Finite Group Databasebeta

← #13138637

Subgroups — #13138637

Order 1728.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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×D3×Dic18C211yesmaximal, characteristic, complemented
2C2×C3×#3664C211yesmaximal, characteristic, complemented
3#11721037C211yesmaximal, characteristic
4C2×#88415C211yesmaximal, characteristic, complemented
5#11720539C211yesmaximal, characteristic, complemented
6C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10590822—31nomaximal
9#10588404—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×#6194—21no—
15C2×D3×Dic9—21no—
16C2×#6189—21no—
17D3×Dic18—22no—
18D3×Dic18—22no—
19C2×C3×C8×C9C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C3×#3664—22no—
23(C2×D9)⋊Dic3—21no—
24#88415—22no—
25C2×D3×Dic6—31no—
26C2×C3×#204—31no—
27#63908—31no—
28C2×#3716—31no—
29C22×Dic18—31no—
30#63643—31no—
31C3×(C2×D3)⋊C8D311yescharacteristic
32#63566—31no—
33#63492—31no—
34C2×#3664—31no—
35C2×#3664D311yescharacteristic, complemented
36C9×C22⋊C8—31no—
37#63376—31no—
38#63376—31no—
39#63376—61no—
40C22×D3×C9D411yescharacteristic
41C22×C3×D9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9—21no—
45#6226—21no—
46#6226D412yescomplemented
47C2×C9×Dic3D411yescharacteristic
48C2×C3×Dic9—21no—
49C4×D3×C9—21no—
50C4×D3×C9—21no—
51C3×D36—21no—
52C3×D36—21no—
53C3×Dic18—21no—
54C3×Dic18D412yescomplemented
55#6194—42no—
56D3×Dic9—24no—
57#6189—24no—
58C3×C8×C9—21no—
59C9×C3⋊C8—21no—
60#5170—91no—
61#4738—91no—
62C2×#634—31no—
63C2×C3×D12—31no—
64C2×C3×C4×D3C2×D311yescharacteristic
65C2×C3×Dic6—31no—
66C2×#627—61no—
67C2×#625—61no—
68C2×D3×Dic3—61no—
69D3×Dic6—62no—
70D3×Dic6—62no—
71C2×C32×C8C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)—31no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74C3×#204—62no—
75(C2×D3)⋊Dic3—61no—
76#3716—62no—
77C22×C4×C9—31no—
78C2×#611—61no—
79C22×Dic9—61no—
80C2×D36C2×D311yescharacteristic
81C2×Dic18—31no—
82C2×Dic18—62no—
83C2×Dic18—61no—
84C2×Dic18—62no—
85C2×Dic18—61no—
86C2×Dic18C2×D311yescharacteristic
87C2×C8×C9—21no—
88C2×C8×C9—31no—
89C2×C8×C9C2×D311yescharacteristic
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—31no—
92C4⋊Dic9—61no—
93#3664—22no—
94#3664—62no—
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—21no—
97C2×D3×C9—21no—
98C2×D3×C9SD1612yescomplemented
99C2×C3×D9—41no—
100C2×C3×D9—41no—
101C2×C3×D9—41no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—22no—
105C3⋊Dic9—41no—
106C9×Dic3—21no—
107C3×Dic9—22no—
108C2×D3×Q8—91no—
109C22×Dic6—91no—
110C2×C3×SD16—91no—
111#938—91no—
112C2×#214—91no—
113#930—91no—
114C2×#204—31no—
115C2×#204—91no—
116C3×C22⋊C8—31no—
117#835—91no—
118(C2×D3)⋊C8D911yescharacteristic, complemented
119#816—91no—
120#816—91no—
121#816—91no—
122#816—181no—
123C22×C3×D3—61no—
124C22×C3×D3D1211yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—61no—
128#634—32no—
129#634—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3D1211yescharacteristic
