Finite Group Databasebeta

← #13137577

Subgroups — #13137577

Order 1728.

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

normal subgroups only

This group has 362 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#11721753C211yesmaximal, characteristic
2#11720922C211yesmaximal, characteristic
3C9×#808C211yesmaximal, characteristic, complemented
4C3×#63366C211yesmaximal, characteristic, complemented
5#11720282C211yesmaximal, characteristic
6#11720277C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590450—31nomaximal
9#10588548—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13#88536—21no—
14C4⋊(C3⋊Dic9)—21no—
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic3×Dic9—21no—
21#64106—31no—
22#63849—31no—
23C3×#808D311yescharacteristic
24C3×#807—31no—
25#63567—31no—
26#63567—31no—
27#63557—31no—
28#63459—31no—
29#63459—61no—
30#63454—31no—
31C9×#106—31no—
32(C2×D9)⋊C8—31no—
33#63376—31no—
34#63368—31no—
35#63366D311yescharacteristic, complemented
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C4×C3⋊D9—21no—
42C4×C3⋊D9—21no—
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4888—91no—
50#4868—91no—
51C2×C4×C3⋊D3—31no—
52C2×#634—31no—
53C32×C4⋊C4C2×D311yescharacteristic
54#3752—61no—
55C4⋊(C3⋊Dic3)—61no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60#3722—61no—
61Dic32—61no—
62C2×C4×D9—31no—
63C2×C4×D9—61no—
64C2×Dic18—31no—
65C2×Dic18—61no—
66C2×C8×C9—31no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—31no—
69C9×C4⋊C4C2×D311yescharacteristic
70(C2×D9)⋊C4—61no—
71(C2×D9)⋊C4—61no—
72(C2×D9)⋊C4—62no—
73C4⋊Dic9—61no—
74C4⋊Dic9—62no—
75C4⋊Dic9C2×D311yescharacteristic
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—21no—
80C2×C3⋊D9—21no—
81C2×C3⋊D9—21no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87C9×Dic3—41no—
88C3×Dic9—41no—
89#897—91no—
90#897—91no—
91#897—181no—
92#892—91no—
93#892—91no—
94C3×#107—91no—
95C3×#106—31no—
96(C2×D3)⋊C8—91no—
97(C2×D3)⋊C8—91no—
98#816—91no—
99#816—91no—
100#810—91no—
101#810—91no—
102#808D911yescharacteristic, complemented
103#807—31no—
104C22×C3⋊D3—31no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C4×C3⋊D3—61no—
110C4×C3⋊D3—61no—
111#634—61no—
112#634—61no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—61no—
115C22×D9—31no—
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—61no—
127C2×Dic9—62no—
128C2×Dic9—61no—
129C4×D9—61no—
130C4×D9—61no—
131C4×D9—121no—
132C4×D9—121no—
133Dic18—61no—
134Dic18—61no—
135Dic18—121no—
136Dic18—121no—
137C8×C9—61no—
138C9⋊C8—21no—
139#485—271noSylow
140C2×C3×C9#12311yescharacteristic
141C2×C3×C9#13511yescharacteristic
142C2×C3×C9#13711yescharacteristic
143C3⋊D9—41no—
144C2×C4×D3—91no—
145C2×C4×D3—91no—
146C2×C4×D3—181no—
147C2×Dic6—91no—
148C2×Dic6—91no—
149C2×Dic6—181no—
150C2×C3×C8—31no—
151C2×C3×C8—91no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4C2×D911yescharacteristic
156C3×C4⋊C4D3211yescharacteristic
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—182no—
162C4⋊Dic3—31no—
163C4⋊Dic3—181no—
164C4⋊Dic3—181no—
165C4⋊Dic3—182no—
166C4⋊Dic3C2×D911yescharacteristic
167C4×Dic3—181no—
168C4×Dic3—181no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32D3×D411yescharacteristic
172C2×C3⋊D3—61no—
173C2×C3⋊D3—61no—
174C2×C3⋊D3—61no—
175C32×C4—41no—
176C32×C4#23511yescharacteristic
177C32×C4#23911yescharacteristic
178C3⋊Dic3—61no—
179C3⋊Dic3—121no—
180C3×Dic3—41no—
181C3×Dic3—121no—
182C22×C9—21no—
