Finite Group Databasebeta

← #13137589

Subgroups — #13137589

Order 1728.

392 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 85 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×C4⋊C4×D9C211yesmaximal, characteristic, complemented
2#11721004C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic, complemented
4C9×#807C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10590422—31nomaximal
9#10588562—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×C4×D9—21no—
12C2×C3×C4×D9C2211yescharacteristic
13C2×#6192—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20#88445—21no—
21C3×D3×C4⋊C4—31no—
22#63909—31no—
23#63636—31no—
24C3×#807D311yescharacteristic
25#63576—31no—
26#63564—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28#63457—31no—
29C4⋊C4×D9D311yescharacteristic, complemented
30C9×#107—31no—
31#63380—31no—
32(C2×D9)⋊C8—31no—
33#63369—31no—
34#63369—61no—
35#63366—31no—
36C22×C3⋊D9—21no—
37C22×C3×D9D411yescharacteristic
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C3×C4×D9—21no—
46C3×C4×D9—21no—
47C3×C4×D9—42no—
48#6192—42no—
49C3⋊(C9⋊C8)—21no—
50C9×C3⋊C8—21no—
51#4967—91no—
52#4882—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—61no—
56C2×#626—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×Dic3⋊C4—61no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63#3722—61no—
64C2×D36—31no—
65C2×D36—61no—
66C2×D36—61no—
67C2×C4×D9—21no—
68C2×C4×D9C2×D311yescharacteristic
69C2×C8×C9—31no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4—31no—
72C9×C4⋊C4C2×D311yescharacteristic
73(C2×D9)⋊C4—61no—
74C4⋊Dic9C2×D311yescharacteristic
75Dic9⋊C4—21no—
76C2×C9⋊C8—31no—
77C2×C9⋊C8—61no—
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C2×C3×D9—21no—
83C2×C3×D9—21no—
84C2×C3×D9D812yes—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C9×Dic3—41no—
89C3×Dic9—21no—
90C3×Dic9—41no—
91C2×C3×C4⋊C4—91no—
92#930—91no—
93#895—91no—
94D3×C4⋊C4—31no—
95C3×#107—31no—
96#830—91no—
97#821—91no—
98(C2×D3)⋊C8—91no—
99#809—91no—
100#809—91no—
101#809—91no—
102#809—181no—
103#807—91no—
104#807—91no—
105#807D911yescharacteristic, complemented
106C22×C3⋊D3—61no—
107C22×C3×D3—31no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C4⋊(C3⋊D3)—61no—
111C4⋊(C3⋊D3)—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×C4×D3—122no—
118#626—122no—
119C22×D9—61no—
120C22×D9—121no—
121C22×D9#6711yescharacteristic
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—21no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—41no—
127C2×C4×C9—61no—
128C2×C4×C9C22×D311yescharacteristic
129C2×Dic9—21no—
130C2×Dic9#6711yescharacteristic
131D36—62no—
132D36—61no—
133D36—61no—
134D36—121no—
135D36—122no—
136C4×D9—21no—
137C4×D9—21no—
138C4×D9—42no—
139C8×C9—61no—
140C9⋊C8—61no—
141C9⋊C8—121no—
142#480—271noSylow
143C2×C3×C9#12311yescharacteristic
144C2×C3×C9C2×D811yescharacteristic
145C2×C3×C9#13711yescharacteristic
146C3⋊D9—81no—
147C3×D9—22no—
148C22×C3×C4—91no—
149C22×C3×C4—181no—
150C2×#67—181no—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×D12—181no—
155C2×C4×D3—31no—
156C2×C4×D3—61no—
157C2×C3×C8—31no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4C2×D911yescharacteristic
163C3×C4⋊C4D3211yescharacteristic
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3C2×D911yescharacteristic
168Dic3⋊C4—61no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—91no—
171C2×C3⋊C8—181no—
172C2×C3⋊C8C2×D911yescharacteristic
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—121no—
175C2×C3⋊D3—121no—
176C2×C3⋊D3—121no—
177C2×C3×D3—32no—
178C2×C3×D3—61no—
179C2×C3×D3—61no—
180C32×C4—41no—
181C32×C4#23511yescharacteristic
182C32×C4#23911yescharacteristic
183C3×Dic3—41no—
184C3×Dic3—61no—
185C3×Dic3—121no—
186C22×C9—21no—
187C22×C9C2×#6711yescharacteristic
188C2×D9—21no—
189C2×D9—21no—