132C3×D12—61no—
133C3×D12—61no—
134C3×C4×D3—21no—
135C3×C4×D3—21no—
136C3×Dic6—32no—
137C3×Dic6—61no—
138#627—64no—
139#625—122no—
140D3×Dic3—64no—
141C23×C9—31no—
142C22×D9—21no—
143C32×C8—21no—
144C3×C3⋊C8—21no—
145C2×C4×C9—21no—
146C2×C4×C9—31no—
147C2×C4×C9—61no—
148C2×C4×C9—61no—
149C2×C4×C9C22×D311yescharacteristic
150#611—122no—
151C2×Dic9—21no—
152C2×Dic9—64no—
153C2×Dic9—61no—
154C2×Dic9—61no—
155C2×Dic9—121no—
156C2×Dic9—121no—
157D36—21no—
158D36—21no—
159Dic18—21no—
160Dic18—32no—
161Dic18—62no—
162Dic18—64no—
163Dic18—61no—
164Dic18—121no—
165Dic18#6712yescomplemented
166C8×C9—21no—
167C8×C9—41no—
168C8×C9—61no—
169#448—271noSylow
170C2×C3×C9#12011yescharacteristic
171C2×C3×C9C2×SD1611yescharacteristic
172C2×C3×C9#13711yescharacteristic
173D3×C9—22no—
174C3×D9—81no—
175C2×C3×Q8—91no—
176C2×C3×D4—91no—
177C22×C3×C4—31no—
178C2×#67—181no—
179C2×#67—181no—
180C22×Dic3—181no—
181D3×Q8—182no—
182D3×Q8—182no—
183C2×D12—31no—
184C2×C4×D3—181no—
185C2×C4×D3C2×D911yescharacteristic
186C2×Dic6—31no—
187C2×Dic6—91no—
188C2×Dic6—91no—
189C2×Dic6—181no—
190C2×Dic6—182no—
191C2×Dic6—182no—
192C2×Dic6—181no—
193C2×Dic6—181no—
194C3×SD16—182no—
195C2×C3×C8—21no—
196C2×C3×C8—31no—
197C2×C3×C8C2×D911yescharacteristic
198C2×C3×C8D3211yescharacteristic
199#217—181no—
200#214—182no—
201(C2×D3)⋊C4—181no—
202C4⋊Dic3—91no—
203C4⋊Dic3—91no—
204C4⋊Dic3—181no—
205C2×C3⋊C8C2×D911yescharacteristic
206#204—62no—
207#204—182no—
208C22×C32C2×D1211yescharacteristic
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3#20412yes—
215C32×C4D3×D411yescharacteristic
216C32×C4D3×D411yescharacteristic
217C3⋊Dic3—62no—
218C3⋊Dic3—121no—
219C3×Dic3—21no—
220C3×Dic3—62no—
221C22×C9—21no—
222C22×C9—32no—
223C22×C9—61no—
224C22×C9—61no—
225C22×C9D3×D411yescharacteristic
226C2×D9—41no—
227C2×D9—41no—
228C2×D9—41no—
229C4×C9—21no—
230C4×C9—21no—
231C4×C9—61no—
232C4×C9D3×D411yescharacteristic
233C4×C9C2×#6711yescharacteristic
234Dic9—22no—
235Dic9—62no—
236Dic9—122no—
237Dic9—121no—
238Dic9—241no—
239C22×Q8—271no—
240C2×SD16—91no—
241C2×SD16—271no—
242#121—271no—
243#103—271no—
244#103—271no—
245C22⋊C8—31no—
246C3×C9#44811yescharacteristic, complemented, Sylow
247C23×C3—31no—
248C23×C3—181no—
249C22×D3—61no—
250C22×D3D3611yescharacteristic
251C3×Q8—92no—
252C3×Q8—181no—
253C3×D4—181no—
254C3×D4—181no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262#67—362no—
263#67—362no—
264C2×Dic3—61no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C2×Dic3—181no—
270C2×Dic3—361no—
271C2×Dic3—361no—
272C2×Dic3D3611yescharacteristic
273D12—61no—
274D12—61no—
275C4×D3—21no—
276C4×D3—21no—
277C4×D3—184no—
278Dic6—32no—
279Dic6—61no—
280Dic6—92no—
281Dic6—92no—
282Dic6—182no—
283Dic6—184no—
284Dic6—184no—
285Dic6—181no—
286Dic6—181no—
287Dic6—361no—
288C3×C8—21no—
289C3×C8—21no—
290C3×C8—41no—
291C3×C8—61no—
292C3⋊C8—21no—
293C2×C32#88211yescharacteristic
