Finite Group Databasebeta

← #86024

Subgroups — #86024

Order 384.

1124 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#525C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic, complemented
3#5827C211yesmaximal, characteristic, complemented
4#5791C211yesmaximal, characteristic, complemented
5C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#5665C214yesmaximal, complemented
7#5642C212yesmaximal, complemented
8#5635C212yesmaximal, complemented
9#5618C212yesmaximal, complemented
10#5611C212yesmaximal, complemented
11#5586C214yesmaximal, complemented
12#5565C212yesmaximal, complemented
13#5555C212yesmaximal, complemented
14#5543C212yesmaximal, complemented
15#5541C212yesmaximal, complemented
16#5523C211yesmaximal, characteristic, complemented
17#5519C211yesmaximal, characteristic, complemented
18#3492—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C23×Dic3C2212yes—
22C2×#237C2214yes—
23C2×#235C2211yescharacteristic
24C3×#123C2212yes—
25C3×C22⋊Q8C2211yescharacteristic
26C3×C22⋊Q8C2211yescharacteristic
27C3×#121C2211yescharacteristic
28C3×#121C2211yescharacteristic
29C3×#120C2212yes—
30C3×C4×D4C2212yes—
31C3×C4×D4C2212yes—
32C3×#117C2211yescharacteristic
33#953C2212yes—
34#946C2212yescomplemented
35Q8×Dic3C2212yescomplemented
36#939C2214yes—
37#939C2214yes—
38#938C2212yescomplemented
39#935C2214yes—
40D4×Dic3C2214yes—
41D4×Dic3C2214yes—
42D4×Dic3C2212yescomplemented
43#930C2211yescharacteristic
44#930C2211yescharacteristic
45#930C2212yescomplemented
46#929C2212yes—
47C4×#67C2212yes—
48C4×#67C2212yes—
49#926C2211yescharacteristic
50#926C2212yescomplemented
51C2×C4⋊Dic3C2212yes—
52C2×C4⋊Dic3C2212yes—
53C2×C4⋊Dic3C2212yes—
54C2×C4⋊Dic3C2211yescharacteristic
55C2×C4⋊Dic3C2211yescharacteristic
56C2×C4⋊Dic3C2212yes—
57C2×C4⋊Dic3C2212yes—
58C2×C4⋊Dic3C2212yescomplemented
59#924C2211yescharacteristic
60#924C2211yescharacteristic
61#924C2212yescomplemented
62#898C2214yescomplemented
63#897C2214yescomplemented
64#892C2212yescomplemented
65D3×C4⋊C4C2212yescomplemented
66#890C2214yescomplemented
67#890C2212yescomplemented
68#889C2214yescomplemented
69#888C2212yescomplemented
70#886C2214yes—
71#886C2214yescomplemented
72#883C2214yescomplemented
73#67⋊C4C2214yes—
74#67⋊C4C2214yes—
75#879C2214yescomplemented
76#878C2214yes—
77#878C2214yes—
78#878C2214yescomplemented
79#877C2214yes—
80C4×D12C2212yescomplemented
81#870C2212yescomplemented
82#869C2212yescomplemented
83C4×Dic6C2212yescomplemented
84#554—32no—
85#554—32no—
86#553—32no—
87#553—32no—
88#552—32no—
89#552—32no—
90#547—32no—
91#547—32no—
92#544—31no—
93#539—34no—
94#537—31no—
95#525—31no—
96#525—34no—
97#525D311yescharacteristic, complemented
98#520—31no—
99#520—31no—
100C22×C4⋊C4—31no—
101C24×C3C2311yescharacteristic
102C3×#41—24no—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2312yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C22×C3×C4—24no—
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2311yescharacteristic
111C22×C3×C4C2312yes—
112C22×C3×C4C2311yescharacteristic
113C22×C3×C4C2311yescharacteristic
114C22×C3×C4C2311yescharacteristic
115C22×C3×C4C2311yescharacteristic
116C2×#67C2312yes—
117C2×#67C2312yes—
118C2×#67C2312yes—
119C2×#67C2314yes—
120C22×Dic3—44no—
121C22×Dic3C2314yes—
122C22×Dic3C2312yes—
123C22×Dic3C2314yes—
124C22×Dic3C2314yes—
125C22×Dic3C2312yes—
126#237—48no—
127#235—24no—
128C2×D12C2311yescharacteristic
129C2×C4×D3C2312yes—
130C2×C4×D3C2314yes—
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2314yes—
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2312yes—
