Finite Group Databasebeta

← #13137852

Subgroups — #13137852

Order 1728.

518 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 91 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)⋊Q8C211yesmaximal, characteristic
2#11721753C211yesmaximal, characteristic
3C2×#88428C211yesmaximal, characteristic, complemented
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×#810C211yesmaximal, characteristic, complemented
6C3×#63368C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic
8#10590605—31nomaximal
9#10588489—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×(C3⋊D9)⋊C4—21no—
15C2×#6189—21no—
16(C3⋊D9)⋊Q8—22no—
17(C3⋊D9)⋊Q8—22no—
18C3×C9×C4⋊C4C2211yescharacteristic
19#88536—21no—
20C4⋊(C3⋊Dic9)—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88428—22no—
24#88418—22no—
25C2×#3797—31no—
26#64106—31no—
27C2×#3719—31no—
28C2×#3719—31no—
29C2×Q8×D9—31no—
30C3×#810—31no—
31C3×#810D311yescharacteristic
32#63557—31no—
33C2×#3674—31no—
34C2×Dic36—31no—
35#63459—31no—
36#63459—61no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63368D311yescharacteristic, complemented
40C22×C3⋊D9D411yescharacteristic
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C4×C3⋊D9—21no—
46C4×C3⋊D9—21no—
47#6226—21no—
48#6226—21no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9—21no—
51C9×Dic6—21no—
52C9×Dic6D412yes—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55(C3⋊D9)⋊C4—24no—
56#6189—24no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59#4809—91no—
60#4809—91no—
61C2×C4×C3⋊D3—31no—
62C2×#634—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×#627—61no—
66C2×#624—61no—
67#3797—62no—
68#3797—62no—
69C32×C4⋊C4C2×D311yescharacteristic
70#3752—61no—
71C4⋊(C3⋊Dic3)—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74#3719—62no—
75#3719—62no—
76C2×Q8×C9—31no—
77Q8×D9—62no—
78Q8×D9—62no—
79C2×C4×D9—31no—
80C2×C4×D9—61no—
81C2×C4×D9—61no—
82C2×Dic18—31no—
83C2×Dic18—61no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C2×C8×C9—31no—
87C9×C4⋊C4—21no—
88C9×C4⋊C4C2×D311yescharacteristic
89#3674—62no—
90(C2×D9)⋊C4—61no—
91(C2×D9)⋊C4—62no—
92C4⋊Dic9—61no—
93C4⋊Dic9—62no—
94C2×C9⋊C8C2×D311yescharacteristic
95Dic36—62no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—21no—
98C2×C3⋊D9—21no—
99C2×C3⋊D9Q1612yescomplemented
100C3×C4×C9—41no—
101C3×C4×C9C2×D411yescharacteristic
102C3×C4×C9C2×D411yescharacteristic
103C3⋊Dic9—21no—
104C3⋊Dic9—41no—
105C9×Dic3—22no—
106C3×Dic9—22no—
107C2×D3×Q8—91no—
108C2×D3×Q8—91no—
109C2×#216—91no—
110C2×#216—91no—
111C2×Dic12—91no—
112C2×Dic12—91no—
113#897—91no—
114#897—91no—
115#897—181no—
116C3×#103—31no—
117C3×#103—91no—
118(C2×D3)⋊C8—91no—
119(C2×D3)⋊C8—91no—
120#810—31no—
121#810D911yescharacteristic, complemented
122C22×C3⋊D3—31no—
123C2×C32×C4—21no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—61no—
127C4×C3⋊D3—61no—
128C4×C3⋊D3—61no—
