Finite Group Databasebeta

← #13160968

Subgroups — #13160968

Order 1728.

1400 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1400 conjugacy classes of subgroups — too many to draw. Only its 176 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×D4×C3⋊D3C211yesmaximal, characteristic
2C2×#88861C211yesmaximal, characteristic, complemented
3#11723431C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592475—31nomaximal
9#10591459—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic
12C2×#6308C2211yescharacteristic, complemented
13C2×C3×#638—21no—
14C3×D4×C3⋊D3—22no—
15C3×D4×C3⋊D3—22no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18#88861—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C32×C4⋊Dic3C2211yescharacteristic, complemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88796—22no—
25C2×D4×C3⋊D3D311yescharacteristic
26C2×C3×D3×D4—34no—
27C2×#3754—31no—
28C2×#3754—34no—
29C2×#3754—64no—
30C2×#3744—31no—
31#64106—31no—
32#63857—34no—
33C2×#3716—34no—
34C32×#102—31no—
35#63647—31no—
36#63636—31no—
37C3×#832D314yescomplemented
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—42no—
44C22×C3×C3⋊D3—42no—
45C22×C3×C3⋊D3D411yescharacteristic
46C33×D4—21no—
47C33×D4D412yes—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308—21no—
52C3×#638—24no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)D412yes—
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C2×C32×Dic3—21no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C32×C3⋊C8—21no—
61#5218—91no—
62#4776—94no—
63C23×C3⋊D3—21no—
64C23×C3×D3—64no—
65C2×C32×D4—24no—
66C2×C32×D4C2×D314yes—
67C2×C32×D4C2×D311yescharacteristic
68C2×#638—21no—
69D4×C3⋊D3—22no—
70D4×C3⋊D3—22no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634—31no—
74C2×#634—34no—
75C2×#634—64no—
76C2×C3×#67—64no—
77C3×D3×D4—68no—
78C3×D3×D4—68no—
79C2×C3×D12—34no—
80C2×C3×C4×D3—34no—
81C2×C32×C8—31no—
82C32×C4⋊C4—31no—
83#3754—616no—
84#3754—68no—
85#3754—62no—
86#3752—61no—
87C4⋊(C3⋊Dic3)—61no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91#3744—62no—
92C3×C4⋊Dic3C2×D314yescomplemented
93C2×C3×C3⋊C8C2×D314yescomplemented
94Dic3⋊Dic3—64no—
95(C2×D3)⋊Dic3—64no—
96#3716—68no—
97C22×C33—22no—
98C22×C33—41no—
99C22×C33—41no—
100C22×C33C2×D411yescharacteristic
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—22no—
104C2×C3×C3⋊D3—48no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3SD1612yescomplemented
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3⋊(C3⋊Dic3)—41no—
111C3×C3⋊Dic3—21no—
112C32×Dic3—41no—
113C22×C3×D4—91no—
114C2×D3×D4—34no—
115C2×#214—94no—
116C2×#214—91no—
117C2×#214—91no—
118C2×#214—184no—
119#924—91no—
120C2×#204—94no—
121#897—94no—
122C3×#102—34no—
123#832—91no—
124#832C3⋊D311yescharacteristic, complemented
125#830—91no—
126(C2×D3)⋊C8—94no—
127#809—94no—
128C23×C32—21no—
129C23×C32—24no—
130C23×C32—44no—
131C22×C3⋊D3—24no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—42no—
134C22×C3⋊D3—42no—
135C22×C3⋊D3#6711yescharacteristic
136C22×C3×D3—34no—
137C22×C3×D3—616no—
138C22×C3×D3—64no—
139C22×C3×D3—128no—
140C22×C3×D3—128no—
141C32×D4—21no—
142C32×D4—28no—
143C32×D4—24no—
144C32×D4—44no—
145C32×D4D1218yes—
146C32×D4#6712yes—
147C2×C32×C4—24no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D314yes—
150C2×C32×C4C22×D311yescharacteristic
