Finite Group Databasebeta

← #13137814

Subgroups — #13137814

Order 1728.

492 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 117 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
2C9×#892C211yesmaximal, characteristic
3C2×#88420C211yesmaximal, characteristic
4#11720527C211yesmaximal, characteristic, complemented
5C3×#63368C211yesmaximal, characteristic, complemented
6#11720269C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590600—31nomaximal
9#10588681—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×D3×Dic9—21no—
14C2×#6189—21no—
15D3×Dic18—22no—
16D3×Dic18C414yescomplemented
17C3×C9×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C9×(C2×D3)⋊C4—21no—
20C9×C4⋊Dic3C2211yescharacteristic
21C4×C9×Dic3—21no—
22C2×C3×C9⋊C8C2211yescharacteristic
23#88420—22no—
24C2×D3×Dic6—31no—
25C3×#892D311yescharacteristic
26C2×#3711—31no—
27C22×Dic18—31no—
28#63637—31no—
29C3×#810—31no—
30#63567—31no—
31#63565—31no—
32C9×#117—31no—
33C2×#3667—31no—
34#63368—31no—
35#63368—31no—
36#63368—31no—
37#63368—61no—
38#63368D311yescharacteristic, complemented
39C22×D3×C9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226C2×C412yes—
45C2×C9×Dic3—21no—
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48C4×D3×C9C2×C412yes—
49C4×D3×C9D412yes—
50C3×Dic18—21no—
51C3×Dic18C2×C412yes—
52D3×Dic9—24no—
53#6189—24no—
54C3⋊(C9⋊C8)—21no—
55C3×C9⋊C8—21no—
56#5004—91no—
57#4802—91no—
58C2×#634—31no—
59C2×C3×C4×D3C2×D311yescharacteristic
60C2×C3×Dic6—31no—
61C2×#627—61no—
62C2×D3×Dic3—61no—
63D3×Dic6—34no—
64D3×Dic6—62no—
65C32×C4⋊C4C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C3×(C2×D3)⋊C4—21no—
68C3×C4⋊Dic3C2×D311yescharacteristic
69C3×C4×Dic3—21no—
70C2×C3×C3⋊C8—31no—
71#3711—62no—
72C22×C4×C9—31no—
73C22×Dic9—61no—
74C2×Dic18—31no—
75C2×Dic18—34no—
76C2×Dic18—62no—
77C2×Dic18—61no—
78C2×Dic18—61no—
79C2×Dic18C2×D311yescharacteristic
80C9×C4⋊C4—21no—
81C9×C4⋊C4—31no—
82C9×C4⋊C4C2×D311yescharacteristic
83C9×C22⋊C4—61no—
84C42×C9—61no—
85#3667—62no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9C22⋊C412yes—
93C3×C4×C9—41no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—22no—
97C9×Dic3—41no—
98C9×Dic3C22⋊C412yes—
99C3×Dic9—22no—
100C2×D3×Q8—91no—
101C22×Dic6—91no—
102C3×#117—31no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—91no—
105#892D911yescharacteristic
106C3×#103—91no—
107#835—91no—
108#816—91no—
109#810—31no—
110#810—91no—
111#810—91no—
112#810—91no—
113#810—91no—
114#810—181no—
115C22×C3×D3#6711yescharacteristic
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—61no—
119#634—32no—
120#634—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3#6711yescharacteristic
124C3×C4×D3C4×D312yes—
125C3×C4×D3D1212yes—
126C3×Dic6—32no—
127C3×Dic6—61no—
128#627—64no—
129D3×Dic3—64no—
130C23×C9—31no—
131C3⋊(C3⋊C8)—61no—
132C3×C3⋊C8—61no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—31no—
136C2×C4×C9—32no—
137C2×C4×C9—32no—
138C2×C4×C9—41no—
139C2×C4×C9—61no—
140C2×C4×C9C22×D311yescharacteristic
141C2×Dic9—21no—
142C2×Dic9—64no—
143C2×Dic9—61no—
144C2×Dic9—121no—
145Dic18—21no—
146Dic18—32no—
147Dic18—62no—
148Dic18—64no—
149Dic18—61no—
150Dic18—121no—
151Dic18C4×D312yes—
152C9⋊C8—21no—
153C9⋊C8—61no—
154C9⋊C8—121no—
155#419—271noSylow
156C2×C3×C9C2×C22⋊C411yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C2×C3×C9#13711yescharacteristic
159D3×C9—22no—
160C2×C3×Q8—91no—
161C22×C3×C4—31no—
162C22×Dic3—181no—
163D3×Q8—94no—
164D3×Q8—182no—
165C2×C4×D3—181no—
166C2×C4×D3C2×D911yescharacteristic
167C2×Dic6—31no—
168C2×Dic6—94no—
169C2×Dic6—91no—
170C2×Dic6—91no—
171C2×Dic6—181no—
172C2×Dic6—182no—
173C2×Dic6—181no—
174C2×Dic6—181no—
175C2×C3×C8—91no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4C2×D911yescharacteristic
179C3×C4⋊C4D3211yescharacteristic
180C3×C22⋊C4—61no—
181C3×C42—61no—
182(C2×D3)⋊C4—21no—
183C4⋊Dic3C2×D911yescharacteristic
184C4×Dic3—21no—
185#208—182no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—181no—
190ΓL(1,𝔽25)—182no—
191C22×C32C2×#6711yescharacteristic
192C2×C3×D3—21no—
193C2×C3×D3—21no—
194C2×C3×D3(C2×D3)⋊C412yes—
