Finite Group Databasebeta

← #13137849

Subgroups — #13137849

Order 1728.

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

normal subgroups only

This group has 516 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
1C2×D9×Dic6C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic
3C2×#88425C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5#11720527C211yesmaximal, characteristic, complemented
6C9×#810C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590603—31nomaximal
9#10588486—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×D36—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C3×C4×D9C2211yescharacteristic
14C2×C9×Dic6C2211yescharacteristic
15C2×Dic3×D9—21no—
16C2×#6189—21no—
17D9×Dic6—22no—
18D9×Dic6—22no—
19C3×C9×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C3×(C2×D9)⋊C4—21no—
22C2×C9×C3⋊C8C2211yescharacteristic
23#88425—22no—
24#88415—22no—
25C2×D3×Dic6—31no—
26C3×#895—31no—
27C2×#3716—31no—
28C2×#3715—31no—
29C2×Q8×D9—31no—
30#63637—31no—
31C3×#810D311yescharacteristic
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3675—31no—
34C2×#3664—31no—
35#63457D311yescharacteristic
36C9×#103—31no—
37(C2×D9)⋊C8—31no—
38#63368—31no—
39#63368—61no—
40C22×C3×D9—21no—
41C22×C3×D9D411yescharacteristic
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45#6226—21no—
46#6226D412yescomplemented
47C2×C9×Dic3—21no—
48C2×C3×Dic9D411yescharacteristic
49C3×D36—21no—
50C3×D36—21no—
51C3×D36—42no—
52C3×C4×D9—21no—
53C3×C4×D9—21no—
54C9×Dic6—21no—
55C9×Dic6D412yescomplemented
56Dic3×D9—24no—
57#6189—24no—
58C3⋊(C9⋊C8)—21no—
59C9×C3⋊C8—21no—
60#5040—91no—
61#4806—91no—
62C2×#634—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—31no—
66C2×C3×Dic6C2×D311yescharacteristic
67C2×#627—61no—
68C2×D3×Dic3—61no—
69D3×Dic6—62no—
70D3×Dic6—62no—
71C32×C4⋊C4C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C3×(C2×D3)⋊C4—61no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3716—62no—
76#3715—62no—
77C2×Q8×C9—31no—
78Q8×D9—62no—
79Q8×D9—62no—
80C2×D36—21no—
81C2×D36C2×D311yescharacteristic
82C2×C4×D9—61no—
83C2×C4×D9C2×D311yescharacteristic
84C2×Dic18—31no—
85C2×Dic18—61no—
86C2×Dic18—61no—
87C2×C8×C9—31no—
88C9×C4⋊C4—21no—
89C9×C4⋊C4C2×D311yescharacteristic
90#3675—62no—
91(C2×D9)⋊C4—21no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94#3664—62no—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3×D9—21no—
97C2×C3×D9—21no—
98C2×C3×D9—41no—
99C2×C3×D9—41no—
100C2×C3×D9—41no—
101C2×C3×D9SD1612yescomplemented
102C3×C4×C9—41no—
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9—22no—
106C9×Dic3—22no—
107C3×Dic9—21no—
108C2×D3×Q8—91no—
109C22×Dic6—91no—
110C3×#121—91no—
111C2×#215—91no—
112C2×#214—91no—
113C2×#214—91no—
114C2×#204—91no—
115#895—31no—
116C3×#103—31no—
117#830—91no—
118(C2×D3)⋊C8—91no—
119#810—91no—
120#810—91no—
121#810—181no—
122#810D911yescharacteristic, complemented
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—61no—
128#634—32no—
129#634—61no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—31no—
132C3×D12—61no—
133C3×D12—61no—
134C3×D12—122no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×Dic6—21no—
138C3×Dic6D1212yes—
139#627—64no—
140D3×Dic3—64no—
141C22×D9—21no—
142C22×D9#6711yescharacteristic
143C3⋊(C3⋊C8)—61no—
144C3×C3⋊C8—21no—
145Q8×C9—32no—
146Q8×C9—61no—
147C2×C4×C9—21no—
148C2×C4×C9—21no—
149C2×C4×C9—41no—
150C2×C4×C9—61no—
151C2×C4×C9C22×D311yescharacteristic
152C2×Dic9—61no—
153C2×Dic9—121no—
154C2×Dic9#6711yescharacteristic
155D36—21no—
156D36—21no—
157D36—42no—
158C4×D9—21no—
159C4×D9—21no—
160C4×D9—64no—
161Dic18—32no—
162Dic18—62no—
163Dic18—64no—
164Dic18—61no—
165Dic18—121no—
166C8×C9—61no—
167C9⋊C8—61no—
168C9⋊C8—121no—
169#448—271noSylow
170C2×C3×C9#12011yescharacteristic
171C2×C3×C9C2×SD1611yescharacteristic
172C2×C3×C9#13711yescharacteristic
173C3×D9—22no—
174C3×D9—81no—
175C2×C3×Q8—31no—
176C2×C3×D4—91no—
177C2×C3×D4—181no—
178C22×C3×C4—91no—
179C22×Dic3—181no—
180D3×Q8—182no—
181D3×Q8—182no—
182C2×D12—31no—
183C2×D12—61no—
184C2×C4×D3—31no—
185C2×C4×D3—181no—
186C2×Dic6—91no—
187C2×Dic6—91no—
188C2×Dic6—181no—
189C2×Dic6—182no—
190C2×Dic6—181no—
191C2×Dic6—181no—
192C2×Dic6—182no—
193C2×Dic6C2×D911yescharacteristic
194C2×C3×C8—31no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4C2×D911yescharacteristic
197C3×C4⋊C4D3211yescharacteristic
198C3×C22⋊C4—181no—
199#215—182no—
200#214—182no—
201#214—182no—
