Finite Group Databasebeta

← #13139514

Subgroups — #13139514

Order 1728.

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

normal subgroups only

This group has 830 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, complemented
2C3×#63492C211yesmaximal, characteristic, complemented
3C2×#88417C212yesmaximal, complemented
4C9×#830C211yesmaximal, characteristic, complemented
5#11720284C212yesmaximal, complemented
6#10591168—31nomaximal
7#10588414—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×C3×#611—21no—
14C2×C3×D36C2211yescharacteristic, complemented
15C3×(C2×D9)⋊C4—21no—
16C3×C4⋊Dic9C2211yescharacteristic, complemented
17C2×C9×C3⋊C8C2212yescomplemented
18#88417—24no—
19C22×C4⋊(C3⋊D3)—31no—
20C3×#930—31no—
21C2×#3714—32no—
22C22×D36—31no—
23C22×D36—61no—
24C3×#830D311yescharacteristic
25#63564—32no—
26#63492D311yescharacteristic, complemented
27C2×D72—32no—
28C9×C22⋊C8—31no—
29#63380—32no—
30C23×C3×C9D411yescharacteristic
31C22×C3⋊D9—24no—
32C22×C3⋊D9—22no—
33C22×C3⋊D9—42no—
34C22×C3⋊D9—42no—
35C22×C3×D9—21no—
36C2×C3×C4×C9—22no—
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C4⋊(C3⋊D9)—24no—
40C4⋊(C3⋊D9)—22no—
41C4⋊(C3⋊D9)D414yescomplemented
42C3×#611—42no—
43C2×C3×Dic9—21no—
44C3×D36—22no—
45C9×C3⋊C8—22no—
46#5036—91no—
47#4732—91no—
48C23×C3⋊D3—61no—
49C22×C32×C4C2×D311yescharacteristic
50C2×C4⋊(C3⋊D3)—32no—
51C2×C4⋊(C3⋊D3)—64no—
52C2×C4⋊(C3⋊D3)—61no—
53C2×C3×#67—61no—
54C2×C3×D12—31no—
55C23×D9—62no—
56C23×D9—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C2×C3×C3⋊C8C2×D312yes—
60#3714—64no—
61C22×C4×C9—21no—
62C22×C4×C9C2×D311yescharacteristic
63C2×#611—21no—
64C2×D36—32no—
65C2×D36—64no—
66C2×D36—62no—
67C2×D36—68no—
68C2×D36—62no—
69C2×D36—61no—
70C2×D36C2×D311yescharacteristic, complemented
71C2×C8×C9—32no—
72(C2×D9)⋊C4—21no—
73C4⋊Dic9C2×D311yescharacteristic, complemented
74D72—64no—
75C22×C3×C9—21no—
76C22×C3×C9—21no—
77C22×C3×C9D812yescomplemented
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—24no—
80C2×C3⋊D9—48no—
81C2×C3⋊D9—42no—
82C2×C3⋊D9—42no—
83C2×C3×D9—41no—
84C2×C3×D9—41no—
85C2×C3×D9—41no—
86C3×C4×C9—21no—
87C3×C4×C9C2×D412yes—
88C3×Dic9—41no—
89C22×D12—91no—
90C22×D12—91no—
91C22×D12—181no—
92C3×#121—91no—
93C2×#213—92no—
94#930—31no—
95C2×D24—92no—
96C3×C22⋊C8—31no—
97#830D911yescharacteristic, complemented
98#821—92no—
99#809—92no—
100C23×C32D1211yescharacteristic
101C22×C3⋊D3—64no—
102C22×C3⋊D3—62no—
103C22×C3⋊D3—122no—
104C22×C3⋊D3—122no—
105C22×C3×D3—61no—
106C2×C32×C4—22no—
107C2×C32×C4D1211yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C4⋊(C3⋊D3)—34no—
110C4⋊(C3⋊D3)—64no—
111C4⋊(C3⋊D3)—62no—
112C3×#67—122no—
113C2×C3×Dic3—61no—
114C3×D12—62no—
115C23×C9—21no—
116C23×C9#6711yescharacteristic
117C22×D9—21no—
118C22×D9—68no—
119C22×D9—64no—
120C22×D9—64no—
121C22×D9—62no—
122C22×D9—124no—
123C22×D9—122no—
124C22×D9—124no—
