Finite Group Databasebeta

← #80335

Subgroups — #80335

Order 384.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#912C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5715C211yesmaximal, characteristic, complemented
4#5546C211yesmaximal, characteristic, complemented
5C3×#459C211yesmaximal, characteristic, complemented
6#5153C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic, complemented
8#5075C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4771C211yesmaximal, characteristic, complemented
13D3×#102C211yesmaximal, characteristic, complemented
14#4767C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3045—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic, complemented
22C3×#127C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic, complemented
24#940C2211yescharacteristic
25#939—21no—
26#939—21no—
27#937—21no—
28#937—21no—
29#936C2211yescharacteristic
30D4×Dic3C2211yescharacteristic, complemented
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic, complemented
33C2×#214C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic
35C4×#67—21no—
36C2×Dic3⋊C4—21no—
37#912—24no—
38#911—24no—
39C2×Dic12C2211yescharacteristic, complemented
40C2×#204C2211yescharacteristic
41C2×ΓL(1,𝔽25)C2211yescharacteristic
42C2×D3×C8C2211yescharacteristic
43D3×C4⋊C4C2211yescharacteristic
44#887C2211yescharacteristic, complemented
45#67⋊C4—21no—
46#877—21no—
47#875C2211yescharacteristic
48#872C2211yescharacteristic
49C4×Dic6C2211yescharacteristic, complemented
50C3×C4⋊C8C2211yescharacteristic, complemented
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic, complemented
53#832C2211yescharacteristic
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#816C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#535—31no—
65#520—31no—
66#464—31no—
67#464—31no—
68#463—31no—
69#463—31no—
70#460—31no—
71#460—31no—
72#459—31no—
73#459D311yescharacteristic, complemented
74#424—31no—
75#417—31no—
76C2×#102—31no—
77C23×D3—21no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C22×Dic3—21no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92#237—22no—
93#237—22no—
94#237—22no—
95#237D414yescomplemented
96D3×D4—22no—
97D3×D4—22no—
98C2×D12C2311yescharacteristic
99C2×C4×D3—21no—
100C2×C4×D3C2311yescharacteristic
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C3×D8—22no—
104C3×D8—22no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C42C2311yescharacteristic
110#217—21no—
111#217—21no—
112#217—21no—
113#214—22no—
114#214—22no—
115#214—22no—
116#213—22no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4D411yescharacteristic
120(C2×D3)⋊C4D411yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125Dic3⋊C4D411yescharacteristic
126Dic3⋊C4D411yescharacteristic
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132Dic12—22no—
133#204—22no—
134ΓL(1,𝔽25)—22no—
135D3×C8—22no—
136C2×#41—31no—
137C2×#41—31no—
138C22×D4—31no—
139AGL(1,ℤ/8)—64no—
140#135—64no—
141C2×Q16—31no—
142C2×SD16—31no—
143C2×SD16—31no—
144C2×SD16—31no—
145C2×SD16—31no—
146C2×D8—31no—
147C2×D8C2×D311yescharacteristic
148C2×D8C2×D311yescharacteristic, complemented
149C2×M16—31no—
150C22×C8—31no—
151#127—31no—
152#127C2×D311yescharacteristic
153#124—31no—
154#124—31no—
155#121—61no—
156#121—61no—
157#120—61no—
158#120—61no—
159C4×Q8—31no—
160C4×Q8—31no—
161C4×D4—31no—
162C4×D4—61no—
163C4×D4—61no—
164C4×D4C2×D311yescharacteristic, complemented
165#117—31no—
166#117—61no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8C2×D311yescharacteristic, complemented
173#103—31no—
174#103—31no—
175#102—31no—
176#102—31no—
177#102—31no—
178#102—31no—
179#102C2×D311yescharacteristic
180#102C2×D311yescharacteristic, complemented
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C22×D3—21no—
185C22×D3—42no—
186C22×D3—81no—
187C22×D3—81no—
188C22×D3C2×D411yescharacteristic
189C3×D4—21no—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—21no—
194C3×D4—81no—
195C3×D4—81no—
196C3×D4C2×D412yes—
197C2×C3×C4—21no—
198C2×C3×C4—81no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—24no—
203#67—42no—
204#67—42no—
205#67—81no—
206#67—81no—
207C2×Dic3—21no—
208C2×Dic3—24no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—42no—
