Finite Group Databasebeta

← #13137842

Subgroups — #13137842

Order 1728.

498 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#88638C211yesmaximal, characteristic
2C3×#63459C211yesmaximal, characteristic, complemented
3C2×#88423C211yesmaximal, characteristic, complemented
4C2×#88418C211yesmaximal, characteristic, complemented
5#11720527C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590614—31nomaximal
9#10588463—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×#6194—21no—
15C2×D3×Dic9—21no—
16#88638—22no—
17#88638—22no—
18C3×C9×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C3×(C2×D9)⋊C4—21no—
21C3×C4⋊Dic9—21no—
22C2×C9×C3⋊C8C2211yescharacteristic
23#88423—22no—
24#88418—22no—
25C2×#3795—31no—
26C3×#897—31no—
27C2×#3719—31no—
28C2×#3715—31no—
29C2×#3699—31no—
30#63637—31no—
31C3×#809D311yescharacteristic
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3672—31no—
34C2×Dic36—31no—
35#63459D311yescharacteristic, complemented
36C9×#102—31no—
37(C2×D9)⋊C8—31no—
38#63368—31no—
39#63368—61no—
40C22×D3×C9—21no—
41C22×C3×D9D411yescharacteristic
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45#6226—21no—
46#6226D412yes—
47C2×C3×Dic9—21no—
48C2×C3×Dic9D411yescharacteristic
49C9×D12—21no—
50C9×D12D412yes—
51C3×C4×D9—21no—
52C3×C4×D9—21no—
53C3×Dic18—21no—
54C3×Dic18—21no—
55#6194—24no—
56D3×Dic9—24no—
57C3⋊(C9⋊C8)—21no—
58C9×C3⋊C8—21no—
59#5047—91no—
60#4783—91no—
61C2×#634—31no—
62C2×C3×D12C2×D311yescharacteristic
63C2×C3×C4×D3—31no—
64C2×C3×Dic6—31no—
65C2×#625—61no—
66C2×D3×Dic3—61no—
67#3795—62no—
68#3795—62no—
69C32×C4⋊C4C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—61no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74#3719—62no—
75#3715—62no—
76C2×D4×C9—31no—
77C2×#611—61no—
78C22×Dic9—61no—
79#3699—62no—
80#3699—62no—
81C2×C4×D9C2×D311yescharacteristic
82C2×Dic18—31no—
83C2×Dic18—61no—
84C2×Dic18C2×D311yescharacteristic
85C2×C8×C9—31no—
86C9×C4⋊C4—21no—
87C9×C4⋊C4C2×D311yescharacteristic
88#3672—62no—
89(C2×D9)⋊C4—21no—
90C4⋊Dic9—21no—
91C2×C9⋊C8—31no—
92C2×C9⋊C8—61no—
93Dic36—62no—
94C22×C3×C9C2×D411yescharacteristic
95C2×D3×C9—22no—
96C2×D3×C9—41no—
97C2×D3×C9—41no—
98C2×C3×D9—21no—
99C2×C3×D9—21no—
100C2×C3×D9—21no—
101C3×C4×C9—41no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—22no—
105C3×Dic9—21no—
106C3×Dic9—41no—
107C2×#237—91no—
108C2×#235—91no—
109C3×C22⋊Q8—91no—
110C2×#216—91no—
111C2×#215—91no—
112C2×#214—91no—
113C2×Dic12—91no—
114#897—31no—
115C3×#102—31no—
116#830—91no—
117(C2×D3)⋊C8—91no—
118#810—91no—
119#810—91no—
120#810—181no—
121#809D911yescharacteristic, complemented
122C22×C3×D3—21no—
123C22×C3×D3—31no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127#634—32no—
128#634—61no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C3×D12—21no—
132C3×D12D1212yes—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137#625—64no—
138D3×Dic3—64no—
139C23×C9—61no—
140C22×D9#6711yescharacteristic
141C3⋊(C3⋊C8)—61no—
142C3×C3⋊C8—21no—
143D4×C9—32no—
144D4×C9—61no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—41no—
148C2×C4×C9C22×D311yescharacteristic
149#611—64no—
150C2×Dic9—21no—
151C2×Dic9—64no—
152C2×Dic9—61no—
153C2×Dic9—121no—
154C2×Dic9#6711yescharacteristic
155C4×D9—21no—
156C4×D9—21no—
157Dic18—21no—
158Dic18—21no—
159Dic18—32no—
160Dic18—62no—
161Dic18—61no—
162Dic18—121no—
163C8×C9—61no—
164C9⋊C8—61no—
165C9⋊C8—121no—
166#452—271noSylow
167C2×C3×C9#12011yescharacteristic
168C2×C3×C9#13511yescharacteristic
169C2×C3×C9#13711yescharacteristic
170D3×C9—42no—
171C3×D9—41no—
172C2×C3×Q8—91no—
173C2×C3×D4—31no—
174C22×C3×C4—91no—
175C2×#67—181no—
176C2×#67—181no—
177C22×Dic3—181no—
178#237—182no—
179#237—182no—
180#235—182no—
181#235—182no—
182C2×D12C2×D911yescharacteristic
183C2×C4×D3—31no—
184C2×C4×D3—181no—
185C2×Dic6—31no—
186C2×Dic6—91no—
187C2×Dic6—91no—
188C2×Dic6—181no—
189C2×C3×C8—31no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—181no—
192C3×C4⋊C4C2×D911yescharacteristic
193C3×C4⋊C4D3211yescharacteristic
194C3×C22⋊C4—181no—
195#216—182no—
196#215—182no—
197#214—182no—
198(C2×D3)⋊C4—61no—
199C4⋊Dic3—61no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—181no—
203C2×C3⋊C8C2×D911yescharacteristic
204Dic12—182no—
205C22×C32D3×D411yescharacteristic
