Finite Group Databasebeta

← #13139502

Subgroups — #13139502

Order 1728.

446 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 107 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#11721813C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic
3C22×C9×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic, complemented
6#11720278C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10591175—31nomaximal
9#10588795—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×#611—21no—
13C2×C3×D36C2211yescharacteristic
14#88551—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16#88534—21no—
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8—22no—
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22#88415—22no—
23#64133—31no—
24C3×#930—31no—
25C22×C3×C3⋊C8D311yescharacteristic
26C2×#3716—31no—
27#63573—31no—
28#63567—31no—
29#63563—31no—
30C22×C8×C9—31no—
31#63492D311yescharacteristic
32#63486—31no—
33#63486—61no—
34C2×#3664—31no—
35#63380—31no—
36#63377—31no—
37#63376—31no—
38C23×C3×C9D411yescharacteristic
39C22×C3×D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9—21no—
46#6226—21no—
47#6226—21no—
48C3×#611—42no—
49C2×C3×Dic9—21no—
50C3×D36—21no—
51C3×D36—21no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8D412yescomplemented
54#5114—91no—
55#5041—91no—
56C22×C32×C4C2×D311yescharacteristic
57C2×#634—31no—
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60#3757—61no—
61C4⋊(C3⋊Dic3)—31no—
62#3750—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×C4⋊Dic3—31no—
65C2×C3×C3⋊C8—22no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67C2×C3×C3⋊C8C2×D311yescharacteristic
68#3716—62no—
69C22×C4×C9—21no—
70C22×C4×C9C2×D311yescharacteristic
71C2×#611—21no—
72C2×D36C2×D311yescharacteristic
73C2×Dic18—31no—
74C2×Dic18—61no—
75C2×C8×C9—31no—
76C2×C8×C9—31no—
77C2×C8×C9—62no—
78#3676—62no—
79#3676—61no—
80(C2×D9)⋊C4—21no—
81C4⋊Dic9—31no—
82C4⋊Dic9—61no—
83C4⋊Dic9C2×D311yescharacteristic
84Dic9⋊C4—62no—
85Dic9⋊C4—61no—
86#3664—62no—
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9SD1612yescomplemented
90C22×C3×C9C2×D411yescharacteristic
91C2×C3×D9—41no—
92C2×C3×D9—41no—
93C2×C3×D9—41no—
94C3×C4×C9—21no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9#4111yescharacteristic
97C3⋊Dic9—41no—
98C3⋊Dic9—41no—
99C3×Dic9—41no—
100C22×C3×C8—31no—
101C3×#121—91no—
102C2×#214—91no—
103#930—31no—
104#924—91no—
105#924—91no—
106#924—181no—
107C22×C3⋊C8D911yescharacteristic, complemented
108C2×#204—91no—
109#832—91no—
110#821—91no—
111#817—91no—
112#816—91no—
113#810—91no—
114#808—91no—
115C23×C32D1211yescharacteristic
116C22×C3×D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4D1211yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—61no—
123#634—61no—
124#634—61no—
125C3×#67—122no—
126C2×C3×Dic3—61no—
127C3×D12—61no—
128C3×D12—61no—
129C23×C9—21no—
130C23×C9#6711yescharacteristic
131C22×D9—21no—
132C3×C3⋊C8—21no—
133C3×C3⋊C8#6712yes—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—41no—
139C2×C4×C9—41no—
140C2×C4×C9#6711yescharacteristic
141C2×C4×C9C22×D311yescharacteristic
142#611—42no—
143C2×Dic9—21no—
144C2×Dic9—62no—
145C2×Dic9—62no—
146C2×Dic9—61no—
147C2×Dic9—61no—
148D36—21no—
149D36—21no—
150Dic18—61no—
151Dic18—61no—
152Dic18—121no—
153Dic18—121no—
154C8×C9—32no—
155C8×C9—61no—
156#465—271noSylow
157C2×C3×C9—22no—
158C2×C3×C9#12111yescharacteristic
159C2×C3×C9C2×SD1611yescharacteristic
160C2×C3×C9#13511yescharacteristic
161C3×D9—81no—
162C2×C3×D4—91no—
163C2×C3×D4—181no—
164C22×C3×C4—21no—
165C22×C3×C4C2×D911yescharacteristic
166C22×C3×C4D3211yescharacteristic
167C2×#67—61no—
168C2×D12—31no—
169C2×Dic6—91no—
170C2×Dic6—91no—
171C2×Dic6—181no—
172C2×C3×C8—31no—
173C2×C3×C8—31no—
174C2×C3×C8—62no—
175C3×C4⋊C4—91no—
176C3×C22⋊C4—181no—
177#217—182no—
178#217—181no—
179#217—181no—
180#214—182no—
181(C2×D3)⋊C4—61no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186Dic3⋊C4—182no—
187Dic3⋊C4—181no—
188Dic3⋊C4—181no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8C2×D911yescharacteristic
191C2×C3⋊C8C2×D911yescharacteristic
192#204—182no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32#20412yes—
196C22×C32C2×D1211yescharacteristic
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—21no—
201C32×C4#23511yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—121no—
