Finite Group Databasebeta

← #13137698

Subgroups — #13137698

Order 1728.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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
1D3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2#11721038C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic, complemented
4C3×#63367C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
6#11720267C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10590423—31nomaximal
9#10588651—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×D3×C9—21no—
12C2×C4×D3×C9C2211yescharacteristic
13C2×#6195—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C9×Dic3⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88445—21no—
21C3×D3×C4⋊C4D311yescharacteristic
22#63909—31no—
23#63636—31no—
24C3×#808—31no—
25#63574—31no—
26#63564—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28C2×C9×C4⋊C4—31no—
29#63492—31no—
30#63389—31no—
31#63369—31no—
32#63369—31no—
33#63369—61no—
34#63367—31no—
35#63367D311yescharacteristic, complemented
36C22×C3⋊D9—21no—
37C22×D3×C9D411yescharacteristic
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)—21no—
42C2×C9×Dic3—21no—
43C2×C9×Dic3D411yescharacteristic
44C2×C3×Dic9—21no—
45C4×D3×C9—21no—
46C4×D3×C9—21no—
47C4×D3×C9—42no—
48#6195—42no—
49C3⋊(C9⋊C8)—21no—
50C3×C9⋊C8—21no—
51#4968—91no—
52#4863—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C3×C4×D3—21no—
55C2×C3×C4×D3C2×D311yescharacteristic
56C2×#626—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×Dic3⋊C4—21no—
62C2×C3×C3⋊C8—31no—
63#3722—61no—
64C22×C4×C9—31no—
65C22×C4×C9—61no—
66C2×#611—61no—
67C2×D36—31no—
68C2×D36—61no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4—31no—
71C9×C4⋊C4—61no—
72C9×C4⋊C4C2×D311yescharacteristic
73(C2×D9)⋊C4—61no—
74C4⋊Dic9C2×D311yescharacteristic
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C2×D3×C9—21no—
83C2×D3×C9—21no—
84C2×D3×C9SD1612yes—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C9×Dic3—21no—
89C9×Dic3—41no—
90C3×Dic9—41no—
91C2×C3×C4⋊C4—31no—
92#930—91no—
93#895—91no—
94D3×C4⋊C4D911yescharacteristic, complemented
95C3×#106—91no—
96#830—91no—
97#821—91no—
98(C2×D3)⋊C8—91no—
99#809—91no—
100#809—91no—
101#809—91no—
102#809—181no—
103#808—31no—
104#808—91no—
105#808—91no—
106C22×C3⋊D3—61no—
107C22×C3×D3#6711yescharacteristic
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C4⋊(C3⋊D3)—61no—
111C4⋊(C3⋊D3)—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3#6711yescharacteristic
115C3×C4×D3—21no—
116C3×C4×D3—21no—
117C3×C4×D3—42no—
118#626—122no—
119C23×C9—31no—
120C22×D9—61no—
121C22×D9—121no—
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—61no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—31no—
127C2×C4×C9—41no—
128C2×C4×C9—61no—
129C2×C4×C9—61no—
130C2×C4×C9—61no—
131C2×C4×C9—122no—
132C2×C4×C9C22×D311yescharacteristic
133#611—122no—
134C2×Dic9—21no—
135D36—62no—
136D36—61no—
137D36—61no—
138D36—121no—
139C9⋊C8—21no—
140C9⋊C8—61no—
141C9⋊C8—121no—
142#481—271noSylow
143C2×C3×C9#12311yescharacteristic
144C2×C3×C9C2×SD1611yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C3⋊D9—81no—
147D3×C9—22no—
148C22×C3×C4—31no—
149C22×C3×C4—61no—
150C2×#67—181no—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×D12—181no—
155C2×C4×D3—21no—
156C2×C4×D3C2×D911yescharacteristic
157C2×C3×C8—91no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—91no—
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—21no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—91no—
171C2×C3⋊C8—91no—
172C2×C3⋊C8—181no—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—121no—
175C2×C3⋊D3—121no—
176C2×C3⋊D3—121no—
177C2×C3×D3—21no—
178C2×C3×D3—21no—
179C2×C3×D3#21512yes—
180C32×C4—41no—
181C32×C4#23511yescharacteristic
182C32×C4#23511yescharacteristic
183C3×Dic3—21no—
184C3×Dic3—41no—
185C3×Dic3—121no—
186C22×C9—21no—
187C22×C9—32no—
188C22×C9—61no—
189C22×C9—61no—
