Finite Group Databasebeta

← #13140753

Subgroups — #13140753

Order 1728.

428 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721854C211yesmaximal, characteristic, complemented
2C2×C3×#3675C211yesmaximal, characteristic
3#11720998C211yesmaximal, characteristic
4C2×#88415C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic, complemented
6#11720273C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10588498—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×Dic3×D9—21no—
14#88534—21no—
15#88534—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C3×#3675—22no—
18C9×C4⋊Dic3C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D9)⋊Dic3—21no—
22#88415—22no—
23#64153—31no—
24C2×C3×#215—31no—
25#63898—31no—
26C2×#3716—31no—
27C3×#835D311yescharacteristic
28#63572—31no—
29#63567—31no—
30#63506—31no—
31#63506—61no—
32C2×#3675D311yescharacteristic
33C2×#3664—31no—
34#63455—31no—
35C9×#103—31no—
36Dic9⋊C8—31no—
37#63368—31no—
38C22×C3×D9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46#6226—21no—
47#6226—21no—
48C2×C9×Dic3—21no—
49C3×D36—21no—
50C3×D36D412yes—
51Dic3×D9—42no—
52C9×C3⋊C8—21no—
53C3×C9⋊C8—21no—
54#5162—91no—
55#4818—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×#634—31no—
58C2×C3×D12—31no—
59C2×D3×Dic3—61no—
60#3750—61no—
61#3750—61no—
62C4×C3⋊Dic3—31no—
63C3×#215—62no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67(C2×D3)⋊Dic3—61no—
68#3716—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D36C2×D311yescharacteristic
72C2×C4×D9—61no—
73C2×Dic18—31no—
74C2×Dic18—61no—
75C2×C8×C9—31no—
76C9×C4⋊C4—31no—
77#3675—22no—
78(C2×D9)⋊C4—61no—
79Dic9⋊C4—61no—
80Dic9⋊C4—61no—
81Dic9⋊C4—62no—
82Dic9⋊C4—62no—
83C4×Dic9—31no—
84C4×Dic9—61no—
85C2×C9⋊C8C2×D311yescharacteristic
86#3664—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3×D9—22no—
89C2×C3×D9—41no—
90C2×C3×D9—41no—
91C3×C4×C9—41no—
92C3×C4×C9C2×D411yescharacteristic
93C3×C4×C9#4111yescharacteristic
94C3⋊Dic9—21no—
95C3⋊Dic9—41no—
96C3⋊Dic9SD1612yescomplemented
97C9×Dic3—41no—
98C2×C3×SD16—91no—
99#944—91no—
100#944—91no—
101#944—181no—
102C2×#215—31no—
103D4×Dic3—91no—
104C2×#214—91no—
105C2×#204—91no—
106#893—91no—
107C3×#103—31no—
108#835D911yescharacteristic, complemented
109#816—91no—
110Dic3⋊C8—91no—
111Dic3⋊C8—91no—
112#810—91no—
113C22×C3×D3—61no—
114C32×Q8—21no—
115C32×Q8—21no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—61no—
121#634—61no—
122#634—61no—
123C2×C3×Dic3—21no—
124C3×D12—32no—
125C3×D12—61no—
126D3×Dic3—122no—
127C22×D9—21no—
128C3×C3⋊C8—21no—
129C3×C3⋊C8—61no—
130Q8×C9—21no—
131Q8×C9—21no—
132Q8×C9—41no—
133Q8×C9—41no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9—31no—
140C2×Dic9—31no—
141C2×Dic9—61no—
142C2×Dic9—61no—
143C2×Dic9—62no—
144C2×Dic9—61no—
145D36—21no—
146D36#6712yes—
147C4×D9—122no—
148Dic18—61no—
149Dic18—61no—
150Dic18—121no—
151Dic18—121no—
152C8×C9—61no—
153C9⋊C8—21no—
154#461—271noSylow
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9C2×SD1611yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C3×D9—42no—
159C2×C3×Q8—21no—
160C2×C3×Q8C2×D911yescharacteristic
161C2×C3×Q8D3211yescharacteristic
162C2×C3×D4—91no—
163C22×Dic3—181no—
164C2×D12—31no—
165C2×C4×D3—181no—
166C2×Dic6—91no—
167C2×Dic6—91no—
168C2×Dic6—181no—
169C3×SD16—182no—
170C2×C3×C8—31no—
171C2×C3×C8—91no—
172C3×C4⋊C4—31no—
173#217—181no—
174#215—62no—
175#214—182no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3C2×D911yescharacteristic
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—182no—
183Dic3⋊C4—182no—
184C4×Dic3—91no—
185C4×Dic3—91no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8C2×D911yescharacteristic
189#204—182no—
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—62no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C32×C4—41no—
