Finite Group Databasebeta

← #83506

Subgroups — #83506

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
1C3×#546C211yesmaximal, characteristic
2C2×#951C211yesmaximal, characteristic, complemented
3C22×#214C211yesmaximal, characteristic, complemented
4D4×Dic6C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5218C211yesmaximal, characteristic, complemented
7#5060C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#5050C211yesmaximal, characteristic, complemented
11#5042C211yesmaximal, characteristic, complemented
12#5040C211yesmaximal, characteristic, complemented
13#5019C211yesmaximal, characteristic, complemented
14#5018C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2211yescharacteristic, complemented
21C3×#124C2211yescharacteristic
22C3×#123—21no—
23C3×C22⋊Q8C2211yescharacteristic
24C3×#121—21no—
25C3×#121C2211yescharacteristic
26C3×#120—21no—
27C3×C4×D4C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic
29C2×C3×C22⋊C4—21no—
30#951—24no—
31C2×#216C2211yescharacteristic, complemented
32C2×#216C2211yescharacteristic, complemented
33C2×#216C2211yescharacteristic, complemented
34D4×Dic3—21no—
35C2×#214—24no—
36C2×#214C2211yescharacteristic
37C2×#214C2211yescharacteristic
38C2×#214C2211yescharacteristic
39C2×#214C2211yescharacteristic
40C2×#214C2211yescharacteristic
41C2×#214C2211yescharacteristic
42#924C2211yescharacteristic
43#924C2211yescharacteristic, complemented
44C2×#208C2211yescharacteristic, complemented
45C22×C3⋊C8C2211yescharacteristic
46#888—21no—
47#878—21no—
48#878—21no—
49#869C2211yescharacteristic, complemented
50C4×Dic6C2211yescharacteristic, complemented
51#835C2211yescharacteristic, complemented
52#832C2211yescharacteristic
53#832C2211yescharacteristic
54#830C2211yescharacteristic
55#830C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60C4⋊(C3⋊C8)C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×#137—31no—
63C22×SD16—31no—
64D4×Q8—31no—
65#546D311yescharacteristic
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—
77C24×C3—21no—
78C3×#41—22no—
79C3×#41—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—22no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—41no—
86C2×C3×D4D411yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—21no—
98C2×Dic6—21no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×C4⋊C4—21no—
105C3×C4⋊C4D411yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4D411yescharacteristic
113C3×C22⋊C4D411yescharacteristic
114C3×C42C2311yescharacteristic
115#217—21no—
116#217—21no—
117#216—22no—
118#216—22no—
119#216—41no—
120#214—22no—
121#214—22no—
122#214—22no—
123#214—22no—
124#214—41no—
125#214D414yescomplemented
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3C2311yescharacteristic
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132C4×Dic3—21no—
133#208—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137C2×C3⋊C8C2311yescharacteristic
138C2×C3⋊C8C2311yescharacteristic
139C2×#41C2×D311yescharacteristic
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—31no—
152C2×SD16—31no—
153C2×SD16—64no—
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#124C2×D311yescharacteristic
159#123—21no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163C22⋊Q8—61no—
164C22⋊Q8C2×D311yescharacteristic
165#121—21no—
166#121C2×D311yescharacteristic
167#120—21no—
168C4×Q8—31no—
169C4×D4—61no—
170C4×D4C2×D311yescharacteristic
171C4×D4C2×D311yescharacteristic
172C2×C22⋊C4—21no—
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×C3—21no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—81no—
189C23×C3C2×D411yescharacteristic
190C23×C3C2×D411yescharacteristic
191C3×Q8—21no—
192C3×Q8—21no—
193C3×D4—22no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—81no—
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—21no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2×D411yescharacteristic
213C2×C3×C4C2411yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—42no—
219C2×Dic3—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—22no—
234#41—41no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—121no—
243C2×Q8—121no—
244C2×Q8C22×D311yescharacteristic
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4#6711yescharacteristic
251C2×D4C22×D311yescharacteristic
252C2×D4C22×D311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4C22×D311yescharacteristic
258C22×C4C22×D311yescharacteristic
259Q16—62no—
260Q16—62no—
261Q16—121no—
262SD16—34no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268M16—62no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—31no—
273C2×C8—62no—
274C4⋊C4—21no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4#6711yescharacteristic
282C4⋊C4C22×D311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4#6711yescharacteristic
291C22⋊C4#6711yescharacteristic
292C42—61no—
293C42C22×D311yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—84no—
306C22×C3#13712yescomplemented
307C22×C3C22×D411yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4C22×D411yescharacteristic
315C3×C4H(5,𝔽2)11yescharacteristic
316Dic3—22no—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320C23—21no—
321C23—21no—
322C23—42no—
323C23—42no—
324C23—81no—
325C23C2×#6711yescharacteristic
326C23C2×#6711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—32no—
330Q8—61no—
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8—122no—
337D4—22no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—81no—
344D4D3×D412yes—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—81no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4C2×#6711yescharacteristic
360C2×C4C2×#6711yescharacteristic
361C2×C4C2×#6711yescharacteristic
362C2×C4C2×#6711yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—32no—
365C8—61no—
366C8—61no—
367C8—61no—
368C2×C3—22no—
369C2×C3—41no—
370C2×C3—42no—
371C2×C3—81no—
372C2×C3D4211yescharacteristic
373C2×C3C2×#13711yescharacteristic
374C2×C3#57711yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—84no—
387C22#95112yescomplemented
388C22C22×#6711yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—62no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4C2×D3×D411yescharacteristic
400C4#100411yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×#6711yescharacteristic
407C2C2×#95111yescharacteristic
408C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#546C2
socleC22×C3C22×D4
hypercenter#546D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
2. C2×#951C2semidirect
3. C22×#214C2semidirect
4. D4×Dic6C2semidirect
5. #5239C2semidirect
6. #5218C2semidirect
7. #5060C2semidirect
8. #5054C2semidirect
9. #5051C2semidirect
10. #5050C2semidirect
11. #5042C2semidirect
12. #5040C2semidirect
13. #5019C2semidirect
14. #5018C2semidirect
15. #5011C2semidirect
20. C22×Dic6C22semidirect
31. C2×#216C22semidirect
32. C2×#216C22semidirect
33. C2×#216C22semidirect
43. #924C22semidirect
44. C2×#208C22semidirect
49. #869C22semidirect
50. C4×Dic6C22semidirect
51. #835C22semidirect
55. #830C22semidirect
56. #810C22semidirect
57. #810C22semidirect
58. #810C22semidirect
59. #807C22semidirect
60. C4⋊(C3⋊C8)C22semidirect
61. #803C22semidirect
125. #214D4semidirect
306. C22×C3#137semidirect
387. C22#951semidirect
401. C3#3266semidirect

Maximal subgroups

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

subgroupindex
C3×#5462
C2×#9512
C22×#2142
D4×Dic62
#52392
#52182
#50602
#50542
#50512
#50502
#50422
#50402
#50192
#50182
#50112
#32663