190C2×D9—122no—
191C2×D9—121no—
192C2×D9—121no—
193C2×D9—121no—
194C2×D9—241no—
195C2×D9—241no—
196C2×D9#21312yes—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—41no—
200C4×C9—81no—
201C4×C9—121no—
202C4×C9#23511yescharacteristic
203C4×C9#23511yescharacteristic
204Dic9—21no—
205Dic9—41no—
206#121—271no—
207C2×C4⋊C4—91no—
208#107—31no—
209#107—271no—
210#102—271no—
211#102—271no—
212C22⋊C8—271no—
213C3×C9#48011yescharacteristic, complemented, Sylow
214C23×C3—91no—
215C22×D3—31no—
216C22×D3—181no—
217C22×D3—181no—
218C22×D3—361no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—362no—
229C2×C3×C4C22×D911yescharacteristic
230C2×C3×C4C2×D3211yescharacteristic
231#67—362no—
232C2×Dic3—21no—
233C2×Dic3—31no—
234C2×Dic3—61no—
235D12—182no—
236D12—181no—
237D12—181no—
238D12—181no—
239D12—181no—
240D12—361no—
241D12—362no—
242C4×D3—61no—
243C4×D3—61no—
244C4×D3—122no—
245C3×C8—61no—
246C3⋊C8—21no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C2×C32#89411yescharacteristic
251C2×C32D3×D811yescharacteristic
252C2×C32#91811yescharacteristic
253C3⋊D3—241no—
254C3×D3—62no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9#92911yescharacteristic
259C2×C9C2×#21311yescharacteristic
260C2×C9#94211yescharacteristic
261D9—22no—
262D9—242no—
263D9—241no—
264C2×D4—271no—
265C2×D4—541no—
266C22×C4—91no—
267C22×C4—181no—
268C2×C8—31no—
269C2×C8—271no—
270C4⋊C4—31no—
271C4⋊C4—91no—
272C4⋊C4—181no—
273C4⋊C4D3×D911yescharacteristic
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—92no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D4×D911yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—32no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—362no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—361no—
289C2×D3—361no—
290C2×D3—361no—
291C2×D3—721no—
292C2×D3—721no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—81no—
298C3×C4—121no—
299C3×C4—181no—
300C3×C4—361no—
301C3×C4#369711yescharacteristic
302C3×C4#370111yescharacteristic
303C3×C4#379811yescharacteristic
304C3×C4#379911yescharacteristic
305Dic3—41no—
306Dic3—61no—
307Dic3—121no—
308C32#488211yescharacteristic
309C9—21no—
310C9#496711yescharacteristic, complemented
311C23—91no—
312C23—541no—
313D4—541no—
314D4—541no—
315D4—1082no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—91no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—362no—
323C2×C4C2×D3×D911yescharacteristic
324C8—61no—
325C8—541no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—182no—
330C2×C3#6345611yescharacteristic
331C2×C3D8×D911yescharacteristic
332C2×C3#6348011yescharacteristic
333C2×C3#6385011yescharacteristic
334C2×C3D3×#21311yescharacteristic
335C2×C3#6394811yescharacteristic
336D3—62no—
337D3—722no—
338D3—721no—
339D3—721no—
340C22—92no—
341C22—181no—
342C22—181no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D9×#6711yescharacteristic
347C4—41no—
348C4—121no—
349C4—181no—
350C4—361no—
351C4#8863511yescharacteristic
352C4#8864011yescharacteristic
353C3—21no—
354C3#1058856211yescharacteristic, complemented
355C3#1059042211yescharacteristic
356C2—182no—
357C2—2161no—
358C2#1172093811yescharacteristic
359C2D9×#21311yescharacteristic
360C2#1172122211yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721004C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#480
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
1. C3×C4⋊C4×D9C2semidirect
3. #11720528C2semidirect
4. C9×#807C2semidirect
5. #11720284C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720263C2semidirect
29. C4⋊C4×D9D3semidirect
105. #807D9semidirect
213. C3×C9#480semidirect
310. C9#4967semidirect
354. C3#10588562semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊C4×D92
#117210042
#117205282
C9×#8072
#117202842
(C2×D9)⋊(C3⋊C8)2
#117202632
#105904223
#105885623