135C3×C4⋊C4C2311yescharacteristic
136C3×C4⋊C4C2312yes—
137C3×C22⋊C4C2312yes—
138C3×C22⋊C4C2312yes—
139C3×C22⋊C4C2312yes—
140C3×C22⋊C4C2312yes—
141C3×C22⋊C4C2312yes—
142C3×C42C2312yes—
143#217C2312yes—
144#217C2312yes—
145#217C2312yes—
146#217C2314yes—
147#217C2314yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2311yescharacteristic
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2314yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2314yes—
158C4⋊Dic3C2314yes—
159C4⋊Dic3C2314yes—
160C4⋊Dic3C2312yes—
161C4⋊Dic3C2312yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166Dic3⋊C4C2312yes—
167Dic3⋊C4C2312yes—
168C4×Dic3C2314yes—
169C4×Dic3C2314yes—
170C4×Dic3C2312yes—
171C2×#41—34no—
172C2×#41—31no—
173C2×#41C2×D311yescharacteristic
174C23×C4—32no—
175C23×C4C2×D311yescharacteristic
176C4⋊Q8—32no—
177C4⋊Q8—32no—
178#126—34no—
179#126—34no—
180#125—34no—
181#125—34no—
182#125—32no—
183#125—32no—
184#123—34no—
185#123—32no—
186#123—34no—
187#123—34no—
188#123—34no—
189#123C2×D312yes—
190C22⋊Q8—32no—
191C22⋊Q8—31no—
192C22⋊Q8—31no—
193C22⋊Q8—32no—
194C22⋊Q8—34no—
195C22⋊Q8—34no—
196C22⋊Q8—34no—
197C22⋊Q8—34no—
198C22⋊Q8C2×D311yescharacteristic
199C22⋊Q8C2×D311yescharacteristic
200#121—34no—
201#121—34no—
202#121—32no—
203#121—31no—
204#121—31no—
205#121—32no—
206#121C2×D311yescharacteristic
207#121C2×D311yescharacteristic
208#120—32no—
209#120C2×D312yes—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×D4—34no—
213C4×D4—32no—
214C4×D4—34no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—34no—
219C4×D4—32no—
220C4×D4C2×D312yes—
221C4×D4C2×D312yes—
222#117—31no—
223#117—32no—
224#117—32no—
225#117—34no—
226#117C2×D311yescharacteristic
227C2×C4⋊C4—31no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—32no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3C2412yes—
240C23×C3C2411yescharacteristic
241C23×C3C2411yescharacteristic
242C23×C3C2411yescharacteristic
243C22×D3C2412yes—
244C3×Q8—22no—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4—24no—
248C3×D4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4#4114yescomplemented
256C2×C3×C4C2412yes—
257C2×C3×C4C2412yes—
258C2×C3×C4C2412yes—
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261C2×C3×C4C2411yescharacteristic
262C2×C3×C4C2411yescharacteristic
263#67—24no—
264#67—24no—
265#67—24no—
266#67—44no—
267C2×Dic3—28no—
268C2×Dic3—44no—
269C2×Dic3—42no—
270C2×Dic3—44no—
271C2×Dic3—42no—
272C2×Dic3C2412yes—
273C2×Dic3C2412yes—
274C2×Dic3C2414yes—
275C2×Dic3C2412yes—
276C2×Dic3C2414yes—
277D12—22no—
278C4×D3—24no—
279C4×D3—44no—
280Dic6—22no—
281Dic6—44no—
282C24C22×D311yescharacteristic
283#41—24no—
284#41—64no—
285#41—128no—
286C2×Q8—31no—
287C2×Q8—34no—
288C2×Q8C22×D311yescharacteristic
289C2×D4—32no—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—31no—
294C2×D4C22×D312yes—
295C2×D4C22×D312yes—
296C2×D4C22×D311yescharacteristic
297C2×D4C22×D312yes—
298C22×C4—24no—
299C22×C4—34no—
300C22×C4—32no—
301C22×C4—34no—
302C22×C4—32no—
303C22×C4—34no—
304C22×C4—32no—
305C22×C4—34no—
306C22×C4—124no—
307C22×C4C22×D311yescharacteristic
308C22×C4C22×D311yescharacteristic
309C22×C4C22×D311yescharacteristic
310C22×C4C22×D311yescharacteristic
311C22×C4C22×D311yescharacteristic
312C22×C4C22×D312yes—
313C22×C4C22×D311yescharacteristic
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—34no—
319C4⋊C4—34no—
320C4⋊C4—31no—
321C4⋊C4—34no—
322C4⋊C4—34no—
323C4⋊C4—31no—
324C4⋊C4—34no—
325C4⋊C4—34no—
326C4⋊C4—32no—
327C4⋊C4—34no—
328C4⋊C4—34no—
329C4⋊C4—32no—
330C4⋊C4C22×D311yescharacteristic
