Finite Group Databasebeta

← #83987

Subgroups — #83987

Order 384.

494 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×C3×#137C211yesmaximal, characteristic, complemented
2#5829C211yesmaximal, characteristic
3Q8×#67C211yesmaximal, characteristic, complemented
4C2×D3×SD16C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5227C211yesmaximal, characteristic, complemented
7#5190C211yesmaximal, characteristic, complemented
8#5188C211yesmaximal, characteristic, complemented
9#5186C211yesmaximal, characteristic, complemented
10#5185C211yesmaximal, characteristic, complemented
11#5169C211yesmaximal, characteristic, complemented
12#5164C211yesmaximal, characteristic, complemented
13#5162C211yesmaximal, characteristic, complemented
14#5134C211yesmaximal, characteristic, complemented
15#5127C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19C2×D3×Q8C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C3×#137—24no—
22C2×C3×Q16C2211yescharacteristic, complemented
23C2×C3×Q16C2211yescharacteristic, complemented
24C2×C3×SD16C2211yescharacteristic
25C2×C3×SD16C2211yescharacteristic
26C2×C3×M16C2211yescharacteristic, complemented
27#947C2211yescharacteristic
28#946—21no—
29#946C2211yescharacteristic
30#946C2211yescharacteristic
31Q8×Dic3C2211yescharacteristic, complemented
32#944—21no—
33#944C2211yescharacteristic, complemented
34C2×#216C2211yescharacteristic, complemented
35C2×#215C2211yescharacteristic
36C2×#215C2211yescharacteristic
37#939—21no—
38#937—21no—
39D4×Dic3C2211yescharacteristic
40C2×#214C2211yescharacteristic
41#930C2211yescharacteristic
42#929—21no—
43C4×#67—21no—
44C4×#67C2211yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46#924—21no—
47#924C2211yescharacteristic, complemented
48D3×SD16—24no—
49C2×#204C2211yescharacteristic
50C2×D3×C8C2211yescharacteristic
51#835C2211yescharacteristic, complemented
52#835C2211yescharacteristic, complemented
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#830C2211yescharacteristic, complemented
56#821C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic
58#818C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#815C2211yescharacteristic, complemented
61Dic3⋊C8C2211yescharacteristic, complemented
62C2×#137D311yescharacteristic, complemented
63C22×SD16—31no—
64D4×Q8—31no—
65#546—31no—
66#497—31no—
67#468—31no—
68#467—31no—
69#464—31no—
70#461—31no—
71#452—31no—
72#451—31no—
73#450—31no—
74#448—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C3×#41—22no—
79C3×#41—41no—
80C2×C3×Q8—21no—
81C2×C3×Q8—22no—
82C2×C3×Q8—41no—
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4—21no—
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4C2311yescharacteristic
91C2×#67—21no—
92C2×#67—21no—
93C2×#67D411yescharacteristic
94C22×Dic3—21no—
95D3×Q8—22no—
96D3×Q8—41no—
97D3×D4—22no—
98D3×D4—41no—
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
103C2×Dic6C2311yescharacteristic
104C3×Q16—22no—
105C3×Q16—22no—
106C3×SD16—22no—
107C3×SD16D414yescomplemented
108C3×M16—22no—
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111#217—21no—
112#217—21no—
113#217D411yescharacteristic
114#216—41no—
115#215—22no—
116#215—41no—
117#214—22no—
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—
123(C2×D3)⋊C4D411yescharacteristic
124C4⋊Dic3—21no—
125C4⋊Dic3C2311yescharacteristic
126C4⋊Dic3C2311yescharacteristic
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132Dic3⋊C4D411yescharacteristic
133C4×Dic3—21no—
134C4×Dic3C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137#204—22no—
138D3×C8—22no—
139C2×#41C2×D311yescharacteristic
140C22×Q8—31no—
141C22×Q8C2×D311yescharacteristic, complemented
142C22×D4—31no—
143#137—24no—
144C2×Q16—31no—
145C2×Q16C2×D311yescharacteristic, complemented
146C2×Q16C2×D311yescharacteristic, complemented
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—64no—
152C2×SD16C2×D311yescharacteristic
153C2×SD16C2×D311yescharacteristic
154C2×M16C2×D311yescharacteristic, complemented
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#124—31no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165#121—31no—
166#121—61no—
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#103—31no—
177#103—31no—
