Finite Group Databasebeta

← #13160969

Subgroups — #13160969

Order 1728.

1016 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 150 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×#88861C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic
3#11723431C211yesmaximal, characteristic, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592477—31nomaximal
9#10591460—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14#88861—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21#88803—21no—
22C3⋊D3×C3⋊C8—22no—
23#64147D311yescharacteristic
24C2×#3754—31no—
25C2×#3754—34no—
26C2×#3754—64no—
27C2×C8×C3⋊D3—31no—
28#64106—31no—
29C3×#938—34no—
30#63857—34no—
31C2×D3×C3⋊C8—34no—
32C32×#102—31no—
33#63643—31no—
34#63635—31no—
35C3×#832D314yescomplemented
36#63574—34no—
37#63567—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46C3×#638—42no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3—21no—
52C3⋊(C3⋊(C3⋊C8))D412yescomplemented
53C32×C3⋊C8—21no—
54#5041—91no—
55#4778—94no—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C2×D311yescharacteristic
59C2×#638—21no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×#67—64no—
65C2×C3×C4×D3—34no—
66C2×C32×C8—31no—
67C32×C4⋊C4—31no—
68#3757—21no—
69#3754—62no—
70#3754—68no—
71#3754—616no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C8×C3⋊D3—62no—
79C3×#217—64no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3C2×D314yes—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84(C2×D3)⋊Dic3—64no—
85D3×C3⋊C8—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3SD1612yescomplemented
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—41no—
98C32×Dic3—41no—
99C3×#121—91no—
100#938—34no—
101C2×#214—94no—
102C2×#214—91no—
103C2×#214—184no—
104#924—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—94no—
107#897—94no—
108C3×#102—34no—
109#832C3⋊D311yescharacteristic, complemented
110#816—94no—
111#810—91no—
112#808—94no—
113#808—91no—
114C23×C32—21no—
115C23×C32—24no—
116C23×C32—44no—
117C22×C3⋊D3#6711yescharacteristic
118C22×C3×D3—34no—
119C32×D4—28no—
120C32×D4—24no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—44no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129#638—42no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—124no—
134C2×C3⋊Dic3#6711yescharacteristic
135C4×C3⋊D3—21no—
136C4×C3⋊D3—21no—
137#634—68no—
138#634—64no—
139#634—61no—
140#634—61no—
141#634—64no—
142#634—124no—
143C3×#67—128no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—64no—
147C3×C4×D3—64no—
148C3×C4×D3—64no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8—24no—
154#465—271noSylow
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C2×C33C2×SD1611yescharacteristic
158C2×C33#13511yescharacteristic
159C3×C3⋊D3—22no—
160C2×C3×D4—24no—
161C2×C3×D4—181no—
162C2×C3×D4D3214yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C22×C3×C4—91no—
165C2×#67—64no—
166C2×C4×D3—34no—
167C2×Dic6—94no—
168C2×Dic6—91no—
169C2×Dic6—184no—
170C2×C3×C8—34no—
171C3×C4⋊C4—34no—
172C3×C4⋊C4—91no—
173C3×C22⋊C4—181no—
174#217—64no—
175#217—181no—
176#214—188no—
177#214—182no—
178#214—1816no—
179(C2×D3)⋊C4—184no—
180C4⋊Dic3—34no—
181C4⋊Dic3—184no—
182C4⋊Dic3C2×C3⋊D311yescharacteristic
183Dic3⋊C4—181no—
184C2×C3⋊C8—94no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—184no—
187C2×C3⋊C8—182no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189D3×C8—188no—
190C22×C32—24no—
191C22×C32—48no—
192C22×C32—44no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—44no—
197C22×C32—44no—
198C22×C32—84no—
199C22×C32—84no—
200C22×C32D3×D414yes—
201C22×C32C2×#6711yescharacteristic
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3#21412yescomplemented
205C2×C3×D3—38no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—121no—
211C32×C4#23514yes—
212C32×C4D3×D414yes—
213C32×C4D3×D411yescharacteristic