212C2×Dic3—81no—
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216D12—21no—
217D12—21no—
218C4×D3—81no—
219C4×D3C2×D412yes—
220C4×D3#4112yes—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6C2×D412yes—
225C3×C8—21no—
226C3×C8—21no—
227C3⋊C8—21no—
228C3⋊C8—21no—
229C24—61no—
230#41—34no—
231#41—62no—
232#41—62no—
233#41—62no—
234C2×Q8—31no—
235C2×Q8—31no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—31no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4C22×D311yescharacteristic
247C2×D4C22×D311yescharacteristic
248C2×D4C22×D311yescharacteristic
249C22×C4—21no—
250C22×C4—31no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255Q16—62no—
256SD16—62no—
257SD16—62no—
258SD16—62no—
259SD16—62no—
260D8—22no—
261D8—22no—
262D8—62no—
263M16—62no—
264C2×C8—31no—
265C2×C8—31no—
266C2×C8—62no—
267C2×C8C22×D311yescharacteristic
268C2×C8C22×D311yescharacteristic
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4C22×D311yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—31no—
278C22⋊C4—31no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C42—31no—
285C42—61no—
286C42—61no—
287C42C22×D311yescharacteristic
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3C22×D411yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—84no—
304C2×D3#12112yes—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—41no—
308C3×C4C22×D411yescharacteristic
309C3×C4C2×#4111yescharacteristic
310Dic3—22no—
311Dic3—21no—
312Dic3—21no—
313Dic3—41no—
314Dic3—41no—
315Dic3#12112yes—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—31no—
320C23—61no—
321C23—122no—
322C23—241no—
323C23—241no—
324Q8—32no—
325Q8—61no—
326Q8—61no—
327Q8—61no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—64no—
334D4—61no—
335D4—61no—
336D4—81no—
337D4—81no—
338D4—122no—
339D4—122no—
340D4—241no—
341D4—241no—
342D4D3×D412yes—
343C2×C4—21no—
344C2×C4—31no—
345C2×C4—32no—
346C2×C4—32no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—64no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—81no—
354C2×C4—122no—
355C2×C4—241no—
356C2×C4—241no—
357C2×C4D3×D411yescharacteristic
358C2×C4D3×D411yescharacteristic
359C2×C4C23×D311yescharacteristic
360C8—21no—
361C8—21no—
362C8—61no—
363C8—61no—
364C2×C3—42no—
365C2×C3—81no—
366C2×C3—81no—
367C2×C3C2×#12111yescharacteristic
368C2×C3#57611yescharacteristic
369C2×C3#57711yescharacteristic
370D3—22no—
371D3—81no—
372C22—22no—
373C22—32no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—61no—
383C22—61no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—244no—
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—21no—
391C4—32no—
392C4—41no—
393C4—61no—
394C4—61no—
395C4—62no—
396C4—121no—
397C4—121no—
398C4C2×D3×D411yescharacteristic
399C4D3×#4111yescharacteristic
400C3#304511yescharacteristic, complemented, Sylow
401C2—42no—
402C2—62no—
403C2—81no—
404C2—81no—
405C2—241no—
406C2D3×#12111yescharacteristic
407C2#575711yescharacteristic
408C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3045
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#912C2semidirect
2. C2×#911C2semidirect
3. #5715C2semidirect
4. #5546C2semidirect
5. C3×#459C2semidirect
6. #5153C2semidirect
7. #5150C2semidirect
8. #5075C2semidirect
9. #5016C2semidirect
10. #4845C2semidirect
11. #4839C2semidirect
12. #4771C2semidirect
13. D3×#102C2semidirect
14. #4767C2semidirect
15. #4762C2semidirect
21. C2×C3×D8C22semidirect
23. C3×C4×D4C22semidirect
30. D4×Dic3C22semidirect
32. C2×#214C22semidirect
33. C2×#214C22semidirect
39. C2×Dic12C22semidirect
44. #887C22semidirect
49. C4×Dic6C22semidirect
50. C3×C4⋊C8C22semidirect
52. C3×#102C22semidirect
54. #832C22semidirect
56. #816C22semidirect
57. Dic3⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. #810C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
73. #459D3semidirect
95. #237D4semidirect
148. C2×D8C2×D3semidirect
164. C4×D4C2×D3semidirect
172. C4⋊C8C2×D3semidirect
180. #102C2×D3semidirect
400. C3#3045semidirect

Maximal subgroups

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

subgroupindex
C2×#9122
C2×#9112
#57152
#55462
C3×#4592
#51532
#51502
#50752
#50162
#48452
#48392
#47712
D3×#1022
#47672
#47622
#30453