Finite Group Databasebeta

← #80728

Subgroups — #80728

Order 384.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic, complemented
2C2×#904C211yesmaximal, characteristic, complemented
3#5612C211yesmaximal, characteristic, complemented
4#5599C211yesmaximal, characteristic, complemented
5C3×#466C211yesmaximal, characteristic, complemented
6#5154C211yesmaximal, characteristic, complemented
7#5111C211yesmaximal, characteristic, complemented
8#5037C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4889C211yesmaximal, characteristic, complemented
11#4886C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4783C211yesmaximal, characteristic, complemented
14#4773C211yesmaximal, characteristic, complemented
15#4767C211yesmaximal, characteristic, complemented
16#3268—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×#235C2211yescharacteristic, complemented
21C2×C3×D8C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic, complemented
25#940—21no—
26#940C2211yescharacteristic, complemented
27#939—21no—
28#938—21no—
29#937—21no—
30#936C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic, complemented
33C2×#213C2211yescharacteristic
34C2×#213C2211yescharacteristic
35C4×#67C2211yescharacteristic
36#911—24no—
37C2×Dic12C2211yescharacteristic, complemented
38#904—24no—
39C2×D24C2211yescharacteristic
40C2×#204C2211yescharacteristic
41C2×#204C2211yescharacteristic, complemented
42C2×ΓL(1,𝔽25)C2211yescharacteristic
43#897C2211yescharacteristic
44#895C2211yescharacteristic
45#893C2211yescharacteristic
46#887C2211yescharacteristic, complemented
47#884—21no—
48#883—21no—
49#67⋊C4—21no—
50D3×C22⋊C4—21no—
51C3×#107C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic, complemented
54#830C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic
56#815C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic
61#809C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#550—31no—
65#546—31no—
66#496—31no—
67#468—31no—
68#466D311yescharacteristic, complemented
69#464—31no—
70#459—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67D411yescharacteristic
90C22×Dic3—21no—
91#237—22no—
92#237—41no—
93D3×D4—22no—
94D3×D4—41no—
95#235—22no—
96#235—22no—
97#235—41no—
98#235—41no—
99C2×D12—21no—
100C2×D12C2311yescharacteristic
101C2×D12C2311yescharacteristic
102C2×C4×D3—21no—
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C2×Dic6C2311yescharacteristic
107C2×Dic6C2311yescharacteristic
108C3×D8—22no—
109C2×C3×C8—22no—
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116#217—21no—
117#217—21no—
118#217D411yescharacteristic
119#214—22no—
120#214—41no—
121#213—22no—
122#213—41no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4D411yescharacteristic
127C4⋊Dic3—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4D411yescharacteristic
130C4×Dic3—21no—
131C4×Dic3C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134Dic12—22no—
135D24—22no—
136#204—22no—
137#204D414yescomplemented
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8C2×D311yescharacteristic, complemented
156#127—31no—
157#127—61no—
158#124—31no—
159#123—61no—
160C22⋊Q8—31no—
161#121—31no—
162#121—61no—
163#121—61no—
164#121—61no—
165#121C2×D311yescharacteristic
166#121C2×D311yescharacteristic, complemented
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#107C2×D311yescharacteristic, complemented
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102C2×D311yescharacteristic
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—81no—
192C22×D3C2×D411yescharacteristic
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—81no—
198C3×D4—81no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2411yescharacteristic
205#67—41no—
206#67—42no—
207#67—42no—
208#67—42no—
209#67—42no—
210#67—42no—
211#67—81no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—42no—
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218D12—21no—
219D12—21no—
220D12—21no—
221D12—81no—
