Finite Group Databasebeta

← #13137746

Subgroups — #13137746

Order 1728.

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

normal subgroups only

This group has 426 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×#63457C211yesmaximal, characteristic, complemented
2#11721036C211yesmaximal, characteristic, complemented
3C2×#88421C211yesmaximal, characteristic, complemented
4C2×#88413C211yesmaximal, characteristic, complemented
5#11720526C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590504—31nomaximal
9#10588462—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×D36—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C3×C4×D9C2211yescharacteristic
14C2×#6194—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C2×C3⋊(C9⋊C8)C2211yescharacteristic
18C3×(C2×D9)⋊C4—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21#88421—22no—
22#88413—22no—
23C3×#895—31no—
24#63908—31no—
25C2×#3713—31no—
26C2×#3711—31no—
27#63635—31no—
28C3×#809D311yescharacteristic
29#63563—31no—
30#63500—31no—
31C2×#3673—31no—
32C2×#3663—31no—
33#63457D311yescharacteristic, complemented
34C9×#102—31no—
35#63380—31no—
36#63367—31no—
37#63367—61no—
38C22×D3×C9—21no—
39C22×C3×D9—21no—
40C22×C3×D9D411yescharacteristic
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C9×D12—21no—
46C9×D12—21no—
47C3×D36—21no—
48C3×D36—21no—
49C3×D36—42no—
50C3×C4×D9—21no—
51C3×C4×D9—21no—
52#6194—42no—
53C3⋊(C9⋊C8)D412yescomplemented
54C9×C3⋊C8—21no—
55#5039—91no—
56#4782—91no—
57C2×C3×D12—31no—
58C2×C3×D12—61no—
59C2×C3×D12C2×D311yescharacteristic
60C2×C3×C4×D3—31no—
61C2×#625—61no—
62C32×C4⋊C4C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)—31no—
64C2×C3⋊(C3⋊C8)—31no—
65C3×(C2×D3)⋊C4—61no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67(C2×D3)⋊Dic3—61no—
68#3713—62no—
69#3711—62no—
70C2×D4×C9—31no—
71C2×#611—61no—
72C2×D36—21no—
73C2×D36C2×D311yescharacteristic
74C2×C4×D9C2×D311yescharacteristic
75C2×C8×C9—31no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4C2×D311yescharacteristic
78#3676—61no—
79#3673—62no—
80(C2×D9)⋊C4—21no—
81C4⋊Dic9—31no—
82C4⋊Dic9—61no—
83C2×C9⋊C8—31no—
84C2×C9⋊C8—61no—
85#3663—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×D3×C9—41no—
88C2×D3×C9—41no—
89C2×D3×C9—41no—
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9—21no—
93C2×C3×D9—41no—
94C2×C3×D9—41no—
95C2×C3×D9—41no—
96C3×C4×C9—41no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9#4111yescharacteristic
99C3⋊Dic9—41no—
100C3×Dic9—21no—
101C3×#121—91no—
102#938—91no—
103C2×#213—91no—
104C2×#213—91no—
105#930—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—91no—
108#895—31no—
109C3×#102—31no—
110#832—91no—
111#821—91no—
112#809D911yescharacteristic, complemented
113#808—91no—
114#808—91no—
115#808—181no—
116C22×C3×D3—21no—
117C22×C3×D3—31no—
118C22×C3×D3—61no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—61no—
122C2×C3×Dic3—31no—
123C3×D12—21no—
124C3×D12—21no—
125C3×D12—61no—
126C3×D12—61no—
127C3×D12—122no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130#625—122no—
131C23×C9—61no—
132C22×D9—21no—
133C22×D9#6711yescharacteristic
134C3⋊(C3⋊C8)—32no—
135C3×C3⋊C8—21no—
136D4×C9—61no—
137D4×C9—61no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—41no—
141C2×C4×C9C22×D311yescharacteristic
142#611—122no—
143C2×Dic9—61no—
144C2×Dic9—121no—
145C2×Dic9#6711yescharacteristic
146D36—21no—
147D36—21no—
148D36—42no—
149C4×D9—21no—
150C4×D9—21no—
151C8×C9—61no—
152C9⋊C8—32no—
153C9⋊C8—62no—
154#469—271noSylow
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
158D3×C9—81no—
159C3×D9—41no—
160C3×D9—81no—
161C2×C3×D4—31no—
162C2×C3×D4—91no—
163C2×C3×D4—181no—
164C22×C3×C4—91no—
165C2×#67—181no—
166C2×#67—181no—
167C2×D12—31no—
168C2×D12—61no—
169C2×D12C2×D911yescharacteristic
170C2×C4×D3—31no—
171C2×C3×C8—31no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4C2×D911yescharacteristic
174C3×C4⋊C4D3211yescharacteristic
175C3×C22⋊C4—181no—
176#217—181no—
177#213—182no—
178#213—182no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—91no—
182C4⋊Dic3—91no—
183C4⋊Dic3—181no—
184#208—182no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—181no—
