Finite Group Databasebeta

← #13140792

Subgroups — #13140792

Order 1728.

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

normal subgroups only

This group has 440 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×#88548C211yesmaximal, characteristic, complemented
2C9×#946C211yesmaximal, characteristic, complemented
3#11721038C211yesmaximal, characteristic
4C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720267C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10591585—31nomaximal
9#10588731—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13C2×#6195—21no—
14#88548—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×(C2×D3)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88445—21no—
22D3×C9⋊C8—22no—
23C2×#3755—31no—
24C3×#946D311yescharacteristic
25#63909—31no—
26C2×D3×C3⋊C8—31no—
27C3×#835—31no—
28#63574—31no—
29#63564—31no—
30C9×C22⋊Q8—31no—
31C2×#3675—31no—
32C2×#3675—61no—
33#63492—31no—
34C22×C9⋊C8—31no—
35#63395D311yescharacteristic, complemented
36#63369—31no—
37#63367—31no—
38C22×C3⋊D9—21no—
39C22×D3×C9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46C2×C9×Dic3—21no—
47C2×C9×Dic3D411yescharacteristic
48C2×C3×Dic9—21no—
49C4×D3×C9—21no—
50C4×D3×C9—21no—
51#6195—42no—
52C3⋊(C9⋊C8)D412yescomplemented
53C3×C9⋊C8—21no—
54#5048—91no—
55#4807—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)—31no—
58C2×C3×C4×D3C2×D311yescharacteristic
59C2×#626—61no—
60#3755—62no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×(C2×D3)⋊C4—21no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65C3×Dic3⋊C4—21no—
66C2×C3×C3⋊C8—31no—
67#3722—61no—
68D3×C3⋊C8—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C22×C4×C9—31no—
72C2×#611—61no—
73C2×D36—31no—
74C2×D36—61no—
75C9×C4⋊C4—31no—
76C9×C4⋊C4—61no—
77C9×C22⋊C4—61no—
78#3675—62no—
79#3675—64no—
80(C2×D9)⋊C4—61no—
81C4⋊Dic9C2×D311yescharacteristic
82C2×C9⋊C8—31no—
83C2×C9⋊C8—62no—
84C2×C9⋊C8—61no—
85C2×C9⋊C8C2×D311yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—41no—
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9SD1612yescomplemented
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—21no—
97C9×Dic3—41no—
98C3×Dic9—41no—
99C3×C22⋊Q8—31no—
100#946D911yescharacteristic, complemented
101C2×#215—91no—
102C2×#215—91no—
103C2×#215—181no—
104#930—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—91no—
107#895—91no—
108C3×#103—91no—
109#835—31no—
110#821—91no—
111#809—91no—
112#808—91no—
113#808—91no—
114C22×C3⋊D3—61no—
115C22×C3×D3#6711yescharacteristic
116C32×Q8—21no—
117C32×Q8—21no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—61no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3#6711yescharacteristic
125C3×C4×D3—21no—
126C3×C4×D3—21no—
127#626—122no—
128C23×C9—31no—
129C22×D9—61no—
130C22×D9—121no—
131C3⋊(C3⋊C8)—32no—
132C3×C3⋊C8—61no—
133Q8×C9—22no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—31no—
140C2×C4×C9—41no—
141C2×C4×C9—61no—
142C2×C4×C9—61no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145#611—122no—
146C2×Dic9—21no—
147D36—62no—
148D36—61no—
149D36—61no—
150D36—121no—
151C9⋊C8—21no—
152C9⋊C8—32no—
153C9⋊C8—62no—
154#465—271noSylow
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9C2×SD1611yescharacteristic
157C2×C3×C9#13511yescharacteristic
158C3⋊D9—81no—
159D3×C9—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D911yescharacteristic
162C2×C3×Q8D3211yescharacteristic
163C22×C3×C4—31no—
164C2×#67—181no—
165C2×D12—91no—
166C2×D12—91no—
167C2×D12—181no—
168C2×D12—181no—
169C2×C4×D3C2×D911yescharacteristic
170C2×C3×C8—91no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—91no—
174C3×C22⋊C4—61no—
175#215—182no—
176#215—182no—
177#215—184no—
178(C2×D3)⋊C4—21no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3C2×D911yescharacteristic
183Dic3⋊C4—21no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—181no—
188C2×C3⋊C8—182no—
189D3×C8—182no—
190C22×C32C2×#6711yescharacteristic
191C2×C3⋊D3—121no—
192C2×C3⋊D3—121no—
193C2×C3⋊D3—121no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3#21512yes—
197C32×C4—41no—
198C32×C4D3×D411yescharacteristic
