Finite Group Databasebeta

← #13138695

Subgroups — #13138695

Order 1728.

562 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C2×C8×C3⋊D9C211yesmaximal, characteristic
3#11721000C211yesmaximal, characteristic
4C9×#816C211yesmaximal, characteristic, complemented
5C3×#63376C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590672—31nomaximal
9D9×#103—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C2×(C3⋊D9)⋊C4—21no—
15C2×#6189—21no—
16(C3⋊D9)⋊Q8—22no—
17(C3⋊D9)⋊Q8C414yescomplemented
18C2×C3×C8×C9C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C8×C3⋊D9—22no—
21C9×C4⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic9C2211yescharacteristic
23#88442—21no—
24C2×#3797—31no—
25C2×C8×C3⋊D3—31no—
26#63900—31no—
27C2×Q8×D9—31no—
28C3×#816—31no—
29C3×#816D311yescharacteristic
30#63565—31no—
31#63565—31no—
32C2×C8×D9—31no—
33C2×C8×D9—61no—
34C4⋊C4×D9—31no—
35C9×#103—31no—
36#63395—31no—
37#63376D311yescharacteristic, complemented
38#63368—31no—
39C22×C3⋊D9D411yescharacteristic
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C4×C3⋊D9C2×C412yes—
43C4×C3⋊D9D412yes—
44C2×C9×Dic3—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9—21no—
48C9×Dic6—21no—
49C9×Dic6C2×C412yescomplemented
50C3×Dic18—21no—
51C3×Dic18C2×C412yescomplemented
52(C3⋊D9)⋊C4—24no—
53(C3⋊D9)⋊C4—42no—
54#6189—24no—
55C3×C8×C9—21no—
56C3⋊(C9⋊C8)—21no—
57D3×#103—91no—
58D3×#103—91no—
59C2×C4×C3⋊D3—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6C2×D311yescharacteristic
62C2×#627—61no—
63C2×#624—61no—
64C2×#624—61no—
65#3797—34no—
66#3797—62no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C8×C3⋊D3—62no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72#3724—61no—
73C2×Q8×C9—31no—
74Q8×D9—34no—
75Q8×D9—62no—
76C2×C4×D9—31no—
77C2×C4×D9—61no—
78C2×C4×D9—61no—
79C2×C4×D9—61no—
80C2×Dic18—61no—
81C2×Dic18C2×D311yescharacteristic
82C2×C8×C9—21no—
83C2×C8×C9C2×D311yescharacteristic
84C9×C4⋊C4—31no—
85C4⋊Dic9C2×D311yescharacteristic
86Dic9⋊C4—61no—
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89C8×D9—62no—
90C8×D9—64no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9C22⋊C412yes—
93C2×C3⋊D9SD1612yes—
94C2×C3⋊D9Q1612yes—
95C3×C4×C9C22×C411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C3⋊Dic9C22⋊C412yes—
98C9×Dic3—22no—
99C9×Dic3—41no—
100C3×Dic9—22no—
101C3×Dic9—41no—
102C2×D3×Q8—91no—
103C2×D3×Q8—91no—
104C2×D3×C8—91no—
105C2×D3×C8—91no—
106C2×D3×C8—181no—
107D3×C4⋊C4—91no—
108D3×C4⋊C4—91no—
109C3×#103—31no—
110C3×#103—91no—
111#835—91no—
112#835—91no—
113#816—31no—
114#816D911yescharacteristic, complemented
115#810—91no—
116#810—91no—
117C22×C3⋊D3—31no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—31no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—32no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×Dic6—21no—
127C3×Dic6—32no—
128C3×Dic6—61no—
129C3×Dic6C4×D312yes—
130#627—64no—
131#624—64no—
132#624—122no—
133C22×D9—31no—
134C22×D9—61no—
135C32×C8—21no—
136C3⋊(C3⋊C8)—61no—
137Q8×C9—32no—
138Q8×C9—61no—
139C2×C4×C9—21no—
140C2×C4×C9—61no—
141C2×C4×C9—61no—
142C2×C4×C9C22×D311yescharacteristic
143C2×Dic9—21no—
144C2×Dic9—21no—
145C2×Dic9—31no—
146C2×Dic9—61no—
147C4×D9—32no—
148C4×D9—32no—
149C4×D9—62no—
150C4×D9—62no—
151C4×D9—64no—
152C4×D9—122no—
153Dic18—21no—
154Dic18—64no—
155Dic18C4×D312yescomplemented
156C8×C9—22no—
157C8×C9—21no—
158C9⋊C8—62no—
159C9⋊C8—61no—
160C2×#103—271noSylow
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C2×C3×C9C2×SD1611yescharacteristic
163C2×C3×C9C2×Q1611yescharacteristic
164C3⋊D9#10314yescomplemented
165C2×C3×Q8—31no—
166C2×C3×Q8—91no—
167D3×Q8—94no—
168D3×Q8—94no—
169D3×Q8—182no—
170D3×Q8—182no—
171C2×C4×D3—91no—
172C2×C4×D3—91no—
173C2×C4×D3—181no—
174C2×C4×D3—181no—
175C2×C4×D3—181no—
176C2×C4×D3—181no—
177C2×C4×D3—181no—
178C2×Dic6—31no—
179C2×Dic6—181no—
180C2×Dic6—181no—
181C2×Dic6C2×D911yescharacteristic
182C2×C3×C8—21no—
183C2×C3×C8C2×D911yescharacteristic
184C2×C3×C8D3211yescharacteristic
185C3×C4⋊C4—31no—
186C3×C4⋊C4—91no—
187C4⋊Dic3—31no—
188C4⋊Dic3C2×D911yescharacteristic
189Dic3⋊C4—181no—
190Dic3⋊C4—181no—
191C2×C3⋊C8—91no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—181no—
194D3×C8—182no—
