Finite Group Databasebeta

← #79496

Subgroups — #79496

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#908C211yesmaximal, characteristic, complemented
2C2×#904C211yesmaximal, characteristic, complemented
3#5551C211yesmaximal, characteristic, complemented
4#5550C211yesmaximal, characteristic, complemented
5C3×#435C211yesmaximal, characteristic, complemented
6#5137C211yesmaximal, characteristic, complemented
7#5114C211yesmaximal, characteristic, complemented
8#4844C211yesmaximal, characteristic, complemented
9#4841C211yesmaximal, characteristic, complemented
10#4734C211yesmaximal, characteristic, complemented
11#4733C211yesmaximal, characteristic, complemented
12#4733C211yesmaximal, characteristic, complemented
13#4731C211yesmaximal, characteristic, complemented
14#4715C211yesmaximal, characteristic, complemented
15#4711C211yesmaximal, characteristic, complemented
16#3267—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×#235C2211yescharacteristic, complemented
20C22×D12C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic, complemented
23C3×C4×D4C2211yescharacteristic
24#938—21no—
25#937—21no—
26#930C2211yescharacteristic
27#924C2211yescharacteristic
28#924C2211yescharacteristic
29#924C2211yescharacteristic, complemented
30#908—24no—
31C2×Dic12C2211yescharacteristic, complemented
32C2×Dic12C2211yescharacteristic, complemented
33#904—24no—
34C2×D24C2211yescharacteristic
35C2×D24C2211yescharacteristic
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×#204C2211yescharacteristic
39C2×#204C2211yescharacteristic
40C2×#204C2211yescharacteristic, complemented
41#894—21no—
42D3×C4⋊C4—21no—
43#884—21no—
44#883—21no—
45#883—21no—
46D3×C22⋊C4—21no—
47#875C2211yescharacteristic
48C4×D12C2211yescharacteristic
49C4×D12C2211yescharacteristic, complemented
50#869C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×C22⋊C8C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic, complemented
54C3×#97C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57#821C2211yescharacteristic, complemented
58#817C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#816C2211yescharacteristic, complemented
61#816C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#547—31no—
65#546—31no—
66#497—31no—
67#470—31no—
68#465—31no—
69#463—31no—
70#461—31no—
71#452—31no—
72#452—31no—
73#450—31no—
74#449—31no—
75#442—31no—
76#435D311yescharacteristic, complemented
77C23×D3—21no—
78C2×C3×D4D411yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85#235—22no—
86#235—22no—
87#235—22no—
88#235—41no—
89#235—41no—
90#235—41no—
91C2×D12—22no—
92C2×D12—21no—
93C2×D12—41no—
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×M16—22no—
106C2×C3×C8—22no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C3×C4⋊C4D411yescharacteristic
112C3×C22⋊C4D411yescharacteristic
113C3×C22⋊C4D411yescharacteristic
114C3×C42C2311yescharacteristic
115#217—21no—
116#217—21no—
117#217—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3C2311yescharacteristic
126C4⋊Dic3C2311yescharacteristic
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic12—22no—
131Dic12—41no—
132D24—22no—
133D24D414yescomplemented
134#204—22no—
135#204—22no—
136#204—22no—
137#204—22no—
138#204—41no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16C2×D311yescharacteristic
155C22×C8C2×D311yescharacteristic, complemented
156C4⋊Q8—31no—
157#124—31no—
158#123—61no—
159#123—61no—
160#123—61no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4C2×D311yescharacteristic
171C2×C4⋊C4—61no—
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8C2×D311yescharacteristic
182C22⋊C8C2×D311yescharacteristic, complemented
183#97C2×D311yescharacteristic, complemented
184C23×C3C2×D411yescharacteristic
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—81no—
192C3×D4—41no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—42no—
203#67—42no—
204#67—42no—
205#67—81no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—42no—
216D12C2×D412yes—
217D12C2×D412yes—
218C4×D3—42no—
219C4×D3—42no—
220C4×D3—42no—
221C4×D3—42no—
222C4×D3—81no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—21no—
230C3×C8—21no—
231C3×C8—21no—
232C3×C8C2×D412yes—
233C24—61no—
234#41—62no—
235#41—62no—
236#41—62no—
237#41—121no—
238#41—121no—
239#41—121no—
240C2×Q8—31no—
241C2×Q8—31no—
242C2×Q8—31no—
243C2×D4—31no—
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—62no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—121no—
253C2×D4D1211yescharacteristic
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4C22×D311yescharacteristic
260C22×C4C22×D311yescharacteristic
261Q16—62no—
262Q16—121no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268D8—34no—
269D8—62no—
270M16—22no—
271C2×C8—22no—
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4D1211yescharacteristic
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4D1211yescharacteristic
293C22⋊C4D1211yescharacteristic
294C42C22×D311yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3C22×D411yescharacteristic
302C2×D3—22no—
303C2×D3—22no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—84no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4C22×D411yescharacteristic
317C3×C4H(5,𝔽2)11yescharacteristic
318Dic3—41no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—122no—
327C23—122no—
328C23—241no—
329C23C2×D1211yescharacteristic
330C23C2×D1211yescharacteristic
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337D4—32no—
338D4—32no—
339D4—41no—
340D4—61no—
341D4—61no—
342D4—61no—
343D4—61no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—122no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—241no—
363C2×C4C2×D1211yescharacteristic
364C2×C4C2×D1211yescharacteristic
365C2×C4C2×D1211yescharacteristic
366C2×C4C2×D1211yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—21no—
370C8—21no—
371C8D3×D412yes—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3D4211yescharacteristic
375C2×C3#57511yescharacteristic
376C2×C3#57711yescharacteristic
377D3—42no—
378D3—42no—
379D3—81no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—62no—
387C22—62no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C22×D1211yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—121no—
402C4—121no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100911yescharacteristic
408C3#326711yescharacteristic, complemented, Sylow
409C2—41no—
410C2—41no—
411C2—122no—
412C2—122no—
413C2—241no—
414C2D4×D1211yescharacteristic
415C2#574811yescharacteristic
416C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3267
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#908C2semidirect
2. C2×#904C2semidirect
3. #5551C2semidirect
4. #5550C2semidirect
5. C3×#435C2semidirect
6. #5137C2semidirect
7. #5114C2semidirect
8. #4844C2semidirect
9. #4841C2semidirect
10. #4734C2semidirect
11. #4733C2semidirect
12. #4733C2semidirect
13. #4731C2semidirect
14. #4715C2semidirect
15. #4711C2semidirect
19. C2×#235C22semidirect
22. C22×C3×C8C22semidirect
29. #924C22semidirect
31. C2×Dic12C22semidirect
32. C2×Dic12C22semidirect
40. C2×#204C22semidirect
49. C4×D12C22semidirect
50. #869C22semidirect
51. C3×C4⋊C8C22semidirect
53. C3×C22⋊C8C22semidirect
54. C3×#97C22semidirect
57. #821C22semidirect
58. #817C22semidirect
59. #816C22semidirect
60. #816C22semidirect
61. #816C22semidirect
76. #435D3semidirect
133. D24D4semidirect
155. C22×C8C2×D3semidirect
174. C4⋊C8C2×D3semidirect
182. C22⋊C8C2×D3semidirect
183. #97C2×D3semidirect
408. C3#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
C2×#9042
#55512
#55502
C3×#4352
#51372
#51142
#48442
#48412
#47342
#47332
#47312
#47152
#47112
#32673