Finite Group Databasebeta

← #80228

Subgroups — #80228

Order 384.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#41C211yesmaximal, characteristic
2C2×#918C211yesmaximal, characteristic, complemented
3C2×#915C211yesmaximal, characteristic, complemented
4D3×C22⋊Q8C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5239C211yesmaximal, characteristic, complemented
7#5186C211yesmaximal, characteristic, complemented
8#5169C211yesmaximal, characteristic, complemented
9#5047C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4803C211yesmaximal, characteristic, complemented
12#4776C211yesmaximal, characteristic, complemented
13#4771C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4726C211yesmaximal, characteristic, complemented
16#3007—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18D3×#41—24no—
19D3×#41—24no—
20C2×#239C2211yescharacteristic
21C2×D3×Q8C2211yescharacteristic
22C2×D3×Q8C2211yescharacteristic, complemented
23C2×#237—21no—
24C2×#237C2211yescharacteristic
25C2×D3×D4—21no—
26C2×D3×D4C2211yescharacteristic
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×C4×D3—21no—
30C22×C4×D3C2211yescharacteristic
31C2×C3×Q16C2211yescharacteristic, complemented
32C2×C3×SD16C2211yescharacteristic, complemented
33C3×C22⋊Q8C2211yescharacteristic, complemented
34#946C2211yescharacteristic
35C2×#216C2211yescharacteristic, complemented
36C2×#216C2211yescharacteristic, complemented
37C2×#215C2211yescharacteristic, complemented
38C2×#214C2211yescharacteristic, complemented
39#924C2211yescharacteristic, complemented
40#918—24no—
41#915—24no—
42C2×Dic12C2211yescharacteristic, complemented
43C2×#204C2211yescharacteristic, complemented
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
46#897C2211yescharacteristic
47#896—21no—
48D3×C4⋊C4—21no—
49D3×C4⋊C4C2211yescharacteristic, complemented
50D3×C22⋊C4—21no—
51#878—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59(C2×D3)⋊C8C2211yescharacteristic
60(C2×D3)⋊C8C2211yescharacteristic
61#816C2211yescharacteristic, complemented
62#810C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C22×#41—31no—
65C2×#137—31no—
66C2×#137—31no—
67C2×C22⋊Q8—31no—
68#452—31no—
69#452—31no—
70#452—31no—
71#452D311yescharacteristic, complemented
72#451—31no—
73#451—31no—
74#450—31no—
75#450—31no—
76#417—31no—
77#417—31no—
78#407—31no—
79C23×D3—21no—
80C23×D3D411yescharacteristic
81C3×#41—22no—
82C3×#41—22no—
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4—21no—
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4C2311yescharacteristic
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C22×Dic3—21no—
93C22×Dic3D411yescharacteristic
94#239—22no—
95#239D414yes—
96D3×Q8—22no—
97D3×Q8—22no—
98D3×Q8—22no—
99D3×Q8—22no—
100#237—28no—
101#237—22no—
102#237D414yes—
103D3×D4—28no—
104D3×D4—22no—
105D3×D4—22no—
106#235—28no—
107#235—22no—
108#235—22no—
109C2×D12—21no—
110C2×D12C2311yescharacteristic
111C2×C4×D3—28no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3D411yescharacteristic
120C2×C4×D3D411yescharacteristic
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2311yescharacteristic
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6C2311yescharacteristic
126C2×Dic6C2311yescharacteristic
127C3×Q16—22no—
128C3×SD16—22no—
129C2×C3×C8C2311yescharacteristic
130C2×C3×C8C2311yescharacteristic
