Finite Group Databasebeta

← #13138562

Subgroups — #13138562

Order 1728.

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

normal subgroups only

This group has 500 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×#88524C211yesmaximal, characteristic, complemented
2C2×C9×D24C211yesmaximal, characteristic, complemented
3#11721040C211yesmaximal, characteristic, complemented
4#11720964C211yesmaximal, characteristic, complemented
5C2×#88424C211yesmaximal, characteristic, complemented
6C2×#88422C211yesmaximal, characteristic, complemented
7C3×#63375C211yesmaximal, characteristic, complemented
8#10590883—31nomaximal
9#10588834—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C9×D12C2211yescharacteristic
14C2×#6195—22no—
15C2×C3×C8×C9C2211yescharacteristic
16#88524—22no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19C9×D24—22no—
20(C2×D3)⋊Dic9—22no—
21#88424—22no—
22#88422—22no—
23C2×#3744—31no—
24C2×C3×D24D311yescharacteristic
25#63911—31no—
26#63860—31no—
27C2×#3716—31no—
28C2×#3714—31no—
29C3×#815—31no—
30C2×C9×D8—31no—
31#63502—31no—
32#63498—31no—
33C2×#3673—31no—
34C2×#3672—31no—
35C2×#3664—31no—
36C2×#3664—61no—
37#63375D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×D3×C9—21no—
40C22×D3×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45#6226—21no—
46#6226—21no—
47C2×C3×Dic9D412yes—
48C9×D12—21no—
49C9×D12—21no—
50C9×D12—21no—
51C9×D12—21no—
52#6195—44no—
53C3×C8×C9D412yescomplemented
54C3×C9⋊C8D412yescomplemented
55#5154—91no—
56#4712—91no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×#634—31no—
59C2×C3×D12C2×D311yescharacteristic
60C2×C3×D12C2×D311yescharacteristic
61C2×#626—62no—
62C2×C32×C8C2×D311yescharacteristic
63#3744—62no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C3×D24—22no—
67(C2×D3)⋊Dic3—62no—
68#3716—62no—
69#3714—62no—
70C2×D4×C9—31no—
71C2×D4×C9—31no—
72C2×#611—62no—
73C2×D36—31no—
74C2×D36—61no—
75C2×Dic18—31no—
76C2×Dic18—61no—
77C9×D8—62no—
78C2×C8×C9—21no—
79C2×C8×C9C2×D311yescharacteristic
80#3676—62no—
81#3673—62no—
82#3672—62no—
83C4×Dic9C2×D311yescharacteristic
84C2×C9⋊C8C2×D311yescharacteristic
85#3664—64no—
86#3664—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—41no—
89C2×C3⋊D9—42no—
90C2×D3×C9—41no—
91C2×D3×C9—41no—
92C2×D3×C9—42no—
93C2×D3×C9—42no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—41no—
97C3×Dic9—22no—
98C2×C3×D8—31no—
99#940—91no—
100#936—91no—
101C2×#214—91no—
102C2×#213—91no—
103C2×D24—91no—
104C2×D24D911yescharacteristic, complemented
105C2×#204—91no—
106C2×#204—91no—
107C2×#204—91no—
108C2×#204—181no—
109#875—91no—
110#874—91no—
111C3×#97—91no—
112#815—31no—
113C22×C3⋊D3—61no—
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—61no—
120#634—61no—
121#634—61no—
122C2×C3×Dic3—32no—
123C3×D12—21no—
124C3×D12—21no—
125C3×D12—21no—
126C3×D12—21no—
127#626—124no—
128C23×C9—61no—
129C23×C9—61no—
130C22×D9—61no—
131C22×D9—121no—
132C32×C8#6712yes—
133C3×C3⋊C8—32no—
134D4×C9—61no—
135D4×C9—61no—
136D4×C9—61no—
137D4×C9—61no—
138C2×C4×C9—21no—
139C2×C4×C9C22×D311yescharacteristic
140#611—124no—
141C2×Dic9—61no—
142C2×Dic9—121no—
143C2×Dic9D1212yes—
144D36—62no—
145D36—61no—
146D36—61no—
147D36—121no—
148Dic18—62no—
149Dic18—61no—
150Dic18—61no—
151Dic18—121no—
152C8×C9—22no—
153C8×C9D1212yescomplemented
154C9⋊C8D1212yescomplemented
155#496—271noSylow
156C2×C3×C9#12711yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)12yes—
158C3⋊D9—81no—
159D3×C9—81no—
160D3×C9—81no—
161C2×C3×D4—31no—
162C2×C3×D4—31no—
163C2×#67—182no—
164C2×D12—91no—
165C2×D12—91no—
166C2×D12—181no—
167C2×D12—182no—
168C2×D12C2×D911yescharacteristic
169C2×D12C2×D911yescharacteristic
170C2×Dic6—91no—
171C2×Dic6—91no—
172C2×Dic6—181no—
173C3×D8—62no—
174C2×C3×C8—21no—
175C2×C3×C8—91no—
176C2×C3×C8C2×D911yescharacteristic
177C2×C3×C8D3211yescharacteristic
178C3×C42—91no—
179#217—182no—
180#214—182no—
181#213—182no—
182(C2×D3)⋊C4—182no—
183C4×Dic3—31no—
184C2×C3⋊C8—31no—
185D24—22no—
186D24—182no—
187#204—184no—
188#204—182no—
189#204—182no—
190#204—182no—
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—121no—