214C32×C4#23711yescharacteristic
215C3⋊Dic3—21no—
216C3⋊Dic3—41no—
217C3⋊Dic3—128no—
218C3⋊Dic3—124no—
219C3⋊Dic3—121no—
220C3×Dic3—44no—
221C3×Dic3—64no—
222C3×Dic3—124no—
223C2×SD16—271no—
224C22×C8—271no—
225C22⋊Q8—271no—
226#121—91no—
227#106—271no—
228#103—271no—
229#102—31no—
230C33#46511yescharacteristic, complemented, Sylow
231C23×C3—24no—
232C23×C3—21no—
233C23×C3—44no—
234C23×C3—91no—
235C22×D3—34no—
236C3×D4—28no—
237C3×D4—24no—
238C3×D4—24no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—44no—
242C3×D4—362no—
243C2×C3×C4—24no—
244C2×C3×C4—64no—
245C2×C3×C4—91no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3214yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251#67—128no—
252C2×Dic3—21no—
253C2×Dic3—34no—
254C2×Dic3—64no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—364no—
258C4×D3—64no—
259C4×D3—64no—
260Dic6—188no—
261Dic6—184no—
262Dic6—184no—
263Dic6—181no—
264Dic6—181no—
265Dic6—364no—
266C3×C8—64no—
267C3⋊C8—21no—
268C3⋊C8—98no—
269C3⋊C8—92no—
270C3⋊C8—188no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—88no—
275C2×C32—84no—
276C2×C32—81no—
277C2×C32#88414yes—
278C2×C32#91214yes—
279C2×C32D3×SD1614yes—
280C2×C32C2×#21411yescharacteristic
281C2×C32#93911yescharacteristic
282C2×C32#95011yescharacteristic
283C3⋊D3—22no—
284C3×D3—68no—
285C2×Q8—271no—
286C2×D4—181no—
287C2×D4D3×C3⋊D311yescharacteristic
288C22×C4—91no—
289SD16—542no—
290C2×C8—31no—
291C2×C8—271no—
292C2×C8—542no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—541no—
296C22⋊C4—181no—
297C22⋊C4—541no—
298C22×C3—24no—
299C22×C3—48no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—84no—
307C22×C3—84no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D3×#6714yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—38no—
314C2×D3—64no—
315C2×D3—64no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—124no—
319C3×C4—181no—
320C3×C4—361no—
321C3×C4#380414yes—
322C3×C4#381114yes—
323C3×C4#382811yescharacteristic
324C3×C4D4×C3⋊D311yescharacteristic
325Dic3—41no—
326Dic3—64no—
327Dic3—124no—
328Dic3—368no—
329Dic3—361no—
330Dic3—364no—
331C32—24no—
332C32#477814yescomplemented
333C32#504111yescharacteristic, complemented
334C23—21no—
335C23—91no—
336Q8—541no—
337Q8—541no—
338D4—21no—
339D4—21no—
340D4—362no—
341C2×C4—61no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—541no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—61no—
349C8—272no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—88no—
354C2×C3—81no—
355C2×C3—84no—
356C2×C3—182no—
357C2×C3D3×#21414yes—
358C2×C3#6393314yes—
359C2×C3#6397114yes—
360C2×C3#6409311yescharacteristic
361C2×C3#6412111yescharacteristic
362C2×C3SD16×C3⋊D311yescharacteristic
363D3—68no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22C3⋊D3×#6711yescharacteristic
371C4—121no—
372C4—181no—
373C4—361no—
374C4—1081no—
375C4#8903111yescharacteristic
376C4#8903911yescharacteristic
377C3—24no—
378C3#1059146014yescomplemented
379C3#1059247711yescharacteristic, complemented
380C2—81no—
381C2—182no—
382C2C3⋊D3×#21411yescharacteristic
383C2#1172355911yescharacteristic
384C2#1172362411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#465
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88861C2semidirect
3. #11723431C2semidirect
4. C2×C3⋊D3×C3⋊C8C2semidirect
5. C32×#832C2semidirect
6. #11722053C2semidirect
7. #11722042C2semidirect
35. C3×#832D3semidirect
52. C3⋊(C3⋊(C3⋊C8))D4semidirect
92. C2×C3×C3⋊D3SD16semidirect
109. #832C3⋊D3semidirect
204. C2×C3⋊D3#214semidirect
230. C33#465semidirect
332. C32#4778semidirect
333. C32#5041semidirect
378. C3#10591460semidirect
379. C3#10592477semidirect

Maximal subgroups

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

subgroupindex
C2×#888612
C3×#641472
#117234312
C2×C3⋊D3×C3⋊C82
C32×#8322
#117220532
#117220422
#105924773
#105914603