Finite Group Databasebeta

← #13137759

Subgroups — #13137759

Order 1728.

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

normal subgroups only

This group has 454 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
1#11721743C211yesmaximal, characteristic, complemented
2#11721043C211yesmaximal, characteristic
3C2×#88424C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5C9×#809C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10590553—31nomaximal
9#10588442—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6194—21no—
14C2×#6194—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16#88534—21no—
17C4×C3⋊Dic9—21no—
18C4×C3⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88424—22no—
22#88415—22no—
23#64096—31no—
24#63914—31no—
25C2×#3716—31no—
26C2×#3716—31no—
27C3×#809—31no—
28C3×#809D311yescharacteristic
29#63572—31no—
30#63502—31no—
31C2×#3672—31no—
32C2×#3664—31no—
33#63449—31no—
34#63449—61no—
35C9×#102—31no—
36Dic9⋊C8—31no—
37#63369D311yescharacteristic, complemented
38C22×D3×C9—21no—
39C22×C3×D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45#6226—21no—
46#6226D412yescomplemented
47C9×D12—21no—
48C9×D12—21no—
49C3×D36—21no—
50C3×D36—21no—
51#6194—42no—
52#6194—42no—
53C9×C3⋊C8—21no—
54C3×C9⋊C8—21no—
55#4762—91no—
56#4762—91no—
57C2×#634—31no—
58C2×C3×D12—31no—
59C2×C3×D12C2×D311yescharacteristic
60C2×#625—61no—
61C2×#625—61no—
62C32×C4⋊C4C2×D311yescharacteristic
63#3750—61no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8C2×D311yescharacteristic
68#3716—62no—
69#3716—62no—
70C2×D4×C9—31no—
71C2×#611—61no—
72C2×#611—61no—
73C2×D36C2×D311yescharacteristic
74C2×Dic18—31no—
75C2×Dic18—61no—
76C2×C8×C9—31no—
77C9×C4⋊C4—21no—
78C9×C4⋊C4C2×D311yescharacteristic
79#3672—62no—
80Dic9⋊C4—61no—
81Dic9⋊C4—62no—
82C4×Dic9—31no—
83C4×Dic9—61no—
84C4×Dic9—61no—
85C4×Dic9—62no—
86C2×C9⋊C8C2×D311yescharacteristic
87#3664—62no—
88C22×C3×C9C2×D411yescharacteristic
89C2×D3×C9—41no—
90C2×D3×C9—41no—
91C2×D3×C9—41no—
92C2×C3×D9—41no—
93C2×C3×D9—41no—
94C2×C3×D9—41no—
95C3×C4×C9—41no—
96C3×C4×C9C2×D411yescharacteristic
97C3×C4×C9#4111yescharacteristic
98C3⋊Dic9—22no—
99C3⋊Dic9—21no—
100C3⋊Dic9SD1612yescomplemented
101#940—91no—
102#940—91no—
103C2×#214—91no—
104C2×#214—91no—
105C2×#204—91no—
106C2×#204—91no—
107#887—91no—
108#887—91no—
109#887—181no—
110C3×#102—31no—
111C3×#102—91no—
112Dic3⋊C8—91no—
113Dic3⋊C8—91no—
114#809—31no—
115#809D911yescharacteristic, complemented
116C22×C3×D3—21no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—61no—
123#634—32no—
124#634—61no—
125C3×D12—21no—
126C3×D12—21no—
127C3×D12—61no—
128C3×D12—61no—
129#625—122no—
130#625—122no—
131C23×C9—61no—
132C22×D9—21no—
133C3×C3⋊C8—21no—
134C3×C3⋊C8—61no—
135D4×C9—61no—
136D4×C9—61no—
137C2×C4×C9—22no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9C22×D311yescharacteristic
141#611—122no—
142#611—122no—
143C2×Dic9—31no—
144C2×Dic9—31no—
145C2×Dic9—61no—
146C2×Dic9—61no—
147C2×Dic9—62no—
148C2×Dic9—61no—
149D36—21no—
150D36—21no—
151Dic18—32no—
152Dic18—62no—
153Dic18—62no—
154Dic18—61no—
155C8×C9—61no—
156C9⋊C8—21no—
157#460—271noSylow
158C2×C3×C9#12111yescharacteristic
159C2×C3×C9C2×SD1611yescharacteristic
160C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
161D3×C9—81no—
162C3×D9—81no—
163C2×C3×D4—31no—
164C2×C3×D4—91no—
165C2×#67—181no—
166C2×#67—181no—
167C2×#67—181no—
168C2×#67—181no—
169C2×D12—31no—
170C2×D12C2×D911yescharacteristic
171C2×Dic6—91no—
172C2×Dic6—91no—
173C2×Dic6—181no—
174C2×C3×C8—31no—
175C2×C3×C8—91no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4C2×D911yescharacteristic
178C3×C4⋊C4D3211yescharacteristic
179#214—182no—
180#214—182no—
181Dic3⋊C4—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—182no—
184C4×Dic3—91no—
185C4×Dic3—91no—
186C4×Dic3—181no—
187C4×Dic3—181no—
188C4×Dic3—181no—
189C4×Dic3—182no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8C2×D911yescharacteristic
192#204—182no—
193#204—182no—
194C22×C32D3×D411yescharacteristic
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—41no—
202C32×C4#23511yescharacteristic
