Finite Group Databasebeta

← #13137846

Subgroups — #13137846

Order 1728.

492 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C3×#63454C211yesmaximal, characteristic
3C2×#88413C211yesmaximal, characteristic
4#11720527C211yesmaximal, characteristic, complemented
5C9×#810C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10590600—31nomaximal
9#10588482—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×Dic3×D9—21no—
14C2×#6189—21no—
15D9×Dic6—22no—
16D9×Dic6C414yescomplemented
17C3×C9×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C3×(C2×D9)⋊C4—21no—
20C3×C4⋊Dic9C2211yescharacteristic
21C3×C4×Dic9—21no—
22C2×C9×C3⋊C8C2211yescharacteristic
23#88413—22no—
24C2×D3×Dic6—31no—
25C3×#892—31no—
26C2×#3711—31no—
27C2×Q8×D9—31no—
28#63637—31no—
29C3×#810D311yescharacteristic
30#63567—31no—
31#63565—31no—
32C2×#3663—31no—
33#63454D311yescharacteristic
34C9×#103—31no—
35#63395—31no—
36#63376—31no—
37#63368—31no—
38#63368—61no—
39C22×C3×D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226C2×C412yes—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×C4×D9C2×C412yes—
49C3×C4×D9D412yes—
50C9×Dic6—21no—
51C9×Dic6C2×C412yes—
52Dic3×D9—24no—
53#6189—24no—
54C3⋊(C9⋊C8)—21no—
55C9×C3⋊C8—21no—
56#5004—91no—
57#4802—91no—
58C2×#634—31no—
59C2×C3×C4×D3—31no—
60C2×C3×Dic6C2×D311yescharacteristic
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—61no—
68C3×C4⋊Dic3—31no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71#3711—62no—
72C2×Q8×C9—31no—
73Q8×D9—34no—
74Q8×D9—62no—
75C2×C4×D9—61no—
76C2×C4×D9C2×D311yescharacteristic
77C2×Dic18—31no—
78C2×Dic18—61no—
79C2×Dic18—61no—
80C2×C8×C9—31no—
81C9×C4⋊C4—21no—
82C9×C4⋊C4C2×D311yescharacteristic
83(C2×D9)⋊C4—21no—
84C4⋊Dic9C2×D311yescharacteristic
85C4×Dic9—21no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88#3663—62no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9C22⋊C412yes—
93C3×C4×C9—41no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—22no—
97C9×Dic3—22no—
98C3×Dic9—41no—
99C3×Dic9C22⋊C412yes—
100C2×D3×Q8—91no—
101C22×Dic6—91no—
102C3×#117—91no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—91no—
105#892—31no—
106C3×#103—31no—
107#835—91no—
108#816—91no—
109#810—91no—
110#810—91no—
111#810—91no—
112#810—91no—
113#810—181no—
114#810D911yescharacteristic, complemented
115C22×C3×D3—31no—
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—31no—
123C2×C3×Dic3—61no—
124C3×C4×D3—32no—
125C3×C4×D3—32no—
126C3×Dic6—21no—
127C3×Dic6C4×D312yes—
128#627—64no—
129D3×Dic3—64no—
130C22×D9#6711yescharacteristic
131C3⋊(C3⋊C8)—61no—
132C3×C3⋊C8—21no—
133Q8×C9—32no—
134Q8×C9—61no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—41no—
138C2×C4×C9—61no—
139C2×C4×C9C22×D311yescharacteristic
140C2×Dic9—21no—
141C2×Dic9—61no—
142C2×Dic9—121no—
143C2×Dic9#6711yescharacteristic
144C4×D9—64no—
145C4×D9C4×D312yes—
146C4×D9D1212yes—
147Dic18—32no—
148Dic18—62no—
149Dic18—61no—
150Dic18—64no—
151Dic18—121no—
152C8×C9—61no—
153C9⋊C8—61no—
154C9⋊C8—121no—
155#419—271noSylow
156C2×C3×C9C2×C22⋊C411yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C2×C3×C9#13711yescharacteristic
159C3×D9—22no—
160C2×C3×Q8—31no—
161C22×C3×C4—91no—
162C22×Dic3—181no—
163D3×Q8—94no—
164D3×Q8—182no—
165C2×C4×D3—31no—
166C2×C4×D3—181no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×Dic6—94no—
170C2×Dic6—181no—
171C2×Dic6—182no—
172C2×Dic6—181no—
173C2×Dic6—181no—
174C2×Dic6C2×D911yescharacteristic
175C2×C3×C8—31no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4C2×D911yescharacteristic
179C3×C4⋊C4D3211yescharacteristic
180C3×C22⋊C4—181no—
181C3×C42—181no—
182(C2×D3)⋊C4—61no—
183C4⋊Dic3—31no—
184C4×Dic3—61no—
185#208—182no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—181no—
189C2×C3⋊C8C2×D911yescharacteristic
190ΓL(1,𝔽25)—182no—
191C22×C32D3×D411yescharacteristic
