Finite Group Databasebeta

← #80123

Subgroups — #80123

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
1#5834C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5612C211yesmaximal, characteristic, complemented
5C3×#449C211yesmaximal, characteristic, complemented
6#5166C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5154C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4817C211yesmaximal, characteristic, complemented
11#4808C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4780C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3268—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C2×#239C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#950—24no—
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic
28#940—21no—
29#940C2211yescharacteristic, complemented
30#938—21no—
31#938—21no—
32#938C2211yescharacteristic
33#937—21no—
34#936C2211yescharacteristic
35C2×#214C2211yescharacteristic, complemented
36C2×#213C2211yescharacteristic
37C2×#213C2211yescharacteristic
38#930—21no—
39#930C2211yescharacteristic, complemented
40C4×#67—21no—
41C4×#67C2211yescharacteristic
42C22×C3⋊C8C2211yescharacteristic, complemented
43#914—24no—
44C2×D24C2211yescharacteristic
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic
47#897C2211yescharacteristic
48#893C2211yescharacteristic
49#883—21no—
50D3×C22⋊C4—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#816C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#550—31no—
65#546—31no—
66#496—31no—
67#468—31no—
68#466—31no—
69#464—31no—
70#459—31no—
71#452—31no—
72#449—31no—
73#449D311yescharacteristic, complemented
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C3×#41—22no—
79C3×#41—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4C2311yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67D411yescharacteristic
94#239—22no—
95#239—41no—
96D3×D4—22no—
97D3×D4—41no—
98#235—22no—
99#235—41no—
100C2×D12—21no—
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3—21no—
106C2×C4×D3C2311yescharacteristic
107C2×Dic6C2311yescharacteristic
108C3×SD16—22no—
109C3×D8—41no—
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C22⋊C4—21no—
114#217—21no—
115#217—21no—
116#217—21no—
117#217D411yescharacteristic
118#216—22no—
119#215D414yescomplemented
120#214—22no—
121#213—22no—
122#213—22no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4D411yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137#204—41no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41C2×D311yescharacteristic, complemented
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16C2×D311yescharacteristic
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156#127—31no—
157#127—61no—
158#124—31no—
159#123—61no—
160C22⋊Q8—31no—
161#121—31no—
162#121—31no—
163#121—61no—
164#121—61no—
165#121—61no—
166#121C2×D311yescharacteristic
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic, complemented
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×Q8—21no—
194C3×Q8C2×D412yes—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—42no—
200C3×D4—81no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2411yescharacteristic
208#67—41no—
209#67—42no—
210#67—42no—
211#67—42no—
212#67—81no—
213#67—81no—
214#67—81no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3C2×D411yescharacteristic
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220D12—21no—
221D12—21no—
222D12—21no—
223D12—42no—
224D12C2×D412yes—
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—42no—
228C4×D3—42no—
229C4×D3—81no—
230Dic6—21no—
231Dic6—21no—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—61no—
237#41—22no—
238#41—41no—
239#41—62no—
240#41—62no—
241#41—121no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8C22×D311yescharacteristic
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—21no—
262C22×C4—31no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4C22×D311yescharacteristic
267Q16—62no—
268SD16—22no—
269SD16—34no—
270SD16—62no—
271SD16—121no—
272D8—41no—
273D8—62no—
274D8—62no—
275D8—62no—
276M16—62no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—62no—
280C2×C8C22×D311yescharacteristic
281C2×C8C22×D311yescharacteristic
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4C22×D311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—31no—
289C22⋊C4—31no—
290C22⋊C4—61no—
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—22no—
320C3×C4—41no—
321C3×C4C22×D411yescharacteristic
322C3×C4H(5,𝔽2)11yescharacteristic
323Dic3—21no—
324Dic3—21no—
325Dic3—21no—
326Dic3—41no—
327Dic3—41no—
328C23—21no—
329C23—21no—
330C23—31no—
331C23—61no—
332C23—61no—
333C23—122no—
334C23—122no—
335C23—241no—
336C23D3×D411yescharacteristic
337Q8—21no—
338Q8—61no—
339Q8—61no—
340Q8D3×D412yes—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—32no—
346D4—42no—
347D4—61no—
348D4—61no—
349D4—61no—
350D4—81no—
351D4—122no—
352D4—121no—
353D4—122no—
354D4—122no—
355D4—122no—
356D4—241no—
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—42no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4D3×D411yescharacteristic
375C2×C4C23×D311yescharacteristic
376C8—21no—
377C8—21no—
378C8—32no—
379C8—61no—
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—22no—
411C4—41no—
412C4—61no—
413C4—61no—
414C4—61no—
415C4—121no—
416C4—121no—
417C4C2×D3×D411yescharacteristic
418C4#100411yescharacteristic
419C3#326811yescharacteristic, complemented, Sylow
420C2—41no—
421C2—81no—
422C2—81no—
423C2—121no—
424C2—122no—
425C2—241no—
426C2#559011yescharacteristic
427C2#576811yescharacteristic
428C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3268
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #5834C2semidirect
2. C2×#950C2semidirect
3. C2×#914C2semidirect
4. #5612C2semidirect
5. C3×#449C2semidirect
6. #5166C2semidirect
7. #5156C2semidirect
8. #5154C2semidirect
9. #5038C2semidirect
10. #4817C2semidirect
11. #4808C2semidirect
12. #4786C2semidirect
13. #4780C2semidirect
14. #4734C2semidirect
15. #4730C2semidirect
17. C2×C3×#41C22semidirect
25. Q8×Dic3C22semidirect
26. C2×#216C22semidirect
29. #940C22semidirect
35. C2×#214C22semidirect
39. #930C22semidirect
42. C22×C3⋊C8C22semidirect
45. C2×#204C22semidirect
51. C3×#103C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
57. #816C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #809C22semidirect
61. #807C22semidirect
73. #449D3semidirect
119. #215D4semidirect
141. C2×#41C2×D3semidirect
176. #103C2×D3semidirect
180. #102C2×D3semidirect
182. C22⋊C8C2×D3semidirect
419. C3#3268semidirect

Maximal subgroups

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

subgroupindex
#58342
C2×#9502
C2×#9142
#56122
C3×#4492
#51662
#51562
#51542
#50382
#48172
#48082
#47862
#47802
#47342
#47302
#32683