Finite Group Databasebeta

← #13140737

Subgroups — #13140737

Order 1728.

516 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×#3701C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic, complemented
3#11720945C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10591599—31nomaximal
9#10588488—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C3×#3701—22no—
13C3×#3701—22no—
14C2×C3×D36—21no—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9—21no—
17C2×C3×C4×D9C2211yescharacteristic
18#88547—22no—
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23Dic3⋊Dic9—21no—
24#88415—22no—
25C2×C3×#239—31no—
26C2×#3756—31no—
27#63857—31no—
28C2×#3716—31no—
29C2×#3701D311yescharacteristic
30C3×#835D311yescharacteristic
31#63562—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3674—31no—
34C2×#3674—61no—
35C2×#3664—31no—
36#63459—31no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63369—31no—
40C22×C3×D9—21no—
41C22×C3×D9D411yescharacteristic
42C3×Q8×C9—21no—
43C3×Q8×C9D412yes—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9—21no—
47#6226—21no—
48#6226—21no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9D411yescharacteristic
51C3×D36—24no—
52C3×D36—21no—
53C3×D36D412yes—
54C3×C4×D9—24no—
55C3×C4×D9—21no—
56C3×C4×D9—21no—
57C3⋊(C9⋊C8)—21no—
58C9×C3⋊C8—21no—
59#5239—91no—
60#4808—91no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×#634—31no—
63C3×#239—62no—
64C3×#239—62no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—61no—
69#3756—62no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75#3716—62no—
76C2×Q8×C9—21no—
77C2×Q8×C9C2×D311yescharacteristic
78#3701—22no—
79#3701—22no—
80C2×D36—21no—
81C2×D36C2×D311yescharacteristic
82C2×C4×D9—21no—
83C2×C4×D9C2×D311yescharacteristic
84C2×Dic18—31no—
85C2×Dic18—61no—
86C2×C8×C9—31no—
87C9×C4⋊C4—31no—
88#3674—64no—
89#3674—62no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—61no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94#3664—62no—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3×D9—21no—
97C2×C3×D9—22no—
98C2×C3×D9—21no—
99C2×C3×D9—21no—
100C2×C3×D9—41no—
101C2×C3×D9—41no—
102C3×C4×C9—22no—
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9—41no—
106C9×Dic3—41no—
107C3×Dic9—21no—
108C2×C3×#41—91no—
109C2×#239—31no—
110C2×#216—91no—
111C2×#216—91no—
112C2×#216—181no—
113C2×#214—91no—
114#924—91no—
115C2×#204—91no—
116#897—91no—
117C3×#103—31no—
118#835D911yescharacteristic, complemented
119#832—91no—
120#830—91no—
121(C2×D3)⋊C8—91no—
122#809—91no—
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C32×Q8—21no—
126C32×Q8D1212yes—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—61no—
130#634—61no—
131#634—61no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—31no—
134C3×D12—32no—
135C3×D12—64no—
136C3×D12—61no—
137C3×C4×D3—64no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C22×D9—21no—
141C22×D9#6711yescharacteristic
142C3⋊(C3⋊C8)—61no—
143C3×C3⋊C8—21no—
144Q8×C9—21no—
145Q8×C9—22no—
146Q8×C9—41no—
147Q8×C9#6712yes—
148C2×C4×C9—21no—
149C2×C4×C9—21no—
150C2×C4×C9—41no—
151C2×C4×C9—61no—
152C2×C4×C9C22×D311yescharacteristic
153C2×Dic9—61no—
154C2×Dic9—121no—
155C2×Dic9#6711yescharacteristic
156D36—24no—
157D36—21no—
158D36#6712yes—
159C4×D9—24no—
160C4×D9—21no—
161C4×D9—21no—
162Dic18—62no—
163Dic18—61no—
164Dic18—61no—
165Dic18—121no—
166C8×C9—61no—
167C9⋊C8—62no—
168C9⋊C8—61no—
169#452—271noSylow
170C2×C3×C9#12011yescharacteristic
171C2×C3×C9#13511yescharacteristic
172C2×C3×C9#13711yescharacteristic
173C3×D9—41no—
174C3×D9—42no—
175C3×#41—182no—
176C3×#41—182no—
177C2×C3×Q8—21no—
178C2×C3×Q8C2×D911yescharacteristic
179C2×C3×Q8D3211yescharacteristic
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184#239—62no—
185#239—62no—
186C2×D12—31no—
187C2×D12—61no—
188C2×C4×D3—31no—
189C2×C4×D3—61no—
190C2×Dic6—91no—
191C2×Dic6—91no—
192C2×Dic6—181no—
193C2×C3×C8—31no—
194C3×C4⋊C4—31no—
195#217—181no—
196#216—184no—
197#216—182no—
198#216—182no—
199#214—182no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—181no—
202C4⋊Dic3C2×D911yescharacteristic
203Dic3⋊C4—181no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—181no—
207C2×C3⋊C8C2×D911yescharacteristic
208#204—182no—
209C22×C32D3×D411yescharacteristic
