Finite Group Databasebeta

← #13140707

Subgroups — #13140707

Order 1728.

364 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721857C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic, complemented
3C9×#835C211yesmaximal, characteristic, complemented
4C3×#63395C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic
6#11720271C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic
8#10591539—31nomaximal
9#10588480—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12#88536—21no—
13#88536—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64156—31no—
22#63872—31no—
23C3×#835—31no—
24C3×#835D311yescharacteristic
25#63564—31no—
26#63564—31no—
27#63553—31no—
28#63509—31no—
29#63509—61no—
30#63452—31no—
31C9×#103—31no—
32#63395D311yescharacteristic, complemented
33#63380—31no—
34C8×Dic9—31no—
35#63369—31no—
36C22×C3⋊D9—21no—
37C3×Q8×C9—21no—
38C3×Q8×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4800—91no—
50#4800—91no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C4⋊(C3⋊D3)—31no—
53#3752—61no—
54#3752—61no—
55C4×C3⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62C2×Q8×C9—21no—
63C2×Q8×C9C2×D311yescharacteristic
64C2×D36—31no—
65C2×D36—61no—
66C2×C8×C9—31no—
67C9×C4⋊C4—31no—
68(C2×D9)⋊C4—61no—
69(C2×D9)⋊C4—61no—
70(C2×D9)⋊C4—62no—
71(C2×D9)⋊C4—62no—
72C4⋊Dic9—61no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—21no—
87C9×Dic3—41no—
88C3×Dic9—41no—
89#947—91no—
90#947—91no—
91#947—181no—
92#890—91no—
93#890—91no—
94C3×#103—31no—
95C3×#103—91no—
96#835—31no—
97#835D911yescharacteristic, complemented
98#821—91no—
99#821—91no—
100C8×Dic3—91no—
101C8×Dic3—91no—
102#809—91no—
103#809—91no—
104C22×C3⋊D3—61no—
105C32×Q8—21no—
106C32×Q8—21no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—31no—
111C4⋊(C3⋊D3)—61no—
112C4⋊(C3⋊D3)—61no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—61no—
115C22×D9—62no—
116C22×D9—61no—
117C3×C3⋊C8—21no—
118C3×C3⋊C8—61no—
119Q8×C9—21no—
120Q8×C9—21no—
121Q8×C9—41no—
122Q8×C9—41no—
123C2×C4×C9—22no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128C2×Dic9—21no—
129C2×Dic9—31no—
130C2×Dic9—31no—
131C2×Dic9—61no—
132C2×Dic9—61no—
133D36—61no—
134D36—61no—
135D36—121no—
136D36—121no—
137C8×C9—61no—
138C9⋊C8—21no—
139#488—271noSylow
140C2×C3×C9#12411yescharacteristic
141C2×C3×C9#13511yescharacteristic
142C2×C3×C9#13511yescharacteristic
143C3⋊D9—81no—
144C2×C3×Q8—21no—
145C2×C3×Q8C2×D911yescharacteristic
146C2×C3×Q8D3211yescharacteristic
147C2×D12—91no—
148C2×D12—91no—
149C2×D12—181no—
150C2×C3×C8—31no—
151C2×C3×C8—91no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—91no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—182no—
159(C2×D3)⋊C4—182no—
160C4⋊Dic3—31no—
161C4⋊Dic3—181no—
162C4⋊Dic3—181no—
163C4⋊Dic3C2×D911yescharacteristic
164Dic3⋊C4—181no—
165Dic3⋊C4—181no—
166C4×Dic3—91no—
167C4×Dic3—91no—
168C4×Dic3—181no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32D3×D411yescharacteristic
172C2×C3⋊D3—121no—
173C2×C3⋊D3—121no—
174C2×C3⋊D3—121no—
175C32×C4—41no—
176C32×C4#23511yescharacteristic
177C32×C4#23711yescharacteristic
178C3⋊Dic3—61no—
179C3⋊Dic3—61no—
180C3×Dic3—41no—
181C3×Dic3—121no—
182C22×C9—21no—
183C22×C9D3×D411yescharacteristic
184C2×D9—122no—
185C2×D9—122no—
186C2×D9—122no—
187C2×D9—121no—
188C2×D9—121no—
189C2×D9—121no—
190C4×C9—21no—
191C4×C9—21no—
192C4×C9—42no—
193C4×C9—41no—
194C4×C9—121no—
