Finite Group Databasebeta

← #80807

Subgroups — #80807

Order 384.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×SD16C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5628C211yesmaximal, characteristic, complemented
4#5607C211yesmaximal, characteristic
5C3×#468C211yesmaximal, characteristic, complemented
6#5174C211yesmaximal, characteristic, complemented
7#5134C211yesmaximal, characteristic, complemented
8#5048C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4886C211yesmaximal, characteristic, complemented
11#4880C211yesmaximal, characteristic, complemented
12#4807C211yesmaximal, characteristic, complemented
13#4802C211yesmaximal, characteristic, complemented
14#4778C211yesmaximal, characteristic, complemented
15#4770C211yesmaximal, characteristic, complemented
16#3057—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic
26#940—21no—
27#938C2211yescharacteristic
28#937—21no—
29C2×#214C2211yescharacteristic
30#930C2211yescharacteristic
31#924C2211yescharacteristic, complemented
32C22×C3⋊C8C2211yescharacteristic, complemented
33D3×SD16—24no—
34#907—24no—
35C2×#204C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
38C2×D3×C8C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic, complemented
40#897C2211yescharacteristic, complemented
41#895C2211yescharacteristic
42#894—21no—
43#892C2211yescharacteristic
44#892C2211yescharacteristic
45#888—21no—
46#885—21no—
47#885—21no—
48#883—21no—
49#882—21no—
50C3×#107C2211yescharacteristic, complemented
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56#818C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#808C2211yescharacteristic, complemented
61#808C2211yescharacteristic, complemented
62C22×SD16—31no—
63C2×#131—31no—
64#536—31no—
65#535—31no—
66#468—31no—
67#468—31no—
68#468—31no—
69#468D311yescharacteristic, complemented
70#465—31no—
71#465—31no—
72#465—31no—
73#465—31no—
74#427—31no—
75#419—31no—
76#418—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67D411yescharacteristic
86C2×#67D411yescharacteristic
87D3×Q8—22no—
88D3×Q8—22no—
89D3×D4—22no—
90D3×D4—22no—
91#235—22no—
92#235—22no—
93C2×D12—21no—
94C2×D12D411yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C2×C4×D3C2311yescharacteristic
99C2×Dic6—21no—
100C2×Dic6D411yescharacteristic
101C2×Dic6C2311yescharacteristic
102C3×SD16—22no—
103C3×M16—22no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111#217—21no—
112#217—21no—
113#215—22no—
114#214—22no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3C2311yescharacteristic
123C4⋊Dic3C2311yescharacteristic
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131#204—22no—
132ΓL(1,𝔽25)—22no—
133ΓL(1,𝔽25)—22no—
134D3×C8—22no—
135D3×C8D414yescomplemented
136C2×#41—31no—
137C22×Q8—31no—
138C22×D4—31no—
139C2×SD16—31no—
140C2×SD16—31no—
141C2×SD16—31no—
142C2×SD16—64no—
143C2×SD16C2×D311yescharacteristic
144#131—64no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16C2×D311yescharacteristic, complemented
148C22×C8—31no—
149C22×C8—31no—
150C22×C8—31no—
151C4⋊Q8—61no—
152#127—61no—
153#124—61no—
154#124—61no—
155#123—61no—
156#123—61no—
157C22⋊Q8—31no—
158C22⋊Q8—31no—
159C22⋊Q8—31no—
160C22⋊Q8C2×D311yescharacteristic, complemented
161#121—31no—
162#121—31no—
163#121—31no—
164#121C2×D311yescharacteristic
165#120—61no—
166#120—61no—
167#117—31no—
168#117—31no—
169#107—31no—
170#107C2×D311yescharacteristic, complemented
171#106—31no—
172#106—31no—
173#103—31no—
174#103—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181C23×C3—21no—
182C23×C3C2×D411yescharacteristic
183C22×D3—21no—
184C22×D3—42no—
185C22×D3—81no—
186C22×D3—81no—
187C22×D3C2×D411yescharacteristic
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—81no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—22no—
202#67—22no—
203#67—42no—
204#67—81no—
205#67—81no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211D12—22no—
212D12—21no—
213D12—21no—
214D12—81no—
215C4×D3—21no—
216C4×D3—21no—
217C4×D3—42no—
218C4×D3C2×D412yes—