125C22×D9—122no—
126C3×C3⋊C8—22no—
127C2×C4×C9—24no—
128C2×C4×C9—22no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9#6711yescharacteristic
132C2×C4×C9C22×D311yescharacteristic
133#611—42no—
134C2×Dic9—21no—
135D36—22no—
136D36—34no—
137D36—64no—
138D36—68no—
139D36—64no—
140D36—64no—
141D36—62no—
142C8×C9—62no—
143#447—271noSylow
144C2×C3×C9—22no—
145C2×C3×C9#12011yescharacteristic
146C2×C3×C9C2×D811yescharacteristic
147C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
148C3⋊D9—44no—
149C3×D9—81no—
150C23×D3—182no—
151C23×D3—181no—
152C23×D3—181no—
153C2×C3×D4—91no—
154C2×C3×D4—181no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D911yescharacteristic
157C22×C3×C4D3211yescharacteristic
158C2×#67—61no—
159C2×D12—31no—
160C2×D12—92no—
161C2×D12—92no—
162C2×D12—182no—
163C2×D12—184no—
164C2×D12—184no—
165C2×D12—188no—
166C2×D12—182no—
167C2×D12—181no—
168C2×D12—181no—
169C2×C3×C8—32no—
170C3×C4⋊C4—91no—
171C3×C22⋊C4—181no—
172#213—184no—
173(C2×D3)⋊C4—61no—
174C4⋊Dic3—31no—
175C2×C3⋊C8C2×D912yescomplemented
176D24—184no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32D2412yes—
180C22×C32C2×D1211yescharacteristic
181C2×C3⋊D3—64no—
182C2×C3⋊D3—128no—
183C2×C3⋊D3—122no—
184C2×C3⋊D3—122no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C32×C4—21no—
189C32×C4D3×D412yes—
190C3×Dic3—121no—
191C22×C9—22no—
192C22×C9—22no—
193C22×C9—21no—
194C22×C9—21no—
195C22×C9—22no—
196C22×C9—21no—
197C22×C9#21312yescomplemented
198C22×C9C2×#6711yescharacteristic
199C2×D9—41no—
200C2×D9—41no—
201C2×D9—41no—
202C2×D9—68no—
203C2×D9—64no—
204C2×D9—1216no—
205C2×D9—128no—
206C2×D9—124no—
207C2×D9—124no—
208C2×D9—122no—
209C2×D9—122no—
210C4×C9—22no—
211C4×C9—21no—
212C4×C9—22no—
213C4×C9D3×D412yes—
214Dic9—41no—
215C22×D4—271no—
216C2×D8—272no—
217#121—91no—
218#102—272no—
219C22⋊C8—31no—
220C3×C9#44711yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—181no—
223C23×C3D3611yescharacteristic
224C23×C3#62611yescharacteristic
225C22×D3—61no—
226C22×D3—184no—
227C22×D3—188no—
228C22×D3—184no—
229C22×D3—184no—
230C22×D3—182no—
231C22×D3—182no—
232C22×D3—362no—
233C22×D3—364no—
234C22×D3—362no—
235C22×D3—364no—
236C22×D3—362no—
237C22×D3—362no—
238C3×D4—182no—
239C3×D4—362no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—181no—
246C2×C3×C4D3611yescharacteristic
247C2×C3×C4C22×D911yescharacteristic
248C2×C3×C4#62611yescharacteristic
249C2×C3×C4C2×D3211yescharacteristic
250#67—122no—
251C2×Dic3—61no—
252D12—62no—
253D12—94no—
254D12—94no—
255D12—184no—
256D12—184no—
257D12—188no—
258D12—184no—
259D12—184no—
260D12—182no—
261D12—182no—
262C3×C8—62no—
263C3⋊C8—22no—
264C2×C32—22no—
265C2×C32#88211yescharacteristic
266C2×C32C2×D2411yescharacteristic
267C2×C32#90811yescharacteristic
268C3⋊D3—124no—
269C3×D3—241no—
270C2×C9—24no—
271C2×C9—22no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9C2×#21311yescharacteristic
