Finite Group Databasebeta

← #13140457

Subgroups — #13140457

Order 1728.

492 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 87 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#11721840C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720911C211yesmaximal, characteristic, complemented
4C2×#88426C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10591444—31nomaximal
9#10588843—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×C9×Dic6C2211yescharacteristic
15C4×C3⋊Dic9C2211yescharacteristic
16C9×#214—22no—
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20Dic3⋊Dic9—21no—
21Dic3×Dic9—21no—
22#88426—22no—
23#64149—31no—
24C2×C3×#214D311yescharacteristic
25#63843—31no—
26C2×#3717—31no—
27C3×#832—31no—
28#63572—31no—
29#63564—31no—
30C2×C9×SD16—31no—
31Q8×Dic9—31no—
32C2×#3675—31no—
33#63502—31no—
34#63502—61no—
35#63392D311yescharacteristic, complemented
36#63380—31no—
37Dic9⋊C8—31no—
38C23×C3×C9—21no—
39C22×C3⋊D9—21no—
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—42no—
44#6233—42no—
45C2×C3⋊Dic9D411yescharacteristic
46C2×C3⋊Dic9D411yescharacteristic
47C4⋊(C3⋊D9)—21no—
48C4⋊(C3⋊D9)—21no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9—21no—
51C9×Dic6—21no—
52C9×Dic6D412yescomplemented
53C9×C3⋊C8—21no—
54C3×C9⋊C8—21no—
55#5163—91no—
56#4762—91no—
57C2×C32×D4C2×D311yescharacteristic
58C2×#638—61no—
59C2×#638—61no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C3×Dic6C2×D311yescharacteristic
62C4×C3⋊Dic3—31no—
63C3×#214—22no—
64C3×C4⋊Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67Dic3⋊Dic3—61no—
68Dic32—61no—
69#3717—62no—
70C2×Q8×C9—31no—
71C2×D4×C9—21no—
72C2×D4×C9C2×D311yescharacteristic
73C2×#611—61no—
74C2×#611—61no—
75C2×#611—62no—
76C2×#611—62no—
77C2×D36—31no—
78C2×D36—61no—
79C9×SD16—62no—
80C2×C8×C9—31no—
81#3675—62no—
82C4⋊Dic9—61no—
83C4⋊Dic9C2×D311yescharacteristic
84C4×Dic9—31no—
85C4×Dic9—61no—
86C4×Dic9—61no—
87C2×C9⋊C8C2×D311yescharacteristic
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9—41no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C2×C3⋊D9—41no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9#4111yescharacteristic
97C3⋊Dic9—21no—
98C3⋊Dic9SD1612yescomplemented
99C9×Dic3—22no—
100C3×Dic9—41no—
101C2×C3×SD16—31no—
102Q8×Dic3—91no—
103C2×#215—91no—
104#940—91no—
105#940—91no—
106#940—181no—
107C2×#214D911yescharacteristic, complemented
108C2×#204—91no—
109#887—91no—
110C3×#102—91no—
111#832—31no—
112#821—91no—
113Dic3⋊C8—91no—
114Dic3⋊C8—91no—
115#809—91no—
116C23×C32—21no—
117C22×C3⋊D3—61no—
118C32×D4—21no—
119C32×D4—21no—
120C2×C32×C4C22×D311yescharacteristic
121#638—122no—
122#638—122no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—31no—
125C4⋊(C3⋊D3)—61no—
126C4⋊(C3⋊D3)—61no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—61no—
129C3×Dic6—21no—
130C3×Dic6#6712yes—
131C23×C9—22no—
132C23×C9—21no—
133C22×D9—62no—
134C22×D9—61no—
135C3×C3⋊C8—21no—
136C3×C3⋊C8—61no—
137Q8×C9—32no—
138Q8×C9—61no—
139D4×C9—21no—
140D4×C9—21no—
141D4×C9—41no—
142D4×C9—41no—
143C2×C4×C9—21no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146#611—122no—
147#611—124no—
148#611—124no—
149#611—122no—
150C2×Dic9—21no—
151C2×Dic9—31no—
152C2×Dic9—31no—
153C2×Dic9—61no—
154C2×Dic9—61no—
155D36—61no—
156D36—61no—
157D36—121no—
158D36—121no—
159C8×C9—61no—
160C9⋊C8—21no—
161#460—271noSylow
162C2×C3×C9—81no—
163C2×C3×C9#12111yescharacteristic
164C2×C3×C9C2×SD1611yescharacteristic
165C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
166C3⋊D9—81no—
167C2×C3×Q8—31no—
168C2×C3×D4—21no—
169C2×C3×D4C2×D911yescharacteristic
170C2×C3×D4D3211yescharacteristic
171C2×#67—181no—
172C2×#67—181no—
173C2×#67—181no—
174C2×#67—181no—
175C2×#67—182no—
176C2×#67—182no—
177C2×D12—91no—
178C2×D12—91no—
179C2×D12—181no—
180C2×Dic6C2×D911yescharacteristic
181C3×SD16—62no—
182C2×C3×C8—31no—
183C2×C3×C8—91no—
184C3×C4⋊C4—91no—
185#215—182no—
186#214—22no—
187C4⋊Dic3—31no—
188C4⋊Dic3—181no—
189Dic3⋊C4—181no—
190C4×Dic3—91no—
191C4×Dic3—91no—
192C4×Dic3—181no—
193C4×Dic3—181no—
194C4×Dic3—181no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8C2×D911yescharacteristic
197#204—182no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32D3×D411yescharacteristic
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C32×C4#23711yescharacteristic
