Finite Group Databasebeta

← #13140440

Subgroups — #13140440

Order 1728.

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

normal subgroups only

This group has 558 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
1C9×#1004C211yesmaximal, characteristic
2#11721835C211yesmaximal, characteristic, complemented
3#11721266C212yesmaximal, complemented
4C3×#63393C211yesmaximal, characteristic, complemented
5#11720332C212yesmaximal, complemented
6#10591534—31nomaximal
7#10588923—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C22×C3⋊Dic9C2211yescharacteristic, complemented
10C2×C9×#67—21no—
11C2×C9×#67C2212yes—
12C9×#237—22no—
13D3×D4×C9—22no—
14C9×#235—21no—
15#88551—21no—
16#88551C2211yescharacteristic, complemented
17#88534—21no—
18C3×#3676C2212yescomplemented
19(C2×D3)⋊Dic9—22no—
20Dic9⋊Dic3—22no—
21C3×#1004D311yescharacteristic
22#64144—31no—
23#63962—32no—
24C9×H(5,𝔽2)—31no—
25C3×#834—31no—
26#63580—32no—
27#63497—31no—
28#63497—32no—
29#63497—61no—
30#63393—32no—
31#63393D311yescharacteristic, complemented
32C23×C3×C9D411yescharacteristic
33C23×C3×C9C2311yescharacteristic
34C22×D3×C9—21no—
35C22×D3×C9D412yes—
36C3×D4×C9—22no—
37C2×C3×C4×C9D411yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9—22no—
41C9×#67—22no—
42C9×#67—22no—
43C9×#67—22no—
44C2×C9×Dic3—21no—
45C2×C9×Dic3D412yes—
46C2×C3×Dic9—22no—
47C9×D12—21no—
48C4×D3×C9—22no—
49C9×Dic6—21no—
50#5243—91no—
51#4742—91no—
52C2×C32×D4C2×D311yescharacteristic
53C22×C3⋊Dic3—31no—
54C2×C3×#67—21no—
55C2×C3×#67C2×D312yes—
56C3×#237—22no—
57C3×D3×D4—22no—
58C3×#235—21no—
59#3757—31no—
60#3757—61no—
61#3750—61no—
62C3×#217—32no—
63Dic3⋊Dic3—62no—
64(C2×D3)⋊Dic3—62no—
65C9×#41—62no—
66C9×#41—61no—
67C2×D4×C9—21no—
68C2×D4×C9—32no—
69C2×D4×C9—62no—
70C2×D4×C9—61no—
71C2×D4×C9C2×D311yescharacteristic
72C22×Dic9—31no—
73C22×Dic9—61no—
74#3676—31no—
75#3676—62no—
76#3676—61no—
77#3676—62no—
78#3676—61no—
79#3676C2×D312yescomplemented
80Dic9⋊C4—62no—
81Dic9⋊C4—62no—
82Dic9⋊C4—61no—
83C22×C3×C9—21no—
84C22×C3×C9—41no—
85C22×C3×C9—41no—
86C22×C3×C9C2×D411yescharacteristic
87C22×C3×C9C2×D412yes—
88C2×D3×C9—22no—
89C2×D3×C9—42no—
90C2×D3×C9—81no—
91C3×C4×C9—21no—
92C3⋊Dic9—41no—
93C3⋊Dic9—41no—
94C9×Dic3—22no—
95C3×Dic9—42no—
96C3×H(5,𝔽2)—31no—
97#1004D911yescharacteristic
98#935—92no—
99#935—91no—
100#935—91no—
101#935—181no—
102#886—92no—
103C3×#99—91no—
104#834—31no—
105#834—92no—
106#806—92no—
107C23×C32#6711yescharacteristic
108C23×C32C22×D311yescharacteristic
109C22×C3×D3—21no—
110C22×C3×D3#6712yes—
111C32×D4—22no—
112C2×C32×C4#6711yescharacteristic
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—62no—
116C3×#67—22no—
117C3×#67—22no—
118C3×#67—22no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3#6712yes—
122C3×D12—21no—
123C3×C4×D3—22no—
124C3×Dic6—21no—
125C23×C9—21no—
126C23×C9—21no—
127C23×C9—32no—
128C23×C9—61no—
129C23×C9D1211yescharacteristic
130C23×C9C22×D311yescharacteristic
131Q8×C9—61no—
132D4×C9—22no—
133D4×C9—42no—
134D4×C9—61no—
135D4×C9—62no—
136D4×C9—62no—
137D4×C9—62no—
138C2×C4×C9—21no—
139C2×C4×C9—32no—
140C2×C4×C9—62no—
141C2×C4×C9—61no—
142C2×C4×C9D1211yescharacteristic
143C2×Dic9—22no—
144C2×Dic9—62no—
145C2×Dic9—62no—
146C2×Dic9—61no—
147C2×Dic9—61no—
148C2×Dic9—62no—
149C2×Dic9—122no—
150#458—271noSylow
151C2×C3×C9—21no—
152C2×C3×C9—22no—
153C2×C3×C9—41no—
154C2×C3×C9#12011yescharacteristic
155D3×C9—42no—
156C3×#41—62no—
157C3×#41—61no—
158C2×C3×D4—21no—
159C2×C3×D4—32no—
160C2×C3×D4—62no—
161C2×C3×D4—61no—
162C2×C3×D4C2×D911yescharacteristic
163C2×C3×D4D3211yescharacteristic
164C2×#67—21no—
165C2×#67C2×D912yes—
166C22×Dic3—91no—
167C22×Dic3—91no—
168C22×Dic3—181no—
169#237—22no—
170D3×D4—22no—
171#235—21no—
172C3×C22⋊C4—92no—
173#217—32no—
174#217—91no—
175#217—91no—
176#217—181no—
177#217—182no—
178#217—182no—
179#217—181no—
180#217—181no—
181(C2×D3)⋊C4—182no—
182C4⋊Dic3—182no—
183Dic3⋊C4—182no—
184Dic3⋊C4—182no—
185Dic3⋊C4—181no—
186Dic3⋊C4—181no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32C2×#6711yescharacteristic
191C22×C32C2×#6712yes—
192C2×C3×D3—22no—
193C2×C3×D3—42no—
