Finite Group Databasebeta

← #80126

Subgroups — #80126

Order 384.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5831C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×D3×SD16C211yesmaximal, characteristic, complemented
4#5612C211yesmaximal, characteristic, complemented
5C3×#449C211yesmaximal, characteristic, complemented
6#5166C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5155C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4788C211yesmaximal, characteristic, complemented
12#4787C211yesmaximal, characteristic, complemented
13#4778C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16#3282—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C2×D3×Q8C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#950—24no—
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic
28#938—21no—
29#938—21no—
30#938C2211yescharacteristic
31#937—21no—
32#936—21no—
33#936C2211yescharacteristic
34#936C2211yescharacteristic, complemented
35C2×#214C2211yescharacteristic
36C2×#214C2211yescharacteristic, complemented
37C2×#213C2211yescharacteristic
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40#924—21no—
41#924C2211yescharacteristic, complemented
42C22×C3⋊C8C2211yescharacteristic, 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×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#816C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#810C2211yescharacteristic, complemented
61#808C2211yescharacteristic, complemented
62C2×#135—31no—
63C22×SD16—31no—
64#548—31no—
65#546—31no—
66#495—31no—
67#465—31no—
68#465—31no—
69#464—31no—
70#463—31no—
71#452—31no—
72#451—31no—
73#449D311yescharacteristic, complemented
74#447—31no—
75C4×SD16—31no—
76C8×D4—31no—
77C23×D3—21no—
78C3×#41—22no—
79C3×#41—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4C2311yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67D411yescharacteristic
91D3×Q8—22no—
92D3×Q8—41no—
93D3×D4—22no—
94D3×D4—41no—
95#235—22no—
96#235—41no—
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×SD16—22no—
106C3×D8—41no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111#217—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#217—21no—
117#217D411yescharacteristic
118#216—22no—
119#215D414yescomplemented
120#214—22no—
121#214—22no—
122#213—22no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4D411yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3C2311yescharacteristic
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—41no—
137#204—22no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic, complemented
141C22×Q8—31no—
142C22×D4—31no—
143#135—64no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—64no—
151C2×SD16C2×D311yescharacteristic
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C22×C8—31no—
155C22×C8—31no—
156#124—31no—
157#124—31no—
158#124—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—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#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic, complemented
183C4×C8—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—81no—
191C22×D3C2×D411yescharacteristic
192C3×Q8—21no—
193C3×Q8C2×D412yes—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—42no—
199C3×D4—81no—
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—
210#67—42no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217D12—21no—
218D12C2×D412yes—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—42no—
222C4×D3—42no—
223C4×D3—81no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—21no—
228Dic6—42no—
229C3×C8—21no—
230C3×C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233C24—61no—
234#41—22no—
235#41—41no—
236#41—62no—
237#41—121no—
238C2×Q8—31no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—61no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—121no—
253C2×D4C22×D311yescharacteristic
254C2×D4C22×D311yescharacteristic
255C22×C4—21no—
256C22×C4—31no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4C22×D311yescharacteristic
261Q16—62no—
262Q16—121no—
263SD16—22no—
264SD16—34no—
265SD16—62no—
266SD16—62no—
267SD16—62no—
268D8—41no—
269D8—62no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4C22×D311yescharacteristic
282C22⋊C4—21no—
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—
292C22⋊C4—61no—
293C42—31no—
294C42—61no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3C22×D411yescharacteristic
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—22no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—84no—
311C2×D3#13512yescomplemented
312C3×C4—22no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4C22×D411yescharacteristic
316C3×C4H(5,𝔽2)11yescharacteristic
317Dic3—21no—
318Dic3—21no—
319Dic3—21no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323C23—21no—
324C23—21no—
325C23—31no—
326C23—61no—
327C23—122no—
328C23—122no—
329C23—241no—
330C23D3×D411yescharacteristic
331Q8—21no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—122no—
337Q8D3×D412yes—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—32no—
343D4—42no—
344D4—62no—
345D4—61no—
346D4—81no—
347D4—122no—
348D4—122no—
349D4—122no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—42no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—241no—
366C2×C4D3×D411yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—21no—
370C8—32no—
371C8—61no—
372C2×C3—41no—
373C2×C3—81no—
374C2×C3—81no—
375C2×C3D4211yescharacteristic
376C2×C3C2×#13511yescharacteristic
377C2×C3#57711yescharacteristic
378D3—22no—
379D3—42no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—32no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—61no—
391C22—61no—
392C22—62no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—22no—
399C4—41no—
400C4—61no—
401C4—61no—
402C4—61no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100411yescharacteristic
408C3#328211yescharacteristic, complemented, Sylow
409C2—41no—
410C2—62no—
411C2—81no—
412C2—81no—
413C2—122no—
414C2#559011yescharacteristic
415C2D3×#13511yescharacteristic
416C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3282
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #5831C2semidirect
2. C2×#950C2semidirect
3. C2×D3×SD16C2semidirect
4. #5612C2semidirect
5. C3×#449C2semidirect
6. #5166C2semidirect
7. #5156C2semidirect
8. #5155C2semidirect
9. #5041C2semidirect
10. #4809C2semidirect
11. #4788C2semidirect
12. #4787C2semidirect
13. #4778C2semidirect
14. #4733C2semidirect
15. #67⋊C8C2semidirect
17. C2×C3×#41C22semidirect
25. Q8×Dic3C22semidirect
26. C2×#216C22semidirect
34. #936C22semidirect
36. C2×#214C22semidirect
41. #924C22semidirect
42. C22×C3⋊C8C22semidirect
44. C2×Dic12C22semidirect
51. C3×#103C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
57. #816C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #810C22semidirect
61. #808C22semidirect
73. #449D3semidirect
119. #215D4semidirect
140. C2×#41C2×D3semidirect
177. #103C2×D3semidirect
180. #102C2×D3semidirect
182. C22⋊C8C2×D3semidirect
311. C2×D3#135semidirect
408. C3#3282semidirect

Maximal subgroups

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

subgroupindex
#58312
C2×#9502
C2×D3×SD162
#56122
C3×#4492
#51662
#51562
#51552
#50412
#48092
#47882
#47872
#47782
#47332
#67⋊C82
#32823