Finite Group Databasebeta

← #13140479

Subgroups — #13140479

Order 1728.

410 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#63500C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720944C211yesmaximal, characteristic, complemented
4C2×#88413C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720279C211yesmaximal, characteristic, complemented
8#10591502—31nomaximal
9#10588854—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C2×C3×C4×D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×#3676—21no—
17C9×#214—22no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21#88442—21no—
22#88413—22no—
23C3×#938—31no—
24C2×C3×#214D311yescharacteristic
25#63856—31no—
26C2×#3711—31no—
27#63659—31no—
28#63575—31no—
29#63565—31no—
30C2×C9×SD16—31no—
31#63508—31no—
32#63500D311yescharacteristic
33C2×#3663—31no—
34#63395—31no—
35#63392—31no—
36#63392—61no—
37#63378—31no—
38C23×C3×C9—21no—
39C22×C3×D9D411yescharacteristic
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C3×#611—42no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×C4×D9—21no—
49C3×C4×D9—21no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8D412yescomplemented
54#5174—91no—
55#5042—91no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C3×#67—61no—
58C2×C3×C4×D3—31no—
59C2×C3×Dic6C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)—31no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×#217—61no—
63C3×#214—22no—
64C3×C4⋊Dic3—31no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66#3724—61no—
67(C2×D3)⋊Dic3—61no—
68#3711—62no—
69C2×Q8×C9—31no—
70C2×D4×C9—21no—
71C2×D4×C9C2×D311yescharacteristic
72C2×#611—21no—
73C2×C4×D9C2×D311yescharacteristic
74C9×SD16—62no—
75C2×C8×C9—31no—
76#3676—21no—
77(C2×D9)⋊C4—61no—
78C4⋊Dic9—31no—
79C4⋊Dic9—61no—
80C4⋊Dic9C2×D311yescharacteristic
81Dic9⋊C4—61no—
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84#3663—62no—
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9C2×D411yescharacteristic
89C2×C3×D9—21no—
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C3×C4×C9C2×D411yescharacteristic
93C3×C4×C9#4111yescharacteristic
94C3⋊Dic9—41no—
95C9×Dic3—41no—
96C3×Dic9—21no—
97C3×Dic9—41no—
98C2×C3×SD16—31no—
99C3×#121—91no—
100#946—91no—
101#938—31no—
102C2×#214D911yescharacteristic, complemented
103#924—91no—
104C2×#208—91no—
105C2×ΓL(1,𝔽25)—91no—
106#835—91no—
107#832—91no—
108#832—91no—
109#832—181no—
110#818—91no—
111#810—91no—
112#807—91no—
113C23×C32—21no—
114C22×C3×D3—31no—
115C32×D4—21no—
116C32×D4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—61no—
119C3×#67—122no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—61no—
123C3×C4×D3—61no—
124C3×C4×D3—61no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C23×C9—21no—
128C23×C9—41no—
129C22×D9#6711yescharacteristic
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8#6712yes—
132Q8×C9—61no—
133Q8×C9—61no—
134D4×C9—22no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9—41no—
138C2×C4×C9—21no—
139C2×C4×C9—61no—
140C2×C4×C9C22×D311yescharacteristic
141#611—42no—
142C2×Dic9—21no—
143C2×Dic9—61no—
144C2×Dic9—121no—
145C2×Dic9#6711yescharacteristic
146C4×D9—21no—
147C4×D9—21no—
148C8×C9—32no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#468—271noSylow
152C2×C3×C9—81no—
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
155C2×C3×C9#13711yescharacteristic
156C3×D9—41no—
157C2×C3×Q8—31no—
158C2×C3×D4—21no—
159C2×C3×D4—181no—
160C2×C3×D4C2×D911yescharacteristic
161C2×C3×D4D3211yescharacteristic
162C22×C3×C4—91no—
163C2×#67—61no—
164C2×C4×D3—31no—
165C2×Dic6C2×D911yescharacteristic
166C3×SD16—62no—
167C2×C3×C8—31no—
168C3×C4⋊C4—91no—
169C3×C22⋊C4—181no—
170#217—61no—
171#217—181no—
172#214—22no—
173(C2×D3)⋊C4—181no—
174C4⋊Dic3—31no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177C4⋊Dic3—181no—
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180#208—182no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—181no—
184C2×C3⋊C8C2×D911yescharacteristic
185ΓL(1,𝔽25)—182no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32D3×D411yescharacteristic
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4#23711yescharacteristic
194C32×C4C2×#6711yescharacteristic
