Finite Group Databasebeta

← #13140843

Subgroups — #13140843

Order 1728.

486 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 101 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×Q8×D9C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic, complemented
3#11720944C211yesmaximal, characteristic
4C2×#88427C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720276C211yesmaximal, characteristic, complemented
8#10591594—31nomaximal
9#10588866—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C3×Q8×D9—22no—
12C3×Q8×D9—22no—
13C2×C3×C4×D9—21no—
14C2×C3×C4×D9C2211yescharacteristic
15C2×C9×Dic6C2211yescharacteristic
16C2×C3×Dic18—21no—
17C2×C3×Dic18C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×#216—22no—
21C2×C9×C3⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23#88442—21no—
24#88427—22no—
25C2×C3×D3×Q8—31no—
26C2×C3×#216D311yescharacteristic
27#63856—31no—
28C2×#3718—31no—
29C2×Q8×D9D311yescharacteristic
30#63662—31no—
31#63566—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×C9×Q16—31no—
34#63508—31no—
35C2×#3674—31no—
36#63395—31no—
37#63395—61no—
38(C2×D9)⋊C8—31no—
39#63376—31no—
40C22×C3×D9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9D412yes—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9D411yescharacteristic
49C3×C4×D9—24no—
50C3×C4×D9—21no—
51C3×C4×D9—21no—
52C9×Dic6—21no—
53C9×Dic6—21no—
54C3×Dic18—24no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C3⋊(C9⋊C8)—21no—
58C9×C3⋊C8—21no—
59#5228—91no—
60#5186—91no—
61C2×C32×Q8C2×D311yescharacteristic
62C3×D3×Q8—62no—
63C3×D3×Q8—62no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—61no—
68C2×C3×Dic6C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×#216—22no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73#3724—61no—
74(C2×D3)⋊Dic3—61no—
75#3718—62no—
76C2×Q8×C9—21no—
77C2×Q8×C9—31no—
78C2×Q8×C9C2×D311yescharacteristic
79Q8×D9—22no—
80Q8×D9—22no—
81C2×C4×D9—21no—
82C2×C4×D9C2×D311yescharacteristic
83C2×Dic18—21no—
84C2×Dic18C2×D311yescharacteristic
85C9×Q16—62no—
86C2×C8×C9—31no—
87#3674—62no—
88(C2×D9)⋊C4—61no—
89C4⋊Dic9—31no—
90C4⋊Dic9—61no—
91Dic9⋊C4—61no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94C22×C3×C9C2×D411yescharacteristic
95C2×C3×D9—21no—
96C2×C3×D9—21no—
97C2×C3×D9Q1612yescomplemented
98C3×C4×C9—22no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—41no—
102C9×Dic3—41no—
103C3×Dic9—21no—
104C3×Dic9—22no—
105C22×C3×Q8—91no—
106C2×D3×Q8—31no—
107C2×C3×Q16—31no—
108#946—91no—
109C2×#216—91no—
110C2×#216—91no—
111C2×#216D911yescharacteristic, complemented
112#924—91no—
113#835—91no—
114#835—91no—
115#835—91no—
116#835—181no—
117#830—91no—
118(C2×D3)⋊C8—91no—
119#816—91no—
120C22×C3×D3—31no—
121C32×Q8—21no—
122C32×Q8#6712yes—
123C2×C32×C4—21no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—61no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C3×C4×D3—64no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×Dic6—21no—
133C3×Dic6—21no—
134C3×Dic6—32no—
135C3×Dic6—64no—
136C3×Dic6—61no—
137C22×D9#6711yescharacteristic
138C3⋊(C3⋊C8)—61no—
139C3×C3⋊C8—21no—
140Q8×C9—21no—
141Q8×C9—22no—
142Q8×C9—41no—
143Q8×C9—61no—
144Q8×C9—61no—
145Q8×C9#6712yes—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—41no—
149C2×C4×C9—61no—
150C2×C4×C9C22×D311yescharacteristic
151C2×Dic9—21no—
152C2×Dic9—61no—
153C2×Dic9—121no—
154C2×Dic9#6711yescharacteristic
155C4×D9—24no—
156C4×D9—21no—
157C4×D9—21no—
158Dic18—24no—
159Dic18—21no—
160Dic18#6712yes—
161C8×C9—61no—
162C9⋊C8—61no—
163C9⋊C8—121no—
164#451—271noSylow
165C2×C3×C9#12011yescharacteristic
166C2×C3×C9C2×Q1611yescharacteristic
167C2×C3×C9#13711yescharacteristic
168C3×D9—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—31no—
171C2×C3×Q8—91no—
172C2×C3×Q8—181no—
173C2×C3×Q8—182no—
174C2×C3×Q8—182no—
175C2×C3×Q8C2×D911yescharacteristic
176C2×C3×Q8D3211yescharacteristic
177C22×C3×C4—91no—
178C22×C3×C4—181no—
179D3×Q8—62no—
180D3×Q8—62no—
181C2×C4×D3—31no—
182C2×C4×D3—61no—
183C2×Dic6—31no—
184C2×Dic6—61no—
185C2×Dic6C2×D911yescharacteristic
186C3×Q16—62no—
187C2×C3×C8—31no—
188#217—181no—
189#216—22no—
190#216—182no—
191#216—182no—
192(C2×D3)⋊C4—181no—
