Finite Group Databasebeta

← #13137547

Subgroups — #13137547

Order 1728.

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

normal subgroups only

This group has 418 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
1#11721749C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic, complemented
3C9×#808C211yesmaximal, characteristic, complemented
4C3×#63367C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic
8#10590433—31nomaximal
9#10588550—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×#6189—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14#88536—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64102—31no—
22#63919—31no—
23C3×#808—31no—
24C3×#808D311yescharacteristic
25#63572—31no—
26#63564—31no—
27#63564—31no—
28#63455—31no—
29#63455—61no—
30#63450—31no—
31C9×#106—31no—
32#63380—31no—
33Dic9⋊C8—31no—
34#63369—31no—
35#63367D311yescharacteristic, complemented
36C22×C3⋊D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)Q812yescomplemented
43C4×C3⋊D9—42no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46#6189—42no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4870—91no—
50#4870—91no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4×C3⋊D3—61no—
53C2×#627—61no—
54C32×C4⋊C4C2×D311yescharacteristic
55#3752—61no—
56C4×C3⋊Dic3—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3724—61no—
62C2×D36—31no—
63C2×D36—61no—
64C2×C4×D9—61no—
65C2×C4×D9—62no—
66C2×Dic18—61no—
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4C2×D311yescharacteristic
71(C2×D9)⋊C4—61no—
72(C2×D9)⋊C4—62no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—22no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C3×C4×C9—41no—
83C3×C4×C9C2×Q811yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9SD1612yes—
87C9×Dic3—41no—
88C3×Dic9—41no—
89#893—91no—
90#893—91no—
91#893—181no—
92#888—91no—
93#888—91no—
94C3×#106—31no—
95C3×#106—91no—
96#821—91no—
97#821—91no—
98Dic3⋊C8—91no—
99Dic3⋊C8—91no—
100#809—91no—
101#809—91no—
102#808—31no—
103#808D911yescharacteristic, complemented
104C22×C3⋊D3—61no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C4⋊(C3⋊D3)—32no—
110C4⋊(C3⋊D3)—61no—
111C4×C3⋊D3—122no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114#627—122no—
115C22×D9—62no—
116C22×D9—61no—
117C3×C3⋊C8—21no—
118C3×C3⋊C8—61no—
119C2×C4×C9—22no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—61no—
123C2×C4×C9C22×D311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—31no—
126C2×Dic9—31no—
127C2×Dic9—61no—
128C2×Dic9—61no—
129D36—32no—
130D36—62no—
131D36—62no—
132D36—61no—
133C4×D9—122no—
134C4×D9—124no—
135Dic18—122no—
136C8×C9—61no—
137C9⋊C8—21no—
138#476—271noSylow
139C2×C3×C9C22⋊Q811yescharacteristic
140C2×C3×C9C2×SD1611yescharacteristic
141C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
142C3⋊D9—42no—
143C2×D12—91no—
144C2×D12—91no—
145C2×D12—181no—
146C2×C4×D3—181no—
147C2×C4×D3—181no—
148C2×C4×D3—182no—
149C2×Dic6—181no—
150C2×Dic6—181no—
151C2×C3×C8—31no—
152C2×C3×C8—91no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4C2×D911yescharacteristic
157C3×C4⋊C4D3211yescharacteristic
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—182no—
161C4⋊Dic3—31no—
162C4⋊Dic3C2×D911yescharacteristic
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165C4×Dic3—91no—
166C4×Dic3—91no—
167C4×Dic3—181no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8C2×D911yescharacteristic
170C22×C32D3×D411yescharacteristic
171C2×C3⋊D3—62no—
172C2×C3⋊D3—121no—
173C2×C3⋊D3—121no—
174C32×C4—41no—
175C32×C4#23511yescharacteristic
176C32×C4D3×Q811yescharacteristic
177C3⋊Dic3—32no—
178C3⋊Dic3—61no—
