Finite Group Databasebeta

← #13140363

Subgroups — #13140363

Order 1728.

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

normal subgroups only

This group has 524 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×#3699C211yesmaximal, characteristic
2C2×#88543C211yesmaximal, characteristic, complemented
3#11720945C211yesmaximal, characteristic, complemented
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10591463—31nomaximal
9#10588463—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×#611—21no—
13C22×C3×Dic9—21no—
14C3×#3699—22no—
15C3×#3699—22no—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18#88543—22no—
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23Dic3⋊Dic9—21no—
24#88418—22no—
25C2×C3×#237—31no—
26C2×#3754—31no—
27#63857—31no—
28C2×#3719—31no—
29C2×#3699D311yescharacteristic
30C3×#832D311yescharacteristic
31#63565—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3672—31no—
34C2×#3672—61no—
35C2×Dic36—31no—
36#63459—31no—
37C9×#102—31no—
38(C2×D9)⋊C8—31no—
39#63368—31no—
40C23×C3×C9—21no—
41C22×C3×D9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9D412yes—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46#6226—21no—
47#6226—21no—
48C3×#611—24no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9—24no—
51C2×C3×Dic9—21no—
52C2×C3×Dic9D411yescharacteristic
53C3×C4×D9—21no—
54C3×C4×D9—21no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C3⋊(C9⋊C8)—21no—
58C9×C3⋊C8—21no—
59#5239—91no—
60#4783—91no—
61C2×C32×D4C2×D311yescharacteristic
62C2×#634—31no—
63C2×C3×#67—61no—
64C22×C3×Dic3—61no—
65C3×#237—62no—
66C3×#237—62no—
67C2×C3×C4×D3—31no—
68C2×C3×Dic6—31no—
69#3754—62no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75#3719—62no—
76C2×D4×C9—21no—
77C2×D4×C9C2×D311yescharacteristic
78C2×#611—21no—
79C22×Dic9—21no—
80#3699—22no—
81#3699—22no—
82C2×C4×D9C2×D311yescharacteristic
83C2×Dic18—31no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C2×C8×C9—31no—
87C9×C4⋊C4—31no—
88#3672—64no—
89#3672—62no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—61no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94Dic36—62no—
95C22×C3×C9—22no—
96C22×C3×C9—41no—
97C22×C3×C9—41no—
98C22×C3×C9C2×D411yescharacteristic
99C2×C3×D9—21no—
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—41no—
105C9×Dic3—41no—
106C3×Dic9—22no—
107C3×Dic9—21no—
108C2×C3×#41—91no—
109C2×#237—31no—
110C2×#216—91no—
111C2×#214—91no—
112C2×#214—91no—
113C2×#214—181no—
114#924—91no—
115C2×Dic12—91no—
116#897—91no—
117C3×#102—31no—
118#835—91no—
119#832D911yescharacteristic, complemented
120#830—91no—
121(C2×D3)⋊C8—91no—
122#810—91no—
123C23×C32—21no—
124C22×C3×D3—31no—
125C32×D4—21no—
126C32×D4D1212yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—61no—
129#634—61no—
130#634—61no—
131C3×#67—64no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×Dic6—32no—
139C3×Dic6—61no—
140C23×C9—21no—
141C23×C9—41no—
142C22×D9#6711yescharacteristic
143C3⋊(C3⋊C8)—61no—
144C3×C3⋊C8—21no—
145D4×C9—21no—
146D4×C9—22no—
147D4×C9—41no—
148D4×C9#6712yes—
149C2×C4×C9—21no—
150C2×C4×C9—61no—
151C2×C4×C9C22×D311yescharacteristic
152#611—24no—
153C2×Dic9—24no—
154C2×Dic9—21no—
155C2×Dic9—61no—
156C2×Dic9—121no—
157C2×Dic9#6711yescharacteristic
158C4×D9—21no—
159C4×D9—21no—
160Dic18—21no—
161Dic18—62no—
162Dic18—61no—
163Dic18—61no—
164Dic18—121no—
165Dic18#6712yes—
166C8×C9—61no—
167C9⋊C8—62no—
168C9⋊C8—61no—
169#452—271noSylow
170C2×C3×C9—42no—
171C2×C3×C9#12011yescharacteristic
172C2×C3×C9#13511yescharacteristic
173C2×C3×C9#13711yescharacteristic
174C3×D9—41no—
175C3×#41—182no—
176C3×#41—182no—
177C2×C3×Q8—91no—
178C2×C3×D4—21no—
179C2×C3×D4—181no—
180C2×C3×D4C2×D911yescharacteristic
181C2×C3×D4D3211yescharacteristic
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184C2×#67—61no—
185C22×Dic3—61no—
186#237—62no—
187#237—62no—
188C2×C4×D3—31no—
189C2×Dic6—31no—
190C2×Dic6—91no—
191C2×Dic6—91no—
192C2×Dic6—181no—
193C2×C3×C8—31no—
194C3×C4⋊C4—31no—
195#217—181no—
196#216—182no—
197#214—184no—
198#214—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
208Dic12—182no—
209C22×C32—22no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32D3×D411yescharacteristic