129#634—61no—
130#634—61no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—61no—
133C3×Dic6—21no—
134C3×Dic6—32no—
135C3×Dic6—61no—
136C3×Dic6D1212yes—
137#627—64no—
138#624—64no—
139C22×D9—31no—
140C22×D9—61no—
141C3×C3⋊C8—21no—
142C3×C3⋊C8—61no—
143Q8×C9—32no—
144Q8×C9—61no—
145C2×C4×C9—22no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—61no—
149C2×C4×C9C22×D311yescharacteristic
150C2×Dic9—21no—
151C2×Dic9—31no—
152C2×Dic9—61no—
153C2×Dic9—62no—
154C2×Dic9—61no—
155C4×D9—64no—
156C4×D9—61no—
157C4×D9—61no—
158C4×D9—121no—
159C4×D9—121no—
160Dic18—21no—
161Dic18—64no—
162Dic18—61no—
163Dic18—61no—
164Dic18—121no—
165Dic18—121no—
166Dic18D1212yes—
167C8×C9—61no—
168C9⋊C8—21no—
169#451—271noSylow
170C2×C3×C9#12011yescharacteristic
171C2×C3×C9C2×Q1611yescharacteristic
172C2×C3×C9#13711yescharacteristic
173C3⋊D9—22no—
174C2×C3×Q8—31no—
175C2×C3×Q8—91no—
176D3×Q8—182no—
177D3×Q8—182no—
178D3×Q8—182no—
179D3×Q8—182no—
180C2×C4×D3—91no—
181C2×C4×D3—91no—
182C2×C4×D3—181no—
183C2×C4×D3—181no—
184C2×C4×D3—181no—
185C2×Dic6—31no—
186C2×Dic6—91no—
187C2×Dic6—91no—
188C2×Dic6—181no—
189C2×Dic6—181no—
190C2×Dic6—181no—
191C2×Dic6C2×D911yescharacteristic
192C2×C3×C8—31no—
193C2×C3×C8—91no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D911yescharacteristic
196C3×C4⋊C4D3211yescharacteristic
197#216—182no—
198#216—182no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201(C2×D3)⋊C4—182no—
202C4⋊Dic3—181no—
203C4⋊Dic3—181no—
204C4⋊Dic3—182no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8C2×D911yescharacteristic
207Dic12—182no—
208Dic12—182no—
209C22×C32D3×D411yescharacteristic
210C2×C3⋊D3—32no—
211C2×C3⋊D3—61no—
212C2×C3⋊D3—61no—
213C32×C4—41no—
214C32×C4C2×D1211yescharacteristic
215C32×C4D3×D411yescharacteristic
216C3⋊Dic3—61no—
217C3⋊Dic3—121no—
218C3×Dic3—22no—
219C3×Dic3—62no—
220C22×C9—21no—
221C22×C9D3×D411yescharacteristic
222C2×D9—32no—
223C2×D9—62no—
224C2×D9—61no—
225C2×D9—61no—
226C2×D9—121no—
227C2×D9—121no—
228C4×C9—21no—
229C4×C9—21no—
230C4×C9—42no—
231C4×C9—41no—
232C4×C9—62no—
233C4×C9C2×D1211yescharacteristic
234C4×C9D3×D411yescharacteristic
235Dic9—22no—
236Dic9—61no—
237Dic9—121no—
238Dic9—122no—
239Dic9—121no—
240C22×Q8—271no—
241C2×Q16—271no—
242C2×Q16—271no—
243C22⋊Q8—271no—
244#103—31no—
245#103—91no—
246C22⋊C8—271no—
247C3×C9#45111yescharacteristic, complemented, Sylow
248C22×D3—91no—
249C22×D3—91no—
250C22×D3—181no—
251C3×Q8—32no—
252C3×Q8—61no—
253C3×Q8—92no—
254C3×Q8—181no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—61no—
260C2×C3×C4—181no—
261C2×C3×C4C22×D911yescharacteristic
262C2×C3×C4C2×D3211yescharacteristic
263C2×Dic3—21no—
264C2×Dic3—61no—
265C2×Dic3—91no—
266C2×Dic3—91no—
267C2×Dic3—181no—
268C2×Dic3—182no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C4×D3—184no—
272C4×D3—184no—
273C4×D3—181no—
274C4×D3—181no—
275C4×D3—181no—
276C4×D3—181no—