193C4⋊Dic3—91no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196Dic3⋊C4—181no—
197Dic3⋊C4—181no—
198C2×C3⋊C8—91no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—181no—
201C2×C3⋊C8C2×D911yescharacteristic
202C22×C32D3×D411yescharacteristic
203C2×C3×D3—32no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C32×C4—22no—
207C32×C4D3×D411yescharacteristic
208C32×C4C2×#6711yescharacteristic
209C3⋊Dic3—121no—
210C3×Dic3—41no—
211C3×Dic3—62no—
212C3×Dic3—61no—
213C22×C9—21no—
214C22×C9C2×#6711yescharacteristic
215C2×D9—21no—
216C2×D9—21no—
217C2×D9#21612yescomplemented
218C4×C9—22no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—42no—
222C4×C9—121no—
223C4×C9C2×#6711yescharacteristic
224C4×C9C2×#6711yescharacteristic
225Dic9—21no—
226Dic9—22no—
227Dic9—121no—
228Dic9—241no—
229C22×Q8—91no—
230C2×Q16—31no—
231C2×Q16—271no—
232C22⋊Q8—271no—
233#103—271no—
234#103—271no—
235C22⋊C8—271no—
236C3×C9#45111yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C22×D3—31no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—22no—
242C3×Q8—41no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—92no—
246C3×Q8—184no—
247C3×Q8—181no—
248C3×Q8#61112yes—
249C3×Q8#62512yes—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—61no—
255C2×C3×C4—91no—
256C2×C3×C4—184no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—31no—
264C2×Dic3—61no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—361no—
268C4×D3—64no—
269C4×D3—61no—
270C4×D3—61no—
271Dic6—21no—
272Dic6—21no—
273Dic6—32no—
274Dic6—64no—
275Dic6—61no—
276C3×C8—61no—
277C3⋊C8—21no—
278C3⋊C8—181no—
279C3⋊C8—181no—
280C3⋊C8—361no—
281C2×C32D3×Q1611yescharacteristic
282C2×C32#91811yescharacteristic
283C2×C32#93711yescharacteristic
284C3×D3—62no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9#94211yescharacteristic
289C2×C9C2×#21611yescharacteristic
290C2×C9#95311yescharacteristic
291D9—22no—
292C2×Q8—31no—
293C2×Q8—91no—
294C2×Q8—181no—
295C2×Q8—182no—
296C2×Q8—182no—
297C2×Q8D3×D911yescharacteristic
298C22×C4—91no—
299C22×C4—181no—
300Q16—62no—
301Q16—542no—
302C2×C8—31no—
303C2×C8—271no—
304C4⋊C4—271no—
305C4⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—21no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D4×D911yescharacteristic
312C22×C3D3×#6711yescharacteristic
313C2×D3—32no—
314C2×D3—61no—
315C2×D3—61no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—42no—
321C3×C4—121no—
322C3×C4—181no—
323C3×C4—182no—
324C3×C4D4×D911yescharacteristic
325C3×C4C2×#61111yescharacteristic
326C3×C4C2×#62511yescharacteristic
327C3×C4D3×#6711yescharacteristic
328Dic3—41no—
329Dic3—62no—
330Dic3—61no—
331Dic3—361no—
332Dic3—361no—
333Dic3—721no—
334C32#518611yescharacteristic
335C9—21no—
336C9#522811yescharacteristic, complemented
337C23—91no—
338Q8—21no—
339Q8—61no—
340Q8—61no—
341Q8—92no—
342Q8—184no—
343Q8—181no—
344Q8#619412yes—
345C2×C4—21no—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—184no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4C2×D3×D911yescharacteristic
354C8—61no—
355C8—541no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—182no—
360C2×C3Q16×D911yescharacteristic
361C2×C3#6348011yescharacteristic
362C2×C3#6349911yescharacteristic
363C2×C3D3×#21611yescharacteristic
364C2×C3#6394811yescharacteristic
365C2×C3#6397611yescharacteristic
366D3—62no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22D9×#6711yescharacteristic
371C4—22no—
372C4—121no—
373C4—181no—
374C4—182no—
375C4—1081no—
376C4C2×#619411yescharacteristic
377C4D9×#6711yescharacteristic
378C3—21no—
379C3#1058886611yescharacteristic, complemented
380C3#1059159411yescharacteristic
381C2—182no—
382C2D9×#21611yescharacteristic
383C2#1172122211yescharacteristic
384C2#1172128511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C9×#216C2semidirect
4. C2×#88427C2semidirect
5. #11720582C2semidirect
7. #11720276C2semidirect
97. C2×C3×D9Q16semidirect
111. C2×#216D9semidirect
217. C2×D9#216semidirect
236. C3×C9#451semidirect
336. C9#5228semidirect
379. C3#10588866semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
C2×C9×#2162
#117209442
C2×#884272
#117205822
(C2×D9)⋊(C3⋊C8)2
#117202762
#105915943
#105888663