193C2×C3⋊D3—122no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C32×C4D3×D411yescharacteristic
199C32×C4C2×#6711yescharacteristic
200C3⋊Dic3—121no—
201C3×Dic3—62no—
202C22×C9—21no—
203C22×C9—121no—
204C22×C9—121no—
205C22×C9—122no—
206C22×C9—122no—
207C22×C9C2×D1211yescharacteristic
208C2×D9—122no—
209C2×D9—121no—
210C2×D9—122no—
211C2×D9—242no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9C2×D1211yescharacteristic
215C4×C9C2×D1211yescharacteristic
216Dic9—22no—
217Dic9—122no—
218Dic9—121no—
219C2×SD16—271no—
220C2×SD16—271no—
221C2×D8—31no—
222C2×D8—271no—
223#127—271no—
224#124—271no—
225#97—91no—
226C3×C9#49611yescharacteristic, complemented, Sylow
227C23×C3—61no—
228C23×C3—61no—
229C22×D3—21no—
230C22×D3—21no—
231C22×D3—181no—
232C22×D3—181no—
233C22×D3—361no—
234C3×D4—61no—
235C3×D4—61no—
236C3×D4—61no—
237C3×D4—61no—
238C2×C3×C4—21no—
239C2×C3×C4—92no—
240C2×C3×C4C22×D911yescharacteristic
241C2×C3×C4C2×D3211yescharacteristic
242#67—364no—
243C2×Dic3—32no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C2×Dic3—361no—
247D12—21no—
248D12—21no—
249D12—21no—
250D12—21no—
251D12—182no—
252D12—181no—
253D12—181no—
254D12—181no—
255D12—181no—
256D12—361no—
257D12—364no—
258Dic6—182no—
259Dic6—181no—
260Dic6—181no—
261Dic6—181no—
262Dic6—181no—
263Dic6—361no—
264C3×C8—22no—
265C3×C8—92no—
266C3×C8#61112yescomplemented
267C3×C8#62612yes—
268C3⋊C8—32no—
269C2×C32#91112yes—
270C2×C32#94011yescharacteristic
271C3⋊D3—241no—
272C3×D3—81no—
273C3×D3—81no—
274C2×C9—22no—
275C2×C9—21no—
276C2×C9—241no—
277C2×C9—241no—
278C2×C9#87411yescharacteristic
279C2×C9#90812yes—
280D9—242no—
281D9—241no—
282C2×Q8—271no—
283C2×D4—31no—
284C2×D4—31no—
285C2×D4—271no—
286C2×D4—542no—
287SD16—542no—
288SD16—542no—
289D8—62no—
290D8—542no—
291C2×C8—91no—
292C2×C8D3×D911yescharacteristic
293C22⋊C4—542no—
294C42—91no—
295C22×C3—21no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—121no—
299C22×C3—121no—
300C22×C3D4×D911yescharacteristic
301C22×C3D3×D1211yescharacteristic
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—362no—
307C2×D3—361no—
308C2×D3—362no—
309C2×D3—361no—
310C2×D3—362no—
311C2×D3—722no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—182no—
315C3×C4D4×D911yescharacteristic
316C3×C4C2×#61111yescharacteristic
317C3×C4D3×D1211yescharacteristic
318C3×C4C2×#62611yescharacteristic
319Dic3—62no—
320Dic3—362no—
321Dic3—361no—
322Dic3—361no—
323C32#515411yescharacteristic
324C9—21no—
325C9#471211yescharacteristic, complemented
326C23—61no—
327C23—61no—
328C23—541no—
329Q8—541no—
330Q8—541no—
331D4—61no—
332D4—61no—
333D4—61no—
334D4—61no—
335D4—541no—
336D4—541no—
337D4—1084no—
338C2×C4—92no—
339C2×C4—541no—
340C2×C4C2×D3×D911yescharacteristic
341C8—92no—
342C8#619512yescomplemented
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—241no—
346C2×C3—241no—
347C2×C3#6347312yes—
348C2×C3#6350211yescharacteristic
349C2×C3#6379512yes—
350C2×C3#6391111yescharacteristic
351D3—81no—
352D3—81no—
353D3—722no—
354D3—721no—
355D3—721no—
356C22—121no—
357C22—121no—
358C22—122no—
359C22—122no—
360C22—1081no—
361C22—1082no—
362C22D9×D1211yescharacteristic
363C4—182no—
364C4—1081no—
365C4D9×D1211yescharacteristic
366C4C2×#619511yescharacteristic
367C3—21no—
368C3#1058883411yescharacteristic, complemented
369C3#1059088311yescharacteristic
370C2—241no—
371C2—241no—
372C2—2161no—
373C2#1172081612yes—
374C2#1172104011yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#496
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88524C2semidirect
2. C2×C9×D24C2semidirect
3. #11721040C2semidirect
4. #11720964C2semidirect
5. C2×#88424C2semidirect
6. C2×#88422C2semidirect
7. C3×#63375C2semidirect
37. #63375D3semidirect
53. C3×C8×C9D4semidirect
54. C3×C9⋊C8D4semidirect
104. C2×D24D9semidirect
153. C8×C9D12semidirect
154. C9⋊C8D12semidirect
226. C3×C9#496semidirect
266. C3×C8#611semidirect
325. C9#4712semidirect
342. C8#6195semidirect
368. C3#10588834semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
C2×C9×D242
#117210402
#117209642
C2×#884242
C2×#884222
C3×#633752
#105908833
#105888343