206C2×C3×D3—22no—
207C2×C3×D3—41no—
208C2×C3×D3—41no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C32×C4—41no—
213C32×C4C2×D1211yescharacteristic
214C32×C4D3×D411yescharacteristic
215C3⋊Dic3—62no—
216C3×Dic3—61no—
217C3×Dic3—121no—
218C22×C9—21no—
219C22×C9—62no—
220C22×C9—121no—
221C22×C9—121no—
222C22×C9C2×#6711yescharacteristic
223C2×D9—21no—
224C2×D9—21no—
225C2×D9—21no—
226C4×C9—21no—
227C4×C9—21no—
228C4×C9—41no—
229C4×C9—81no—
230C4×C9D3×D411yescharacteristic
231C4×C9D3×D411yescharacteristic
232Dic9—21no—
233Dic9—41no—
234Dic9—62no—
235Dic9—122no—
236C2×#41—271no—
237C2×Q16—271no—
238C2×SD16—271no—
239C22⋊Q8—91no—
240#103—271no—
241#102—31no—
242C22⋊C8—271no—
243C3×C9#45211yescharacteristic, complemented, Sylow
244C23×C3—61no—
245C23×C3—91no—
246C22×D3—21no—
247C22×D3—31no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×D4—32no—
251C3×D4—61no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262#67—184no—
263#67—184no—
264C2×Dic3—31no—
265C2×Dic3—61no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C2×Dic3—361no—
270D12—21no—
271D12D3612yes—
272C4×D3—61no—
273C4×D3—61no—
274C4×D3—184no—
275Dic6—61no—
276Dic6—61no—
277Dic6—92no—
278Dic6—92no—
279Dic6—182no—
280Dic6—181no—
281Dic6—181no—
282Dic6—361no—
283C3×C8—61no—
284C3⋊C8—21no—
285C3⋊C8—181no—
286C3⋊C8—181no—
287C3⋊C8—361no—
288C2×C32#88211yescharacteristic
289C2×C32#91211yescharacteristic
290C2×C32#91511yescharacteristic
291C3×D3—42no—
292C3×D3—121no—
293C2×C9—21no—
294C2×C9—21no—
295C2×C9—21no—
296C2×C9—122no—
297C2×C9#93711yescharacteristic
298C2×C9#94211yescharacteristic
299C2×C9#95011yescharacteristic
300D9—41no—
301#41—542no—
302#41—542no—
303C2×Q8—91no—
304C2×Q8—271no—
305C2×D4—31no—
306C2×D4—541no—
307C22×C4—91no—
308C22×C4—541no—
309Q16—542no—
310SD16—542no—
311C2×C8—31no—
312C2×C8—271no—
313C4⋊C4—181no—
314C4⋊C4D3×D911yescharacteristic
315C22⋊C4—181no—
316C22×C3—21no—
317C22×C3—62no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3—181no—
323C22×C3D4×D911yescharacteristic
324C22×C3D3×#6711yescharacteristic
325C2×D3—22no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—61no—
329C2×D3—61no—
330C2×D3—61no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—81no—
336C3×C4—181no—
337C3×C4—361no—
338C3×C4C2×D3611yescharacteristic
339C3×C4D4×D911yescharacteristic
340C3×C4D3×D1211yescharacteristic
341C3×C4#380211yescharacteristic
342Dic3—61no—
343Dic3—121no—
344Dic3—182no—
345Dic3—182no—
346Dic3—362no—
347C32#478311yescharacteristic
348C9—21no—
349C9#504711yescharacteristic, complemented
350C23—61no—
351C23—91no—
352Q8—181no—
353Q8—181no—
354Q8—272no—
355Q8—541no—
356D4—32no—
357D4—61no—
358D4—544no—
359C2×C4—21no—
360C2×C4—91no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4—544no—
365C2×C4—541no—
366C2×C4C2×D3×D911yescharacteristic
367C8—61no—
368C8—541no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—122no—
373C2×C3—361no—
374C2×C3#6344411yescharacteristic
375C2×C3#6347411yescharacteristic
376C2×C3#6347711yescharacteristic
377C2×C3#6392211yescharacteristic
378C2×C3#6393311yescharacteristic
379C2×C3#6394611yescharacteristic
380D3—42no—
381D3—121no—
382C22—62no—
383C22—121no—
384C22—121no—
385C22—181no—
386C22—181no—
387C22—181no—
388C22D9×#6711yescharacteristic
389C4—41no—
390C4—181no—
391C4—361no—
392C4—542no—
393C4D3×D3611yescharacteristic
394C4#8864611yescharacteristic
395C3—21no—
396C3#1058846311yescharacteristic, complemented
397C3#1059061411yescharacteristic
398C2—122no—
399C2—361no—
400C2#1172104911yescharacteristic
401C2#1172117711yescharacteristic
402C2#1172121411yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88638C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#452
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
2. C3×#63459C2semidirect
3. C2×#88423C2semidirect
4. C2×#88418C2semidirect
5. #11720527C2semidirect
6. C9×#809C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
35. #63459D3semidirect
121. #809D9semidirect
243. C3×C9#452semidirect
349. C9#5047semidirect
396. C3#10588463semidirect

Maximal subgroups

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

subgroupindex
C2×#886382
C3×#634592
C2×#884232
C2×#884182
#117205272
C9×#8092
(C2×D9)⋊(C3⋊C8)2
#105906143
#105884633