Finite Group Databasebeta

← #85990

Subgroups — #85990

Order 384.

1134 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2#5817C212yesmaximal, complemented
3#5791C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5#5647C214yesmaximal, complemented
6#5644C212yesmaximal, complemented
7#5643C212yesmaximal, complemented
8#5635C212yesmaximal, complemented
9#5634C212yesmaximal, complemented
10#5630C212yesmaximal, complemented
11#5629C212yesmaximal, complemented
12#5515C212yesmaximal, complemented
13#5510C212yesmaximal, complemented
14#5507C212yesmaximal, complemented
15#5490C214yesmaximal, complemented
16#3536—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic, complemented
18C23×C3×C4C2211yescharacteristic
19C22×#67C2212yes—
20C2×#239C2212yescomplemented
21C2×#235C2214yescomplemented
22C2×#235C2211yescharacteristic, complemented
23C22×D12C2211yescharacteristic
24C3×C22⋊Q8C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C22⋊Q8C2212yescomplemented
27C3×C22⋊Q8C2212yescomplemented
28C2×C3×C4⋊C4C2211yescharacteristic, complemented
29C2×C3×C4⋊C4C2212yescomplemented
30C2×C3×C22⋊C4C2212yes—
31#953C2212yes—
32#947C2212yescomplemented
33#947C2214yescomplemented
34#946C2214yescomplemented
35#946C2212yescomplemented
36Q8×Dic3C2212yescomplemented
37#930C2212yes—
38#930C2211yescharacteristic
39#930C2212yes—
40#930C2212yes—
41#930C2212yes—
42#930C2214yescomplemented
43#930C2211yescharacteristic, complemented
44#929C2212yescomplemented
45#929C2214yescomplemented
46C4×#67C2212yes—
47C4×#67C2214yescomplemented
48C4×#67C2212yescomplemented
49C2×(C2×D3)⋊C4C2212yes—
50#926C2211yescharacteristic, complemented
51C2×C4⋊Dic3C2211yescharacteristic
52#924C2211yescharacteristic
53#924C2211yescharacteristic, complemented
54#898C2214yescomplemented
55#897C2214yescomplemented
56#897C2214yescomplemented
57#896C2214yescomplemented
58#896C2214yescomplemented
59#895C2214yescomplemented
60#895C2214yescomplemented
61#895C2212yescomplemented
62#894C2214yescomplemented
63#894C2214yescomplemented
64#893C2212yescomplemented
65D3×C4⋊C4C2212yescomplemented
66#889C2214yescomplemented
67#888C2212yescomplemented
68#885C2214yescomplemented
69#884C2214yescomplemented
70#884C2214yescomplemented
71#883C2214yescomplemented
72#882C2214yes—
73#67⋊C4C2214yes—
74D3×C22⋊C4C2214yes—
75#878C2214yes—
76#562—34no—
77#554—32no—
78#548—32no—
79#547—32no—
80#546—32no—
81#546—34no—
82#543—32no—
83#542—32no—
84#539—32no—
85#538—32no—
86#537—32no—
87#537—32no—
88#525—31no—
89C2×C22⋊Q8D311yescharacteristic, complemented
90C2×#121—31no—
91C24×C3C2311yescharacteristic
92C23×D3C2312yes—
93C2×C3×Q8—42no—
94C2×C3×Q8C2312yes—
95C2×C3×Q8C2312yes—
96C22×C3×C4—42no—
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2312yes—
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2311yescharacteristic
103C22×C3×C4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C2×#67—44no—
106C2×#67C2312yes—
107C2×#67C2314yes—
108C2×#67C2312yes—
109C2×#67C2314yes—
110C2×#67C2312yes—
111C22×Dic3C2312yes—
112#239—44no—
113#235—42no—
114#235—48no—
115C2×D12—42no—
116C2×D12C2312yes—
117C2×D12C2314yes—
118C2×D12C2314yes—
119C2×D12C2312yes—
120C2×D12C2311yescharacteristic
121C2×C4×D3C2312yes—
122C2×C4×D3C2314yes—
123C2×C4×D3C2314yes—
124C2×C4×D3C2314yes—
125C2×Dic6C2314yes—
126C2×Dic6C2311yescharacteristic
127C3×C4⋊C4C2312yes—
128C3×C4⋊C4C2314yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2314yes—