219C4×D3#4112yes—
220Dic6—22no—
221Dic6—21no—
222Dic6—21no—
223Dic6—42no—
224C3×C8—21no—
225C3×C8C2×D412yes—
226C3⋊C8—21no—
227C3⋊C8C2×D412yes—
228C24—61no—
229#41—62no—
230#41—62no—
231C2×Q8—31no—
232C2×Q8—31no—
233C2×Q8—61no—
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8C22×D311yescharacteristic
237C2×D4—21no—
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—31no—
241C2×D4—31no—
242C2×D4—61no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4C22×D311yescharacteristic
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—61no—
252C22×C4C22×D311yescharacteristic
253SD16—22no—
254SD16—62no—
255SD16—62no—
256SD16—62no—
257M16—22no—
258M16—62no—
259M16—62no—
260C2×C8—31no—
261C2×C8—34no—
262C2×C8—31no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8C22×D311yescharacteristic
266C2×C8C22×D311yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4C22×D311yescharacteristic
274C4⋊C4C22×D311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C42—61no—
286C42—61no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3C22×D411yescharacteristic
294C2×D3—21no—
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—84no—
303C2×D3#12112yes—
304C3×C4—21no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4C22×D411yescharacteristic
309C3×C4C2×#4111yescharacteristic
310Dic3—21no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314Dic3#12112yes—
315C23—21no—
316C23—31no—
317C23—31no—
318C23—61no—
319C23—122no—
320C23—241no—
321C23—241no—
322C23D3×D411yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—62no—
326Q8—61no—
327Q8—61no—
328Q8—122no—
329D4—21no—
330D4—21no—
331D4—62no—
332D4—62no—
333D4—62no—
334D4—61no—
335D4—61no—
336D4—81no—
337D4—122no—
338D4—241no—
339D4—241no—
340D4—241no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—31no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—122no—
356C2×C4D3×D411yescharacteristic
357C2×C4C23×D311yescharacteristic
358C8—21no—
359C8—32no—
360C8—61no—
361C8D3×D412yes—
362C2×C3—41no—
363C2×C3—81no—
364C2×C3C2×#12111yescharacteristic
365C2×C3#57711yescharacteristic
366C2×C3#57711yescharacteristic
367D3—22no—
368D3—41no—
369D3—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—32no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—61no—
378C22—61no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—244no—
386C22C2×D3×D411yescharacteristic
387C4—21no—
388C4—32no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—61no—
393C4—121no—
394C4—121no—
395C4—121no—
396C4C2×D3×D411yescharacteristic
397C4D3×#4111yescharacteristic
398C3#305711yescharacteristic, complemented, Sylow
399C2—41no—
400C2—62no—
401C2—81no—
402C2—121no—
403C2—241no—
404C2D3×#12111yescharacteristic
405C2#577511yescharacteristic
406C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radical#5607C2
hypercenter#468D3
hypocenterC3#3057
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×D3×SD16C2semidirect
2. C2×#907C2semidirect
3. #5628C2semidirect
5. C3×#468C2semidirect
6. #5174C2semidirect
7. #5134C2semidirect
8. #5048C2semidirect
9. #5041C2semidirect
10. #4886C2semidirect
11. #4880C2semidirect
12. #4807C2semidirect
13. #4802C2semidirect
14. #4778C2semidirect
15. #4770C2semidirect
21. C2×C3×M16C22semidirect
22. C3×C22⋊Q8C22semidirect
24. #946C22semidirect
31. #924C22semidirect
32. C22×C3⋊C8C22semidirect
37. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
40. #897C22semidirect
50. C3×#107C22semidirect
51. C3×#103C22semidirect
53. #835C22semidirect
56. #818C22semidirect
57. #816C22semidirect
58. #810C22semidirect
60. #808C22semidirect
61. #808C22semidirect
69. #468D3semidirect
135. D3×C8D4semidirect
147. C2×M16C2×D3semidirect
160. C22⋊Q8C2×D3semidirect
170. #107C2×D3semidirect
176. #103C2×D3semidirect
398. C3#3057semidirect

Maximal subgroups

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

subgroupindex
C2×D3×SD162
C2×#9072
#56282
#56072
C3×#4682
#51742
#51342
#50482
#50412
#48862
#48802
#48072
#48022
#47782
#47702
#30573