188C2×C3⋊C8C2×D911yescharacteristic
189ΓL(1,𝔽25)—182no—
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—41no—
201C32×C4#23511yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—121no—
204C3×Dic3—61no—
205C22×C9—21no—
206C22×C9—121no—
207C22×C9—121no—
208C22×C9—121no—
209C22×C9C2×#6711yescharacteristic
210C2×D9—21no—
211C2×D9—21no—
212C2×D9—21no—
213C2×D9—41no—
214C2×D9—41no—
215C2×D9—41no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—41no—
219C4×C9—81no—
220C4×C9D3×D411yescharacteristic
221C4×C9#23711yescharacteristic
222Dic9—21no—
223Dic9—121no—
224Dic9—241no—
225C2×D8—271no—
226C2×M16—271no—
227#121—91no—
228#121—271no—
229#106—271no—
230#102—31no—
231#102—271no—
232C3×C9#46911yescharacteristic, complemented, Sylow
233C23×C3—61no—
234C23×C3—91no—
235C23×C3—181no—
236C22×D3—21no—
237C22×D3—31no—
238C22×D3—61no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—181no—
242C3×D4—181no—
243C3×D4—362no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253#67—362no—
254#67—362no—
255C2×Dic3—31no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C2×Dic3—361no—
259D12—21no—
260D12—21no—
261D12—61no—
262D12—61no—
263D12—122no—
264C4×D3—61no—
265C4×D3—61no—
266C3×C8—61no—
267C3⋊C8—21no—
268C3⋊C8—92no—
269C3⋊C8—92no—
270C3⋊C8—182no—
271C2×C32#88411yescharacteristic
272C2×C32#91111yescharacteristic
273C2×C32#91411yescharacteristic
274C3×D3—81no—
275C3×D3—121no—
276C3×D3—241no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—241no—
281C2×C9#93211yescharacteristic
282C2×C9#93911yescharacteristic
283C2×C9#94911yescharacteristic
284D9—41no—
285D9—81no—
286C2×D4—31no—
287C2×D4—91no—
288C2×D4—181no—
289C2×D4—541no—
290C22×C4—91no—
291D8—542no—
292M16—542no—
293C2×C8—31no—
294C2×C8—271no—
295C4⋊C4—271no—
296C4⋊C4D3×D911yescharacteristic
297C22⋊C4—181no—
298C22⋊C4—541no—
299C22×C3—21no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3—181no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3D4×D911yescharacteristic
310C22×C3D3×#6711yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—61no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—121no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—81no—
325C3×C4—181no—
326C3×C4#369711yescharacteristic
327C3×C4D4×D911yescharacteristic
328C3×C4#379511yescharacteristic
329C3×C4#380211yescharacteristic
330Dic3—61no—
331Dic3—361no—
332Dic3—361no—
333Dic3—721no—
334C32#478211yescharacteristic
335C9—21no—
336C9#503911yescharacteristic, complemented
337C23—61no—
338C23—91no—
339C23—181no—
340D4—61no—
341D4—61no—
342D4—181no—
343D4—181no—
344D4—362no—
345D4—1082no—
346C2×C4—21no—
347C2×C4—91no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—541no—
351C2×C4C2×D3×D911yescharacteristic
352C8—61no—
353C8—272no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—241no—
358C2×C3—361no—
359C2×C3—721no—
360C2×C3#6344611yescharacteristic
361C2×C3#6347311yescharacteristic
362C2×C3#6347611yescharacteristic
363C2×C3#6388511yescharacteristic
364C2×C3#6393211yescharacteristic
365C2×C3#6393911yescharacteristic
366D3—81no—
367D3—121no—
368D3—241no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22—181no—
373C22—181no—
374C22—181no—
375C22—361no—
376C22—361no—
377C22—361no—
378C22D9×#6711yescharacteristic
379C4—41no—
380C4—181no—
381C4—1081no—
382C4#8864111yescharacteristic
383C4#8864611yescharacteristic
384C3—21no—
385C3#1058846211yescharacteristic, complemented
386C3#1059050411yescharacteristic
387C2—241no—
388C2—361no—
389C2—721no—
390C2#1172098211yescharacteristic
391C2#1172117611yescharacteristic
392C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#469
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×#63457C2semidirect
2. #11721036C2semidirect
3. C2×#88421C2semidirect
4. C2×#88413C2semidirect
5. #11720526C2semidirect
6. C9×#809C2semidirect
7. #11720283C2semidirect
33. #63457D3semidirect
53. C3⋊(C9⋊C8)D4semidirect
112. #809D9semidirect
232. C3×C9#469semidirect
336. C9#5039semidirect
385. C3#10588462semidirect

Maximal subgroups

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

subgroupindex
C3×#634572
#117210362
C2×#884212
C2×#884132
#117205262
C9×#8092
#117202832
#105905043
#105884623