195C32×C4#23511yescharacteristic
196C32×C4D3×D411yescharacteristic
197C3⋊Dic3—32no—
198C3⋊Dic3—61no—
199C3⋊Dic3—121no—
200C3×Dic3—41no—
201C22×C9—21no—
202C22×C9D3×D411yescharacteristic
203C2×D9—22no—
204C2×D9—41no—
205C2×D9—41no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—42no—
209C4×C9—41no—
210C4×C9—121no—
211C4×C9#23711yescharacteristic
212C4×C9C2×#6711yescharacteristic
213Dic9—32no—
214Dic9—62no—
215Dic9—62no—
216Dic9—61no—
217Dic9—122no—
218Dic9—121no—
219C2×SD16—91no—
220C2×SD16—271no—
221C4⋊Q8—271no—
222C4×D4—271no—
223C4⋊C8—271no—
224#103—31no—
225#103—271no—
226C3×C9#46111yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×D4—92no—
236C3×D4—181no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—61no—
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—21no—
245C2×Dic3—91no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—182no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—362no—
255D12—32no—
256D12—61no—
257C4×D3—362no—
258Dic6—181no—
259Dic6—181no—
260Dic6—181no—
261Dic6—181no—
262Dic6—361no—
263Dic6—361no—
264C3×C8—61no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—61no—
268C2×C32#88411yescharacteristic
269C2×C32D3×SD1611yescharacteristic
270C2×C32#91811yescharacteristic
271C3×D3—122no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9D3×SD1611yescharacteristic
276C2×C9#91511yescharacteristic
277C2×C9#93911yescharacteristic
278D9—42no—
279C2×Q8—271no—
280C2×Q8D3×D911yescharacteristic
281C2×D4—91no—
282C22×C4—541no—
283SD16—182no—
284SD16—542no—
285C2×C8—31no—
286C2×C8—91no—
287C4⋊C4—31no—
288C4⋊C4—541no—
289C4⋊C4—541no—
290C22⋊C4—541no—
291C42—271no—
292C22×C3—21no—
293C22×C3—182no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D4×D911yescharacteristic
297C22×C3#381111yescharacteristic
298C2×D3—62no—
299C2×D3—121no—
300C2×D3—121no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—42no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—121no—
307C3×C4#369711yescharacteristic
308C3×C4D4×D911yescharacteristic
309C3×C4#380411yescharacteristic
310C3×C4D3×#6711yescharacteristic
311Dic3—41no—
312Dic3—92no—
313Dic3—92no—
314Dic3—182no—
315Dic3—182no—
316Dic3—181no—
317Dic3—181no—
318Dic3—362no—
319Dic3—361no—
320Dic3—361no—
321C32#481811yescharacteristic
322C9—21no—
323C9#516211yescharacteristic, complemented
324C23—181no—
325Q8—21no—
326Q8—21no—
327Q8—541no—
328Q8—541no—
329D4—92no—
330D4—181no—
331C2×C4—21no—
332C2×C4—61no—
333C2×C4—271no—
334C2×C4—271no—
335C2×C4—541no—
336C2×C4—1082no—
337C2×C4C2×D3×D911yescharacteristic
338C8—61no—
339C8—181no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—362no—
344C2×C3#6344611yescharacteristic
345C2×C3SD16×D911yescharacteristic
346C2×C3#6348011yescharacteristic
347C2×C3#6394011yescharacteristic
348C2×C3#6395111yescharacteristic
349C2×C3#6397111yescharacteristic
350D3—122no—
351C22—182no—
352C22—361no—
353C22—361no—
354C22#8866811yescharacteristic
355C4—41no—
356C4—121no—
357C4—272no—
358C4—541no—
359C4—1081no—
360C4#8865811yescharacteristic
361C4D9×#6711yescharacteristic
362C3—21no—
363C3#1058849811yescharacteristic, complemented
364C3#1059156911yescharacteristic
365C2—362no—
366C2#1172121211yescharacteristic
367C2#1172122511yescharacteristic
368C2#1172127111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#461
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11721854C2semidirect
5. C9×#835C2semidirect
6. #11720273C2semidirect
7. #11720269C2semidirect
96. C3⋊Dic9SD16semidirect
108. #835D9semidirect
226. C3×C9#461semidirect
323. C9#5162semidirect
363. C3#10588498semidirect

Maximal subgroups

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

subgroupindex
#117218542
C2×C3×#36752
#117209982
C2×#884152
C9×#8352
#117202732
#117202692
#105915693
#105884983