Finite Group Databasebeta

← #13137756

Subgroups — #13137756

Order 1728.

380 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 81 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×#63452C211yesmaximal, characteristic, complemented
2#11721040C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic, complemented
4C9×#809C211yesmaximal, characteristic, complemented
5#11720294C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720274C211yesmaximal, characteristic, complemented
8#10590550—31nomaximal
9#10588444—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×#6195—21no—
13C2×#6195—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×C4⋊Dic9—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C3×C4×Dic9C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C3×#890—31no—
22#63911—31no—
23#63636—31no—
24C3×#809D311yescharacteristic
25#63564—31no—
26#63562—31no—
27#63554—31no—
28#63502—31no—
29#63452D311yescharacteristic, complemented
30C9×#102—31no—
31#63392—31no—
32#63380—31no—
33#63375—31no—
34#63369—31no—
35#63369—61no—
36C22×C3⋊D9—21no—
37C22×D3×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45C9×D12—21no—
46C9×D12—21no—
47#6195—42no—
48#6195—42no—
49C3⋊(C9⋊C8)—21no—
50C9×C3⋊C8—21no—
51#5067—91no—
52#4764—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C3×D12C2×D311yescharacteristic
55C2×#626—61no—
56C2×#626—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—61no—
62C3×C4×Dic3—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64C2×D4×C9—31no—
65C2×#611—61no—
66C2×#611—61no—
67C2×D36—31no—
68C2×D36—61no—
69C2×C8×C9—31no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4C2×D311yescharacteristic
72C4⋊Dic9—21no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—21no—
75C4×Dic9C2×D311yescharacteristic
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×D3×C9—41no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3×Dic9—21no—
89C3×Dic9—21no—
90C3×Dic9—41no—
91C3×#125—91no—
92#940—91no—
93#890—31no—
94#874—91no—
95C3×#102—31no—
96#832—91no—
97#821—91no—
98#815—91no—
99#809—91no—
100#809—91no—
101#809—91no—
102#809—91no—
103#809—181no—
104#809D911yescharacteristic, complemented
105#803—91no—
106C22×C3⋊D3—61no—
107C22×C3×D3—21no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C4⋊(C3⋊D3)—61no—
111C4⋊(C3⋊D3)—61no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—61no—
115C3×D12—21no—
116C3×D12—21no—
117#626—122no—
118#626—122no—
119C23×C9—61no—
120C22×D9—61no—
121C22×D9—121no—
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—21no—
124D4×C9—61no—
125D4×C9—61no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—41no—
129C2×C4×C9C22×D311yescharacteristic
130#611—122no—
131#611—122no—
132C2×Dic9—21no—
133C2×Dic9#6711yescharacteristic
134C2×Dic9#6711yescharacteristic
135D36—62no—
136D36—61no—
137D36—61no—
138D36—121no—
139C8×C9—61no—
140C9⋊C8—61no—
141C9⋊C8—121no—
142#490—271noSylow
143C2×C3×C9#12411yescharacteristic
144C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C3⋊D9—81no—
147D3×C9—81no—
148C2×C3×D4—31no—
149C2×#67—181no—
150C2×#67—181no—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×D12—181no—
155C2×D12—181no—
156C2×D12C2×D911yescharacteristic
157C2×C3×C8—31no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4C2×D911yescharacteristic
163C3×C4⋊C4D3211yescharacteristic
164C3×C42—91no—
165C4⋊Dic3—31no—
166C4⋊Dic3—61no—
167Dic3⋊C4—61no—
168C4×Dic3—31no—
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—41no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C32×C4—41no—
181C32×C4#23511yescharacteristic
182C32×C4#23711yescharacteristic
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—121no—
186C22×C9—21no—
187C22×C9—121no—
188C22×C9—121no—
189C22×C9—121no—
190C22×C9C2×#6711yescharacteristic
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—81no—
201C4×C9#23911yescharacteristic
202C4×C9#23911yescharacteristic
203Dic9—21no—
204Dic9—21no—
205Dic9—41no—
206#127—271no—
207#125—91no—
208#102—31no—
209#102—271no—
210#102—271no—
211#102—271no—
212#97—271no—
213C3×C9#49011yescharacteristic, complemented, Sylow
214C23×C3—61no—
215C22×D3—21no—
216C22×D3—181no—
217C22×D3—181no—
218C22×D3—361no—
219C3×D4—61no—
220C3×D4—61no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230#67—362no—
231#67—362no—
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—61no—
235D12—21no—
236D12—21no—
237D12—182no—
238D12—181no—
239D12—181no—
240D12—181no—
241D12—181no—
242D12—361no—
243D12—362no—
244D12—362no—
245C3×C8—61no—
246C3⋊C8—21no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C2×C32#88511yescharacteristic
251C2×C32#91111yescharacteristic
252C2×C32#91411yescharacteristic
253C3⋊D3—241no—
254C3×D3—81no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—241no—
259C2×C9#94711yescharacteristic
260C2×C9#94911yescharacteristic
261C2×C9#94911yescharacteristic
262D9—242no—
263D9—241no—
264C2×D4—31no—
265C2×D4—271no—
266C2×D4—541no—
267C2×D4—541no—
268C2×C8—31no—
269C2×C8—271no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4—181no—
273C4⋊C4D3×D911yescharacteristic
274C42—91no—
275C22×C3—21no—
276C22×C3—121no—
277C22×C3—121no—
278C22×C3—121no—
279C22×C3D4×D911yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
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—181no—
299C3×C4—181no—
300C3×C4—361no—
301C3×C4#369711yescharacteristic
302C3×C4#369911yescharacteristic
303C3×C4#379611yescharacteristic
304C3×C4#379911yescharacteristic
305Dic3—61no—
306Dic3—61no—
307Dic3—121no—
308C32#476411yescharacteristic
309C9—21no—
310C9#506711yescharacteristic, complemented
311C23—61no—
312C23—541no—
313D4—61no—
314D4—61no—
315D4—541no—
316D4—541no—
317D4—1082no—
318D4—1082no—
319C2×C4—21no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—181no—
323C2×C4C2×D3×D911yescharacteristic
324C8—61no—
325C8—541no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—241no—
330C2×C3#6344711yescharacteristic
331C2×C3#6347311yescharacteristic
332C2×C3#6347611yescharacteristic
333C2×C3#6390711yescharacteristic
334C2×C3#6392511yescharacteristic
335C2×C3#6393911yescharacteristic
336D3—81no—
337D3—722no—
338D3—721no—
339D3—721no—
340C22—121no—
341C22—121no—
342C22—121no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D9×#6711yescharacteristic
347C4—41no—
348C4—181no—
349C4—181no—
350C4—361no—
351C4#8863911yescharacteristic
352C4#8864211yescharacteristic
353C3—21no—
354C3#1058844411yescharacteristic, complemented
355C3#1059055011yescharacteristic
356C2—241no—
357C2—2161no—
358C2#1172103511yescharacteristic
359C2#1172117311yescharacteristic
360C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721040C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#490
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×#63452C2semidirect
3. #11720528C2semidirect
4. C9×#809C2semidirect
5. #11720294C2semidirect
6. #11720284C2semidirect
7. #11720274C2semidirect
29. #63452D3semidirect
104. #809D9semidirect
213. C3×C9#490semidirect
310. C9#5067semidirect
354. C3#10588444semidirect

Maximal subgroups

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

subgroupindex
C3×#634522
#117210402
#117205282
C9×#8092
#117202942
#117202842
#117202742
#105905503
#105884443