190C22×C9D3×D411yescharacteristic
191C2×D9—122no—
192C2×D9—121no—
193C2×D9—121no—
194C2×D9—121no—
195C2×D9—241no—
196C2×D9—241no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—41no—
200C4×C9—61no—
201C4×C9—81no—
202C4×C9—121no—
203C4×C9#23511yescharacteristic
204C4×C9#23911yescharacteristic
205Dic9—41no—
206#121—271no—
207C2×C4⋊C4—31no—
208#106—91no—
209#106—271no—
210#102—271no—
211#102—271no—
212C22⋊C8—271no—
213C3×C9#48111yescharacteristic, complemented, Sylow
214C23×C3—31no—
215C22×D3—181no—
216C22×D3—181no—
217C22×D3—361no—
218C22×D3#61111yescharacteristic
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—31no—
223C2×C3×C4—41no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—122no—
228C2×C3×C4—181no—
229C2×C3×C4C22×D911yescharacteristic
230C2×C3×C4C2×D3211yescharacteristic
231#67—362no—
232C2×Dic3—21no—
233C2×Dic3—61no—
234C2×Dic3#61111yescharacteristic
235D12—182no—
236D12—181no—
237D12—181no—
238D12—181no—
239D12—181no—
240D12—361no—
241D12—362no—
242C4×D3—21no—
243C4×D3—21no—
244C4×D3—42no—
245C3×C8—181no—
246C3⋊C8—61no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C2×C32#92911yescharacteristic
251C2×C32#93211yescharacteristic
252C2×C32C2×#21511yescharacteristic
253C3⋊D3—241no—
254C3×D3—22no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—62no—
259C2×C9#89411yescharacteristic
260C2×C9D3×SD1611yescharacteristic
261C2×C9#91411yescharacteristic
262D9—242no—
263D9—241no—
264C2×D4—271no—
265C2×D4—541no—
266C22×C4—31no—
267C22×C4—61no—
268C2×C8—91no—
269C2×C8—271no—
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—91no—
273C4⋊C4D3×D911yescharacteristic
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—32no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3C2×#61111yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—362no—
284C2×D3—361no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—361no—
289C2×D3—361no—
290C2×D3—721no—
291C2×D3—721no—
292C2×D3#367512yes—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—61no—
298C3×C4—81no—
299C3×C4—121no—
300C3×C4—361no—
301C3×C4#369711yescharacteristic
302C3×C4#369711yescharacteristic
303C3×C4#379811yescharacteristic
304C3×C4#379911yescharacteristic
305Dic3—21no—
306Dic3—41no—
307Dic3—121no—
308C32#496811yescharacteristic
309C9—21no—
310C9#486311yescharacteristic, complemented
311C23—31no—
312C23—541no—
313D4—541no—
314D4—541no—
315D4—1082no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—181no—
323C2×C4C2×D3×D911yescharacteristic
324C8—181no—
325C8—541no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—62no—
330C2×C3#6349111yescharacteristic
331C2×C3#6349511yescharacteristic
332C2×C3C2×#367511yescharacteristic
333C2×C3#6385011yescharacteristic
334C2×C3#6393411yescharacteristic
335C2×C3D3×#21511yescharacteristic
336D3—22no—
337D3—722no—
338D3—721no—
339D3—721no—
340C22—32no—
341C22—61no—
342C22—61no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D3×#61111yescharacteristic
347C4—41no—
348C4—61no—
349C4—121no—
350C4—361no—
351C4#8864011yescharacteristic
352C4#8864211yescharacteristic
353C3—21no—
354C3#1058865111yescharacteristic, complemented
355C3#1059042311yescharacteristic
356C2—62no—
357C2—2161no—
358C2#1172095411yescharacteristic
359C2#1172117011yescharacteristic
360C2D3×#367511yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radical#11721038C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#481
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. D3×C9×C4⋊C4C2semidirect
3. #11720528C2semidirect
4. C3×#63367C2semidirect
5. (C2×D3)⋊(C9⋊C8)C2semidirect
6. #11720267C2semidirect
7. #11720265C2semidirect
35. #63367D3semidirect
94. D3×C4⋊C4D9semidirect
213. C3×C9#481semidirect
310. C9#4863semidirect
354. C3#10588651semidirect

Maximal subgroups

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

subgroupindex
D3×C9×C4⋊C42
#117210382
#117205282
C3×#633672
(C2×D3)⋊(C9⋊C8)2
#117202672
#117202652
#105904233
#105886513