183C22×C9D3×D411yescharacteristic
184C2×D9—62no—
185C2×D9—61no—
186C2×D9—61no—
187C2×D9—61no—
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#23911yescharacteristic
197Dic9—41no—
198Dic9—61no—
199Dic9—121no—
200Dic9—122no—
201Dic9—121no—
202C22⋊Q8—271no—
203#117—271no—
204#107—91no—
205#106—31no—
206#103—271no—
207#103—271no—
208C22⋊C8—271no—
209C3×C9#48511yescharacteristic, complemented, Sylow
210C22×D3—91no—
211C22×D3—91no—
212C22×D3—181no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—61no—
218C2×C3×C4—181no—
219C2×C3×C4C22×D911yescharacteristic
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—61no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—181no—
226C2×Dic3—182no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C4×D3—181no—
230C4×D3—181no—
231C4×D3—181no—
232C4×D3—181no—
233C4×D3—361no—
234C4×D3—361no—
235Dic6—181no—
236Dic6—181no—
237Dic6—181no—
238Dic6—181no—
239Dic6—361no—
240Dic6—361no—
241C3×C8—61no—
242C3×C8—181no—
243C3⋊C8—21no—
244C3⋊C8—61no—
245C2×C32#89411yescharacteristic
246C2×C32#91511yescharacteristic
247C2×C32#91611yescharacteristic
248C3⋊D3—121no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9#89411yescharacteristic
253C2×C9#91211yescharacteristic
254C2×C9#91811yescharacteristic
255D9—122no—
256D9—121no—
257C2×Q8—271no—
258C22×C4—271no—
259C2×C8—31no—
260C2×C8—91no—
261C4⋊C4—31no—
262C4⋊C4—91no—
263C4⋊C4—541no—
264C4⋊C4D3×D911yescharacteristic
265C22⋊C4—541no—
266C22⋊C4—541no—
267C42—541no—
268C22×C3—21no—
269C22×C3D4×D911yescharacteristic
270C22×C3#381111yescharacteristic
271C2×D3—182no—
272C2×D3—181no—
273C2×D3—181no—
274C2×D3—181no—
275C2×D3—181no—
276C2×D3—181no—
277C2×D3—181no—
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#370111yescharacteristic
289C3×C4#379911yescharacteristic
290C3×C4#379911yescharacteristic
291Dic3—41no—
292Dic3—121no—
293Dic3—181no—
294Dic3—181no—
295Dic3—361no—
296Dic3—362no—
297Dic3—361no—
298Dic3—361no—
299C32#486811yescharacteristic
300C9—21no—
301C9#488811yescharacteristic, complemented
302C23—271no—
303Q8—541no—
304Q8—541no—
305C2×C4—21no—
306C2×C4—61no—
307C2×C4—181no—
308C2×C4—271no—
309C2×C4—541no—
310C2×C4—541no—
311C2×C4—541no—
312C2×C4C2×D3×D911yescharacteristic
313C8—61no—
314C8—181no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#6345611yescharacteristic
319C2×C3#6347711yescharacteristic
320C2×C3#6347811yescharacteristic
321C2×C3#6389711yescharacteristic
322C2×C3#6393611yescharacteristic
323C2×C3#6395111yescharacteristic
324D3—362no—
325D3—361no—
326D3—361no—
327C22—541no—
328C22—541no—
329C22—541no—
330C22#8866811yescharacteristic
331C4—41no—
332C4—121no—
333C4—361no—
334C4—541no—
335C4—1081no—
336C4#8863511yescharacteristic
337C4#8864211yescharacteristic
338C3—21no—
339C3#1058854811yescharacteristic, complemented
340C3#1059045011yescharacteristic
341C2—1081no—
342C2#1172099711yescharacteristic
343C2#1172118611yescharacteristic
344C2#1172121711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#485
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
3. C9×#808C2semidirect
4. C3×#63366C2semidirect
6. #11720277C2semidirect
7. #11720269C2semidirect
35. #63366D3semidirect
102. #808D9semidirect
209. C3×C9#485semidirect
301. C9#4888semidirect
339. C3#10588548semidirect

Maximal subgroups

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

subgroupindex
#117217532
#117209222
C9×#8082
C3×#633662
#117202822
#117202772
#117202692
#105904503
#105885483