Finite Group Databasebeta

← #13140861

Subgroups — #13140861

Order 1728.

504 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 95 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
2C2×#88547C211yesmaximal, characteristic, complemented
3C9×#946C211yesmaximal, characteristic
4C2×C3×#3674C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic
6#11720269C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10591653—31nomaximal
9#10588726—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×D3×Dic9—21no—
15C2×#6189—21no—
16D3×Dic18—22no—
17D3×Dic18—22no—
18#88547—22no—
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C3×#3674—22no—
21C9×(C2×D3)⋊C4—21no—
22C9×C4⋊Dic3C2211yescharacteristic
23C9×Dic3⋊C4—21no—
24C2×C3×C9⋊C8C2211yescharacteristic
25C2×D3×Dic6—31no—
26C2×#3756—31no—
27C3×#946D311yescharacteristic
28C2×C3×#216—31no—
29C22×Dic18—31no—
30#63567—31no—
31#63565—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C9×C22⋊Q8—31no—
34C2×#3674—31no—
35C2×#3674—61no—
36C2×#3674D311yescharacteristic, complemented
37#63389—31no—
38#63368—31no—
39#63368—31no—
40C22×D3×C9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46#6226—21no—
47#6226D412yes—
48C2×C9×Dic3—21no—
49C2×C9×Dic3D411yescharacteristic
50C2×C3×Dic9—21no—
51C4×D3×C9—21no—
52C4×D3×C9—21no—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55D3×Dic9—24no—
56#6189—24no—
57C3⋊(C9⋊C8)—21no—
58C3×C9⋊C8—21no—
59#5186—91no—
60#5050—91no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×#634—31no—
63C2×C3×C4×D3C2×D311yescharacteristic
64C2×C3×Dic6—31no—
65C2×#627—61no—
66C2×D3×Dic3—61no—
67D3×Dic6—62no—
68D3×Dic6—62no—
69#3756—62no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×#216—62no—
72C3×(C2×D3)⋊C4—21no—
73C3×C4⋊Dic3C2×D311yescharacteristic
74C3×Dic3⋊C4—21no—
75C2×C3×C3⋊C8—31no—
76C2×Q8×C9—21no—
77C2×Q8×C9C2×D311yescharacteristic
78C22×C4×C9—31no—
79C22×Dic9—61no—
80C2×Dic18—31no—
81C2×Dic18—62no—
82C2×Dic18—61no—
83C2×Dic18—62no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C9×C4⋊C4—31no—
87C9×C4⋊C4—61no—
88C9×C22⋊C4—61no—
89#3674—22no—
90#3674—62no—
91#3674—64no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94C2×C9⋊C8C2×D311yescharacteristic
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—21no—
97C2×D3×C9—21no—
98C2×D3×C9Q1612yescomplemented
99C3×C4×C9—41no—
100C3×C4×C9C2×D411yescharacteristic
101C3×C4×C9C2×D411yescharacteristic
102C3⋊Dic9—22no—
103C9×Dic3—21no—
104C9×Dic3—41no—
105C3×Dic9—22no—
106C2×D3×Q8—91no—
107C22×Dic6—91no—
108C2×C3×Q16—91no—
109C3×C22⋊Q8—31no—
110#946D911yescharacteristic
111C2×#216—31no—
112C2×#216—91no—
113C2×#216—91no—
114C2×#216—181no—
115#835—91no—
116#830—91no—
117(C2×D3)⋊C8—91no—
118#816—91no—
119#810—91no—
120#810—91no—
121C22×C3×D3#6711yescharacteristic
122C32×Q8—21no—
123C32×Q8—21no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127#634—32no—
128#634—61no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3#6711yescharacteristic
132C3×C4×D3—21no—
133C3×C4×D3—21no—
134C3×Dic6—32no—
135C3×Dic6—61no—
136#627—64no—
137D3×Dic3—64no—
138C23×C9—31no—
139C3⋊(C3⋊C8)—61no—
140C3×C3⋊C8—61no—
141Q8×C9—22no—
142Q8×C9—21no—
143Q8×C9—21no—
144Q8×C9—41no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—31no—
148C2×C4×C9—41no—
149C2×C4×C9—61no—
150C2×C4×C9—61no—
151C2×C4×C9—61no—
152C2×C4×C9C22×D311yescharacteristic
153C2×Dic9—21no—
154C2×Dic9—64no—
155C2×Dic9—61no—
156C2×Dic9—121no—
157Dic18—21no—
158Dic18—32no—
159Dic18—62no—
160Dic18—64no—
161Dic18—61no—
162Dic18—121no—
163Dic18#6712yes—
164C9⋊C8—21no—
165C9⋊C8—62no—
166C9⋊C8—61no—
167#451—271noSylow
168C2×C3×C9#12011yescharacteristic
169C2×C3×C9C2×Q1611yescharacteristic
170C2×C3×C9#13711yescharacteristic
171D3×C9—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—91no—
174C2×C3×Q8C2×D911yescharacteristic
175C2×C3×Q8D3211yescharacteristic
176C22×C3×C4—31no—
177C22×Dic3—181no—
178D3×Q8—182no—
179D3×Q8—182no—
180C2×C4×D3—181no—
181C2×C4×D3C2×D911yescharacteristic
182C2×Dic6—31no—
183C2×Dic6—91no—
184C2×Dic6—91no—
185C2×Dic6—181no—
186C2×Dic6—182no—
187C2×Dic6—182no—
188C2×Dic6—181no—
189C2×Dic6—181no—
190C3×Q16—182no—
191C2×C3×C8—91no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—61no—
194C3×C22⋊C4—61no—
195#216—62no—