178#103—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—81no—
190C22×D3C2×D411yescharacteristic
191C3×Q8—21no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—42no—
197C3×Q8C2×D412yes—
198C3×D4—21no—
199C3×D4—42no—
200C3×D4C2×D412yes—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2411yescharacteristic
210#67—22no—
211#67—42no—
212#67—81no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—81no—
217C2×Dic3C2×D411yescharacteristic
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220D12—21no—
221D12—21no—
222C4×D3—21no—
223C4×D3—21no—
224C4×D3—42no—
225Dic6—21no—
226Dic6—21no—
227Dic6—42no—
228C3×C8—21no—
229C3×C8C2×D412yes—
230C3⋊C8—21no—
231C3⋊C8—21no—
232C24—61no—
233#41—22no—
234#41—41no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—31no—
238C2×Q8—41no—
239C2×Q8—61no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—21no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—61no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4C22×D311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4C22×D311yescharacteristic
259Q16—22no—
260Q16—22no—
261Q16—121no—
262SD16—22no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—121no—
267SD16#6714yescomplemented
268M16—22no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—62no—
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4—31no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C22⋊C4—31no—
284C22⋊C4—31no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—31no—
293C42—61no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C22×D411yescharacteristic
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—84no—
307C2×D3#13712yescomplemented
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4C22×D411yescharacteristic
313C3×C4H(5,𝔽2)11yescharacteristic
314Dic3—21no—
315Dic3—21no—
316Dic3—21no—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320C23—21no—
321C23—31no—
322C23—61no—
323C23—122no—
324C23—122no—
325C23—241no—
326C23D3×D411yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8C2×#6712yes—
337D4—21no—
338D4—42no—
339D4—62no—
340D4—61no—
341D4—61no—
342D4—122no—
343D4—241no—
344D4C2×#6712yes—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—122no—
361C2×C4—241no—
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—61no—
366C8—61no—
367C8C2×#6712yes—
368C2×C3—42no—
369C2×C3—41no—
370C2×C3D4211yescharacteristic
371C2×C3C2×#13711yescharacteristic
372C2×C3#57711yescharacteristic
373D3—22no—
374D3—81no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—32no—
380C22—41no—
381C22—41no—
382C22—61no—
383C22—61no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—244no—
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—61no—
394C4—61no—
395C4—61no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4#100411yescharacteristic
400C4C22×#6711yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—41no—
403C2—42no—
404C2—62no—
405C2—241no—
406C2D3×#13711yescharacteristic
407C2#577811yescharacteristic
408C2D4×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
prime radical#5829C2
hypercenterC2×#137D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×C3×#137C2semidirect
3. Q8×#67C2semidirect
4. C2×D3×SD16C2semidirect
5. #5239C2semidirect
6. #5227C2semidirect
7. #5190C2semidirect
8. #5188C2semidirect
9. #5186C2semidirect
10. #5185C2semidirect
11. #5169C2semidirect
12. #5164C2semidirect
13. #5162C2semidirect
14. #5134C2semidirect
15. #5127C2semidirect
18. C22×C3×Q8C22semidirect
22. C2×C3×Q16C22semidirect
23. C2×C3×Q16C22semidirect
26. C2×C3×M16C22semidirect
31. Q8×Dic3C22semidirect
33. #944C22semidirect
34. C2×#216C22semidirect
47. #924C22semidirect
51. #835C22semidirect
52. #835C22semidirect
53. #835C22semidirect
55. #830C22semidirect
58. #818C22semidirect
59. #816C22semidirect
60. #815C22semidirect
61. Dic3⋊C8C22semidirect
62. C2×#137D3semidirect
107. C3×SD16D4semidirect
141. C22×Q8C2×D3semidirect
145. C2×Q16C2×D3semidirect
146. C2×Q16C2×D3semidirect
154. C2×M16C2×D3semidirect
267. SD16#67semidirect
307. C2×D3#137semidirect
401. C3#3266semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1372
#58292
Q8×#672
C2×D3×SD162
#52392
#52272
#51902
#51882
#51862
#51852
#51692
#51642
#51622
#51342
#51272
#32663