Finite Group Databasebeta

← #13140321

Subgroups — #13140321

Order 1728.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C3×D4×Dic9C211yesmaximal, characteristic
2#11720969C211yesmaximal, characteristic, complemented
3#11720576C211yesmaximal, characteristic
4C9×#832C211yesmaximal, characteristic
5#11720278C211yesmaximal, characteristic, complemented
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10591408—31nomaximal
9#10588441—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3×Dic9—21no—
12C2×#6189—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×#3676—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21C3×D4×Dic3—31no—
22#63918—31no—
23#63659—31no—
24C3×#832D311yescharacteristic
25#63574—31no—
26#63573—31no—
27Dic3⋊(C3⋊C8)—31no—
28D4×Dic9D311yescharacteristic
29#63450—31no—
30C9×#102—31no—
31#63392—31no—
32#63392—61no—
33#63377—31no—
34Dic9⋊C8—31no—
35#63367—31no—
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9Q812yescomplemented
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9—42no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46#6189—42no—
47C3⋊(C9⋊C8)—21no—
48C9×C3⋊C8—21no—
49#5008—91no—
50#4761—91no—
51C2×C32×D4C2×D311yescharacteristic
52C22×C3×Dic3—61no—
53C2×#627—61no—
54C4⋊(C3⋊Dic3)—31no—
55C2×C3⋊(C3⋊C8)—31no—
56C3×#217—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C3×C4×Dic3—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61Dic3⋊Dic3—61no—
62C2×D4×C9—21no—
63C2×D4×C9C2×D311yescharacteristic
64C22×Dic9—21no—
65C2×Dic18—61no—
66C2×C8×C9—31no—
67C9×C4⋊C4—31no—
68#3676—21no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9C2×D311yescharacteristic
72Dic9⋊C4—61no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9—22no—
77C22×C3×C9—41no—
78C22×C3×C9—41no—
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9C2×Q811yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—41no—
83C9×Dic3—41no—
84C3×Dic9—21no—
85C3×Dic9—41no—
86C3×Dic9SD1612yes—
87C3×C4×D4—91no—
88D4×Dic3—31no—
89#888—91no—
90#869—91no—
91C3×#102—31no—
92#832—91no—
93#832—91no—
94#832—181no—
95#832D911yescharacteristic
96#817—91no—
97Dic3⋊C8—91no—
98#808—91no—
99#808—91no—
100#808—91no—
101C4⋊(C3⋊C8)—91no—
102C23×C32—21no—
103C32×D4—21no—
104C32×D4Dic612yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—122no—
112#627—122no—
113C23×C9—21no—
114C23×C9—41no—
115C3⋊(C3⋊C8)—61no—
116C3×C3⋊C8—21no—
117D4×C9—21no—
118D4×C9—22no—
119D4×C9—41no—
120D4×C9Dic612yescomplemented
121C2×C4×C9—21no—
122C2×C4×C9—61no—
123C2×C4×C9C22×D311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—42no—
126C2×Dic9—61no—
127C2×Dic9—121no—
128C2×Dic9#6711yescharacteristic
129C2×Dic9#6711yescharacteristic
130Dic18—122no—
131C8×C9—61no—
132C9⋊C8—61no—
133C9⋊C8—121no—
134#476—271noSylow
135C2×C3×C9—42no—
136C2×C3×C9C22⋊Q811yescharacteristic
137C2×C3×C9C2×SD1611yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4C2×D911yescharacteristic
141C2×C3×D4D3211yescharacteristic
142C22×C3×C4—181no—
143C22×Dic3—61no—
144C2×Dic6—181no—
145C2×Dic6—181no—
146C2×C3×C8—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—91no—
149C3×C22⋊C4—181no—
150C3×C42—91no—
151#217—61no—
152C4⋊Dic3—31no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3C2×D911yescharacteristic
158Dic3⋊C4—181no—
159C4×Dic3—31no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—181no—
163C2×C3⋊C8C2×D911yescharacteristic
164C22×C32—22no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32D3×D411yescharacteristic
168C32×C4C2×Dic611yescharacteristic
169C32×C4#23711yescharacteristic
170C3⋊Dic3—121no—
171C3×Dic3—32no—
172C3×Dic3—41no—
173C3×Dic3—61no—
174C3×Dic3—121no—
175C22×C9—22no—