195C3⋊Dic3—121no—
196C3×Dic3—41no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C22×C9—21no—
200C22×C9—42no—
201C22×C9—41no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9—81no—
205C22×C9—81no—
206C22×C9C2×#6711yescharacteristic
207C2×D9—21no—
208C2×D9—21no—
209C2×D9—21no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—121no—
213C4×C9D3×D411yescharacteristic
214C4×C9#23711yescharacteristic
215Dic9—21no—
216Dic9—41no—
217Dic9—121no—
218Dic9—241no—
219C2×SD16—31no—
220C2×M16—271no—
221C22⋊Q8—271no—
222#121—91no—
223#107—271no—
224#103—271no—
225#102—271no—
226C3×C9#46811yescharacteristic, complemented, Sylow
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—41no—
230C23×C3—91no—
231C22×D3—31no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×D4—22no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—41no—
240C3×D4—362no—
241C2×C3×C4—21no—
242C2×C3×C4—61no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4C22×D911yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249#67—122no—
250C2×Dic3—21no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255C2×Dic3—361no—
256C4×D3—61no—
257C4×D3—61no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—32no—
261C3⋊C8—181no—
262C3⋊C8—181no—
263C3⋊C8—361no—
264C3⋊C8#61112yescomplemented
265C2×C32—81no—
266C2×C32#91411yescharacteristic
267C2×C32#91511yescharacteristic
268C2×C32#93811yescharacteristic
269C3×D3—121no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—82no—
274C2×C9—81no—
275C2×C9#93211yescharacteristic
276C2×C9#93911yescharacteristic
277C2×C9#95111yescharacteristic
278D9—41no—
279C2×Q8—31no—
280C2×D4—181no—
281C2×D4D3×D911yescharacteristic
282C22×C4—91no—
283SD16—62no—
284M16—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4—91no—
288C4⋊C4—271no—
289C4⋊C4—541no—
290C22⋊C4—181no—
291C22⋊C4—541no—
292C22×C3—21no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—181no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3D4×D911yescharacteristic
306C22×C3D3×#6711yescharacteristic
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—61no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—121no—
313C3×C4—181no—
314C3×C4—361no—
315C3×C4#369911yescharacteristic
316C3×C4C2×#61111yescharacteristic
317C3×C4#380511yescharacteristic
318C3×C4D3×#6711yescharacteristic
319Dic3—41no—
320Dic3—61no—
321Dic3—121no—
322Dic3—361no—
323Dic3—361no—
324Dic3—721no—
325C32#517411yescharacteristic
326C9—21no—
327C9#504211yescharacteristic, complemented
328C23—21no—
329C23—91no—
330Q8—61no—
331Q8—61no—
332D4—21no—
333D4—21no—
334D4—362no—
335C2×C4—61no—
336C2×C4—91no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—541no—
341C2×C4C2×D3×D911yescharacteristic
342C8—32no—
343C8—541no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—82no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3—361no—
351C2×C3#6347611yescharacteristic
352C2×C3#6347711yescharacteristic
353C2×C3#6350011yescharacteristic
354C2×C3#6392911yescharacteristic
355C2×C3#6393411yescharacteristic
356C2×C3#6397011yescharacteristic
357D3—121no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—181no—
362C22—181no—
363C22—181no—
364C22D9×#6711yescharacteristic
365C4—121no—
366C4—181no—
367C4—361no—
368C4—1081no—
369C4#8866311yescharacteristic
370C4D3×#61111yescharacteristic
371C3—21no—
372C3#1058885411yescharacteristic, complemented
373C3#1059150211yescharacteristic
374C2—81no—
375C2—361no—
376C2#1172118111yescharacteristic
377C2#1172118411yescharacteristic
378C2#1172127011yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC3×#63500C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#468
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×C9×#214C2semidirect
3. #11720944C2semidirect
4. C2×#88413C2semidirect
5. #11720576C2semidirect
6. #11720297C2semidirect
7. #11720279C2semidirect
53. C9×C3⋊C8D4semidirect
102. C2×#214D9semidirect
226. C3×C9#468semidirect
264. C3⋊C8#611semidirect
327. C9#5042semidirect
372. C3#10588854semidirect

Maximal subgroups

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

subgroupindex
C3×#635002
C2×C9×#2142
#117209442
C2×#884132
#117205762
#117202972
#117202792
#105915023
#105888543