195C32×C4—41no—
196C32×C4C2×C4×D311yescharacteristic
197C32×C4C2×D1211yescharacteristic
198C3⋊Dic3—62no—
199C3×Dic3—41no—
200C3×Dic3—62no—
201C3×Dic3(C2×D3)⋊C412yes—
202C22×C9—21no—
203C22×C9—32no—
204C22×C9—61no—
205C22×C9—61no—
206C22×C9D3×D411yescharacteristic
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—32no—
210C4×C9—41no—
211C4×C9—81no—
212C4×C9—121no—
213C4×C9C2×C4×D311yescharacteristic
214C4×C9D3×D411yescharacteristic
215Dic9—22no—
216Dic9—62no—
217Dic9—122no—
218C22×Q8—271no—
219C2×M16—271no—
220#117—31no—
221#103—91no—
222#103—271no—
223#103—271no—
224#103—271no—
225C3×C9#41911yescharacteristic, complemented, Sylow
226C23×C3—31no—
227C22×D3#61111yescharacteristic
228C3×Q8—92no—
229C3×Q8—181no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—31no—
234C2×C3×C4—32no—
235C2×C3×C4—32no—
236C2×C3×C4—41no—
237C2×C3×C4—61no—
238C2×C3×C4—181no—
239C2×C3×C4C22×D911yescharacteristic
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—21no—
242C2×Dic3—61no—
243C2×Dic3—184no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C2×Dic3—361no—
247C2×Dic3#61111yescharacteristic
248C4×D3—184no—
249C4×D3C4×D912yes—
250C4×D3D3612yes—
251Dic6—32no—
252Dic6—61no—
253Dic6—92no—
254Dic6—92no—
255Dic6—182no—
256Dic6—184no—
257Dic6—181no—
258Dic6—181no—
259Dic6—184no—
260Dic6—361no—
261C3×C8—181no—
262C3⋊C8—61no—
263C3⋊C8—181no—
264C3⋊C8—181no—
265C3⋊C8—361no—
266C2×C32C2×(C2×D3)⋊C411yescharacteristic
267C2×C32#95111yescharacteristic
268C2×C32#95111yescharacteristic
269C3×D3—22no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—62no—
274C2×C9D3×C22⋊C411yescharacteristic
275C2×C9#91511yescharacteristic
276C2×C9#91811yescharacteristic
277C2×Q8—91no—
278C2×Q8—274no—
279C2×Q8—271no—
280C2×Q8—541no—
281C2×Q8—542no—
282C22×C4—31no—
283C22×C4—541no—
284M16—542no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—31no—
288C4⋊C4D3×D911yescharacteristic
289C22⋊C4—61no—
290C42—61no—
291C22×C3—21no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3C2×#61111yescharacteristic
296C22×C3D3×#6711yescharacteristic
297C2×D3—21no—
298C2×D3—21no—
299C2×D3(C2×D9)⋊C412yes—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—32no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—81no—
306C3×C4—121no—
307C3×C4—182no—
308C3×C4C2×C4×D911yescharacteristic
309C3×C4C2×D3611yescharacteristic
310C3×C4C4×D3211yescharacteristic
311C3×C4D3×D1211yescharacteristic
312Dic3—41no—
313Dic3—62no—
314Dic3—182no—
315Dic3—182no—
316Dic3—362no—
317Dic3(C2×D9)⋊C412yes—
318C32#500411yescharacteristic
319C9—21no—
320C9#480211yescharacteristic, complemented
321C23—31no—
322Q8—92no—
323Q8—181no—
324Q8—272no—
325Q8—544no—
326Q8—541no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—61no—
332C2×C4—181no—
333C2×C4—544no—
334C2×C4—541no—
335C2×C4C2×D3×D911yescharacteristic
336C8—181no—
337C8—541no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—62no—
342C2×C3C2×(C2×D9)⋊C411yescharacteristic
343C2×C3#6351111yescharacteristic
344C2×C3#6351111yescharacteristic
345C2×C3D3×(C2×D3)⋊C411yescharacteristic
346C2×C3#6392911yescharacteristic
347C2×C3#6394311yescharacteristic
348D3—22no—
349C22—32no—
350C22—61no—
351C22—61no—
352C22D3×#61111yescharacteristic
353C4—32no—
354C4—41no—
355C4—121no—
356C4—182no—
357C4—542no—
358C4C4×D3×D911yescharacteristic
359C4D3×D3611yescharacteristic
360C3—21no—
361C3#1058868111yescharacteristic, complemented
362C3#1059060011yescharacteristic
363C2—62no—
364C2D3×(C2×D9)⋊C411yescharacteristic
365C2#1172116511yescharacteristic
366C2#1172120311yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radicalC22×C3×D3#67
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#419
LCSG ▷ 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
4. #11720527C2semidirect
5. C3×#63368C2semidirect
6. #11720269C2semidirect
7. #11720268C2semidirect
16. D3×Dic18C4semidirect
38. #63368D3semidirect
225. C3×C9#419semidirect
320. C9#4802semidirect
361. C3#10588681semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C9×#8922
C2×#884202
#117205272
C3×#633682
#117202692
#117202682
#105906003
#105886813