176C22×C9—21no—
177C22×C9—42no—
178C22×C9—41no—
179C22×C9—41no—
180C22×C9—81no—
181C22×C9—81no—
182C22×C9C2×#6711yescharacteristic
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—121no—
186C4×C9C2×Dic611yescharacteristic
187C4×C9#23511yescharacteristic
188Dic9—21no—
189Dic9—41no—
190Dic9—121no—
191Dic9—241no—
192Dic9#21412yes—
193C4⋊Q8—271no—
194C4×D4—91no—
195#106—271no—
196#106—271no—
197C4⋊C8—271no—
198#102—31no—
199#102—271no—
200C3×C9#47611yescharacteristic, complemented, Sylow
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—41no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—22no—
207C3×D4—41no—
208C3×D4Dic1812yescomplemented
209C3×D4#62712yes—
210C2×C3×C4—21no—
211C2×C3×C4—61no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—181no—
215C2×C3×C4—362no—
216C2×C3×C4C22×D911yescharacteristic
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—61no—
222C2×Dic3—122no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225C2×Dic3—361no—
226Dic6—362no—
227Dic6—362no—
228C3×C8—61no—
229C3⋊C8—21no—
230C3⋊C8—181no—
231C3⋊C8—181no—
232C3⋊C8—361no—
233C2×C32—42no—
234C2×C32#87811yescharacteristic
235C2×C32#91111yescharacteristic
236C2×C32D3×SD1611yescharacteristic
237C2×C9—21no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9—42no—
241C2×C9—82no—
242C2×C9#92411yescharacteristic
243C2×C9C2×#21411yescharacteristic
244C2×C9#94911yescharacteristic
245C2×Q8—541no—
246C2×D4D3×D911yescharacteristic
247C22×C4—181no—
248C2×C8—31no—
249C2×C8—271no—
250C4⋊C4—31no—
251C4⋊C4—91no—
252C4⋊C4—271no—
253C4⋊C4—541no—
254C22⋊C4—181no—
255C42—91no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—42no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3D4×D911yescharacteristic
267C22×C3D3×#6711yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—92no—
271C3×C4—121no—
272C3×C4—181no—
273C3×C4—361no—
274C3×C4C2×Dic1811yescharacteristic
275C3×C4#369911yescharacteristic
276C3×C4#380411yescharacteristic
277C3×C4C2×#62711yescharacteristic
278Dic3—32no—
279Dic3—41no—
280Dic3—61no—
281Dic3—121no—
282Dic3—361no—
283Dic3—361no—
284Dic3—721no—
285C32#476111yescharacteristic
286C9—21no—
287C9#500811yescharacteristic, complemented
288C23—21no—
289Q8—1082no—
290D4—21no—
291D4#618912yescomplemented
292C2×C4—61no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—181no—
296C2×C4—362no—
297C2×C4—541no—
298C2×C4C2×D3×D911yescharacteristic
299C8—61no—
300C8—541no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—42no—
305C2×C3—42no—
306C2×C3—82no—
307C2×C3#6344011yescharacteristic
308C2×C3#6347311yescharacteristic
309C2×C3SD16×D911yescharacteristic
310C2×C3#6392511yescharacteristic
311C2×C3D3×#21411yescharacteristic
312C2×C3#6396411yescharacteristic
313C22—22no—
314C22—41no—
315C22—41no—
316C22D9×#6711yescharacteristic
317C4—92no—
318C4—121no—
319C4—181no—
320C4—361no—
321C4—1081no—
322C4C2×#618911yescharacteristic
323C4#8866211yescharacteristic
324C3—21no—
325C3#1058844111yescharacteristic, complemented
326C3#1059140811yescharacteristic
327C2—42no—
328C2#1172117311yescharacteristic
329C2D9×#21411yescharacteristic
330C2#1172125311yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#476
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720969C2semidirect
5. #11720278C2semidirect
6. Dic9⋊(C3⋊C8)C2semidirect
7. #11720265C2semidirect
38. C3×D4×C9Q8semidirect
120. D4×C9Dic6semidirect
200. C3×C9#476semidirect
208. C3×D4Dic18semidirect
287. C9#5008semidirect
291. D4#6189semidirect
325. C3#10588441semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Dic92
#117209692
#117205762
C9×#8322
#117202782
Dic9⋊(C3⋊C8)2
#117202652
#105914083
#105884413