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C32×C4C2×D1211yescharacteristic
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—121no—
219C3×Dic3—41no—
220C3×Dic3—62no—
221C3×Dic3—61no—
222C22×C9—22no—
223C22×C9—21no—
224C22×C9—42no—
225C22×C9—41no—
226C22×C9—41no—
227C22×C9—81no—
228C22×C9—81no—
229C22×C9C2×#6711yescharacteristic
230C2×D9—21no—
231C2×D9—21no—
232C2×D9—21no—
233C4×C9—21no—
234C4×C9—21no—
235C4×C9—121no—
236C4×C9C2×#6711yescharacteristic
237C4×C9C2×#6711yescharacteristic
238Dic9—22no—
239Dic9—21no—
240Dic9—122no—
241Dic9—121no—
242C2×#41—91no—
243C2×Q16—271no—
244C2×SD16—271no—
245C22⋊Q8—271no—
246#103—271no—
247#102—31no—
248C22⋊C8—271no—
249C3×C9#45211yescharacteristic, complemented, Sylow
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—41no—
253C23×C3—91no—
254C22×D3—31no—
255C3×Q8—92no—
256C3×Q8—181no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—22no—
260C3×D4—41no—
261C3×D4—184no—
262C3×D4D3612yes—
263C3×D4#62612yes—
264C2×C3×C4—21no—
265C2×C3×C4—61no—
266C2×C3×C4—91no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C22×D911yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273#67—64no—
274C2×Dic3—21no—
275C2×Dic3—31no—
276C2×Dic3—64no—
277C2×Dic3—61no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280C2×Dic3—361no—
281C4×D3—61no—
282C4×D3—61no—
283Dic6—32no—
284Dic6—61no—
285Dic6—182no—
286Dic6—181no—
287Dic6—181no—
288Dic6—181no—
289Dic6—181no—
290Dic6—361no—
291C3×C8—61no—
292C3⋊C8—21no—
293C3⋊C8—182no—
294C3⋊C8—181no—
295C3⋊C8—181no—
296C2×C32—42no—
297C2×C32#88211yescharacteristic
298C2×C32#91211yescharacteristic
299C2×C32#91511yescharacteristic
300C3×D3—121no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—42no—
305C2×C9—82no—
306C2×C9#95011yescharacteristic
307C2×C9#95111yescharacteristic
308C2×C9#95311yescharacteristic
309D9—41no—
310#41—182no—
311#41—182no—
312C2×Q8—91no—
313C2×Q8—271no—
314C2×D4—181no—
315C2×D4D3×D911yescharacteristic
316C22×C4—91no—
317C22×C4—181no—
318Q16—542no—
319SD16—542no—
320C2×C8—31no—
321C2×C8—271no—
322C4⋊C4—31no—
323C4⋊C4—541no—
324C22⋊C4—541no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3D4×D911yescharacteristic
339C22×C3D3×#6711yescharacteristic
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—61no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—121no—
346C3×C4—181no—
347C3×C4—182no—
348C3×C4C2×D3611yescharacteristic
349C3×C4D4×D911yescharacteristic
350C3×C4C2×#62611yescharacteristic
351C3×C4D3×#6711yescharacteristic
352Dic3—41no—
353Dic3—62no—
354Dic3—61no—
355Dic3—362no—
356Dic3—361no—
357Dic3—361no—
358C32#478311yescharacteristic
359C9—21no—
360C9#523911yescharacteristic, complemented
361C23—21no—
362C23—91no—
363Q8—92no—
364Q8—181no—
365Q8—541no—
366Q8—541no—
367D4—21no—
368D4—184no—
369D4#619212yes—
370C2×C4—61no—
371C2×C4—91no—
372C2×C4—184no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—541no—
377C2×C4C2×D3×D911yescharacteristic
378C8—61no—
379C8—541no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—82no—
386C2×C3—361no—
387C2×C3#6344411yescharacteristic
388C2×C3#6347411yescharacteristic
389C2×C3#6347711yescharacteristic
390C2×C3#6392911yescharacteristic
391C2×C3#6393311yescharacteristic
392C2×C3#6397711yescharacteristic
393D3—121no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—181no—
398C22—181no—
399C22—181no—
400C22D9×#6711yescharacteristic
401C4—121no—
402C4—181no—
403C4—182no—
404C4—1081no—
405C4C2×#619211yescharacteristic
406C4D9×#6711yescharacteristic
407C3—21no—
408C3#1058846311yescharacteristic, complemented
409C3#1059146311yescharacteristic
410C2—42no—
411C2—361no—
412C2#1172117711yescharacteristic
413C2#1172118111yescharacteristic
414C2#1172127611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#3699C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#452
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×#88543C2semidirect
3. #11720945C2semidirect
4. C2×#88418C2semidirect
5. C9×#832C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720268C2semidirect
119. #832D9semidirect
249. C3×C9#452semidirect
360. C9#5239semidirect
408. C3#10588463semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C2×#885432
#117209452
C2×#884182
C9×#8322
(C2×D9)⋊(C3⋊C8)2
#117202682
#105914633
#105884633