Finite Group Databasebeta

← #13140732

Subgroups — #13140732

Order 1728.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C3×#63508C211yesmaximal, characteristic, complemented
3#11721004C211yesmaximal, characteristic
4C2×#88413C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10591589—31nomaximal
9#10588492—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×#6192—21no—
14#88548—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic9C2211yescharacteristic
19C3×Dic9⋊C4—21no—
20C2×C9×C3⋊C8C2211yescharacteristic
21#88445—21no—
22#88413—22no—
23C2×#3755—31no—
24C3×#946—31no—
25#63909—31no—
26C2×#3711—31no—
27C3×#835D311yescharacteristic
28#63576—31no—
29#63564—31no—
30#63508D311yescharacteristic, complemented
31C2×#3675—31no—
32C2×#3675—61no—
33C2×#3663—31no—
34#63457—31no—
35C9×#103—31no—
36#63380—31no—
37#63366—31no—
38C22×C3⋊D9—21no—
39C22×C3×D9D411yescharacteristic
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×C3×Dic9—21no—
48C2×C3×Dic9D411yescharacteristic
49C3×C4×D9—21no—
50C3×C4×D9—21no—
51#6192—42no—
52C3⋊(C9⋊C8)D412yescomplemented
53C9×C3⋊C8—21no—
54#5049—91no—
55#4812—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)—31no—
58C2×C3×C4×D3—31no—
59C2×#626—61no—
60#3755—62no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65C3×Dic3⋊C4—61no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67#3722—61no—
68#3711—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D36—31no—
72C2×D36—61no—
73C2×D36—61no—
74C2×C4×D9C2×D311yescharacteristic
75C2×C8×C9—31no—
76C9×C4⋊C4—31no—
77#3675—62no—
78#3675—64no—
79(C2×D9)⋊C4—21no—
80(C2×D9)⋊C4—61no—
81C4⋊Dic9C2×D311yescharacteristic
82Dic9⋊C4—21no—
83C2×C9⋊C8—31no—
84C2×C9⋊C8—61no—
85#3663—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—41no—
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9—21no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—41no—
97C3×Dic9—21no—
98C3×Dic9—41no—
99C3×C22⋊Q8—91no—
100#946—31no—
101C2×#215—91no—
102C2×#215—91no—
103C2×#215—181no—
104#930—91no—
105C2×#208—91no—
106C2×ΓL(1,𝔽25)—91no—
107#895—91no—
108C3×#103—31no—
109#835D911yescharacteristic, complemented
110#821—91no—
111#809—91no—
112#807—91no—
113#807—91no—
114C22×C3⋊D3—61no—
115C22×C3×D3—31no—
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—31no—
124C2×C3×Dic3—61no—
125C3×C4×D3—61no—
126C3×C4×D3—61no—
127#626—122no—
128C22×D9—61no—
129C22×D9—121no—
130C22×D9#6711yescharacteristic
131C3⋊(C3⋊C8)—32no—
132C3×C3⋊C8—21no—
133Q8×C9—22no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9—61no—
141C2×C4×C9C22×D311yescharacteristic
142C2×Dic9—21no—
143C2×Dic9#6711yescharacteristic
144D36—62no—
145D36—61no—
146D36—61no—
147D36—121no—
148D36—122no—
149C4×D9—21no—
150C4×D9—21no—
151C8×C9—61no—
152C9⋊C8—32no—
153C9⋊C8—62no—
154#468—271noSylow
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C3⋊D9—81no—
159C3×D9—41no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D911yescharacteristic
162C2×C3×Q8D3211yescharacteristic
163C22×C3×C4—91no—
164C2×#67—181no—
165C2×D12—91no—
166C2×D12—91no—
167C2×D12—181no—
168C2×D12—181no—
169C2×C4×D3—31no—
170C2×C3×C8—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C22⋊C4—181no—
175#215—182no—
176#215—182no—
177#215—184no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3C2×D911yescharacteristic
183Dic3⋊C4—61no—
184#208—182no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—181no—
188C2×C3⋊C8C2×D911yescharacteristic
189ΓL(1,𝔽25)—182no—
190C22×C32D3×D411yescharacteristic
191C2×C3⋊D3—121no—
192C2×C3⋊D3—121no—
193C2×C3⋊D3—121no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C32×C4—41no—
198C32×C4#23511yescharacteristic
199C32×C4D3×D411yescharacteristic
200C3×Dic3—41no—
