Finite Group Databasebeta

← #13137816

Subgroups — #13137816

Order 1728.

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

normal subgroups only

This group has 486 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
2C9×#897C211yesmaximal, characteristic
3C2×#88428C211yesmaximal, characteristic, complemented
4C2×#88427C211yesmaximal, characteristic, complemented
5#11720527C211yesmaximal, characteristic, complemented
6C3×#63368C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic
8#10590604—31nomaximal
9#10588726—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×D3×Dic9—21no—
15C2×#6189—21no—
16D3×Dic18—22no—
17D3×Dic18—22no—
18C3×C9×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×(C2×D3)⋊C4—21no—
21C9×C4⋊Dic3—21no—
22C2×C3×C9⋊C8C2211yescharacteristic
23#88428—22no—
24#88427—22no—
25C2×D3×Dic6—31no—
26C3×#897D311yescharacteristic
27C2×#3719—31no—
28C2×#3718—31no—
29C22×Dic18—31no—
30#63637—31no—
31C3×#810—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C9×C22⋊Q8—31no—
34C2×#3674—31no—
35C2×#3674—31no—
36#63389—31no—
37#63368—31no—
38#63368—61no—
39#63368D311yescharacteristic, complemented
40C22×D3×C9D411yescharacteristic
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44#6226—21no—
45#6226D412yes—
46C2×C9×Dic3—21no—
47C2×C9×Dic3D411yescharacteristic
48C2×C3×Dic9—21no—
49C4×D3×C9—21no—
50C4×D3×C9—21no—
51C9×Dic6—21no—
52C9×Dic6—21no—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55D3×Dic9—24no—
56#6189—24no—
57C3⋊(C9⋊C8)—21no—
58C3×C9⋊C8—21no—
59#5050—91no—
60#4809—91no—
61C2×#634—31no—
62C2×C3×C4×D3C2×D311yescharacteristic
63C2×C3×Dic6—31no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×#627—61no—
66C2×D3×Dic3—61no—
67D3×Dic6—62no—
68D3×Dic6—62no—
69C32×C4⋊C4C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C3×(C2×D3)⋊C4—21no—
72C3×C4⋊Dic3—21no—
73C2×C3×C3⋊C8—31no—
74#3719—62no—
75#3718—62no—
76C2×Q8×C9—31no—
77C22×C4×C9—31no—
78C22×Dic9—61no—
79C2×Dic18—31no—
80C2×Dic18—62no—
81C2×Dic18—61no—
82C2×Dic18—62no—
83C2×Dic18—61no—
84C2×Dic18C2×D311yescharacteristic
85C9×C4⋊C4—21no—
86C9×C4⋊C4—61no—
87C9×C4⋊C4C2×D311yescharacteristic
88C9×C22⋊C4—61no—
89#3674—62no—
90#3674—62no—
91C2×C9⋊C8—31no—
92C2×C9⋊C8—61no—
93C2×C9⋊C8C2×D311yescharacteristic
94C22×C3×C9C2×D411yescharacteristic
95C2×D3×C9—21no—
96C2×D3×C9—21no—
97C2×D3×C9Q1612yescomplemented
98C3×C4×C9—41no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—22no—
102C9×Dic3—21no—
103C9×Dic3—41no—
104C3×Dic9—22no—
105C2×D3×Q8—91no—
106C22×Dic6—91no—
107C3×C22⋊Q8—31no—
108C2×#216—91no—
109C2×#216—91no—
110C2×#216—91no—
111C2×Dic12—91no—
112#897D911yescharacteristic
113C3×#103—91no—
114#830—91no—
115(C2×D3)⋊C8—91no—
116#810—31no—
117#810—91no—
118#810—91no—
119#810—181no—
120C22×C3×D3#6711yescharacteristic
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—61no—
124#634—32no—
125#634—61no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3#6711yescharacteristic
129C3×C4×D3—21no—
130C3×C4×D3—21no—
131C3×Dic6—21no—
132C3×Dic6—21no—
133C3×Dic6—32no—
134C3×Dic6—61no—
135#627—64no—
136D3×Dic3—64no—
137C23×C9—31no—
138C3⋊(C3⋊C8)—61no—
139C3×C3⋊C8—61no—
140Q8×C9—61no—
141Q8×C9—61no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—31no—
145C2×C4×C9—41no—
146C2×C4×C9—61no—
147C2×C4×C9—61no—
148C2×C4×C9—61no—
149C2×C4×C9C22×D311yescharacteristic
150C2×Dic9—21no—
151C2×Dic9—64no—
152C2×Dic9—61no—
153C2×Dic9—121no—
154Dic18—21no—
155Dic18—32no—
156Dic18—62no—
157Dic18—64no—
158Dic18—61no—
159Dic18—121no—
160Dic18D1212yes—
161C9⋊C8—21no—
162C9⋊C8—61no—
163C9⋊C8—121no—
164#451—271noSylow
165C2×C3×C9#12011yescharacteristic
166C2×C3×C9C2×Q1611yescharacteristic
167C2×C3×C9#13711yescharacteristic
168D3×C9—22no—
169C2×C3×Q8—31no—
170C2×C3×Q8—91no—
171C22×C3×C4—31no—
172C22×Dic3—181no—
173D3×Q8—182no—
174D3×Q8—182no—
175C2×C4×D3—181no—
176C2×C4×D3C2×D911yescharacteristic
177C2×Dic6—31no—
178C2×Dic6—91no—
179C2×Dic6—91no—
180C2×Dic6—181no—
181C2×Dic6—182no—
182C2×Dic6—182no—
183C2×Dic6—181no—
184C2×Dic6—181no—
185C2×Dic6C2×D911yescharacteristic
186C2×C3×C8—91no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4C2×D911yescharacteristic
190C3×C4⋊C4D3211yescharacteristic
191C3×C22⋊C4—61no—
192#216—182no—
193#216—182no—
194#216—182no—
195(C2×D3)⋊C4—21no—