151#638—24no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—124no—
155C2×C3⋊Dic3#6711yescharacteristic
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)#6712yes—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160#634—68no—
161#634—64no—
162#634—61no—
163#634—61no—
164#634—64no—
165#634—124no—
166C3×#67—616no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—34no—
169C3×D12—38no—
170C3×D12—64no—
171C3×C4×D3—64no—
172C3×C4×D3—64no—
173C32×C8—61no—
174C3⋊(C3⋊C8)—68no—
175C3⋊(C3⋊C8)—64no—
176C3⋊(C3⋊C8)—61no—
177C3×C3⋊C8—24no—
178#450—271noSylow
179C2×C33—42no—
180C2×C33#12011yescharacteristic
181C2×C33C2×SD1611yescharacteristic
182C2×C33AGL(1,ℤ/8)11yescharacteristic
183C3×C3⋊D3—22no—
184C3×C3⋊D3—42no—
185C24×C3—181no—
186C23×D3—64no—
187C2×C3×D4—24no—
188C2×C3×D4—91no—
189C2×C3×D4—181no—
190C2×C3×D4—182no—
191C2×C3×D4—182no—
192C2×C3×D4D3214yes—
193C2×C3×D4C2×C3⋊D311yescharacteristic
194C22×C3×C4—91no—
195C2×#67—64no—
196D3×D4—68no—
197D3×D4—68no—
198C2×D12—34no—
199C2×C4×D3—34no—
200C2×Dic6—94no—
201C2×Dic6—91no—
202C2×Dic6—184no—
203C2×C3×C8—34no—
204C3×C4⋊C4—34no—
205#217—181no—
206#214—182no—
207#214—1816no—
208#214—188no—
209#214—182no—
210(C2×D3)⋊C4—184no—
211C4⋊Dic3—184no—
212C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
213Dic3⋊C4—181no—
214C2×C3⋊C8—94no—
215C2×C3⋊C8—91no—
216C2×C3⋊C8—184no—
217C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
218#204—188no—
219C22×C32—22no—
220C22×C32—24no—
221C22×C32—28no—
222C22×C32—48no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—44no—
226C22×C32—44no—
227C22×C32—84no—
228C22×C32—84no—
229C22×C32D3×D414yes—
230C22×C32C2×#6711yescharacteristic
231C2×C3⋊D3—21no—
232C2×C3⋊D3—21no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3—48no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3#21412yescomplemented
238C2×C3×D3—38no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—68no—
242C2×C3×D3—1232no—
243C2×C3×D3—124no—
244C2×C3×D3—124no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—121no—
248C32×C4C2×D1214yes—
249C32×C4D3×D414yes—
250C32×C4C2×#6711yescharacteristic
251C32×C4C2×#6711yescharacteristic
252C3⋊Dic3—21no—
253C3⋊Dic3—128no—
254C3⋊Dic3—124no—
255C3⋊Dic3—121no—
256C3×Dic3—44no—
257C3×Dic3—64no—
258C22×D4—91no—
259C2×SD16—271no—
260C2×SD16—271no—
261C22⋊Q8—271no—
262#102—31no—
263#102—271no—
264C22⋊C8—271no—
265C33#45011yescharacteristic, complemented, Sylow
266C23×C3—24no—
267C23×C3—21no—
268C23×C3—44no—
269C23×C3—91no—
270C23×C3—184no—
271C23×C3—181no—
272C23×C3—362no—
273C23×C3—362no—
274C22×D3—34no—
275C22×D3—616no—
276C22×D3—64no—
277C22×D3—128no—
278C22×D3—128no—
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—28no—
282C3×D4—44no—
283C3×D4—92no—
284C3×D4—184no—
285C3×D4—181no—
286C3×D4#62618yes—
287C3×D4C4⋊(C3⋊D3)12yes—
288C2×C3×C4—24no—
289C2×C3×C4—64no—
290C2×C3×C4—91no—
291C2×C3×C4—181no—
292C2×C3×C4—181no—
293C2×C3×C4C2×D3214yes—
294C2×C3×C4C22×C3⋊D311yescharacteristic
295#67—616no—
296C2×Dic3—21no—
297C2×Dic3—34no—
298C2×Dic3—184no—
299C2×Dic3—181no—
300C2×Dic3—364no—
301D12—38no—
302D12—64no—
303C4×D3—64no—
304C4×D3—64no—
305Dic6—188no—
306Dic6—184no—
307Dic6—184no—
308Dic6—181no—
309Dic6—181no—
310Dic6—364no—
311C3×C8—64no—
312C3⋊C8—21no—
313C3⋊C8—188no—
314C3⋊C8—184no—
315C3⋊C8—181no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—24no—
319C2×C32—42no—
320C2×C32—48no—