294C2×C32C2×#20411yescharacteristic
295C2×C32#90911yescharacteristic
296C3×D3—22no—
297C3×D3—241no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—62no—
302C2×C9D3×SD1611yescharacteristic
303C2×C9#91511yescharacteristic
304C2×C9#93711yescharacteristic
305D9—81no—
306C2×Q8—91no—
307C2×Q8—271no—
308C2×Q8—541no—
309C2×Q8—542no—
310C2×Q8—542no—
311C2×D4—91no—
312C2×D4—541no—
313C22×C4—31no—
314C22×C4—541no—
315SD16—182no—
316SD16—542no—
317C2×C8—31no—
318C2×C8D3×D911yescharacteristic
319C4⋊C4—271no—
320C22⋊C4—541no—
321C22×C3—21no—
322C22×C3—32no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3C2×D3611yescharacteristic
329C22×C3D3×D1211yescharacteristic
330C2×D3—21no—
331C2×D3—21no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3#366412yescomplemented
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—61no—
339C3×C4—182no—
340C3×C4D4×D911yescharacteristic
341C3×C4D4×D911yescharacteristic
342C3×C4#380211yescharacteristic
343C3×C4D3×#6711yescharacteristic
344Dic3—21no—
345Dic3—62no—
346Dic3—182no—
347Dic3—182no—
348Dic3—362no—
349Dic3—361no—
350Dic3—361no—
351Dic3—721no—
352C32#473811yescharacteristic
353C9—21no—
354C9#517011yescharacteristic, complemented
355C23—31no—
356C23—181no—
357Q8—92no—
358Q8—181no—
359Q8—272no—
360Q8—544no—
361Q8—541no—
362D4—181no—
363D4—181no—
364D4—1082no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—181no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×D911yescharacteristic
373C8—21no—
374C8—61no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—62no—
379C2×C3—721no—
380C2×C3#6344411yescharacteristic
381C2×C3C2×#366411yescharacteristic
382C2×C3#6347111yescharacteristic
383C2×C3D3×#20411yescharacteristic
384C2×C3#6380011yescharacteristic
385C2×C3#6392211yescharacteristic
386D3—22no—
387D3—241no—
388C22—32no—
389C22—61no—
390C22—61no—
391C22—361no—
392C22—361no—
393C22—361no—
394C22D3×D3611yescharacteristic
395C4—61no—
396C4—182no—
397C4—542no—
398C4—1081no—
399C4#8864611yescharacteristic
400C4D9×#6711yescharacteristic
401C3—21no—
402C3#1058840411yescharacteristic, complemented
403C3#1059082211yescharacteristic
404C2—62no—
405C2—721no—
406C2#1172079611yescharacteristic
407C2D3×#366411yescharacteristic
408C2#1172104911yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radical#11721037C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#448
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×D3×Dic18C2semidirect
2. C2×C3×#3664C2semidirect
4. C2×#88415C2semidirect
5. #11720539C2semidirect
6. C9×(C2×D3)⋊C8C2semidirect
7. #11720276C2semidirect
35. C2×#3664D3semidirect
46. #6226D4semidirect
54. C3×Dic18D4semidirect
98. C2×D3×C9SD16semidirect
118. (C2×D3)⋊C8D9semidirect
165. Dic18#67semidirect
246. C3×C9#448semidirect
335. C2×D3#3664semidirect
354. C9#5170semidirect
402. C3#10588404semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C2×C3×#36642
#117210372
C2×#884152
#117205392
C9×(C2×D3)⋊C82
#117202762
#105908223
#105884043