Finite Group Databasebeta

← #13140825

Subgroups — #13140825

Order 1728.

454 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 93 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3C2×C3×#3675C211yesmaximal, characteristic, complemented
4#11721043C211yesmaximal, characteristic
5#11720295C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10591622—31nomaximal
9#10588843—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6194—21no—
14C2×#6194—21no—
15C4⋊(C3⋊Dic9)—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C4×C3⋊Dic9—21no—
18C4×C3⋊Dic9C2211yescharacteristic
19C9×#215—22no—
20C3×#3675—22no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23Q8×C3⋊Dic3—31no—
24C2×C3×#215—31no—
25C2×C3×#215D311yescharacteristic
26#63914—31no—
27#63572—31no—
28#63563—31no—
29#63563—31no—
30C2×C9×SD16—31no—
31Q8×Dic9—31no—
32Q8×Dic9—61no—
33C2×#3675D311yescharacteristic, complemented
34#63502—31no—
35#63392—31no—
36#63380—31no—
37Dic9⋊C8—31no—
38C22×D3×C9—21no—
39C22×C3×D9—21no—
40C3×Q8×C9—21no—
41C3×Q8×C9D412yescomplemented
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D411yescharacteristic
46C2×C3⋊Dic9D411yescharacteristic
47C9×D12—21no—
48C9×D12—21no—
49C3×D36—21no—
50C3×D36—21no—
51#6194—42no—
52#6194—42no—
53C9×C3⋊C8—21no—
54C3×C9⋊C8—21no—
55#5163—91no—
56#5163—91no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×D12C2×D311yescharacteristic
60C2×#625—61no—
61C2×#625—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—61no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—61no—
66C3×#215—22no—
67C3×#215—62no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8C2×D311yescharacteristic
70C2×Q8×C9—21no—
71C2×Q8×C9C2×D311yescharacteristic
72C2×D4×C9—31no—
73C2×#611—61no—
74C2×#611—61no—
75C2×D36C2×D311yescharacteristic
76C9×SD16—62no—
77C2×C8×C9—31no—
78#3675—22no—
79C4⋊Dic9—31no—
80C4⋊Dic9—61no—
81C4⋊Dic9—61no—
82C4⋊Dic9—62no—
83C4×Dic9—31no—
84C4×Dic9—61no—
85C4×Dic9—61no—
86C4×Dic9—62no—
87C2×C9⋊C8C2×D311yescharacteristic
88C22×C3×C9C2×D411yescharacteristic
89C2×D3×C9—41no—
90C2×D3×C9—41no—
91C2×D3×C9—41no—
92C2×C3×D9—41no—
93C2×C3×D9—41no—
94C2×C3×D9—41no—
95C3×C4×C9—22no—
96C3×C4×C9C2×D411yescharacteristic
97C3×C4×C9#4111yescharacteristic
98C3⋊Dic9—21no—
99C3⋊Dic9—41no—
100C3⋊Dic9SD1612yescomplemented
101C2×C3×SD16—31no—
102C2×C3×SD16—91no—
103Q8×Dic3—91no—
104Q8×Dic3—91no—
105Q8×Dic3—181no—
106C2×#215—31no—
107C2×#215D911yescharacteristic, complemented
108#940—91no—
109#940—91no—
110#832—91no—
111#832—91no—
112#821—91no—
113#821—91no—
114Dic3⋊C8—91no—
115Dic3⋊C8—91no—
116C22×C3×D3—21no—
117C22×C3×D3—61no—
118C32×Q8—21no—
119C32×Q8#6712yes—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—61no—
125C3×D12—21no—
126C3×D12—21no—
127C3×D12—61no—
128C3×D12—61no—
129#625—122no—
130#625—122no—
131C23×C9—61no—
132C22×D9—21no—
133C3×C3⋊C8—21no—
134C3×C3⋊C8—61no—
135Q8×C9—22no—
136Q8×C9—21no—
137Q8×C9—22no—
138Q8×C9#6712yescomplemented
139D4×C9—61no—
140D4×C9—61no—
141C2×C4×C9—22no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9C22×D311yescharacteristic
145#611—122no—
146#611—122no—
147C2×Dic9—31no—
148C2×Dic9—31no—
149C2×Dic9—61no—
150C2×Dic9—61no—
151C2×Dic9—62no—
152C2×Dic9—61no—
153D36—21no—
154D36—21no—
155C8×C9—61no—
156C9⋊C8—21no—
157#460—271noSylow
158C2×C3×C9#12111yescharacteristic
159C2×C3×C9C2×SD1611yescharacteristic
160C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
161D3×C9—81no—
162C3×D9—81no—
163C2×C3×Q8—21no—
164C2×C3×Q8C2×D911yescharacteristic
165C2×C3×Q8D3211yescharacteristic
166C2×C3×D4—31no—
167C2×C3×D4—91no—
168C2×#67—181no—
169C2×#67—181no—
170C2×#67—181no—
171C2×#67—181no—
172C2×D12—31no—
173C2×D12C2×D911yescharacteristic
174C3×SD16—62no—
175C3×SD16—182no—
176C2×C3×C8—31no—
177C2×C3×C8—91no—
178#215—22no—
179#215—62no—
180C4⋊Dic3—91no—
181C4⋊Dic3—91no—
182C4⋊Dic3—181no—
183C4⋊Dic3—181no—
184C4⋊Dic3—181no—
185C4⋊Dic3—182no—
186C4×Dic3—91no—
187C4×Dic3—91no—
188C4×Dic3—181no—
189C4×Dic3—181no—
190C4×Dic3—181no—
191C4×Dic3—182no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8C2×D911yescharacteristic
194C22×C32D3×D411yescharacteristic
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—22no—
202C32×C4#23711yescharacteristic