321C2×C32—88no—
322C2×C32#88214yes—
323C2×C32#91114yes—
324C2×C32D3×SD1614yes—
325C2×C32#93211yescharacteristic
326C2×C32C2×#21411yescharacteristic
327C2×C32#95311yescharacteristic
328C3⋊D3—22no—
329C3⋊D3—42no—
330C3×D3—68no—
331C3×D3—128no—
332C24—181no—
333C2×Q8—271no—
334C2×D4—91no—
335C2×D4—181no—
336C2×D4—182no—
337C2×D4—182no—
338C2×D4D3×C3⋊D311yescharacteristic
339C22×C4—91no—
340SD16—542no—
341SD16—542no—
342C2×C8—31no—
343C2×C8—271no—
344C4⋊C4—31no—
345C4⋊C4—541no—
346C22⋊C4—541no—
347C22×C3—22no—
348C22×C3—28no—
349C22×C3—24no—
350C22×C3—48no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3—84no—
356C22×C3—84no—
357C22×C3—92no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3—182no—
361C22×C3—368no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3D3×#6714yes—
365C22×C3D4×C3⋊D311yescharacteristic
366C2×D3—38no—
367C2×D3—64no—
368C2×D3—64no—
369C2×D3—68no—
370C2×D3—1232no—
371C2×D3—124no—
372C2×D3—124no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—124no—
376C3×C4—181no—
377C3×C4C2×#62614yes—
378C3×C4D3×#6714yes—
379C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
380C3×C4D4×C3⋊D311yescharacteristic
381Dic3—41no—
382Dic3—64no—
383Dic3—368no—
384Dic3—361no—
385Dic3—364no—
386C32—24no—
387C32#477614yescomplemented
388C32#521811yescharacteristic, complemented
389C23—21no—
390C23—91no—
391C23—184no—
392C23—181no—
393C23—362no—
394C23—362no—
395Q8—541no—
396Q8—541no—
397D4—21no—
398D4—92no—
399D4—184no—
400D4—181no—
401D4#629212yes—
402C2×C4—61no—
403C2×C4—91no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—541no—
407C2×C4C2×D3×C3⋊D311yescharacteristic
408C8—61no—
409C8—541no—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—42no—
414C2×C3—48no—
415C2×C3—88no—
416C2×C3—182no—
417C2×C3—362no—
418C2×C3D3×#21414yes—
419C2×C3#6393214yes—
420C2×C3#6397714yes—
421C2×C3#6409111yescharacteristic
422C2×C3#6412011yescharacteristic
423C2×C3SD16×C3⋊D311yescharacteristic
424D3—68no—
425D3—128no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—92no—
430C22—181no—
431C22—181no—
432C22—182no—
433C22—368no—
434C22—361no—
435C22—361no—
436C22C3⋊D3×#6711yescharacteristic
437C4—121no—
438C4—181no—
439C4—1081no—
440C4C2×#629211yescharacteristic
441C4C3⋊D3×#6711yescharacteristic
442C3—24no—
443C3#1059145914yescomplemented
444C3#1059247511yescharacteristic, complemented
445C2—42no—
446C2—182no—
447C2—362no—
448C2C3⋊D3×#21411yescharacteristic
449C2#1172356011yescharacteristic
450C2#1172363611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#450
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
2. C2×#88861C2semidirect
3. #11723431C2semidirect
4. C2×#88796C2semidirect
5. C32×#832C2semidirect
6. #11722033C2semidirect
7. #11722023C2semidirect
12. C2×#6308C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C32×C4⋊Dic3C22semidirect
21. C2×C32×C3⋊C8C22semidirect
37. C3×#832D3semidirect
92. C3×C4⋊Dic3C2×D3semidirect
93. C2×C3×C3⋊C8C2×D3semidirect
107. C2×C3×C3⋊D3SD16semidirect
124. #832C3⋊D3semidirect
212. C4⋊Dic3C2×C3⋊D3semidirect
217. C2×C3⋊C8C2×C3⋊D3semidirect
237. C2×C3⋊D3#214semidirect
265. C33#450semidirect
387. C32#4776semidirect
388. C32#5218semidirect
443. C3#10591459semidirect
444. C3#10592475semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#888612
#117234312
C2×#887962
C32×#8322
#117220332
#117220232
#105924753
#105914593