195D3×C8—182no—
196D3×C8—184no—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—32no—
199C2×C3⋊D3—32no—
200C2×C3⋊D3—32no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—32no—
204C3×Dic3—22no—
205C3×Dic3—41no—
206C3×Dic3—62no—
207C3×Dic3—121no—
208C22×C9—21no—
209C22×C9D3×D411yescharacteristic
210C2×D9—32no—
211C2×D9—32no—
212C2×D9—32no—
213C2×D9—62no—
214C2×D9—62no—
215C2×D9—62no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—62no—
219C4×C9—121no—
220C4×C9C2×C4×D311yescharacteristic
221C4×C9D3×D411yescharacteristic
222Dic9—22no—
223Dic9—32no—
224Dic9—41no—
225Dic9—62no—
226C22×Q8—271no—
227C22×C8—271no—
228C2×C4⋊C4—271no—
229#103—31no—
230#103—91no—
231#103—271no—
232#103—271no—
233C3×C9C2×#10311yescharacteristic, complemented, Sylow
234C22×D3—91no—
235C22×D3—91no—
236C22×D3—181no—
237C3×Q8—32no—
238C3×Q8—61no—
239C3×Q8—92no—
240C3×Q8—181no—
241C2×C3×C4—21no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C22×D911yescharacteristic
247C2×C3×C4C2×D3211yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—181no—
255C4×D3—92no—
256C4×D3—92no—
257C4×D3—92no—
258C4×D3—92no—
259C4×D3—182no—
260C4×D3—182no—
261C4×D3—184no—
262C4×D3—184no—
263C4×D3—362no—
264C4×D3—362no—
265Dic6—21no—
266Dic6—32no—
267Dic6—61no—
268Dic6—184no—
269Dic6—184no—
270Dic6C4×D912yescomplemented
271C3×C8—22no—
272C3×C8—21no—
273C3×C8—21no—
274C3⋊C8—182no—
275C3⋊C8—181no—
276C3⋊C8—181no—
277C2×C32D3×C22⋊C411yescharacteristic
278C2×C32D3×SD1611yescharacteristic
279C2×C32D3×Q1611yescharacteristic
280C3⋊D3—34no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9D3×C22⋊C411yescharacteristic
285C2×C9D3×SD1611yescharacteristic
286C2×C9D3×Q1611yescharacteristic
287D9—34no—
288D9—64no—
289C2×Q8—31no—
290C2×Q8—91no—
291C2×Q8—274no—
292C2×Q8—541no—
293C2×Q8—542no—
294C22×C4—271no—
295C22×C4—541no—
296C22×C4—541no—
297C2×C8—271no—
298C2×C8—542no—
299C2×C8D3×D911yescharacteristic
300C4⋊C4—31no—
301C4⋊C4—91no—
302C4⋊C4—541no—
303C22×C3—21no—
304C22×C3D4×D911yescharacteristic
305C22×C3#380211yescharacteristic
306C2×D3—92no—
307C2×D3—92no—
308C2×D3—92no—
309C2×D3—92no—
310C2×D3—92no—
311C2×D3—92no—
312C2×D3—182no—
313C2×D3—182no—
314C2×D3—182no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—62no—
318C3×C4—121no—
319C3×C4—182no—
320C3×C4—361no—
321C3×C4C2×C4×D911yescharacteristic
322C3×C4D4×D911yescharacteristic
323C3×C4C4×D3211yescharacteristic
324C3×C4#381111yescharacteristic
325Dic3—22no—
326Dic3—41no—
327Dic3—62no—
328Dic3—92no—
329Dic3—92no—
330Dic3—121no—
331Dic3—182no—
332C32D3×#10311yescharacteristic
333C9—21no—
334C9D3×#10311yescharacteristic, complemented
335C23—271no—
336Q8—32no—
337Q8—61no—
338Q8—92no—
339Q8—181no—
340Q8—544no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—271no—
346C2×C4—272no—
347C2×C4—272no—
348C2×C4—544no—
349C2×C4—1082no—
350C2×C4C2×D3×D911yescharacteristic
351C8—21no—
352C8—541no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3C22⋊C4×D911yescharacteristic
357C2×C3SD16×D911yescharacteristic
358C2×C3Q16×D911yescharacteristic
359C2×C3#6379611yescharacteristic
360C2×C3#6380211yescharacteristic
361C2×C3#6392111yescharacteristic
362D3—94no—
363D3—94no—
364D3—184no—
365C22—272no—
366C22—272no—
367C22—272no—
368C22#8864611yescharacteristic
369C4—62no—
370C4—121no—
371C4—182no—
372C4—272no—
373C4—361no—
374C4C4×D3×D911yescharacteristic
375C4#8866811yescharacteristic
376C3—21no—
377C3D9×#10311yescharacteristic, complemented
378C3#1059067211yescharacteristic
379C2—274no—
380C2#1172081711yescharacteristic
381C2#1172082911yescharacteristic
382C2#1172104811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9C2×#103
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C9×#816C2semidirect
5. C3×#63376C2semidirect
6. #11720297C2semidirect
7. #11720268C2semidirect
17. (C3⋊D9)⋊Q8C4semidirect
37. #63376D3semidirect
49. C9×Dic6C2×C4semidirect
51. C3×Dic18C2×C4semidirect
114. #816D9semidirect
155. Dic18C4×D3semidirect
164. C3⋊D9#103semidirect
233. C3×C9C2×#103semidirect
270. Dic6C4×D9semidirect
334. C9D3×#103semidirect
377. C3D9×#103semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C2×C8×C3⋊D92
#117210002
C9×#8162
C3×#633762
#117202972
#117202682
#105906723
D9×#1033