Finite Group Databasebeta

← #13137752

Subgroups — #13137752

Order 1728.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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×#63459C211yesmaximal, characteristic, complemented
2#11721036C211yesmaximal, characteristic
3C2×#88423C211yesmaximal, characteristic, complemented
4C2×#88413C211yesmaximal, characteristic, complemented
5#11720525C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10590503—31nomaximal
9#10588460—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×#6194—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21#88423—22no—
22#88413—22no—
23C3×#897—31no—
24#63908—31no—
25C2×#3715—31no—
26C2×#3711—31no—
27#63634—31no—
28C3×#809D311yescharacteristic
29#63566—31no—
30#63500—31no—
31C2×#3672—31no—
32C2×#3663—31no—
33#63459D311yescharacteristic, complemented
34C9×#102—31no—
35#63376—31no—
36#63366—31no—
37#63366—61no—
38C22×D3×C9—21no—
39C22×C3×D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C9×D12—21no—
46C9×D12—21no—
47C3×C4×D9—21no—
48C3×C4×D9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51#6194—42no—
52C3⋊(C9⋊C8)D412yescomplemented
53C9×C3⋊C8—21no—
54#5049—91no—
55#4780—91no—
56C2×C3×D12C2×D311yescharacteristic
57C2×C3×C4×D3—31no—
58C2×C3×Dic6—31no—
59C2×#625—61no—
60C32×C4⋊C4C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)—31no—
62C2×C3⋊(C3⋊C8)—31no—
63C3×(C2×D3)⋊C4—61no—
64C3×C4⋊Dic3—61no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66(C2×D3)⋊Dic3—61no—
67#3715—62no—
68#3711—62no—
69C2×D4×C9—31no—
70C2×#611—61no—
71C2×C4×D9C2×D311yescharacteristic
72C2×Dic18C2×D311yescharacteristic
73C2×C8×C9—31no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4C2×D311yescharacteristic
76#3676—61no—
77#3672—62no—
78(C2×D9)⋊C4—21no—
79C4⋊Dic9—21no—
80C4⋊Dic9—31no—
81C4⋊Dic9—61no—
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84#3663—62no—
85C22×C3×C9C2×D411yescharacteristic
86C2×D3×C9—41no—
87C2×D3×C9—41no—
88C2×D3×C9—41no—
89C2×C3×D9—21no—
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3⋊Dic9—41no—
96C3×Dic9—21no—
97C3×Dic9—41no—
98C3×C22⋊Q8—91no—
99C2×#215—91no—
100#938—91no—
101C2×#214—91no—
102#930—91no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—91no—
105#897—31no—
106C3×#102—31no—
107#835—91no—
108#816—91no—
109#809D911yescharacteristic, complemented
110#807—91no—
111#807—91no—
112#807—181no—
113C22×C3×D3—21no—
114C22×C3×D3—31no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C3×D12—21no—
121C3×D12—21no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C3×Dic6—61no—
125C3×Dic6—61no—
126#625—122no—
127C23×C9—61no—
128C22×D9#6711yescharacteristic
129C3⋊(C3⋊C8)—32no—
130C3×C3⋊C8—21no—
131D4×C9—61no—
132D4×C9—61no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—41no—
136C2×C4×C9C22×D311yescharacteristic
137#611—122no—
138C2×Dic9—21no—
139C2×Dic9—61no—
140C2×Dic9—121no—
141C2×Dic9#6711yescharacteristic
142C4×D9—21no—
143C4×D9—21no—
144Dic18—21no—
145Dic18—21no—
146C8×C9—61no—
147C9⋊C8—32no—
148C9⋊C8—62no—
149#468—271noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
152C2×C3×C9#13711yescharacteristic
153D3×C9—81no—
154C3×D9—41no—
155C2×C3×Q8—91no—
156C2×C3×D4—31no—
157C22×C3×C4—91no—
158C2×#67—181no—
159C2×#67—181no—
160C2×D12C2×D911yescharacteristic
161C2×C4×D3—31no—
162C2×Dic6—31no—
163C2×C3×C8—31no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4C2×D911yescharacteristic
167C3×C4⋊C4D3211yescharacteristic
168C3×C22⋊C4—181no—
169#217—181no—
170#215—182no—
171#214—182no—
172(C2×D3)⋊C4—61no—
173(C2×D3)⋊C4—181no—
174C4⋊Dic3—61no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177C4⋊Dic3—181no—
178#208—182no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—181no—
182C2×C3⋊C8C2×D911yescharacteristic
183ΓL(1,𝔽25)—182no—
184C22×C32D3×D411yescharacteristic
185C2×C3×D3—41no—
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C32×C4—41no—
192C32×C4#23511yescharacteristic
193C32×C4D3×D411yescharacteristic