179C3×Dic3—41no—
180C3×Dic3—121no—
181C22×C9—21no—
182C22×C9D3×D411yescharacteristic
183C2×D9—64no—
184C2×D9—62no—
185C2×D9—122no—
186C2×D9—122no—
187C2×D9—121no—
188C2×D9—121no—
189C4×C9—21no—
190C4×C9—21no—
191C4×C9—42no—
192C4×C9—41no—
193C4×C9—121no—
194C4×C9#23511yescharacteristic
195C4×C9D3×Q811yescharacteristic
196Dic9—32no—
197Dic9—41no—
198Dic9—62no—
199Dic9—61no—
200Dic9—121no—
201C4⋊Q8—271no—
202C4×D4—271no—
203#106—31no—
204#106—91no—
205C4⋊C8—271no—
206#102—271no—
207#102—271no—
208C3×C9#47611yescharacteristic, complemented, Sylow
209C22×D3—182no—
210C22×D3—181no—
211C22×D3—181no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—61no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—91no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228D12—92no—
229D12—92no—
230D12—182no—
231D12—182no—
232D12—181no—
233D12—181no—
234C4×D3—362no—
235C4×D3—362no—
236C4×D3—364no—
237Dic6—362no—
238Dic6—362no—
239C3×C8—61no—
240C3×C8—181no—
241C3⋊C8—21no—
242C3⋊C8—61no—
243C2×C32#89611yescharacteristic
244C2×C32D3×SD1611yescharacteristic
245C2×C32#91411yescharacteristic
246C3⋊D3—122no—
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—21no—
250C2×C9#89611yescharacteristic
251C2×C9D3×SD1611yescharacteristic
252C2×C9#91411yescharacteristic
253D9—124no—
254D9—122no—
255C2×Q8—541no—
256C2×D4—271no—
257C22×C4—541no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—31no—
261C4⋊C4—91no—
262C4⋊C4—541no—
263C4⋊C4D3×D911yescharacteristic
264C22⋊C4—541no—
265C42—271no—
266C22×C3—21no—
267C22×C3D4×D911yescharacteristic
268C22×C3#381111yescharacteristic
269C2×D3—184no—
270C2×D3—182no—
271C2×D3—182no—
272C2×D3—362no—
273C2×D3—362no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—361no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—42no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—121no—
284C3×C4—361no—
285C3×C4#369711yescharacteristic
286C3×C4Q8×D911yescharacteristic
287C3×C4#379711yescharacteristic
288C3×C4#379811yescharacteristic
289Dic3—41no—
290Dic3—92no—
291Dic3—92no—
292Dic3—121no—
293Dic3—182no—
294Dic3—181no—
295Dic3—181no—
296Dic3—361no—
297C32#487011yescharacteristic
298C9—21no—
299C9#487011yescharacteristic, complemented
300C23—541no—
301Q8—1082no—
302D4—272no—
303D4—541no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—181no—
307C2×C4—271no—
308C2×C4—271no—
309C2×C4—1082no—
310C2×C4C2×D3×D911yescharacteristic
311C8—61no—
312C8—181no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#6345811yescharacteristic
317C2×C3SD16×D911yescharacteristic
318C2×C3#6347611yescharacteristic
319C2×C3#6386511yescharacteristic
320C2×C3#6393011yescharacteristic
321C2×C3#6394111yescharacteristic
322D3—364no—
323D3—362no—
324D3—362no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22#8866811yescharacteristic
329C4—41no—
330C4—121no—
331C4—272no—
332C4—361no—
333C4—541no—
334C4(C3⋊D9)⋊Q811yescharacteristic
335C4#8864011yescharacteristic
336C3—21no—
337C3#1058855011yescharacteristic, complemented
338C3#1059043311yescharacteristic
339C2—1082no—
340C2#1172092311yescharacteristic
341C2#1172118211yescharacteristic
342C2#1172121311yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#476
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
2. #11720971C2semidirect
3. C9×#808C2semidirect
4. C3×#63367C2semidirect
6. #11720273C2semidirect
35. #63367D3semidirect
42. C4⋊(C3⋊D9)Q8semidirect
103. #808D9semidirect
208. C3×C9#476semidirect
299. C9#4870semidirect
337. C3#10588550semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117209712
C9×#8082
C3×#633672
#117202842
#117202732
#117202672
#105904333
#105885503