192C2×C3×D3—32no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C32×C4—41no—
196C32×C4C2×C4×D311yescharacteristic
197C32×C4D3×D411yescharacteristic
198C3⋊Dic3—62no—
199C3×Dic3—22no—
200C3×Dic3—32no—
201C3×Dic3—121no—
202C22×C9—21no—
203C22×C9C2×#6711yescharacteristic
204C2×D9—21no—
205C2×D9—21no—
206C2×D9(C2×D3)⋊C412yes—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—41no—
210C4×C9—62no—
211C4×C9—81no—
212C4×C9C2×C4×D311yescharacteristic
213C4×C9C2×D1211yescharacteristic
214Dic9—41no—
215Dic9—62no—
216Dic9—122no—
217Dic9(C2×D3)⋊C412yes—
218C22×Q8—271no—
219C2×M16—271no—
220#117—91no—
221#103—31no—
222#103—271no—
223#103—271no—
224#103—271no—
225C3×C9#41911yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—31no—
228C3×Q8—32no—
229C3×Q8—61no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—181no—
239C2×C3×C4C22×D911yescharacteristic
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—21no—
242C2×Dic3—31no—
243C2×Dic3—61no—
244C2×Dic3—184no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—361no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—184no—
251Dic6—21no—
252Dic6—92no—
253Dic6—92no—
254Dic6—182no—
255Dic6—184no—
256Dic6—184no—
257Dic6—181no—
258Dic6—181no—
259Dic6—361no—
260Dic6C4×D912yes—
261C3×C8—61no—
262C3⋊C8—21no—
263C3⋊C8—181no—
264C3⋊C8—181no—
265C3⋊C8—361no—
266C2×C32D3×C22⋊C411yescharacteristic
267C2×C32#91511yescharacteristic
268C2×C32#91811yescharacteristic
269C3×D3—62no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9C2×(C2×D3)⋊C411yescharacteristic
274C2×C9#95111yescharacteristic
275C2×C9#95111yescharacteristic
276D9—22no—
277C2×Q8—31no—
278C2×Q8—274no—
279C2×Q8—271no—
280C2×Q8—541no—
281C2×Q8—542no—
282C22×C4—91no—
283C22×C4—541no—
284M16—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4—91no—
288C4⋊C4D3×D911yescharacteristic
289C22⋊C4—181no—
290C42—181no—
291C22×C3—21no—
292C22×C3—92no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3D4×D911yescharacteristic
296C22×C3D3×#6711yescharacteristic
297C2×D3—32no—
298C2×D3—61no—
299C2×D3—61no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—62no—
305C3×C4—81no—
306C3×C4—92no—
307C3×C4—361no—
308C3×C4C2×C4×D911yescharacteristic
309C3×C4D4×D911yescharacteristic
310C3×C4C4×D3211yescharacteristic
311C3×C4D3×D1211yescharacteristic
312Dic3—22no—
313Dic3—32no—
314Dic3—121no—
315Dic3—182no—
316Dic3—182no—
317Dic3—362no—
318C32#480211yescharacteristic
319C9—21no—
320C9#500411yescharacteristic, complemented
321C23—91no—
322Q8—32no—
323Q8—61no—
324Q8—272no—
325Q8—544no—
326Q8—541no—
327C2×C4—21no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—181no—
333C2×C4—544no—
334C2×C4—541no—
335C2×C4C2×D3×D911yescharacteristic
336C8—61no—
337C8—541no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—182no—
342C2×C3C22⋊C4×D911yescharacteristic
343C2×C3#6347711yescharacteristic
344C2×C3#6348011yescharacteristic
345C2×C3D3×(C2×D3)⋊C411yescharacteristic
346C2×C3#6392911yescharacteristic
347C2×C3#6394311yescharacteristic
348D3—62no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22D9×#6711yescharacteristic
353C4—41no—
354C4—62no—
355C4—92no—
356C4—361no—
357C4—542no—
358C4C4×D3×D911yescharacteristic
359C4D9×D1211yescharacteristic
360C3—21no—
361C3#1058848211yescharacteristic, complemented
362C3#1059060011yescharacteristic
363C2—182no—
364C2D9×(C2×D3)⋊C411yescharacteristic
365C2#1172118111yescharacteristic
366C2#1172121911yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
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. C9×#810C2semidirect
6. #11720297C2semidirect
7. #11720277C2semidirect
16. D9×Dic6C4semidirect
114. #810D9semidirect
225. C3×C9#419semidirect
320. C9#5004semidirect
361. C3#10588482semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
C3×#634542
C2×#884132
#117205272
C9×#8102
#117202972
#117202772
#105906003
#105884823