276C2×C9#93711yescharacteristic
277C2×C9#94911yescharacteristic
278D9—81no—
279D9—128no—
280D9—124no—
281C24—541no—
282C2×D4—91no—
283C2×D4—181no—
284C2×D4—272no—
285C2×D4—544no—
286C2×D4—541no—
287C22×C4D3×D911yescharacteristic
288D8—544no—
289C2×C8—32no—
290C4⋊C4—91no—
291C22⋊C4—181no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—22no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3—361no—
303C22×C3D7212yescomplemented
304C22×C3C2×D3611yescharacteristic
305C22×C3#371412yes—
306C22×C3C2×#62611yescharacteristic
307C2×D3—121no—
308C2×D3—121no—
309C2×D3—121no—
310C2×D3—188no—
311C2×D3—184no—
312C2×D3—184no—
313C2×D3—368no—
314C2×D3—3616no—
315C2×D3—368no—
316C2×D3—364no—
317C2×D3—364no—
318C2×D3—362no—
319C2×D3—362no—
320C2×D3—362no—
321C2×D3—362no—
322C3×C4—22no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—22no—
326C3×C4—361no—
327C3×C4D4×D912yes—
328C3×C4#381112yes—
329Dic3—121no—
330C32#473211yescharacteristic
331C9—21no—
332C9#503611yescharacteristic, complemented
333C23—181no—
334C23—544no—
335C23—542no—
336C23—1082no—
337C23—1082no—
338C23#619211yescharacteristic
339D4—182no—
340D4—274no—
341D4—362no—
342D4—544no—
343D4—542no—
344C2×C4—22no—
345C2×C4—181no—
346C2×C4#619211yescharacteristic
347C2×C4C2×D3×D911yescharacteristic
348C8—62no—
349C2×C3—24no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—721no—
356C2×C3#6344411yescharacteristic
357C2×C3C2×D7211yescharacteristic
358C2×C3#6347011yescharacteristic
359C2×C3C2×#371411yescharacteristic
360C2×C3#6382511yescharacteristic
361C2×C3#6398111yescharacteristic
362D3—241no—
363D3—368no—
364D3—364no—
365D3—364no—
366C22—21no—
367C22—21no—
368C22—361no—
369C22—361no—
370C22—361no—
371C22—544no—
372C22—1088no—
373C22—1082no—
374C22—1082no—
375C22#8841712yescomplemented
376C22C2×#619211yescharacteristic
377C4—21no—
378C4—361no—
379C4#8866812yes—
380C3—21no—
381C3#1058841411yescharacteristic, complemented
382C3#1059116811yescharacteristic
383C2—22no—
384C2—721no—
385C2—1084no—
386C2C2×#8841711yescharacteristic
387C2#1172087311yescharacteristic
388C2#1172127811yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#447
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
1. C22×C4⋊(C3⋊D9)C2semidirect
2. C3×#63492C2semidirect
3. C2×#88417C2semidirect
4. C9×#830C2semidirect
5. #11720284C2semidirect
14. C2×C3×D36C22semidirect
16. C3×C4⋊Dic9C22semidirect
17. C2×C9×C3⋊C8C22semidirect
26. #63492D3semidirect
41. C4⋊(C3⋊D9)D4semidirect
70. C2×D36C2×D3semidirect
73. C4⋊Dic9C2×D3semidirect
77. C22×C3×C9D8semidirect
97. #830D9semidirect
175. C2×C3⋊C8C2×D9semidirect
197. C22×C9#213semidirect
220. C3×C9#447semidirect
303. C22×C3D72semidirect
332. C9#5036semidirect
375. C22#88417semidirect
381. C3#10588414semidirect

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
C3×#634922
C2×#884172
C9×#8302
#117202842
#105911683
#105884143