331C4⋊C4C22×D311yescharacteristic
332C4⋊C4C22×D312yes—
333C4⋊C4C22×D312yes—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—34no—
337C22⋊C4—32no—
338C22⋊C4—34no—
339C22⋊C4—34no—
340C22⋊C4—34no—
341C22⋊C4—32no—
342C22⋊C4—32no—
343C22⋊C4C22×D312yes—
344C22⋊C4C22×D312yes—
345C22⋊C4C22×D312yes—
346C22⋊C4C22×D312yes—
347C22⋊C4C22×D312yes—
348C42—34no—
349C42—34no—
350C42—32no—
351C42C22×D312yes—
352C22×C3—28no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3C2×#4112yes—
363C22×C3H(5,𝔽2)12yes—
364C22×C32−1+412yes—
365C22×C3C2511yescharacteristic
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—22no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4C2×#4112yes—
375C3×C4C2×#4112yes—
376Dic3—22no—
377Dic3—22no—
378Dic3—24no—
379Dic3—24no—
380Dic3—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—32no—
385C23C23×D312yes—
386C23C23×D311yescharacteristic
387C23C23×D311yescharacteristic
388C23C23×D311yescharacteristic
389Q8—22no—
390Q8—62no—
391Q8—124no—
392D4—24no—
393D4—24no—
394D4—24no—
395D4—22no—
396D4—64no—
397D4—64no—
398D4—64no—
399D4—62no—
400D4—124no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—34no—
410C2×C4—32no—
411C2×C4—34no—
412C2×C4—68no—
413C2×C4—64no—
414C2×C4—124no—
415C2×C4—122no—
416C2×C4—124no—
417C2×C4—122no—
418C2×C4—124no—
419C2×C4#23714yescomplemented
420C2×C4C23×D312yes—
421C2×C4C23×D312yes—
422C2×C4C23×D312yes—
423C2×C4C23×D311yescharacteristic
424C2×C4C23×D311yescharacteristic
425C2×C4C23×D311yescharacteristic
426C2×C4C23×D311yescharacteristic
427C2×C3—22no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—42no—
431C2×C3C22×#4111yescharacteristic
432C2×C3C2×H(5,𝔽2)11yescharacteristic
433C2×C3C2×2−1+411yescharacteristic
434D3—42no—
435C22—28no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—62no—
446C22—62no—
447C22—62no—
448C22C2×#23712yes—
449C22#100912yes—
450C22#101012yes—
451C22C24×D311yescharacteristic
452C4—21no—
453C4—21no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—62no—
458C4—64no—
459C4—64no—
460C4—62no—
461C4—62no—
462C4C2×#23712yes—
463C4C2×#23712yes—
464C3#349211yescharacteristic, complemented, Sylow
465C2—22no—
466C2—22no—
467C2—22no—
468C2—42no—
469C2—122no—
470C2C22×#23711yescharacteristic
471C2C2×#100911yescharacteristic
472C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3492
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C3×#525C2semidirect
2. #5833C2semidirect
3. #5827C2semidirect
4. #5791C2semidirect
5. C22×C4⋊Dic3C2semidirect
6. #5665C2semidirect
7. #5642C2semidirect
8. #5635C2semidirect
9. #5618C2semidirect
10. #5611C2semidirect
11. #5586C2semidirect
12. #5565C2semidirect
13. #5555C2semidirect
14. #5543C2semidirect
15. #5541C2semidirect
16. #5523C2semidirect
17. #5519C2semidirect
34. #946C22semidirect
35. Q8×Dic3C22semidirect
38. #938C22semidirect
42. D4×Dic3C22semidirect
45. #930C22semidirect
50. #926C22semidirect
58. C2×C4⋊Dic3C22semidirect
61. #924C22semidirect
62. #898C22semidirect
63. #897C22semidirect
64. #892C22semidirect
65. D3×C4⋊C4C22semidirect
66. #890C22semidirect
67. #890C22semidirect
68. #889C22semidirect
69. #888C22semidirect
71. #886C22semidirect
72. #883C22semidirect
75. #879C22semidirect
78. #878C22semidirect
80. C4×D12C22semidirect
81. #870C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
97. #525D3semidirect
255. C2×C3×C4#41semidirect
419. C2×C4#237semidirect
464. C3#3492semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
#58332
#58272
#57912
C22×C4⋊Dic32
#56652
#56422
#56352
#56182
#56112
#55862
#55652
#55552
#55432
#55412
#55232
#55192
#34923