—22no—
253C2×C32#93711yescharacteristic
254C2×C32C2×#21511yescharacteristic
255C2×C32#94911yescharacteristic
256C3⋊D3—124no—
257C2×C9—24no—
258C2×C9—22no—
259C2×C9—21no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9#88211yescharacteristic
263C2×C9C2×#20411yescharacteristic
264C2×C9#90811yescharacteristic
265D9—128no—
266D9—124no—
267C24—541no—
268C2×Q8—31no—
269C2×D4—272no—
270C2×D4—544no—
271C2×D4—541no—
272C22×C4D3×D911yescharacteristic
273SD16—544no—
274C2×C8—92no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C22⋊C4—61no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3#367512yescomplemented
287C22×C3C2×#61111yescharacteristic
288C22×C3#371712yes—
289C22×C3C2×#62611yescharacteristic
290C2×D3—188no—
291C2×D3—184no—
292C2×D3—184no—
293C2×D3—368no—
294C2×D3—3616no—
295C2×D3—368no—
296C2×D3—364no—
297C2×D3—364no—
298C2×D3—362no—
299C2×D3—362no—
300C2×D3—362no—
301C2×D3—362no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—121no—
307C3×C4—121no—
308C3×C4D4×D912yes—
309C3×C4#381112yes—
310Dic3—41no—
311Dic3—41no—
312C32#504611yescharacteristic
313C9—21no—
314C9#473111yescharacteristic, complemented
315C23—544no—
316C23—542no—
317C23—1082no—
318C23—1082no—
319C23#619511yescharacteristic
320Q8—62no—
321D4—274no—
322D4—544no—
323D4—542no—
324C2×C4—22no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4#619511yescharacteristic
328C2×C4C2×D3×D911yescharacteristic
329C8—182no—
330C2×C3—24no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#6349911yescharacteristic
337C2×C3C2×#367511yescharacteristic
338C2×C3#6351211yescharacteristic
339C2×C3#6382511yescharacteristic
340C2×C3C2×#371711yescharacteristic
341C2×C3#6398111yescharacteristic
342D3—368no—
343D3—364no—
344D3—364no—
345C22—21no—
346C22—21no—
347C22—544no—
348C22—1088no—
349C22—1082no—
350C22—1082no—
351C22#8842612yescomplemented
352C22C2×#619511yescharacteristic
353C4—21no—
354C4—121no—
355C4—121no—
356C4#8866812yes—
357C3—21no—
358C3#1058872911yescharacteristic, complemented
359C3#1059118311yescharacteristic
360C2—22no—
361C2—1084no—
362C2#1172088511yescharacteristic
363C2C2×#8842611yescharacteristic
364C2#1172129111yescharacteristic
subgroupG/H
centerC22C2×#6195
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
prime radicalC22×C4⋊(C3⋊D9)C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#450
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C2×#88426C2semidirect
4. C3×#63389C2semidirect
5. #11720267C2semidirect
13. C2×C9×Dic6C22semidirect
15. C9×C4⋊Dic3C22semidirect
17. C2×C3×C9⋊C8C22semidirect
28. #63389D3semidirect
73. C2×C9⋊C8C2×D3semidirect
76. C22×C3×C9SD16semidirect
91. #924D9semidirect
160. C2×Dic6C2×D9semidirect
167. C4⋊Dic3C2×D9semidirect
189. C22×C9#204semidirect
210. C3×C9#450semidirect
286. C22×C3#3675semidirect
314. C9#4731semidirect
351. C22#88426semidirect
358. C3#10588729semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊D9)2
C9×#9242
C2×#884262
C3×#633892
#117202672
#105911833
#105887293