Finite Group Databasebeta

← #83949

Subgroups — #83949

Order 384.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1004C211yesmaximal, characteristic, complemented
2C2×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#5833C211yesmaximal, characteristic, complemented
4C2×#950C211yesmaximal, characteristic, complemented
5C2×#832C211yesmaximal, characteristic, complemented
6#5173C212yesmaximal, complemented
7#5169C212yesmaximal, complemented
8#5158C212yesmaximal, complemented
9#5156C212yesmaximal, complemented
10#5133C211yesmaximal, characteristic, complemented
11#5129C211yesmaximal, characteristic, complemented
12#3012—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic, complemented
14C22×C3×D4C2211yescharacteristic
15C22×#67—22no—
16#1004—24no—
17#1004—24no—
18C2×#237—21no—
19C2×#237C2212yes—
20C2×D3×D4—21no—
21C2×D3×D4C2212yes—
22C2×#235C2211yescharacteristic
23C2×#235C2211yescharacteristic
24C3×AGL(1,ℤ/8)—24no—
25C2×C3×SD16C2212yescomplemented
26C2×C3×D8C2212yescomplemented
27C2×C3×M16C2211yescharacteristic, complemented
28#950—24no—
29#946C2212yescomplemented
30C2×#216C2211yescharacteristic, complemented
31C2×#215C2211yescharacteristic, complemented
32#939—22no—
33#938C2212yescomplemented
34C2×#214C2211yescharacteristic, complemented
35C2×#213C2211yescharacteristic, complemented
36#930—21no—
37C2×C4⋊Dic3C2211yescharacteristic, complemented
38#924—21no—
39C22×C3⋊C8C2211yescharacteristic, complemented
40#832C2212yescomplemented
41#832C2212yescomplemented
42#821C2212yescomplemented
43(C2×D3)⋊C8C2212yescomplemented
44(C2×D3)⋊C8C2212yescomplemented
45#816C2212yescomplemented
46C2×H(5,𝔽2)—31no—
47C2×AGL(1,ℤ/8)D311yescharacteristic, complemented
48C2×#135—31no—
49#537—31no—
50#452—32no—
51#450—32no—
52#449—32no—
53#447—32no—
54#417—31no—
55C2×#102—31no—
56#408—31no—
57C24×C3—21no—
58C23×D3—22no—
59C3×#41—22no—
60C3×#41—22no—
61C2×C3×Q8C2311yescharacteristic
62C2×C3×D4—21no—
63C2×C3×D4—21no—
64C2×C3×D4—22no—
65C2×C3×D4D414yescomplemented
66C2×C3×D4C2311yescharacteristic
67C2×C3×D4C2312yes—
68C22×C3×C4—21no—
69C22×C3×C4C2311yescharacteristic
70C2×#67—216no—
71C2×#67—21no—
72C2×#67—22no—
73C2×#67—22no—
74C2×#67—22no—
75C2×#67D412yes—
76C22×Dic3—22no—
77C22×Dic3—21no—
78#237—28no—
79#237—24no—
80#237—24no—
81D3×D4—28no—
82D3×D4—24no—
83D3×D4—24no—
84#235—22no—
85#235—22no—
86#235—22no—
87#235D414yescomplemented
88C2×D12D411yescharacteristic
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3C2312yes—
92C2×Dic6D411yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×SD16—24no—
95C3×D8—24no—
96C3×M16—22no—
97C2×C3×C8C2312yes—
98#217—22no—
99#216—22no—
100#215—22no—
101#214—22no—
102#213—22no—
103(C2×D3)⋊C4—22no—
104C4⋊Dic3—21no—
105C4⋊Dic3C2312yes—
106Dic3⋊C4—22no—
107C2×C3⋊C8—22no—
108C2×C3⋊C8C2311yescharacteristic
109C2×C3⋊C8C2311yescharacteristic
110H(5,𝔽2)—64no—
111H(5,𝔽2)—64no—
112C2×#41—31no—
113C2×#41—32no—
114C2×#41—31no—
115C2×#41—61no—
116C2×#41C2×D311yescharacteristic, complemented
117C22×D4—32no—
118C22×D4—61no—
119C22×D4—62no—
120C22×D4C2×D311yescharacteristic
121AGL(1,ℤ/8)—24no—
122#135—64no—
123C2×Q16—31no—
124C2×SD16—31no—
125C2×SD16—31no—
126C2×SD16C2×D312yescomplemented
127C2×D8—31no—
128C2×D8C2×D312yescomplemented
129C2×M16C2×D311yescharacteristic, complemented
130C22×C8—31no—
131C22⋊Q8—32no—
132C22⋊Q8—61no—
133#121—32no—
134#121—61no—
135#121—62no—
136C2×C4⋊C4—31no—
137#103—32no—
138#102—32no—
139#102—32no—
140#102—32no—
141C22⋊C8—32no—
142C22⋊C8—32no—
143C23×C3—22no—
144C23×C3—24no—
145C23×C3—21no—
146C23×C3—81no—
147C23×C3—81no—
148C23×C3C2×D411yescharacteristic
149C22×D3—21no—
150C22×D3—28no—
151C22×D3—82no—
152C22×D3—82no—
153C22×D3C2×D412yes—
154C3×Q8—21no—
155C3×Q8—21no—
156C3×D4—24no—
157C3×D4—22no—
158C3×D4—21no—