196C4⋊Dic3—21no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—91no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—181no—
201Dic12—182no—
202C22×C32C2×#6711yescharacteristic
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3#21612yes—
206C32×C4—41no—
207C32×C4D3×D411yescharacteristic
208C32×C4D3×D411yescharacteristic
209C3⋊Dic3—62no—
210C3×Dic3—21no—
211C3×Dic3—41no—
212C3×Dic3—62no—
213C22×C9—21no—
214C22×C9—32no—
215C22×C9—61no—
216C22×C9—61no—
217C22×C9D3×D411yescharacteristic
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—41no—
221C4×C9—61no—
222C4×C9—81no—
223C4×C9—121no—
224C4×C9C2×D1211yescharacteristic
225C4×C9D3×D411yescharacteristic
226Dic9—22no—
227Dic9—62no—
228Dic9—122no—
229C22×Q8—271no—
230C2×Q16—271no—
231C2×Q16—271no—
232C22⋊Q8—31no—
233#103—91no—
234#103—271no—
235C22⋊C8—271no—
236C3×C9#45111yescharacteristic, complemented, Sylow
237C23×C3—31no—
238C22×D3#61111yescharacteristic
239C3×Q8—61no—
240C3×Q8—61no—
241C3×Q8—92no—
242C3×Q8—181no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—41no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—181no—
252C2×C3×C4C22×D911yescharacteristic
253C2×C3×C4C2×D3211yescharacteristic
254C2×Dic3—21no—
255C2×Dic3—61no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258C2×Dic3—181no—
259C2×Dic3—361no—
260C2×Dic3#61111yescharacteristic
261C4×D3—21no—
262C4×D3—21no—
263C4×D3—184no—
264Dic6—21no—
265Dic6—21no—
266Dic6—32no—
267Dic6—61no—
268Dic6—92no—
269Dic6—92no—
270Dic6—182no—
271Dic6—184no—
272Dic6—184no—
273Dic6—181no—
274Dic6—181no—
275Dic6—361no—
276C3×C8—181no—
277C3⋊C8—61no—
278C3⋊C8—181no—
279C3⋊C8—181no—
280C3⋊C8—361no—
281C2×C32#93711yescharacteristic
282C2×C32C2×#21611yescharacteristic
283C2×C32#95111yescharacteristic
284C3×D3—22no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—62no—
289C2×C9#88211yescharacteristic
290C2×C9#91511yescharacteristic
291C2×C9D3×Q1611yescharacteristic
292C2×Q8—31no—
293C2×Q8—91no—
294C2×Q8—271no—
295C2×Q8—541no—
296C2×Q8—542no—
297C2×Q8—542no—
298C22×C4—31no—
299C22×C4—541no—
300Q16—542no—
301Q16—542no—
302C2×C8—91no—
303C2×C8—271no—
304C4⋊C4—61no—
305C4⋊C4D3×D911yescharacteristic
306C22⋊C4—61no—
307C22×C3—21no—
308C22×C3—32no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3C2×#61111yescharacteristic
312C22×C3D3×#6711yescharacteristic
313C2×D3—21no—
314C2×D3—21no—
315C2×D3#367412yescomplemented
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—61no—
321C3×C4—81no—
322C3×C4—121no—
323C3×C4—182no—
324C3×C4D4×D911yescharacteristic
325C3×C4D4×D911yescharacteristic
326C3×C4D3×D1211yescharacteristic
327C3×C4#380211yescharacteristic
328Dic3—21no—
329Dic3—41no—
330Dic3—62no—
331Dic3—182no—
332Dic3—182no—
333Dic3—362no—
334C32#505011yescharacteristic
335C9—21no—
336C9#480911yescharacteristic, complemented
337C23—31no—
338Q8—61no—
339Q8—61no—
340Q8—92no—
341Q8—181no—
342Q8—272no—
343Q8—544no—
344Q8—541no—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—181no—
351C2×C4—544no—
352C2×C4—541no—
353C2×C4C2×D3×D911yescharacteristic
354C8—181no—
355C8—541no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—62no—
360C2×C3#6349911yescharacteristic
361C2×C3C2×#367411yescharacteristic
362C2×C3#6351111yescharacteristic
363C2×C3#6392211yescharacteristic
364C2×C3#6392911yescharacteristic
365C2×C3D3×#21611yescharacteristic
366D3—22no—
367C22—32no—
368C22—61no—
369C22—61no—
370C22D3×#61111yescharacteristic
371C4—41no—
372C4—61no—
373C4—121no—
374C4—182no—
375C4—542no—
376C4D9×D1211yescharacteristic
377C4#8864611yescharacteristic
378C3—21no—
379C3#1058872611yescharacteristic, complemented
380C3#1059060411yescharacteristic
381C2—62no—
382C2#1172105011yescharacteristic
383C2#1172116511yescharacteristic
384C2D3×#367411yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radicalC22×C3×D3#67
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×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
3. C2×#88428C2semidirect
4. C2×#88427C2semidirect
5. #11720527C2semidirect
6. C3×#63368C2semidirect
39. #63368D3semidirect
97. C2×D3×C9Q16semidirect
236. C3×C9#451semidirect
315. C2×D3#3674semidirect
336. C9#4809semidirect
379. 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
C9×#8972
C2×#884282
C2×#884272
#117205272
C3×#633682
(C2×D3)⋊(C9⋊C8)2
#105906043
#105887263