Finite Group Databasebeta

← #13137809

Subgroups — #13137809

Order 1728.

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

normal subgroups only

This group has 406 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
1#11721743C211yesmaximal, characteristic, complemented
2#11720951C211yesmaximal, characteristic
3C2×#88428C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic
5C9×#810C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic
7#11720273C211yesmaximal, characteristic, complemented
8#10590538—31nomaximal
9#10588478—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14#88534—21no—
15C4×C3⋊Dic9—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20(C2×D9)⋊Dic3—21no—
21#88428—22no—
22#88415—22no—
23#64096—31no—
24#63863—31no—
25C2×#3719—31no—
26C2×#3716—31no—
27C3×#810D311yescharacteristic
28C3×#809—31no—
29#63572—31no—
30#63509—31no—
31C2×#3674—31no—
32C2×#3664—31no—
33#63449—31no—
34#63449—61no—
35C9×#103—31no—
36Dic9⋊C8—31no—
37#63369D311yescharacteristic
38C22×C3×D9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9D411yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44#6226—21no—
45#6226D412yes—
46C2×C9×Dic3—21no—
47C3×D36—21no—
48C3×D36—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C9×C3⋊C8—21no—
52C3×C9⋊C8—21no—
53#4798—91no—
54#4767—91no—
55C2×#634—31no—
56C2×C3×D12—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59#3750—61no—
60C4×C3⋊Dic3—31no—
61C4×C3⋊Dic3—61no—
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64(C2×D3)⋊Dic3—61no—
65(C2×D3)⋊Dic3—61no—
66#3719—62no—
67#3716—62no—
68C2×Q8×C9—31no—
69C2×D36C2×D311yescharacteristic
70C2×Dic18—31no—
71C2×Dic18—61no—
72C2×C8×C9—31no—
73C9×C4⋊C4—21no—
74C9×C4⋊C4C2×D311yescharacteristic
75#3674—62no—
76(C2×D9)⋊C4—61no—
77(C2×D9)⋊C4—61no—
78Dic9⋊C4—61no—
79Dic9⋊C4—62no—
80C4×Dic9—31no—
81C4×Dic9—61no—
82C4×Dic9—61no—
83C4×Dic9—62no—
84C2×C9⋊C8C2×D311yescharacteristic
85#3664—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3×D9—41no—
88C2×C3×D9—41no—
89C2×C3×D9—41no—
90C3×C4×C9—41no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—22no—
94C3⋊Dic9—21no—
95C3⋊Dic9—21no—
96C9×Dic3—41no—
97#947—91no—
98C2×#216—91no—
99#936—91no—
100C2×#214—91no—
101C2×Dic12—91no—
102C2×#204—91no—
103#887—91no—
104#887—91no—
105#887—181no—
106C3×#103—31no—
107C3×#102—91no—
108Dic3⋊C8—91no—
109Dic3⋊C8—91no—
110#810D911yescharacteristic, complemented
111#809—31no—
112C22×C3×D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—61no—
118#634—32no—
119#634—61no—
120C2×C3×Dic3—21no—
121C3×D12—61no—
122C3×D12—61no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C22×D9—21no—
126C3×C3⋊C8—21no—
127C3×C3⋊C8—61no—
128Q8×C9—61no—
129Q8×C9—61no—
130C2×C4×C9—22no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—61no—
134C2×C4×C9C22×D311yescharacteristic
135C2×Dic9—31no—
136C2×Dic9—31no—
137C2×Dic9—61no—
138C2×Dic9—61no—
139C2×Dic9—62no—
140C2×Dic9—61no—
141D36—21no—
142D36—21no—
143Dic18—32no—
144Dic18—62no—
145Dic18—62no—
146Dic18—61no—
147C8×C9—61no—
148C9⋊C8—21no—
149#464—271noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9#13511yescharacteristic
152C2×C3×C9#13711yescharacteristic
153C3×D9—81no—
154C2×C3×Q8—31no—
155C2×C3×D4—91no—
156C2×D12—31no—
157C2×Dic6—91no—
158C2×Dic6—91no—
159C2×Dic6—181no—
160C2×Dic6C2×D911yescharacteristic
161C2×C3×C8—31no—
162C2×C3×C8—91no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4C2×D911yescharacteristic
165C3×C4⋊C4D3211yescharacteristic
166#217—181no—
167#217—181no—
168#216—182no—
169#214—182no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—182no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—182no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8C2×D911yescharacteristic
183Dic12—182no—
184#204—182no—
185C22×C32D3×D411yescharacteristic