199C32×C4#23711yescharacteristic
200C3×Dic3—21no—
201C3×Dic3—41no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9—32no—
205C22×C9—61no—
206C22×C9—61no—
207C22×C9D3×D411yescharacteristic
208C2×D9—122no—
209C2×D9—121no—
210C2×D9—121no—
211C2×D9—121no—
212C2×D9—241no—
213C2×D9—241no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—42no—
217C4×C9—41no—
218C4×C9—61no—
219C4×C9—121no—
220C4×C9#23511yescharacteristic
221C4×C9D3×D411yescharacteristic
222Dic9—41no—
223C2×SD16—271no—
224C22×C8—271no—
225C22⋊Q8—31no—
226#121—271no—
227#106—271no—
228#103—91no—
229#102—271no—
230C3×C9#46511yescharacteristic, complemented, Sylow
231C23×C3—31no—
232C22×D3—181no—
233C22×D3—181no—
234C22×D3—361no—
235C22×D3#61111yescharacteristic
236C3×Q8—22no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—41no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—31no—
246C2×C3×C4—41no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—181no—
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253#67—362no—
254C2×Dic3—21no—
255C2×Dic3—61no—
256C2×Dic3#61111yescharacteristic
257D12—182no—
258D12—181no—
259D12—181no—
260D12—181no—
261D12—181no—
262D12—361no—
263D12—362no—
264C4×D3—21no—
265C4×D3—21no—
266C3×C8—181no—
267C3⋊C8—61no—
268C3⋊C8—92no—
269C3⋊C8—92no—
270C3⋊C8—182no—
271C2×C32#93911yescharacteristic
272C2×C32C2×#21511yescharacteristic
273C2×C32#95011yescharacteristic
274C3⋊D3—241no—
275C3×D3—22no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—62no—
280C2×C9#88411yescharacteristic
281C2×C9D3×SD1611yescharacteristic
282C2×C9#91911yescharacteristic
283D9—242no—
284D9—241no—
285C2×Q8D3×D911yescharacteristic
286C2×D4—271no—
287C2×D4—541no—
288C22×C4—31no—
289SD16—542no—
290C2×C8—91no—
291C2×C8—271no—
292C2×C8—542no—
293C4⋊C4—31no—
294C4⋊C4—61no—
295C4⋊C4—91no—
296C22⋊C4—61no—
297C22⋊C4—541no—
298C22×C3—21no—
299C22×C3—32no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3C2×#61111yescharacteristic
303C22×C3D3×#6711yescharacteristic
304C2×D3—21no—
305C2×D3—21no—
306C2×D3—362no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—361no—
313C2×D3—721no—
314C2×D3—721no—
315C2×D3#367512yescomplemented
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—61no—
322C3×C4—121no—
323C3×C4—361no—
324C3×C4D4×D911yescharacteristic
325C3×C4#369911yescharacteristic
326C3×C4#380411yescharacteristic
327C3×C4#381111yescharacteristic
328Dic3—21no—
329Dic3—41no—
330Dic3—121no—
331C32#504811yescharacteristic
332C9—21no—
333C9#480711yescharacteristic, complemented
334C23—31no—
335C23—541no—
336Q8—21no—
337Q8—21no—
338D4—541no—
339D4—541no—
340D4—1082no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—181no—
347C2×C4C2×D3×D911yescharacteristic
348C8—181no—
349C8—272no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—62no—
354C2×C3#6350111yescharacteristic
355C2×C3C2×#367511yescharacteristic
356C2×C3#6351311yescharacteristic
357C2×C3D3×#21511yescharacteristic
358C2×C3#6394911yescharacteristic
359C2×C3#6397111yescharacteristic
360D3—22no—
361D3—722no—
362D3—721no—
363D3—721no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22D3×#61111yescharacteristic
371C4—41no—
372C4—61no—
373C4—121no—
374C4—361no—
375C4#8866211yescharacteristic
376C4#8866811yescharacteristic
377C3—21no—
378C3#1058873111yescharacteristic, complemented
379C3#1059158511yescharacteristic
380C2—62no—
381C2—2161no—
382C2D3×#367511yescharacteristic
383C2#1172120911yescharacteristic
384C2#1172128211yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C2×#67
prime radical#11721038C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#465
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88548C2semidirect
2. C9×#946C2semidirect
4. C2×D3×C9⋊C8C2semidirect
5. C3×#63395C2semidirect
6. #11720267C2semidirect
7. #11720265C2semidirect
35. #63395D3semidirect
52. C3⋊(C9⋊C8)D4semidirect
92. C2×D3×C9SD16semidirect
100. #946D9semidirect
230. C3×C9#465semidirect
315. C2×D3#3675semidirect
333. C9#4807semidirect
378. C3#10588731semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
C9×#9462
#117210382
C2×D3×C9⋊C82
C3×#633952
#117202672
#117202652
#105915853
#105887313