194C2×C3×D3—81no—
195C32×C4—21no—
196C3⋊Dic3—121no—
197C3⋊Dic3—121no—
198C3×Dic3—22no—
199C3×Dic3—122no—
200C22×C9—21no—
201C22×C9—21no—
202C22×C9—22no—
203C22×C9—41no—
204C22×C9—41no—
205C22×C9—41no—
206C22×C9—42no—
207C22×C9—42no—
208C22×C9—62no—
209C22×C9—122no—
210C22×C9—241no—
211C22×C9C2×D1211yescharacteristic
212C22×C9D3×D412yes—
213C4×C9—21no—
214C4×C9—41no—
215C4×C9—62no—
216Dic9—42no—
217Dic9—122no—
218Dic9—121no—
219Dic9—122no—
220Dic9—121no—
221H(5,𝔽2)—31no—
222#123—272no—
223#123—271no—
224#99—91no—
225#99—272no—
226C3×C9#45811yescharacteristic, complemented, Sylow
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—32no—
230C23×C3—61no—
231C23×C3#61111yescharacteristic
232C23×C3C22×D911yescharacteristic
233C23×C3#62611yescharacteristic
234C23×C3C2×D3211yescharacteristic
235C22×D3—21no—
236C22×D3#61112yes—
237C3×Q8—61no—
238C3×D4—22no—
239C3×D4—22no—
240C3×D4—42no—
241C3×D4—61no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—62no—
245C2×C3×C4—21no—
246C2×C3×C4—32no—
247C2×C3×C4—62no—
248C2×C3×C4—61no—
249C2×C3×C4—182no—
250C2×C3×C4#61111yescharacteristic
251C2×C3×C4#62611yescharacteristic
252#67—22no—
253#67—22no—
254#67—22no—
255C2×Dic3—21no—
256C2×Dic3—62no—
257C2×Dic3—182no—
258C2×Dic3—182no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261C2×Dic3—182no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—182no—
265C2×Dic3—362no—
266C2×Dic3#61112yes—
267D12—21no—
268C4×D3—22no—
269Dic6—21no—
270C2×C32—21no—
271C2×C32—22no—
272C2×C32—41no—
273C2×C32#95311yescharacteristic
274C3×D3—42no—
275C2×C9—22no—
276C2×C9—22no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—42no—
280C2×C9—41no—
281C2×C9—81no—
282C2×C9—122no—
283C2×C9#88211yescharacteristic
284#41—62no—
285#41—61no—
286C2×D4—32no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4D3×D911yescharacteristic
290C22×C4—271no—
291C4⋊C4—542no—
292C4⋊C4—541no—
293C22⋊C4—92no—
294C22⋊C4—271no—
295C22⋊C4—542no—
296C22⋊C4—541no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3—62no—
309C22×C3—122no—
310C22×C3—241no—
311C22×C3C2×#61111yescharacteristic
312C22×C3C2×#61112yes—
313C22×C3C2×#62611yescharacteristic
314C22×C3D3×#6712yes—
315C2×D3—22no—
316C2×D3—42no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—41no—
321C3×C4—62no—
322C3×C4—362no—
323Dic3—22no—
324Dic3—122no—
325Dic3—362no—
326Dic3—361no—
327Dic3—362no—
328Dic3—361no—
329Dic3—361no—
330Dic3—361no—
331C32#524311yescharacteristic
332C9—21no—
333C9#474211yescharacteristic, complemented
334C23—32no—
335C23—61no—
336C23#619511yescharacteristic
337C23C2×D3×D911yescharacteristic
338Q8—61no—
339D4—22no—
340D4—61no—
341D4—62no—
342D4—62no—
343D4—62no—
344C2×C4—32no—
345C2×C4—62no—
346C2×C4—61no—
347C2×C4—182no—
348C2×C4—541no—
349C2×C4—541no—
350C2×C4—542no—
351C2×C4#619511yescharacteristic
352C2×C3—22no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—22no—
357C2×C3—21no—
358C2×C3—42no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—81no—
362C2×C3—122no—
363C2×C3#6351511yescharacteristic
364C2×C3#6397711yescharacteristic
365D3—42no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—62no—
370C22—122no—
371C22—241no—
372C22C2×#619511yescharacteristic
373C22D3×#61112yes—
374C4—21no—
375C4—62no—
376C4—362no—
377C4—1081no—
378C4—1081no—
379C3—21no—
380C3#1058892311yescharacteristic, complemented
381C3#1059153411yescharacteristic
382C2—22no—
383C2—21no—
384C2—41no—
385C2—122no—
386C2#1172128711yescharacteristic
subgroupG/H
centerC2#11721287
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#953
prime radicalC3×#1004D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#458
LCSG ▷ C23×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721835C2semidirect
3. #11721266C2semidirect
4. C3×#63393C2semidirect
5. #11720332C2semidirect
9. C22×C3⋊Dic9C22semidirect
16. #88551C22semidirect
18. C3×#3676C22semidirect
31. #63393D3semidirect
79. #3676C2×D3semidirect
226. C3×C9#458semidirect
333. C9#4742semidirect
380. C3#10588923semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
#117218352
#117212662
C3×#633932
#117203322
#105915343
#105889233