194C3⋊Dic3—121no—
195C3×Dic3—61no—
196C3×Dic3—121no—
197C22×C9—21no—
198C22×C9—121no—
199C22×C9—121no—
200C22×C9—121no—
201C22×C9C2×#6711yescharacteristic
202C2×D9—21no—
203C2×D9—21no—
204C2×D9—21no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—41no—
208C4×C9—81no—
209C4×C9D3×D411yescharacteristic
210C4×C9#23711yescharacteristic
211Dic9—21no—
212Dic9—41no—
213Dic9—121no—
214Dic9—241no—
215C2×SD16—271no—
216C2×M16—271no—
217C22⋊Q8—91no—
218#121—271no—
219#107—271no—
220#103—271no—
221#102—31no—
222C3×C9#46811yescharacteristic, complemented, Sylow
223C23×C3—61no—
224C23×C3—91no—
225C22×D3—21no—
226C22×D3—31no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×D4—61no—
230C3×D4—61no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—91no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C22×D911yescharacteristic
240C2×C3×C4C2×D3211yescharacteristic
241#67—362no—
242#67—362no—
243C2×Dic3—31no—
244C2×Dic3—61no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—361no—
248D12—21no—
249D12—21no—
250C4×D3—61no—
251C4×D3—61no—
252Dic6—61no—
253Dic6—61no—
254C3×C8—61no—
255C3⋊C8—21no—
256C3⋊C8—92no—
257C3⋊C8—92no—
258C3⋊C8—182no—
259C2×C32#88411yescharacteristic
260C2×C32#91111yescharacteristic
261C2×C32#91511yescharacteristic
262C3×D3—81no—
263C3×D3—121no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—21no—
267C2×C9—241no—
268C2×C9#93911yescharacteristic
269C2×C9#94211yescharacteristic
270C2×C9#94911yescharacteristic
271D9—41no—
272C2×Q8—91no—
273C2×D4—31no—
274C2×D4—541no—
275C22×C4—91no—
276SD16—542no—
277M16—542no—
278C2×C8—31no—
279C2×C8—271no—
280C4⋊C4—181no—
281C4⋊C4—271no—
282C4⋊C4D3×D911yescharacteristic
283C22⋊C4—181no—
284C22⋊C4—541no—
285C22×C3—21no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3D4×D911yescharacteristic
293C22×C3D3×#6711yescharacteristic
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—61no—
298C2×D3—61no—
299C2×D3—61no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—81no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4#369711yescharacteristic
308C3×C4D4×D911yescharacteristic
309C3×C4#379511yescharacteristic
310C3×C4#380211yescharacteristic
311Dic3—61no—
312Dic3—121no—
313Dic3—361no—
314Dic3—361no—
315Dic3—721no—
316C32#478011yescharacteristic
317C9—21no—
318C9#504911yescharacteristic, complemented
319C23—61no—
320C23—91no—
321Q8—181no—
322Q8—181no—
323D4—61no—
324D4—61no—
325D4—1082no—
326C2×C4—21no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—541no—
332C2×C4C2×D3×D911yescharacteristic
333C8—61no—
334C8—272no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—241no—
339C2×C3—361no—
340C2×C3#6344611yescharacteristic
341C2×C3#6347311yescharacteristic
342C2×C3#6347711yescharacteristic
343C2×C3#6388511yescharacteristic
344C2×C3#6392511yescharacteristic
345C2×C3#6394611yescharacteristic
346D3—81no—
347D3—121no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—181no—
352C22—181no—
353C22—181no—
354C22D9×#6711yescharacteristic
355C4—41no—
356C4—181no—
357C4—361no—
358C4—1081no—
359C4#8864111yescharacteristic
360C4#8864611yescharacteristic
361C3—21no—
362C3#1058846011yescharacteristic, complemented
363C3#1059050311yescharacteristic
364C2—241no—
365C2—361no—
366C2#1172098211yescharacteristic
367C2#1172117311yescharacteristic
368C2#1172121411yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721036C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#468
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×#63459C2semidirect
3. C2×#88423C2semidirect
4. C2×#88413C2semidirect
5. #11720525C2semidirect
6. C9×#809C2semidirect
7. #11720276C2semidirect
33. #63459D3semidirect
52. C3⋊(C9⋊C8)D4semidirect
109. #809D9semidirect
222. C3×C9#468semidirect
318. C9#5049semidirect
362. C3#10588460semidirect

Maximal subgroups

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

subgroupindex
C3×#634592
#117210362
C2×#884232
C2×#884132
#117205252
C9×#8092
#117202762
#105905033
#105884603