131C3×C22⋊C4C2314yes—
132C3×C22⋊C4C2314yes—
133#217C2312yes—
134#217C2312yes—
135#217C2312yes—
136(C2×D3)⋊C4C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145C4⋊Dic3C2314yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155C4×Dic3C2314yes—
156C4×Dic3C2312yes—
157C2×#41—32no—
158C2×#41—31no—
159C2×#41—34no—
160C22×Q8C2×D311yescharacteristic, complemented
161C22×D4—32no—
162C22×D4—31no—
163C23×C4C2×D311yescharacteristic
164C4⋊Q8—32no—
165#126—34no—
166#125—34no—
167#124—34no—
168#124—32no—
169#124—34no—
170#123—32no—
171#123—34no—
172#123—34no—
173#123—34no—
174#123—34no—
175C22⋊Q8—32no—
176C22⋊Q8—34no—
177C22⋊Q8—31no—
178C22⋊Q8—31no—
179C22⋊Q8—34no—
180C22⋊Q8—34no—
181C22⋊Q8—34no—
182C22⋊Q8—34no—
183C22⋊Q8—34no—
184C22⋊Q8C2×D312yes—
185C22⋊Q8C2×D312yes—
186C22⋊Q8C2×D312yescomplemented
187C22⋊Q8C2×D312yescomplemented
188#121—32no—
189#121—32no—
190#121—32no—
191#121—32no—
192#121—31no—
193#121—31no—
194#121—34no—
195#121—34no—
196#121—32no—
197#121—34no—
198#121—34no—
199#121—34no—
200#120—32no—
201#120—34no—
202C4×Q8—32no—
203C4×D4—32no—
204C4×D4—32no—
205C4×D4—34no—
206C4×D4—32no—
207C4×D4—34no—
208#117—31no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4C2×D312yescomplemented
212C2×C4⋊C4C2×D311yescharacteristic, complemented
213C2×C22⋊C4—32no—
214C2×C22⋊C4—34no—
215C2×C22⋊C4C2×D312yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3C2411yescharacteristic
220C23×C3C2411yescharacteristic
221C23×C3C2411yescharacteristic
222C22×D3—24no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3C2414yes—
226C22×D3C2412yes—
227C3×Q8—24no—
228C3×Q8—42no—
229C2×C3×C4—24no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4C2414yes—
238C2×C3×C4C2411yescharacteristic
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2414yes—
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2411yescharacteristic
243#67—24no—
244#67—28no—
245#67—24no—
246#67—42no—
247#67—44no—
248C2×Dic3—42no—
249C2×Dic3C2412yes—
250C2×Dic3C2414yes—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253D12—24no—
254D12—42no—
255D12—44no—
256D12—41no—
257D12—44no—
258C4×D3—24no—
259C4×D3—44no—
260C4×D3—44no—
261C4×D3—44no—
262Dic6—44no—
263Dic6—41no—
264C24—32no—
265C24C22×D311yescharacteristic
266#41—122no—
267#41—128no—
268#41—124no—
269C2×Q8—31no—
270C2×Q8—34no—
271C2×Q8—42no—
272C2×Q8C22×D312yes—
273C2×Q8C22×D312yes—
274C2×D4—32no—
275C2×D4—32no—
276C2×D4—34no—
277C2×D4—32no—
278C2×D4—34no—
279C2×D4—32no—
280C2×D4—34no—
281C2×D4—34no—
282C2×D4—31no—
283C2×D4—32no—
284C2×D4—122no—
285C2×D4—124no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—34no—
289C22×C4—34no—
290C22×C4—34no—
291C22×C4—42no—
292C22×C4C22×D312yes—
293C22×C4C22×D311yescharacteristic
294C22×C4C22×D311yescharacteristic
295C22×C4C22×D311yescharacteristic
296C22×C4C22×D311yescharacteristic
297C22×C4C22×D312yes—
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C4⋊C4—32no—
301C4⋊C4—32no—
302C4⋊C4—34no—
303C4⋊C4—31no—
304C4⋊C4—31no—
305C4⋊C4—34no—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—34no—
309C4⋊C4—34no—
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D314yes—
312C4⋊C4C22×D314yes—
313C4⋊C4C22×D312yes—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4C22×D314yes—
327C22⋊C4C22×D314yes—
328C42—32no—
329C42—34no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—44no—
338C22×C3H(5,𝔽2)12yes—
339C22×C32−1+412yes—
340C22×C3C2511yescharacteristic
341C2×D3—22no—
342C2×D3—22no—