131C3×C4⋊C4—21no—
132C3×C4⋊C4C2311yescharacteristic
133C3×C22⋊C4—21no—
134#217—21no—
135#216—22no—
136#216—22no—
137#215—22no—
138#214—22no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3C2311yescharacteristic
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148Dic12—22no—
149#204—22no—
150ΓL(1,𝔽25)—22no—
151ΓL(1,𝔽25)—22no—
152C2×#41—31no—
153C2×#41—31no—
154C2×#41—31no—
155C2×#41—61no—
156C2×#41—61no—
157C2×#41—64no—
158C2×#41—64no—
159C2×#41C2×D311yescharacteristic
160C22×Q8—31no—
161C22×Q8—31no—
162C22×D4—31no—
163C22×D4—61no—
164C23×C4—31no—
165C23×C4—61no—
166#137—64no—
167#137—64no—
168C2×Q16—31no—
169C2×Q16—31no—
170C2×Q16—31no—
171C2×Q16C2×D311yescharacteristic, complemented
172C2×SD16—31no—
173C2×SD16—31no—
174C2×SD16—31no—
175C2×SD16C2×D311yescharacteristic, complemented
176C2×M16—31no—
177C2×M16—31no—
178C22⋊Q8—31no—
179C22⋊Q8—31no—
180C22⋊Q8—31no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8C2×D311yescharacteristic, complemented
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C22⋊C4—61no—
187#103—31no—
188#103—31no—
189#103—31no—
190#103C2×D311yescharacteristic, complemented
191#102—31no—
192#102—31no—
193#102—31no—
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—21no—
200C23×C3C2×D411yescharacteristic
201C22×D3—24no—
202C22×D3—21no—
203C22×D3—22no—
204C22×D3—22no—
205C22×D3—41no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3C2×D411yescharacteristic
209C22×D3C2×D411yescharacteristic
210C3×Q8—21no—
211C3×Q8—21no—
212C3×Q8—21no—
213C3×Q8C2×D412yes—
214C3×D4—24no—
215C3×D4—21no—
216C3×D4C2×D412yes—
217C2×C3×C4—24no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4C2×D411yescharacteristic
224C2×C3×C4C2411yescharacteristic
225#67—24no—
226#67—24no—
227#67—24no—
228C2×Dic3—21no—
229C2×Dic3—22no—
230C2×Dic3—24no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3C2×D411yescharacteristic
234C2×Dic3C2×D411yescharacteristic
235D12—24no—
236D12—21no—
237D12C2×D412yes—
238C4×D3—24no—
239C4×D3—24no—
240C4×D3—22no—
241C4×D3—21no—
242C4×D3—21no—
243C4×D3—22no—
244C4×D3—24no—
245C4×D3—42no—
246C4×D3—42no—
247C4×D3C2×D412yes—
248C4×D3C2×D412yes—
249Dic6—24no—
250Dic6—21no—
251Dic6—21no—
252Dic6—21no—
253Dic6—42no—
254Dic6C2×D412yes—
255C3×C8—21no—
256C3×C8—21no—
257C3⋊C8—21no—
258C3⋊C8—21no—
259C24—31no—
260C24—61no—
261#41—22no—
262#41—22no—
263#41—34no—
264#41—34no—
265#41—62no—
266#41—62no—
267#41—68no—
268#41—62no—
269#41—62no—
270#41—68no—
271C2×Q8—31no—
272C2×Q8—31no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8C22×D311yescharacteristic
280C2×Q8C22×D311yescharacteristic
281C2×D4—21no—
282C2×D4—31no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—62no—
286C2×D4—68no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—62no—
290C2×D4C22×D311yescharacteristic
291C22×C4—21no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—68no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4C22×D311yescharacteristic
307Q16—22no—
308Q16—62no—
309Q16—62no—
310Q16—62no—
311SD16—22no—
312SD16—62no—
313SD16—62no—
314SD16—62no—
315M16—62no—
316M16—62no—
317C2×C8—31no—
318C2×C8—31no—
319C2×C8C22×D311yescharacteristic
320C2×C8C22×D311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—31no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4C22×D311yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3C22×D411yescharacteristic