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C32×C4—41no—
190C32×C4#23511yescharacteristic
191C32×C4D3×D411yescharacteristic
192C3⋊Dic3—62no—
193C3⋊Dic3—61no—
194C3⋊Dic3—61no—
195C3×Dic3—41no—
196C22×C9—21no—
197C22×C9D3×D411yescharacteristic
198C2×D9—41no—
199C2×D9—41no—
200C2×D9—41no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—42no—
204C4×C9—41no—
205C4×C9—121no—
206C4×C9#23511yescharacteristic
207C4×C9D3×D411yescharacteristic
208Dic9—64no—
209Dic9—62no—
210Dic9—62no—
211Dic9—61no—
212Dic9—61no—
213Dic9—121no—
214C2×Q16—271no—
215C2×SD16—271no—
216#124—271no—
217C4×Q8—271no—
218C4⋊C8—271no—
219#103—31no—
220#102—91no—
221C3×C9#46411yescharacteristic, complemented, Sylow
222C23×C3—181no—
223C22×D3—61no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×D4—181no—
227C3×D4—181no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—61no—
233C2×C3×C4C22×D911yescharacteristic
234C2×C3×C4C2×D3211yescharacteristic
235C2×Dic3—21no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—91no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C2×Dic3—182no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245D12—61no—
246D12—61no—
247Dic6—21no—
248Dic6—21no—
249Dic6—92no—
250Dic6—92no—
251Dic6—182no—
252Dic6—182no—
253Dic6—181no—
254Dic6—181no—
255C3×C8—61no—
256C3×C8—181no—
257C3⋊C8—21no—
258C3⋊C8—61no—
259C2×C32#88411yescharacteristic
260C2×C32#91611yescharacteristic
261C2×C32#91811yescharacteristic
262C3×D3—241no—
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9#88411yescharacteristic
267C2×C9#91211yescharacteristic
268C2×C9#91511yescharacteristic
269D9—81no—
270C2×Q8—31no—
271C2×Q8—271no—
272C2×D4—91no—
273Q16—542no—
274SD16—542no—
275C2×C8—31no—
276C2×C8—91no—
277C4⋊C4—541no—
278C4⋊C4D3×D911yescharacteristic
279C22⋊C4—541no—
280C22⋊C4—541no—
281C42—271no—
282C42—541no—
283C22×C3—21no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3D4×D911yescharacteristic
288C22×C3#381111yescharacteristic
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—42no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—121no—
298C3×C4#369711yescharacteristic
299C3×C4D4×D911yescharacteristic
300C3×C4#379811yescharacteristic
301C3×C4#380211yescharacteristic
302Dic3—41no—
303Dic3—184no—
304Dic3—182no—
305Dic3—182no—
306Dic3—181no—
307Dic3—181no—
308Dic3—182no—
309Dic3—181no—
310Dic3—181no—
311Dic3—361no—
312C32#479811yescharacteristic
313C9—21no—
314C9#476711yescharacteristic, complemented
315C23—181no—
316Q8—61no—
317Q8—61no—
318Q8—272no—
319Q8—541no—
320D4—181no—
321D4—181no—
322C2×C4—21no—
323C2×C4—61no—
324C2×C4—271no—
325C2×C4—271no—
326C2×C4—541no—
327C2×C4C2×D3×D911yescharacteristic
328C8—61no—
329C8—181no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—721no—
334C2×C3#6344611yescharacteristic
335C2×C3#6347811yescharacteristic
336C2×C3#6348011yescharacteristic
337C2×C3#6391211yescharacteristic
338C2×C3#6393611yescharacteristic
339C2×C3#6395111yescharacteristic
340D3—241no—
341C22—361no—
342C22—361no—
343C22—361no—
344C22#8866811yescharacteristic
345C4—41no—
346C4—121no—
347C4—541no—
348C4—542no—
349C4—541no—
350C4#8864011yescharacteristic
351C4#8864611yescharacteristic
352C3—21no—
353C3#1058847811yescharacteristic, complemented
354C3#1059053811yescharacteristic
355C2—721no—
356C2#1172104111yescharacteristic
357C2#1172118611yescharacteristic
358C2#1172122511yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#464
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×#88428C2semidirect
5. C9×#810C2semidirect
7. #11720273C2semidirect
110. #810D9semidirect
221. C3×C9#464semidirect
314. C9#4767semidirect
353. C3#10588478semidirect

Maximal subgroups

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

subgroupindex
#117217432
#117209512
C2×#884282
C2×#884152
C9×#8102
C3×#633692
#117202732
#105905383
#105884783