343C2×D3—24no—
344C2×D3—24no—
345C2×D3—22no—
346C2×D3—24no—
347C2×D3—84no—
348C3×C4—24no—
349C3×C4—21no—
350C3×C4—24no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4H(5,𝔽2)12yes—
354Dic3—22no—
355Dic3—22no—
356Dic3—24no—
357Dic3—22no—
358C23—22no—
359C23—22no—
360C23—34no—
361C23—32no—
362C23—41no—
363C23—64no—
364C23—122no—
365C23—122no—
366C23C23×D311yescharacteristic
367C23C23×D311yescharacteristic
368C23C23×D311yescharacteristic
369Q8—24no—
370Q8—42no—
371Q8—121no—
372Q8—124no—
373D4—64no—
374D4—68no—
375D4—64no—
376D4—64no—
377D4—122no—
378D4—121no—
379D4—124no—
380D4—124no—
381D4—124no—
382D4—122no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—34no—
390C2×C4—32no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—64no—
396C2×C4—124no—
397C2×C4—124no—
398C2×C4—124no—
399C2×C4—122no—
400C2×C4C23×D314yes—
401C2×C4C23×D311yescharacteristic
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D314yes—
404C2×C4C23×D311yescharacteristic
405C2×C4C23×D311yescharacteristic
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3C2×H(5,𝔽2)11yescharacteristic
410C2×C3C2×H(5,𝔽2)11yescharacteristic
411C2×C3C2×2−1+411yescharacteristic
412D3—42no—
413D3—44no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—62no—
423C22—64no—
424C22—62no—
425C22—64no—
426C22—62no—
427C22—64no—
428C22—244no—
429C22#100712yes—
430C22#100912yes—
431C22C24×D311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—24no—
436C4—24no—
437C4—62no—
438C4—62no—
439C4—64no—
440C4—62no—
441C4#100412yes—
442C3#353611yescharacteristic, complemented, Sylow
443C2—22no—
444C2—22no—
445C2—41no—
446C2—124no—
447C2—122no—
448C2C2×#100411yescharacteristic
449C2C2×#100711yescharacteristic
450C2C2×#100911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×C22⋊Q8C2
hypercenterC2×C22⋊Q8D3
hypocenterC3#3536
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
2. #5817C2semidirect
3. #5791C2semidirect
4. C2×#930C2semidirect
5. #5647C2semidirect
6. #5644C2semidirect
7. #5643C2semidirect
8. #5635C2semidirect
9. #5634C2semidirect
10. #5630C2semidirect
11. #5629C2semidirect
12. #5515C2semidirect
13. #5510C2semidirect
14. #5507C2semidirect
15. #5490C2semidirect
17. C22×C3×Q8C22semidirect
20. C2×#239C22semidirect
21. C2×#235C22semidirect
22. C2×#235C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C2×C3×C4⋊C4C22semidirect
29. C2×C3×C4⋊C4C22semidirect
32. #947C22semidirect
33. #947C22semidirect
34. #946C22semidirect
35. #946C22semidirect
36. Q8×Dic3C22semidirect
42. #930C22semidirect
43. #930C22semidirect
44. #929C22semidirect
45. #929C22semidirect
47. C4×#67C22semidirect
48. C4×#67C22semidirect
50. #926C22semidirect
53. #924C22semidirect
54. #898C22semidirect
55. #897C22semidirect
56. #897C22semidirect
57. #896C22semidirect
58. #896C22semidirect
59. #895C22semidirect
60. #895C22semidirect
61. #895C22semidirect
62. #894C22semidirect
63. #894C22semidirect
64. #893C22semidirect
65. D3×C4⋊C4C22semidirect
66. #889C22semidirect
67. #888C22semidirect
68. #885C22semidirect
69. #884C22semidirect
70. #884C22semidirect
71. #883C22semidirect
89. C2×C22⋊Q8D3semidirect
160. C22×Q8C2×D3semidirect
186. C22⋊Q8C2×D3semidirect
187. C22⋊Q8C2×D3semidirect
211. C2×C4⋊C4C2×D3semidirect
212. C2×C4⋊C4C2×D3semidirect
442. C3#3536semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
#58172
#57912
C2×#9302
#56472
#56442
#56432
#56352
#56342
#56302
#56292
#55152
#55102
#55072
#54902
#35363