196#216—182no—
197#216—182no—
198#216—184no—
199(C2×D3)⋊C4—21no—
200C4⋊Dic3C2×D911yescharacteristic
201Dic3⋊C4—21no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—181no—
206C22×C32C2×#6711yescharacteristic
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C2×C3×D3#21612yes—
210C32×C4—41no—
211C32×C4D3×D411yescharacteristic
212C32×C4D3×D411yescharacteristic
213C3⋊Dic3—62no—
214C3×Dic3—21no—
215C3×Dic3—41no—
216C3×Dic3—62no—
217C22×C9—21no—
218C22×C9—32no—
219C22×C9—61no—
220C22×C9—61no—
221C22×C9D3×D411yescharacteristic
222C4×C9—21no—
223C4×C9—21no—
224C4×C9—42no—
225C4×C9—41no—
226C4×C9—61no—
227C4×C9—121no—
228C4×C9D3×D411yescharacteristic
229C4×C9C2×#6711yescharacteristic
230Dic9—22no—
231Dic9—62no—
232Dic9—122no—
233C22×Q8—271no—
234C2×Q16—91no—
235C2×Q16—271no—
236C22⋊Q8—31no—
237#103—271no—
238#103—271no—
239C22⋊C8—271no—
240C3×C9#45111yescharacteristic, complemented, Sylow
241C23×C3—31no—
242C22×D3#61111yescharacteristic
243C3×Q8—22no—
244C3×Q8—21no—
245C3×Q8—21no—
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—41no—
249C3×Q8—92no—
250C3×Q8—181no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—31no—
255C2×C3×C4—41no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—61no—
264C2×Dic3—184no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—361no—
268C2×Dic3#61111yescharacteristic
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—184no—
272Dic6—32no—
273Dic6—61no—
274Dic6—92no—
275Dic6—92no—
276Dic6—182no—
277Dic6—184no—
278Dic6—184no—
279Dic6—181no—
280Dic6—181no—
281Dic6—361no—
282C3×C8—181no—
283C3⋊C8—61no—
284C3⋊C8—182no—
285C3⋊C8—181no—
286C3⋊C8—181no—
287C2×C32#93711yescharacteristic
288C2×C32C2×#21611yescharacteristic
289C2×C32#95111yescharacteristic
290C3×D3—22no—
291C2×C9—21no—
292C2×C9—21no—
293C2×C9—21no—
294C2×C9—62no—
295C2×C9D3×Q1611yescharacteristic
296C2×C9#91811yescharacteristic
297C2×C9#93711yescharacteristic
298C2×Q8—91no—
299C2×Q8—271no—
300C2×Q8—541no—
301C2×Q8—542no—
302C2×Q8—542no—
303C2×Q8D3×D911yescharacteristic
304C22×C4—31no—
305C22×C4—541no—
306Q16—182no—
307Q16—542no—
308C2×C8—91no—
309C2×C8—271no—
310C4⋊C4—31no—
311C4⋊C4—61no—
312C22⋊C4—61no—
313C22×C3—21no—
314C22×C3—32no—
315C22×C3—61no—
316C22×C3—61no—
317C22×C3C2×#61111yescharacteristic
318C22×C3D3×#6711yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3#367412yescomplemented
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4—61no—
328C3×C4—121no—
329C3×C4—182no—
330C3×C4D4×D911yescharacteristic
331C3×C4D4×D911yescharacteristic
332C3×C4D3×#6711yescharacteristic
333C3×C4#381111yescharacteristic
334Dic3—21no—
335Dic3—41no—
336Dic3—62no—
337Dic3—182no—
338Dic3—182no—
339Dic3—362no—
340C32#505011yescharacteristic
341C9—21no—
342C9#518611yescharacteristic, complemented
343C23—31no—
344Q8—21no—
345Q8—21no—
346Q8—92no—
347Q8—181no—
348Q8—272no—
349Q8—544no—
350Q8—541no—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—181no—
357C2×C4—544no—
358C2×C4—541no—
359C2×C4C2×D3×D911yescharacteristic
360C8—181no—
361C8—541no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—62no—
366C2×C3#6349911yescharacteristic
367C2×C3C2×#367411yescharacteristic
368C2×C3#6351111yescharacteristic
369C2×C3D3×#21611yescharacteristic
370C2×C3#6394311yescharacteristic
371C2×C3#6398311yescharacteristic
372D3—22no—
373C22—32no—
374C22—61no—
375C22—61no—
376C22D3×#61111yescharacteristic
377C4—41no—
378C4—61no—
379C4—121no—
380C4—182no—
381C4—542no—
382C4D9×#6711yescharacteristic
383C4#8866811yescharacteristic
384C3—21no—
385C3#1058872611yescharacteristic, complemented
386C3#1059165311yescharacteristic
387C2—62no—
388C2D3×#367411yescharacteristic
389C2#1172120311yescharacteristic
390C2#1172129211yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C2×#67
prime radicalC22×C3×D3#67
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×#88547C2semidirect
4. C2×C3×#3674C2semidirect
6. #11720269C2semidirect
7. #11720268C2semidirect
36. C2×#3674D3semidirect
98. C2×D3×C9Q16semidirect
240. C3×C9#451semidirect
321. C2×D3#3674semidirect
342. C9#5186semidirect
385. C3#10588726semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C2×#885472
C9×#9462
C2×C3×#36742
(C2×D3)⋊(C9⋊C8)2
#117202692
#117202682
#105916533
#105887263