277C4×D3—361no—
278C4×D3—361no—
279Dic6—21no—
280Dic6—32no—
281Dic6—61no—
282Dic6—184no—
283Dic6—184no—
284Dic6—181no—
285Dic6—181no—
286Dic6—181no—
287Dic6—181no—
288Dic6—361no—
289Dic6—361no—
290Dic6D3612yes—
291C3×C8—61no—
292C3×C8—181no—
293C3⋊C8—21no—
294C3⋊C8—61no—
295C2×C32#88211yescharacteristic
296C2×C32#91511yescharacteristic
297C2×C32D3×Q1611yescharacteristic
298C3⋊D3—62no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9#88211yescharacteristic
303C2×C9#91511yescharacteristic
304C2×C9D3×Q1611yescharacteristic
305D9—62no—
306D9—122no—
307C2×Q8—31no—
308C2×Q8—91no—
309C2×Q8—271no—
310C2×Q8—541no—
311C2×Q8—542no—
312C2×Q8—542no—
313C22×C4—271no—
314C22×C4—541no—
315Q16—542no—
316Q16—542no—
317C2×C8—31no—
318C2×C8—91no—
319C4⋊C4—541no—
320C4⋊C4D3×D911yescharacteristic
321C22⋊C4—541no—
322C22×C3—21no—
323C22×C3D4×D911yescharacteristic
324C22×C3#381111yescharacteristic
325C2×D3—92no—
326C2×D3—92no—
327C2×D3—182no—
328C2×D3—181no—
329C2×D3—181no—
330C2×D3—181no—
331C2×D3—181no—
332C2×D3—361no—
333C2×D3—361no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—62no—
340C3×C4—182no—
341C3×C4C2×D3611yescharacteristic
342C3×C4D4×D911yescharacteristic
343C3×C4D3×D1211yescharacteristic
344C3×C4D3×D1211yescharacteristic
345Dic3—22no—
346Dic3—62no—
347Dic3—181no—
348Dic3—181no—
349Dic3—361no—
350Dic3—362no—
351Dic3—361no—
352Dic3—361no—
353C32#480911yescharacteristic
354C9—21no—
355C9#480911yescharacteristic, complemented
356C23—271no—
357Q8—32no—
358Q8—61no—
359Q8—92no—
360Q8—181no—
361Q8—544no—
362Q8—541no—
363Q8—541no—
364C2×C4—21no—
365C2×C4—61no—
366C2×C4—181no—
367C2×C4—271no—
368C2×C4—544no—
369C2×C4—541no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×D911yescharacteristic
373C8—61no—
374C8—181no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3#6344411yescharacteristic
379C2×C3#6347711yescharacteristic
380C2×C3Q16×D911yescharacteristic
381C2×C3#6392311yescharacteristic
382C2×C3#6393111yescharacteristic
383C2×C3#6394411yescharacteristic
384D3—182no—
385D3—182no—
386D3—362no—
387C22—272no—
388C22—541no—
389C22—541no—
390C22#8866811yescharacteristic
391C4—41no—
392C4—62no—
393C4—182no—
394C4—541no—
395C4—1081no—
396C4D3×D3611yescharacteristic
397C4D9×D1211yescharacteristic
398C3—21no—
399C3#1058848911yescharacteristic, complemented
400C3#1059060511yescharacteristic
401C2—542no—
402C2#1172105111yescharacteristic
403C2#1172118311yescharacteristic
404C2#1172122011yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#88428C2semidirect
4. C2×#88418C2semidirect
5. C9×#810C2semidirect
6. C3×#63368C2semidirect
39. #63368D3semidirect
99. C2×C3⋊D9Q16semidirect
121. #810D9semidirect
247. C3×C9#451semidirect
355. C9#4809semidirect
399. C3#10588489semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
#117217532
C2×#884282
C2×#884182
C9×#8102
C3×#633682
#117202822
#105906053
#105884893