Finite Group Databasebeta

← #80061

Subgroups — #80061

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
1Q8×#67C211yesmaximal, characteristic, complemented
2C2×#942C211yesmaximal, characteristic, complemented
3C2×D3×SD16C211yesmaximal, characteristic, complemented
4#5612C211yesmaximal, characteristic
5C3×#448C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5166C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#4818C211yesmaximal, characteristic, complemented
11#4817C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4809C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic, complemented
18C2×D3×Q8C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946—21no—
25#946C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic, complemented
27#944—21no—
28#944C2211yescharacteristic, complemented
29C2×#216C2211yescharacteristic, complemented
30C2×#216C2211yescharacteristic, complemented
31#942—24no—
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic
34#938—21no—
35#937—21no—
36#936C2211yescharacteristic
37C2×#214C2211yescharacteristic
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40#924—21no—
41#924C2211yescharacteristic, complemented
42C2×#208C2211yescharacteristic, complemented
43D3×SD16—24no—
44C2×Dic12C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic
46C2×D3×C8C2211yescharacteristic
47#897C2211yescharacteristic
48#893C2211yescharacteristic
49#883—21no—
50D3×C22⋊C4—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#816C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#810C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×#137—31no—
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#448D311yescharacteristic, complemented
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C2×C3×Q8—21no—
79C2×C3×Q8—22no—
80C2×C3×Q8—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—21no—
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67D411yescharacteristic
90D3×Q8—22no—
91D3×Q8—41no—
92D3×D4—22no—
93D3×D4—41no—
94#235—22no—
95#235—41no—
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3C2311yescharacteristic
101C2×Dic6—21no—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×SD16—22no—
105C3×SD16—41no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110#217—21no—
111#217—21no—
112#217—21no—
113#217D411yescharacteristic
114#216—22no—
115#216—22no—
116#215—22no—
117#215D414yescomplemented
118#214—22no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4D411yescharacteristic
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—41no—
137#204—22no—
138D3×C8—22no—
139C2×#41—31no—
140C22×Q8—31no—
141C22×Q8C2×D311yescharacteristic, complemented
142C22×D4—31no—
143#137—64no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—64no—
152C2×SD16C2×D311yescharacteristic
153C2×SD16C2×D311yescharacteristic
154C2×M16—31no—
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—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#103—31no—
177#103C2×D311yescharacteristic, complemented
178#103C2×D311yescharacteristic, complemented
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic, complemented
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—42no—
195C3×Q8C2×D412yes—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—81no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—42no—
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207#67—22no—
208#67—42no—
209#67—42no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216D12—21no—
217D12C2×D412yes—
218C4×D3—21no—
219C4×D3—21no—
220C4×D3—42no—
221C4×D3—42no—
222C4×D3—81no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—42no—
228C3×C8—21no—
229C3×C8—21no—
230C3⋊C8—21no—
231C3⋊C8C2×D412yes—
232C24—61no—
233#41—62no—
234#41—121no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—31no—
238C2×Q8—31no—
239C2×Q8—41no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
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—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4C22×D311yescharacteristic
259Q16—62no—
260Q16—62no—
261Q16—121no—
262SD16—22no—
263SD16—34no—
264SD16—41no—
265SD16—62no—
266SD16—62no—
267SD16—62no—
268M16—62no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—62no—
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—31no—
285C22⋊C4—31no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—31no—
293C42—61no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C22×D411yescharacteristic
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—22no—
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—42no—
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8D3×D412yes—
337D4—21no—
338D4—21no—
339D4—32no—
340D4—62no—
341D4—61no—
342D4—81no—
343D4—122no—
344D4—122no—
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—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—241no—
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—21no—
366C8—32no—
367C8—61no—
368C2×C3—41no—
369C2×C3—81no—
370C2×C3D4211yescharacteristic
371C2×C3C2×#13711yescharacteristic
372C2×C3#57711yescharacteristic
373D3—22no—
374D3—42no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—32no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—61no—
383C22—61no—
384C22—62no—
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—
399C4C2×D3×D411yescharacteristic
400C4#100411yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—41no—
403C2—62no—
404C2—81no—
405C2—122no—
406C2#559011yescharacteristic
407C2#576211yescharacteristic
408C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5612C2
hypercenter#448D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
2. C2×#942C2semidirect
3. C2×D3×SD16C2semidirect
5. C3×#448C2semidirect
6. #5169C2semidirect
7. #5167C2semidirect
8. #5166C2semidirect
9. #5042C2semidirect
10. #4818C2semidirect
11. #4817C2semidirect
12. #4813C2semidirect
13. #4809C2semidirect
14. #4733C2semidirect
15. #4728C2semidirect
17. C22×C3×Q8C22semidirect
26. Q8×Dic3C22semidirect
28. #944C22semidirect
29. C2×#216C22semidirect
30. C2×#216C22semidirect
41. #924C22semidirect
42. C2×#208C22semidirect
44. C2×Dic12C22semidirect
51. C3×#103C22semidirect
52. C3×#103C22semidirect
53. C3×C22⋊C8C22semidirect
57. #816C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #810C22semidirect
61. #807C22semidirect
74. #448D3semidirect
117. #215D4semidirect
141. C22×Q8C2×D3semidirect
177. #103C2×D3semidirect
178. #103C2×D3semidirect
182. C22⋊C8C2×D3semidirect
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
Q8×#672
C2×#9422
C2×D3×SD162
#56122
C3×#4482
#51692
#51672
#51662
#50422
#48182
#48172
#48132
#48092
#47332
#47282
#32663