Finite Group Databasebeta

← #80169

Subgroups — #80169

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
1#5806C211yesmaximal, characteristic
2C22×#214C211yesmaximal, characteristic, complemented
3C2×#915C211yesmaximal, characteristic, complemented
4#5632C211yesmaximal, characteristic, complemented
5C3×#450C211yesmaximal, characteristic, complemented
6#5170C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5162C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#4783C211yesmaximal, characteristic, complemented
11#4780C211yesmaximal, characteristic, complemented
12#4767C211yesmaximal, characteristic, complemented
13#4757C211yesmaximal, characteristic, complemented
14#4739C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×#237C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#953—21no—
25C2×#217—21no—
26#944C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28#939—21no—
29#938C2211yescharacteristic
30#936C2211yescharacteristic
31#935—21no—
32D4×Dic3C2211yescharacteristic
33C2×#214—24no—
34C2×#214C2211yescharacteristic
35C2×#214C2211yescharacteristic
36C2×#214C2211yescharacteristic
37C2×#214C2211yescharacteristic
38C4×#67C2211yescharacteristic
39#924C2211yescharacteristic
40C22×C3⋊C8C2211yescharacteristic
41#915—24no—
42C2×Dic12C2211yescharacteristic, complemented
43C2×Dic12C2211yescharacteristic, complemented
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45#897C2211yescharacteristic, complemented
46#896—21no—
47#888—21no—
48#887C2211yescharacteristic, complemented
49#67⋊C4—21no—
50#878—21no—
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
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#450D311yescharacteristic, complemented
74#448—31no—
75#440—31no—
76#435—31no—
77C24×C3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—41no—
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88C22×Dic3—21no—
89C22×Dic3—21no—
90C22×Dic3—21no—
91D3×Q8—22no—
92D3×Q8—41no—
93#237—22no—
94#237—41no—
95C2×C4×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—22no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6—41no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×SD16—22no—
105C3×SD16—41no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111#217—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#217D411yescharacteristic
117#216—22no—
118#214—22no—
119#214—22no—
120#214—22no—
121#214D414yescomplemented
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4D411yescharacteristic
124C4⋊Dic3—21no—
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—22no—
137Dic12—41no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C22×Q8—31no—
141C22×Q8—31no—
142C22×D4C2×D311yescharacteristic
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—61no—
163C22⋊Q8—61no—
164C22⋊Q8C2×D311yescharacteristic
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#102C2×D311yescharacteristic
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183#97—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—81no—
189C23×C3C2×D411yescharacteristic
190C22×D3C2×D411yescharacteristic
191C3×Q8—21no—
192C3×Q8—21no—
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—42no—
197C3×D4C2×D412yes—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—22no—
205#67—42no—
206#67—81no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—81no—
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217C4×D3—21no—
218C4×D3—21no—
219C4×D3—42no—
220Dic6—21no—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—42no—
226Dic6—42no—
227Dic6C2×D412yes—
228C3×C8—21no—
229C3×C8—21no—
230C3⋊C8—21no—
231C3⋊C8C2×D412yes—
232C24—21no—
233#41—62no—
234#41—121no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8—121no—
244C2×Q8C22×D311yescharacteristic
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—31no—
248C2×D4—41no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4C22×D311yescharacteristic
252C2×D4C22×D311yescharacteristic
253C22×C4—31no—
254C22×C4—61no—
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—21no—
275C4⋊C4—31no—
276C4⋊C4—31no—
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—22no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—84no—
303C22×C3#13712yescomplemented
304C22×C3C22×D411yescharacteristic
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—21no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4C22×D411yescharacteristic
312C3×C4H(5,𝔽2)11yescharacteristic
313Dic3—22no—
314Dic3—21no—
315Dic3—21no—
316Dic3—21no—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320C23—21no—
321C23—21no—
322C23—31no—
323C23—42no—
324C23—42no—
325C23—81no—
326C23D3×D411yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—32no—
330Q8—61no—
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8—122no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—62no—
342D4—122no—
343D4—241no—
344D4D3×D412yes—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—61no—
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—241no—
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—21no—
366C8—32no—
367C8—61no—
368C2×C3—22no—
369C2×C3—42no—
370C2×C3—81no—
371C2×C3D4211yescharacteristic
372C2×C3C2×#13711yescharacteristic
373C2×C3#57711yescharacteristic
374D3—41no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—84no—
387C22#91512yescomplemented
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—41no—
391C4—41no—
392C4—61no—
393C4—61no—
394C4—61no—
395C4—62no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4C2×D3×D411yescharacteristic
400C4#100411yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—22no—
403C2—42no—
404C2—81no—
405C2—121no—
406C2#559011yescharacteristic
407C2C2×#91511yescharacteristic
408C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radical#5806C2
hypercenter#450D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C22×#214C2semidirect
3. C2×#915C2semidirect
4. #5632C2semidirect
5. C3×#450C2semidirect
6. #5170C2semidirect
7. #5167C2semidirect
8. #5162C2semidirect
9. #5051C2semidirect
10. #4783C2semidirect
11. #4780C2semidirect
12. #4767C2semidirect
13. #4757C2semidirect
14. #4739C2semidirect
15. #4730C2semidirect
18. C2×D3×Q8C22semidirect
26. #944C22semidirect
27. C2×#216C22semidirect
42. C2×Dic12C22semidirect
43. C2×Dic12C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
45. #897C22semidirect
48. #887C22semidirect
54. #835C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. #816C22semidirect
57. #816C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #810C22semidirect
61. #807C22semidirect
73. #450D3semidirect
121. #214D4semidirect
303. C22×C3#137semidirect
387. C22#915semidirect
401. C3#3266semidirect

Maximal subgroups

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

subgroupindex
#58062
C22×#2142
C2×#9152
#56322
C3×#4502
#51702
#51672
#51622
#50512
#47832
#47802
#47672
#47572
#47392
#47302
#32663