203C32×C4C2×#6711yescharacteristic
204C3⋊Dic3—32no—
205C3⋊Dic3—61no—
206C3⋊Dic3—121no—
207C22×C9—21no—
208C22×C9—121no—
209C22×C9—121no—
210C22×C9—121no—
211C22×C9D3×D411yescharacteristic
212C2×D9—41no—
213C2×D9—41no—
214C2×D9—41no—
215C4×C9—24no—
216C4×C9—22no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9#23711yescharacteristic
220C4×C9C2×#6711yescharacteristic
221Dic9—32no—
222Dic9—62no—
223Dic9—61no—
224Dic9—121no—
225Dic9—122no—
226Dic9—121no—
227C2×SD16—31no—
228C2×SD16—91no—
229#127—271no—
230C4×Q8—271no—
231C4⋊C8—271no—
232#102—271no—
233#102—271no—
234C3×C9#46011yescharacteristic, complemented, Sylow
235C23×C3—61no—
236C23×C3—181no—
237C22×D3—21no—
238C22×D3—61no—
239C3×Q8—22no—
240C3×Q8—21no—
241C3×Q8—21no—
242C3×Q8—22no—
243C3×Q8#61112yescomplemented
244C3×Q8#62512yes—
245C3×D4—61no—
246C3×D4—61no—
247C3×D4—181no—
248C3×D4—181no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4C22×D911yescharacteristic
254C2×C3×C4C2×D3211yescharacteristic
255#67—362no—
256#67—362no—
257#67—362no—
258#67—362no—
259C2×Dic3—91no—
260C2×Dic3—91no—
261C2×Dic3—91no—
262C2×Dic3—91no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265C2×Dic3—182no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268D12—21no—
269D12—21no—
270D12—61no—
271D12—61no—
272C3×C8—61no—
273C3×C8—181no—
274C3⋊C8—21no—
275C3⋊C8—61no—
276C2×C32D3×SD1611yescharacteristic
277C2×C32#91411yescharacteristic
278C2×C32#93911yescharacteristic
279C3×D3—81no—
280C3×D3—241no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—241no—
285C2×C9D3×SD1611yescharacteristic
286C2×C9#91411yescharacteristic
287C2×C9#93911yescharacteristic
288D9—81no—
289C2×Q8D3×D911yescharacteristic
290C2×D4—31no—
291C2×D4—91no—
292C2×D4—541no—
293C2×D4—541no—
294SD16—62no—
295SD16—182no—
296C2×C8—31no—
297C2×C8—91no—
298C4⋊C4—271no—
299C4⋊C4—541no—
300C42—271no—
301C42—541no—
302C22×C3—21no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3D4×D911yescharacteristic
310C22×C3#381111yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—121no—
317C3×C4—24no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4#369911yescharacteristic
323C3×C4C2×#61111yescharacteristic
324C3×C4#380511yescharacteristic
325C3×C4C2×#62511yescharacteristic
326Dic3—92no—
327Dic3—92no—
328Dic3—182no—
329Dic3—181no—
330Dic3—181no—
331Dic3—361no—
332Dic3—362no—
333Dic3—361no—
334Dic3—361no—
335C32#516311yescharacteristic
336C9—21no—
337C9#516311yescharacteristic, complemented
338C23—61no—
339C23—181no—
340Q8—21no—
341Q8#619412yescomplemented
342D4—61no—
343D4—61no—
344D4—181no—
345D4—181no—
346D4—1082no—
347D4—1082no—
348C2×C4—21no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—541no—
352C2×C4C2×D3×D911yescharacteristic
353C8—61no—
354C8—181no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—241no—
359C2×C3—721no—
360C2×C3SD16×D911yescharacteristic
361C2×C3#6347611yescharacteristic
362C2×C3#6350111yescharacteristic
363C2×C3#6394011yescharacteristic
364C2×C3#6394111yescharacteristic
365C2×C3#6398011yescharacteristic
366D3—81no—
367D3—241no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#8866811yescharacteristic
375C4—22no—
376C4—272no—
377C4—541no—
378C4—1081no—
379C4#8866311yescharacteristic
380C4C2×#619411yescharacteristic
381C3—21no—
382C3#1058884311yescharacteristic, complemented
383C3#1059162211yescharacteristic
384C2—241no—
385C2—721no—
386C2#1172121211yescharacteristic
387C2#1172121311yescharacteristic
388C2#1172129011yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radical#11721043C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#460
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. Q8×C3⋊Dic9C2semidirect
2. C2×C9×#215C2semidirect
3. C2×C3×#3675C2semidirect
5. #11720295C2semidirect
6. #11720283C2semidirect
7. #11720273C2semidirect
33. C2×#3675D3semidirect
41. C3×Q8×C9D4semidirect
100. C3⋊Dic9SD16semidirect
107. C2×#215D9semidirect
138. Q8×C9#67semidirect
234. C3×C9#460semidirect
243. C3×Q8#611semidirect
337. C9#5163semidirect
341. Q8#6194semidirect
382. C3#10588843semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
C2×C9×#2152
C2×C3×#36752
#117210432
#117202952
#117202832
#117202732
#105916223
#105888433