201C3×Dic3—61no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9C2×#6711yescharacteristic
205C2×D9—21no—
206C2×D9—21no—
207C2×D9—21no—
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—121no—
219C4×C9D3×D411yescharacteristic
220C4×C9#23711yescharacteristic
221Dic9—21no—
222Dic9—41no—
223C2×SD16—271no—
224C2×M16—271no—
225C22⋊Q8—91no—
226#121—271no—
227#107—271no—
228#103—31no—
229#102—271no—
230C3×C9#46811yescharacteristic, complemented, Sylow
231C23×C3—91no—
232C22×D3—31no—
233C22×D3—181no—
234C22×D3—181no—
235C22×D3—361no—
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—41no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253#67—362no—
254C2×Dic3—21no—
255C2×Dic3—31no—
256C2×Dic3—61no—
257D12—182no—
258D12—181no—
259D12—181no—
260D12—181no—
261D12—181no—
262D12—361no—
263D12—362no—
264C4×D3—61no—
265C4×D3—61no—
266C3×C8—61no—
267C3⋊C8—21no—
268C3⋊C8—92no—
269C3⋊C8—92no—
270C3⋊C8—182no—
271C2×C32#88411yescharacteristic
272C2×C32#91411yescharacteristic
273C2×C32#91811yescharacteristic
274C3⋊D3—241no—
275C3×D3—121no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9#93911yescharacteristic
280C2×C9#94211yescharacteristic
281C2×C9#94911yescharacteristic
282D9—41no—
283D9—242no—
284D9—241no—
285C2×Q8D3×D911yescharacteristic
286C2×D4—271no—
287C2×D4—541no—
288C22×C4—91no—
289SD16—542no—
290M16—542no—
291C2×C8—31no—
292C2×C8—271no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—181no—
296C22⋊C4—181no—
297C22⋊C4—541no—
298C22×C3—21no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—181no—
302C22×C3D4×D911yescharacteristic
303C22×C3D3×#6711yescharacteristic
304C2×D3—61no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—362no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—721no—
315C2×D3—721no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—121no—
322C3×C4—181no—
323C3×C4—361no—
324C3×C4#369711yescharacteristic
325C3×C4D4×D911yescharacteristic
326C3×C4#380411yescharacteristic
327C3×C4#381111yescharacteristic
328Dic3—41no—
329Dic3—61no—
330Dic3—121no—
331C32#481211yescharacteristic
332C9—21no—
333C9#504911yescharacteristic, complemented
334C23—91no—
335C23—541no—
336Q8—21no—
337Q8—21no—
338D4—541no—
339D4—541no—
340D4—1082no—
341C2×C4—21no—
342C2×C4—61no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4C2×D3×D911yescharacteristic
348C8—61no—
349C8—272no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—361no—
354C2×C3#6344611yescharacteristic
355C2×C3#6347611yescharacteristic
356C2×C3#6348011yescharacteristic
357C2×C3#6393911yescharacteristic
358C2×C3#6394811yescharacteristic
359C2×C3#6397111yescharacteristic
360D3—121no—
361D3—722no—
362D3—721no—
363D3—721no—
364C22—181no—
365C22—181no—
366C22—181no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22D9×#6711yescharacteristic
371C4—41no—
372C4—121no—
373C4—181no—
374C4—361no—
375C4#8865811yescharacteristic
376C4#8866811yescharacteristic
377C3—21no—
378C3#1058849211yescharacteristic, complemented
379C3#1059158911yescharacteristic
380C2—361no—
381C2—2161no—
382C2#1172121111yescharacteristic
383C2#1172122211yescharacteristic
384C2#1172127111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radical#11721004C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#468
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. C3×#63508C2semidirect
4. C2×#88413C2semidirect
5. C9×#835C2semidirect
6. #11720284C2semidirect
7. #11720263C2semidirect
30. #63508D3semidirect
52. C3⋊(C9⋊C8)D4semidirect
109. #835D9semidirect
230. C3×C9#468semidirect
333. C9#5049semidirect
378. C3#10588492semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
C3×#635082
#117210042
C2×#884132
C9×#8352
#117202842
#117202632
#105915893
#105884923