195C4×C9#23511yescharacteristic
196C4×C9#23711yescharacteristic
197Dic9—41no—
198Dic9—61no—
199Dic9—61no—
200Dic9—121no—
201Dic9—121no—
202#125—271no—
203#124—271no—
204#103—31no—
205#103—91no—
206#102—271no—
207#102—271no—
208C4×C8—271no—
209C3×C9#48811yescharacteristic, complemented, Sylow
210C22×D3—182no—
211C22×D3—181no—
212C22×D3—181no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—41no—
218C3×Q8—41no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—61no—
224C2×C3×C4—181no—
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—61no—
229C2×Dic3—91no—
230C2×Dic3—91no—
231C2×Dic3—91no—
232C2×Dic3—91no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235D12—181no—
236D12—181no—
237D12—181no—
238D12—181no—
239D12—361no—
240D12—361no—
241C3×C8—61no—
242C3×C8—181no—
243C3⋊C8—21no—
244C3⋊C8—61no—
245C2×C32#88511yescharacteristic
246C2×C32#91611yescharacteristic
247C2×C32#91911yescharacteristic
248C3⋊D3—241no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9#88511yescharacteristic
253C2×C9#91611yescharacteristic
254C2×C9#91911yescharacteristic
255D9—242no—
256D9—241no—
257C2×Q8D3×D911yescharacteristic
258C2×D4—271no—
259C2×C8—31no—
260C2×C8—91no—
261C4⋊C4—31no—
262C4⋊C4—91no—
263C4⋊C4—541no—
264C4⋊C4—541no—
265C22⋊C4—541no—
266C22⋊C4—541no—
267C42—271no—
268C22×C3—21no—
269C22×C3D4×D911yescharacteristic
270C22×C3#381111yescharacteristic
271C2×D3—362no—
272C2×D3—362no—
273C2×D3—362no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—361no—
278C2×D3—361no—
279C2×D3—361no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—42no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—121no—
286C3×C4—361no—
287C3×C4#369711yescharacteristic
288C3×C4#369911yescharacteristic
289C3×C4#380411yescharacteristic
290C3×C4#380411yescharacteristic
291Dic3—41no—
292Dic3—121no—
293Dic3—181no—
294Dic3—181no—
295Dic3—181no—
296Dic3—181no—
297Dic3—361no—
298Dic3—361no—
299C32#480011yescharacteristic
300C9—21no—
301C9#480011yescharacteristic, complemented
302C23—541no—
303Q8—21no—
304Q8—21no—
305D4—541no—
306D4—541no—
307C2×C4—21no—
308C2×C4—61no—
309C2×C4—181no—
310C2×C4—271no—
311C2×C4—271no—
312C2×C4C2×D3×D911yescharacteristic
313C8—61no—
314C8—181no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#6344711yescharacteristic
319C2×C3#6347811yescharacteristic
320C2×C3#6348111yescharacteristic
321C2×C3#6395011yescharacteristic
322C2×C3#6395011yescharacteristic
323C2×C3#6395311yescharacteristic
324D3—722no—
325D3—721no—
326D3—721no—
327C22—1081no—
328C22—1081no—
329C22—1081no—
330C22#8866811yescharacteristic
331C4—41no—
332C4—121no—
333C4—361no—
334C4—541no—
335C4—541no—
336C4#8865811yescharacteristic
337C4#8866211yescharacteristic
338C3—21no—
339C3#1058848011yescharacteristic, complemented
340C3#1059153911yescharacteristic
341C2—2161no—
342C2#1172121611yescharacteristic
343C2#1172122411yescharacteristic
344C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#488
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
2. #11720937C2semidirect
3. C9×#835C2semidirect
4. C3×#63395C2semidirect
6. #11720271C2semidirect
32. #63395D3semidirect
97. #835D9semidirect
209. C3×C9#488semidirect
301. C9#4800semidirect
339. C3#10588480semidirect

Maximal subgroups

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

subgroupindex
#117218572
#117209372
C9×#8352
C3×#633952
#117202842
#117202712
#117202672
#105915393
#105884803