202(C2×D3)⋊C4—61no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—181no—
206C2×C3⋊C8C2×D911yescharacteristic
207#204—182no—
208C22×C32D3×D411yescharacteristic
209C2×C3×D3—32no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C32×C4—41no—
216C32×C4C2×D1211yescharacteristic
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—62no—
219C3×Dic3—22no—
220C3×Dic3—61no—
221C22×C9—21no—
222C22×C9C2×#6711yescharacteristic
223C2×D9—21no—
224C2×D9—21no—
225C2×D9—41no—
226C2×D9—41no—
227C2×D9—41no—
228C2×D9#21412yescomplemented
229C4×C9—21no—
230C4×C9—21no—
231C4×C9—41no—
232C4×C9—62no—
233C4×C9—81no—
234C4×C9D3×D411yescharacteristic
235C4×C9D3×D411yescharacteristic
236Dic9—21no—
237Dic9—62no—
238Dic9—122no—
239C22×Q8—271no—
240C2×SD16—271no—
241C2×SD16—271no—
242#121—91no—
243#103—31no—
244#103—271no—
245C22⋊C8—271no—
246C3×C9#44811yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C23×C3—181no—
249C22×D3—31no—
250C22×D3—61no—
251C3×Q8—32no—
252C3×Q8—61no—
253C3×D4—181no—
254C3×D4—181no—
255C3×D4—362no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4C22×D911yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266C2×Dic3—21no—
267C2×Dic3—31no—
268C2×Dic3—184no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C2×Dic3—361no—
272D12—61no—
273D12—61no—
274D12—122no—
275C4×D3—61no—
276C4×D3—61no—
277C4×D3—184no—
278Dic6—21no—
279Dic6—92no—
280Dic6—92no—
281Dic6—182no—
282Dic6—184no—
283Dic6—184no—
284Dic6—181no—
285Dic6—181no—
286Dic6—361no—
287Dic6D3612yescomplemented
288C3×C8—61no—
289C3⋊C8—21no—
290C3⋊C8—181no—
291C3⋊C8—181no—
292C3⋊C8—361no—
293C2×C32#88211yescharacteristic
294C2×C32D3×SD1611yescharacteristic
295C2×C32#91811yescharacteristic
296C3×D3—62no—
297C3×D3—241no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9C2×#21411yescharacteristic
302C2×C9#93711yescharacteristic
303C2×C9#95111yescharacteristic
304D9—22no—
305D9—81no—
306C2×Q8—31no—
307C2×Q8—271no—
308C2×Q8—541no—
309C2×Q8—542no—
310C2×Q8—542no—
311C2×D4—91no—
312C2×D4—181no—
313C22×C4—91no—
314C22×C4—541no—
315SD16—542no—
316SD16—542no—
317C2×C8—31no—
318C2×C8—271no—
319C4⋊C4D3×D911yescharacteristic
320C22⋊C4—181no—
321C22×C3—21no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3D4×D911yescharacteristic
329C22×C3D3×#6711yescharacteristic
330C2×D3—32no—
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—62no—
341C3×C4—81no—
342C3×C4—181no—
343C3×C4C2×D3611yescharacteristic
344C3×C4D4×D911yescharacteristic
345C3×C4D3×D1211yescharacteristic
346C3×C4#380211yescharacteristic
347Dic3—22no—
348Dic3—61no—
349Dic3—182no—
350Dic3—182no—
351Dic3—362no—
352C32#480611yescharacteristic
353C9—21no—
354C9#504011yescharacteristic, complemented
355C23—91no—
356C23—181no—
357Q8—32no—
358Q8—61no—
359Q8—272no—
360Q8—544no—
361Q8—541no—
362D4—181no—
363D4—181no—
364D4—362no—
365C2×C4—21no—
366C2×C4—61no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—544no—
371C2×C4—541no—
372C2×C4C2×D3×D911yescharacteristic
373C8—61no—
374C8—541no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—182no—
379C2×C3—721no—
380C2×C3#6344411yescharacteristic
381C2×C3SD16×D911yescharacteristic
382C2×C3#6348011yescharacteristic
383C2×C3#6392211yescharacteristic
384C2×C3D3×#21411yescharacteristic
385C2×C3#6394311yescharacteristic
386D3—62no—
387D3—241no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22—361no—
392C22—361no—
393C22—361no—
394C22D9×#6711yescharacteristic
395C4—41no—
396C4—62no—
397C4—181no—
398C4—542no—
399C4D3×D3611yescharacteristic
400C4#8864611yescharacteristic
401C3—21no—
402C3#1058848611yescharacteristic, complemented
403C3#1059060311yescharacteristic
404C2—182no—
405C2—721no—
406C2#1172104911yescharacteristic
407C2D9×#21411yescharacteristic
408C2#1172121911yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC3×#63457C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#448
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D9×Dic6C2semidirect
3. C2×#88425C2semidirect
4. C2×#88415C2semidirect
5. #11720527C2semidirect
6. C9×#810C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
46. #6226D4semidirect
55. C9×Dic6D4semidirect
101. C2×C3×D9SD16semidirect
122. #810D9semidirect
228. C2×D9#214semidirect
246. C3×C9#448semidirect
287. Dic6D36semidirect
354. C9#5040semidirect
402. C3#10588486semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
C3×#634572
C2×#884252
C2×#884152
#117205272
C9×#8102
(C2×D9)⋊(C3⋊C8)2
#105906033
#105884863