204C3⋊Dic3—121no—
205C3×Dic3—121no—
206C22×C9—21no—
207C22×C9—21no—
208C22×C9—22no—
209C22×C9—21no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9#21412yescomplemented
213C22×C9C2×#6711yescharacteristic
214C2×D9—41no—
215C2×D9—41no—
216C2×D9—41no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—41no—
221C4×C9D3×D411yescharacteristic
222C4×C9#23711yescharacteristic
223Dic9—41no—
224Dic9—122no—
225Dic9—122no—
226Dic9—121no—
227Dic9—121no—
228C2×SD16—271no—
229C22×C8—31no—
230C22⋊Q8—271no—
231#121—91no—
232#106—271no—
233#103—271no—
234#102—271no—
235C3×C9#46511yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—181no—
238C23×C3D3611yescharacteristic
239C23×C3#62611yescharacteristic
240C22×D3—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—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—181no—
253C2×C3×C4D3611yescharacteristic
254C2×C3×C4C22×D911yescharacteristic
255C2×C3×C4#62611yescharacteristic
256C2×C3×C4C2×D3211yescharacteristic
257#67—122no—
258C2×Dic3—61no—
259C2×Dic3—182no—
260C2×Dic3—182no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265D12—61no—
266D12—61no—
267Dic6—181no—
268Dic6—181no—
269Dic6—181no—
270Dic6—181no—
271Dic6—361no—
272Dic6—361no—
273C3×C8—32no—
274C3×C8—61no—
275C3⋊C8—21no—
276C3⋊C8#61112yescomplemented
277C2×C32—22no—
278C2×C32C2×#20411yescharacteristic
279C2×C32#90411yescharacteristic
280C2×C32#93011yescharacteristic
281C3×D3—241no—
282C2×C9—22no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—42no—
287C2×C9C2×#21411yescharacteristic
288C2×C9#93911yescharacteristic
289C2×C9#95011yescharacteristic
290D9—81no—
291C2×Q8—271no—
292C2×D4—91no—
293C2×D4—181no—
294C22×C4D3×D911yescharacteristic
295SD16—542no—
296C2×C8—31no—
297C2×C8—31no—
298C2×C8—62no—
299C4⋊C4—91no—
300C4⋊C4—271no—
301C4⋊C4—541no—
302C22⋊C4—181no—
303C22⋊C4—541no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3#366412yescomplemented
316C22×C3C2×D3611yescharacteristic
317C22×C3#371612yes—
318C22×C3C2×#62611yescharacteristic
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—121no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—361no—
328C3×C4#369711yescharacteristic
329C3×C4C2×#61111yescharacteristic
330C3×C4#380411yescharacteristic
331C3×C4D3×#6711yescharacteristic
332Dic3—121no—
333Dic3—362no—
334Dic3—362no—
335Dic3—361no—
336Dic3—361no—
337Dic3—361no—
338Dic3—361no—
339C32#511411yescharacteristic
340C9—21no—
341C9#504111yescharacteristic, complemented
342C23—181no—
343C23#619211yescharacteristic
344Q8—541no—
345Q8—541no—
346D4—181no—
347D4—181no—
348D4—362no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4—541no—
354C2×C4#619211yescharacteristic
355C2×C4C2×D3×D911yescharacteristic
356C8—32no—
357C8—61no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—42no—
364C2×C3—721no—
365C2×C3C2×#366411yescharacteristic
366C2×C3#6346711yescharacteristic
367C2×C3#6349211yescharacteristic
368C2×C3C2×#371611yescharacteristic
369C2×C3#6382911yescharacteristic
370C2×C3#6397811yescharacteristic
371D3—241no—
372C22—21no—
373C22—21no—
374C22—361no—
375C22—361no—
376C22—361no—
377C22#8841512yescomplemented
378C22C2×#619211yescharacteristic
379C4—21no—
380C4—361no—
381C4—1081no—
382C4—1081no—
383C4#8865811yescharacteristic
384C4D3×#61111yescharacteristic
385C3—21no—
386C3#1058879511yescharacteristic, complemented
387C3#1059117511yescharacteristic
388C2—22no—
389C2—721no—
390C2C2×#8841511yescharacteristic
391C2#1172086711yescharacteristic
392C2#1172127711yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
prime radicalC3×#63492C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#465
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11721813C2semidirect
3. C22×C9×C3⋊C8C2semidirect
4. C2×#88415C2semidirect
5. #11720283C2semidirect
6. #11720278C2semidirect
7. #11720277C2semidirect
53. C9×C3⋊C8D4semidirect
89. C22×C3×C9SD16semidirect
107. C22×C3⋊C8D9semidirect
212. C22×C9#214semidirect
235. C3×C9#465semidirect
276. C3⋊C8#611semidirect
315. C22×C3#3664semidirect
341. C9#5041semidirect
377. C22#88415semidirect
386. C3#10588795semidirect

Maximal subgroups

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

subgroupindex
#117218132
C3×#634922
C22×C9×C3⋊C82
C2×#884152
#117202832
#117202782
#117202772
#105911753
#105887953