339C2×D3—21no—
340C2×D3—21no—
341C2×D3—21no—
342C2×D3—21no—
343C2×D3—22no—
344C2×D3—44no—
345C2×D3—48no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3#12012yes—
349C2×D3#13712yescomplemented
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4C22×D411yescharacteristic
355C3×C4C22×D411yescharacteristic
356Dic3—21no—
357Dic3—22no—
358Dic3—41no—
359Dic3—41no—
360Dic3#12012yes—
361C23—21no—
362C23—31no—
363C23—31no—
364C23—62no—
365C23—62no—
366C23—64no—
367C23—61no—
368C23—121no—
369C23—122no—
370C23—122no—
371C23D3×D411yescharacteristic
372Q8—21no—
373Q8—21no—
374Q8—21no—
375Q8—32no—
376Q8—64no—
377Q8—61no—
378Q8—61no—
379Q8—61no—
380Q8—122no—
381Q8D3×D412yes—
382D4—24no—
383D4—21no—
384D4—32no—
385D4—64no—
386D4—64no—
387D4—64no—
388D4—64no—
389D4—61no—
390D4D3×D412yes—
391C2×C4—24no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—31no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—31no—
401C2×C4—62no—
402C2×C4—64no—
403C2×C4—62no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4D3×D411yescharacteristic
416C2×C4C23×D311yescharacteristic
417C8—21no—
418C8—21no—
419C8—61no—
420C8—61no—
421C2×C3—41no—
422C2×C3—42no—
423C2×C3C2×#12011yescharacteristic
424C2×C3C2×#13711yescharacteristic
425C2×C3#57511yescharacteristic
426D3—22no—
427D3—22no—
428D3—42no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—32no—
434C22—32no—
435C22—41no—
436C22—41no—
437C22—61no—
438C22—61no—
439C22—61no—
440C22—61no—
441C22—62no—
442C22—124no—
443C22—128no—
444C22—121no—
445C22—121no—
446C22C2×D3×D411yescharacteristic
447C4—21no—
448C4—22no—
449C4—32no—
450C4—41no—
451C4—41no—
452C4—61no—
453C4—62no—
454C4—121no—
455C4—121no—
456C4C2×D3×D411yescharacteristic
457C4C2×D3×D411yescharacteristic
458C3#300711yescharacteristic, complemented, Sylow
459C2—41no—
460C2—42no—
461C2—62no—
462C2—62no—
463C2—122no—
464C2D3×#12011yescharacteristic
465C2#576811yescharacteristic
466C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radicalC2×D3×#41C2
hypercenter#452D3
hypocenterC3#3007
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#918C2semidirect
3. C2×#915C2semidirect
4. D3×C22⋊Q8C2semidirect
5. C3×#452C2semidirect
6. #5239C2semidirect
7. #5186C2semidirect
8. #5169C2semidirect
9. #5047C2semidirect
10. #4809C2semidirect
11. #4803C2semidirect
12. #4776C2semidirect
13. #4771C2semidirect
14. #4733C2semidirect
15. #4726C2semidirect
22. C2×D3×Q8C22semidirect
31. C2×C3×Q16C22semidirect
32. C2×C3×SD16C22semidirect
33. C3×C22⋊Q8C22semidirect
35. C2×#216C22semidirect
36. C2×#216C22semidirect
37. C2×#215C22semidirect
38. C2×#214C22semidirect
39. #924C22semidirect
42. C2×Dic12C22semidirect
43. C2×#204C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
45. C2×ΓL(1,𝔽25)C22semidirect
49. D3×C4⋊C4C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
61. #816C22semidirect
62. #810C22semidirect
63. #809C22semidirect
71. #452D3semidirect
171. C2×Q16C2×D3semidirect
175. C2×SD16C2×D3semidirect
183. C22⋊Q8C2×D3semidirect
190. #103C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
349. C2×D3#137semidirect
458. C3#3007semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#412
C2×#9182
C2×#9152
D3×C22⋊Q82
C3×#4522
#52392
#51862
#51692
#50472
#48092
#48032
#47762
#47712
#47332
#47262
#30073