210C2×C3×D3—62no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C32×C4—22no—
217C32×C4C2×D1211yescharacteristic
218C32×C4D3×D411yescharacteristic
219C3⋊Dic3—121no—
220C3×Dic3—41no—
221C3×Dic3—61no—
222C22×C9—21no—
223C22×C9C2×#6711yescharacteristic
224C2×D9—21no—
225C2×D9—21no—
226C2×D9—21no—
227C2×D9—22no—
228C2×D9—41no—
229C2×D9—41no—
230C4×C9—22no—
231C4×C9—21no—
232C4×C9—21no—
233C4×C9—42no—
234C4×C9—121no—
235C4×C9C2×#6711yescharacteristic
236C4×C9C2×#6711yescharacteristic
237Dic9—21no—
238Dic9—122no—
239Dic9—121no—
240C2×#41—91no—
241C2×Q16—271no—
242C2×SD16—271no—
243C22⋊Q8—271no—
244#103—31no—
245#102—271no—
246C22⋊C8—271no—
247C3×C9#45211yescharacteristic, complemented, Sylow
248C23×C3—91no—
249C23×C3—181no—
250C22×D3—31no—
251C22×D3—61no—
252C3×Q8—21no—
253C3×Q8—21no—
254C3×Q8—22no—
255C3×Q8—41no—
256C3×Q8D3612yes—
257C3×Q8#62612yes—
258C3×D4—92no—
259C3×D4—184no—
260C3×D4—181no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—61no—
266C2×C3×C4—91no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4C22×D911yescharacteristic
271C2×C3×C4C2×D3211yescharacteristic
272C2×Dic3—21no—
273C2×Dic3—31no—
274C2×Dic3—181no—
275C2×Dic3—181no—
276C2×Dic3—361no—
277D12—32no—
278D12—64no—
279D12—61no—
280C4×D3—64no—
281C4×D3—61no—
282C4×D3—61no—
283Dic6—182no—
284Dic6—181no—
285Dic6—181no—
286Dic6—181no—
287Dic6—181no—
288Dic6—361no—
289C3×C8—61no—
290C3⋊C8—21no—
291C3⋊C8—182no—
292C3⋊C8—181no—
293C3⋊C8—181no—
294C2×C32#88211yescharacteristic
295C2×C32#91611yescharacteristic
296C2×C32#91811yescharacteristic
297C3×D3—121no—
298C3×D3—122no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9#95011yescharacteristic
303C2×C9#95111yescharacteristic
304C2×C9#95311yescharacteristic
305D9—41no—
306D9—42no—
307#41—182no—
308#41—182no—
309C2×Q8—271no—
310C2×Q8D3×D911yescharacteristic
311C2×D4—91no—
312C2×D4—181no—
313C22×C4—91no—
314C22×C4—181no—
315Q16—542no—
316SD16—542no—
317C2×C8—31no—
318C2×C8—271no—
319C4⋊C4—31no—
320C4⋊C4—541no—
321C22⋊C4—541no—
322C22×C3—21no—
323C22×C3—182no—
324C22×C3—181no—
325C22×C3—181no—
326C22×C3—181no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3D4×D911yescharacteristic
330C22×C3D3×#6711yescharacteristic
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—62no—
334C2×D3—61no—
335C2×D3—121no—
336C2×D3—121no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—42no—
342C3×C4—121no—
343C3×C4—181no—
344C3×C4C2×D3611yescharacteristic
345C3×C4D4×D911yescharacteristic
346C3×C4C2×#62611yescharacteristic
347C3×C4D3×#6711yescharacteristic
348Dic3—41no—
349Dic3—61no—
350Dic3—362no—
351Dic3—361no—
352Dic3—361no—
353C32#480811yescharacteristic
354C9—21no—
355C9#523911yescharacteristic, complemented
356C23—91no—
357C23—181no—
358Q8—21no—
359Q8—541no—
360Q8—541no—
361Q8#619212yes—
362D4—92no—
363D4—184no—
364D4—181no—
365C2×C4—21no—
366C2×C4—61no—
367C2×C4—91no—
368C2×C4—184no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—541no—
372C2×C4C2×D3×D911yescharacteristic
373C8—61no—
374C8—541no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—361no—
379C2×C3—362no—
380C2×C3#6344411yescharacteristic
381C2×C3#6347811yescharacteristic
382C2×C3#6348011yescharacteristic
383C2×C3#6394311yescharacteristic
384C2×C3#6394711yescharacteristic
385C2×C3#6397711yescharacteristic
386D3—121no—
387D3—122no—
388C22—181no—
389C22—182no—
390C22—181no—
391C22—181no—
392C22—361no—
393C22—361no—
394C22D9×#6711yescharacteristic
395C4—22no—
396C4—121no—
397C4—181no—
398C4—1081no—
399C4C2×#619211yescharacteristic
400C4D9×#6711yescharacteristic
401C3—21no—
402C3#1058848811yescharacteristic, complemented
403C3#1059159911yescharacteristic
404C2—361no—
405C2—362no—
406C2#1172121511yescharacteristic
407C2#1172121911yescharacteristic
408C2#1172127611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#3701C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#452
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
2. C2×#88547C2semidirect
3. #11720945C2semidirect
4. C2×#88415C2semidirect
5. C9×#835C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720266C2semidirect
118. #835D9semidirect
247. C3×C9#452semidirect
355. C9#5239semidirect
402. C3#10588488semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×#885472
#117209452
C2×#884152
C9×#8352
(C2×D9)⋊(C3⋊C8)2
#117202662
#105915993
#105884883