222D12C2×D412yes—
223C4×D3—21no—
224C4×D3—21no—
225C4×D3—42no—
226C4×D3—42no—
227C4×D3—81no—
228Dic6—21no—
229Dic6—21no—
230Dic6—21no—
231Dic6C2×D412yes—
232C3×C8—21no—
233C3×C8C2×D412yes—
234C3⋊C8—21no—
235C3⋊C8—21no—
236C24—61no—
237#41—62no—
238#41—62no—
239#41—62no—
240#41—121no—
241#41—121no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8—31no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—62no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—121no—
259C2×D4C22×D311yescharacteristic
260C2×D4C22×D311yescharacteristic
261C22×C4—31no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4C22×D311yescharacteristic
267Q16—62no—
268SD16—34no—
269SD16—62no—
270SD16—62no—
271SD16—121no—
272D8—22no—
273D8—62no—
274D8—62no—
275D8—121no—
276M16—62no—
277C2×C8—22no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8C22×D311yescharacteristic
281C2×C8C22×D311yescharacteristic
282C4⋊C4—31no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4C22×D311yescharacteristic
286C4⋊C4C22×D311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—31no—
290C22⋊C4—31no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C42—31no—
297C42—61no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3C22×D411yescharacteristic
308C2×D3—21no—
309C2×D3—22no—
310C2×D3—21no—
311C2×D3—21no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—84no—
318C3×C4—21no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4C22×D411yescharacteristic
322C3×C4H(5,𝔽2)11yescharacteristic
323Dic3—22no—
324Dic3—21no—
325Dic3—21no—
326Dic3—21no—
327Dic3—41no—
328C23—21no—
329C23—21no—
330C23—31no—
331C23—61no—
332C23—61no—
333C23—122no—
334C23—122no—
335C23—241no—
336C23D3×D411yescharacteristic
337Q8—32no—
338Q8—61no—
339Q8—61no—
340Q8—61no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—32no—
346D4—61no—
347D4—61no—
348D4—61no—
349D4—81no—
350D4—81no—
351D4—121no—
352D4—122no—
353D4—122no—
354D4—122no—
355D4—122no—
356D4—122no—
357D4—241no—
358D4—241no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4D3×D411yescharacteristic
375C2×C4C23×D311yescharacteristic
376C8—21no—
377C8—61no—
378C8—61no—
379C8D3×D412yes—
380C2×C3—41no—
381C2×C3—81no—
382C2×C3—81no—
383C2×C3D4211yescharacteristic
384C2×C3#57511yescharacteristic
385C2×C3#57611yescharacteristic
386D3—41no—
387D3—42no—
388D3—81no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22—62no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—244no—
408C22C2×D3×D411yescharacteristic
409C4—21no—
410C4—41no—
411C4—41no—
412C4—61no—
413C4—61no—
414C4—61no—
415C4—62no—
416C4—121no—
417C4C2×D3×D411yescharacteristic
418C4#100911yescharacteristic
419C3#326811yescharacteristic, complemented, Sylow
420C2—41no—
421C2—81no—
422C2—81no—
423C2—121no—
424C2—122no—
425C2—241no—
426C2#560911yescharacteristic
427C2#575711yescharacteristic
428C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3268
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×#904C2semidirect
3. #5612C2semidirect
4. #5599C2semidirect
5. C3×#466C2semidirect
6. #5154C2semidirect
7. #5111C2semidirect
8. #5037C2semidirect
9. #5037C2semidirect
10. #4889C2semidirect
11. #4886C2semidirect
12. #4786C2semidirect
13. #4783C2semidirect
14. #4773C2semidirect
15. #4767C2semidirect
20. C2×#235C22semidirect
22. C22×C3×C8C22semidirect
24. C3×#121C22semidirect
26. #940C22semidirect
32. C2×#214C22semidirect
37. C2×Dic12C22semidirect
41. C2×#204C22semidirect
46. #887C22semidirect
51. C3×#107C22semidirect
53. C3×#102C22semidirect
54. #830C22semidirect
56. #815C22semidirect
57. Dic3⋊C8C22semidirect
58. #810C22semidirect
59. #810C22semidirect
61. #809C22semidirect
68. #466D3semidirect
137. #204D4semidirect
155. C22×C8C2×D3semidirect
166. #121C2×D3semidirect
173. #107C2×D3semidirect
180. #102C2×D3semidirect
419. C3#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C2×#9042
#56122
#55992
C3×#4662
#51542
#51112
#50372
#48892
#48862
#47862
#47832
#47732
#47672
#32683