206C32×C4C2×#6711yescharacteristic
207C3⋊Dic3—32no—
208C3⋊Dic3—61no—
209C3×Dic3—22no—
210C3×Dic3—121no—
211C22×C9—21no—
212C22×C9—42no—
213C22×C9—42no—
214C22×C9—42no—
215C22×C9—41no—
216C22×C9—41no—
217C22×C9—41no—
218C22×C9D3×D411yescharacteristic
219C2×D9—122no—
220C2×D9—122no—
221C2×D9—122no—
222C2×D9—121no—
223C2×D9—121no—
224C2×D9—121no—
225C4×C9—21no—
226C4×C9—21no—
227C4×C9—62no—
228C4×C9#23511yescharacteristic
229C4×C9D3×D411yescharacteristic
230Dic9—32no—
231Dic9—41no—
232Dic9—62no—
233Dic9—62no—
234Dic9—61no—
235C2×SD16—31no—
236C2×SD16—271no—
237#127—271no—
238C4×Q8—271no—
239C4⋊C8—271no—
240#102—91no—
241#102—271no—
242C3×C9#46011yescharacteristic, complemented, Sylow
243C23×C3—22no—
244C23×C3—21no—
245C23×C3—21no—
246C22×D3—182no—
247C22×D3—181no—
248C22×D3—181no—
249C3×Q8—32no—
250C3×Q8—61no—
251C3×D4—21no—
252C3×D4—21no—
253C3×D4—21no—
254C3×D4—21no—
255C3×D4—41no—
256C3×D4—41no—
257C2×C3×C4—21no—
258C2×C3×C4—61no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262#67—362no—
263#67—362no—
264#67—362no—
265#67—364no—
266#67—364no—
267#67—362no—
268C2×Dic3—21no—
269C2×Dic3—61no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—181no—
275C2×Dic3—181no—
276D12—181no—
277D12—181no—
278D12—181no—
279D12—181no—
280D12—361no—
281D12—361no—
282Dic6—21no—
283Dic6#61112yescomplemented
284C3×C8—61no—
285C3×C8—181no—
286C3⋊C8—21no—
287C3⋊C8—61no—
288C2×C32—81no—
289C2×C32D3×SD1611yescharacteristic
290C2×C32#91411yescharacteristic
291C2×C32#93911yescharacteristic
292C3⋊D3—241no—
293C2×C9—21no—
294C2×C9—21no—
295C2×C9—21no—
296C2×C9—82no—
297C2×C9—81no—
298C2×C9#88411yescharacteristic
299C2×C9#91111yescharacteristic
300C2×C9D3×SD1611yescharacteristic
301D9—242no—
302D9—241no—
303C2×Q8—31no—
304C2×D4—271no—
305C2×D4—541no—
306C2×D4—541no—
307C2×D4D3×D911yescharacteristic
308SD16—62no—
309SD16—542no—
310C2×C8—31no—
311C2×C8—91no—
312C4⋊C4—91no—
313C4⋊C4—541no—
314C42—271no—
315C42—541no—
316C22×C3—21no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3D4×D911yescharacteristic
327C22×C3#381111yescharacteristic
328C2×D3—362no—
329C2×D3—362no—
330C2×D3—362no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C2×D3—361no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—62no—
340C3×C4—361no—
341C3×C4#369911yescharacteristic
342C3×C4C2×#61111yescharacteristic
343C3×C4#380411yescharacteristic
344C3×C4D3×#6711yescharacteristic
345Dic3—22no—
346Dic3—92no—
347Dic3—92no—
348Dic3—121no—
349Dic3—182no—
350Dic3—182no—
351Dic3—181no—
352Dic3—181no—
353C32#516311yescharacteristic
354C9—21no—
355C9#476211yescharacteristic, complemented
356C23—21no—
357C23—541no—
358Q8—32no—
359Q8—61no—
360D4—21no—
361D4—21no—
362D4—541no—
363D4—541no—
364D4—1082no—
365D4—1082no—
366C2×C4—61no—
367C2×C4—181no—
368C2×C4—271no—
369C2×C4—271no—
370C2×C4C2×D3×D911yescharacteristic
371C8—61no—
372C8—181no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—82no—
377C2×C3—81no—
378C2×C3—81no—
379C2×C3SD16×D911yescharacteristic
380C2×C3#6347611yescharacteristic
381C2×C3#6350111yescharacteristic
382C2×C3#6393011yescharacteristic
383C2×C3#6393711yescharacteristic
384C2×C3#6397111yescharacteristic
385D3—722no—
386D3—721no—
387D3—721no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—1081no—
392C22—1081no—
393C22—1081no—
394C22#8866811yescharacteristic
395C4—62no—
396C4—272no—
397C4—361no—
398C4—541no—
399C4#8866211yescharacteristic
400C4D3×#61111yescharacteristic
401C3—21no—
402C3#1058884311yescharacteristic, complemented
403C3#1059144411yescharacteristic
404C2—81no—
405C2—2161no—
406C2#1172118211yescharacteristic
407C2#1172118711yescharacteristic
408C2#1172128211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721840C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#460
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C9×#214C2semidirect
3. #11720911C2semidirect
4. C2×#88426C2semidirect
5. C3×#63392C2semidirect
6. #11720284C2semidirect
7. #11720273C2semidirect
35. #63392D3semidirect
52. C9×Dic6D4semidirect
98. C3⋊Dic9SD16semidirect
107. C2×#214D9semidirect
242. C3×C9#460semidirect
283. Dic6#611semidirect
355. C9#4762semidirect
402. C3#10588843semidirect

Maximal subgroups

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

subgroupindex
#117218402
C2×C9×#2142
#117209112
C2×#884262
C3×#633922
#117202842
#117202732
#105914443
#105888433