203C32×C4D3×D411yescharacteristic
204C3⋊Dic3—32no—
205C3⋊Dic3—62no—
206C3⋊Dic3—61no—
207C22×C9—21no—
208C22×C9—121no—
209C22×C9—121no—
210C22×C9—121no—
211C22×C9D3×D411yescharacteristic
212C2×D9—41no—
213C2×D9—41no—
214C2×D9—41no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—42no—
218C4×C9—41no—
219C4×C9#23511yescharacteristic
220C4×C9D3×D411yescharacteristic
221Dic9—32no—
222Dic9—62no—
223Dic9—64no—
224Dic9—62no—
225Dic9—61no—
226Dic9—121no—
227C2×SD16—271no—
228C2×SD16—271no—
229#127—271no—
230C4×Q8—271no—
231C4⋊C8—271no—
232#102—31no—
233#102—91no—
234C3×C9#46011yescharacteristic, complemented, Sylow
235C23×C3—61no—
236C23×C3—181no—
237C22×D3—21no—
238C22×D3—61no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—181no—
242C3×D4—181no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4C22×D911yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249#67—362no—
250#67—362no—
251#67—362no—
252#67—362no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C2×Dic3—91no—
256C2×Dic3—91no—
257C2×Dic3—181no—
258C2×Dic3—181no—
259C2×Dic3—182no—
260C2×Dic3—181no—
261C2×Dic3—181no—
262D12—21no—
263D12—21no—
264D12—61no—
265D12—61no—
266Dic6—92no—
267Dic6—92no—
268Dic6—182no—
269Dic6—182no—
270Dic6—181no—
271Dic6—181no—
272C3×C8—61no—
273C3×C8—181no—
274C3⋊C8—21no—
275C3⋊C8—61no—
276C2×C32#88411yescharacteristic
277C2×C32#91111yescharacteristic
278C2×C32D3×SD1611yescharacteristic
279C3×D3—81no—
280C3×D3—241no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—241no—
285C2×C9#88411yescharacteristic
286C2×C9#91111yescharacteristic
287C2×C9D3×SD1611yescharacteristic
288D9—81no—
289C2×Q8—271no—
290C2×D4—31no—
291C2×D4—91no—
292C2×D4—541no—
293C2×D4—541no—
294SD16—542no—
295SD16—542no—
296C2×C8—31no—
297C2×C8—91no—
298C4⋊C4—541no—
299C4⋊C4D3×D911yescharacteristic
300C42—271no—
301C42—541no—
302C22×C3—21no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3D4×D911yescharacteristic
310C22×C3#381111yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—121no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4—41no—
322C3×C4#369711yescharacteristic
323C3×C4D4×D911yescharacteristic
324C3×C4#379811yescharacteristic
325C3×C4#380211yescharacteristic
326Dic3—92no—
327Dic3—92no—
328Dic3—182no—
329Dic3—184no—
330Dic3—182no—
331Dic3—181no—
332Dic3—182no—
333Dic3—181no—
334Dic3—361no—
335C32#476211yescharacteristic
336C9—21no—
337C9#476211yescharacteristic, complemented
338C23—61no—
339C23—181no—
340Q8—272no—
341Q8—541no—
342D4—61no—
343D4—61no—
344D4—181no—
345D4—181no—
346D4—1082no—
347D4—1082no—
348C2×C4—21no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—541no—
352C2×C4C2×D3×D911yescharacteristic
353C8—61no—
354C8—181no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—241no—
359C2×C3—721no—
360C2×C3#6344611yescharacteristic
361C2×C3#6347311yescharacteristic
362C2×C3SD16×D911yescharacteristic
363C2×C3#6391211yescharacteristic
364C2×C3#6392711yescharacteristic
365C2×C3#6394011yescharacteristic
366D3—81no—
367D3—241no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#8866811yescharacteristic
375C4—41no—
376C4—272no—
377C4—542no—
378C4—541no—
379C4#8864011yescharacteristic
380C4#8864611yescharacteristic
381C3—21no—
382C3#1058844211yescharacteristic, complemented
383C3#1059055311yescharacteristic
384C2—241no—
385C2—721no—
386C2#1172104111yescharacteristic
387C2#1172117511yescharacteristic
388C2#1172121211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721043C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#460
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. #11721743C2semidirect
3. C2×#88424C2semidirect
4. C2×#88415C2semidirect
5. C9×#809C2semidirect
6. C3×#63369C2semidirect
7. #11720273C2semidirect
37. #63369D3semidirect
46. #6226D4semidirect
100. C3⋊Dic9SD16semidirect
115. #809D9semidirect
234. C3×C9#460semidirect
337. C9#4762semidirect
382. C3#10588442semidirect

Maximal subgroups

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

subgroupindex
#117217432
#117210432
C2×#884242
C2×#884152
C9×#8092
C3×#633692
#117202732
#105905533
#105884423