159C3×D4—21no—
160C3×D4—42no—
161C3×D4C2×D414yes—
162C2×C3×C4—21no—
163C2×C3×C4—42no—
164C2×C3×C4C2×D411yescharacteristic
165C2×C3×C4C2×D412yes—
166C2×C3×C4C2×D412yes—
167C2×C3×C4C2411yescharacteristic
168#67—28no—
169#67—28no—
170#67—24no—
171#67—24no—
172#67—82no—
173C2×Dic3—28no—
174C2×Dic3—21no—
175C2×Dic3—22no—
176C2×Dic3—81no—
177C2×Dic3C2×D412yes—
178D12—22no—
179D12—21no—
180D12C2×D412yes—
181C4×D3—24no—
182C4×D3—22no—
183C4×D3—22no—
184Dic6—22no—
185Dic6—21no—
186Dic6C2×D412yes—
187C3×C8—22no—
188C3⋊C8—21no—
189C3⋊C8—21no—
190C24—21no—
191C24—62no—
192#41—22no—
193#41—22no—
194#41—34no—
195#41—64no—
196#41—68no—
197#41—64no—
198#41—62no—
199#41—62no—
200#41—62no—
201C2×Q8—31no—
202C2×Q8—31no—
203C2×Q8C22×D311yescharacteristic
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—32no—
208C2×D4—31no—
209C2×D4—31no—
210C2×D4—62no—
211C2×D4—64no—
212C2×D4—68no—
213C2×D4—62no—
214C2×D4—616no—
215C2×D4—61no—
216C2×D4—64no—
217C2×D4—62no—
218C2×D4#6714yescomplemented
219C2×D4C22×D311yescharacteristic
220C2×D4C22×D312yes—
221C22×C4—21no—
222C22×C4—32no—
223C22×C4—62no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4C22×D311yescharacteristic
227Q16—62no—
228SD16—24no—
229SD16—62no—
230SD16—62no—
231D8—24no—
232D8—62no—
233M16—22no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—62no—
237C2×C8C22×D312yes—
238C4⋊C4—32no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C22⋊C4—62no—
242C22⋊C4—62no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—24no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—84no—
252C22×C3#12012yes—
253C22×C3C22×D411yescharacteristic
254C2×D3—22no—
255C2×D3—22no—
256C2×D3—22no—
257C2×D3—22no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—88no—
261C3×C4—41no—
262C3×C4#12012yes—
263C3×C4C22×D411yescharacteristic
264C3×C4C22×D411yescharacteristic
265Dic3—22no—
266Dic3—22no—
267Dic3—42no—
268C23—22no—
269C23—24no—
270C23—21no—
271C23—32no—
272C23—68no—
273C23—61no—
274C23—81no—
275C23—81no—
276C23—242no—
277C23—242no—
278C23D3×D411yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—32no—
282Q8—62no—
283Q8—61no—
284D4—24no—
285D4—22no—
286D4—21no—
287D4—21no—
288D4—32no—
289D4—42no—
290D4—68no—
291D4—68no—
292D4—64no—
293D4—64no—
294D4—62no—
295D4—61no—
296D4—242no—
297D4C2×#6714yes—
298C2×C4—21no—
299C2×C4—32no—
300C2×C4—42no—
301C2×C4—68no—
302C2×C4—61no—
303C2×C4—64no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—241no—
308C2×C4D3×D411yescharacteristic
309C2×C4D3×D412yes—
310C2×C4C2×#6712yes—
311C2×C4C23×D311yescharacteristic
312C8—22no—
313C8—61no—
314C8—61no—
315C2×C3—22no—
316C2×C3—44no—
317C2×C3—81no—
318C2×C3C2×#12011yescharacteristic
319C2×C3#57611yescharacteristic
320C2×C3#57711yescharacteristic
321D3—42no—
322D3—42no—
323C22—21no—
324C22—21no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—62no—
332C22—62no—
333C22—62no—
334C22—62no—
335C22—84no—
336C22—121no—
337C22—121no—
338C22—248no—
339C22#93712yes—
340C22C2×D3×D411yescharacteristic
341C4—41no—
342C4—62no—
343C4—62no—
344C4—122no—
345C4#93712yes—
346C4C2×D3×D411yescharacteristic
347C4C22×#6711yescharacteristic
348C3#301211yescharacteristic, complemented, Sylow
349C2—22no—
350C2—44no—
351C2—81no—
352C2—122no—
353C2—122no—
354C2#577411yescharacteristic
355C2#577511yescharacteristic
356C2C2×#93711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×AGL(1,ℤ/8)C2
socleC22×C3C22×D4
hypercenterC2×AGL(1,ℤ/8)D3
hypocenterC3#3012
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C2×C3×AGL(1,ℤ/8)C2semidirect
3. #5833C2semidirect
4. C2×#950C2semidirect
5. C2×#832C2semidirect
6. #5173C2semidirect
7. #5169C2semidirect
8. #5158C2semidirect
9. #5156C2semidirect
10. #5133C2semidirect
11. #5129C2semidirect
13. C2×C3×#41C22semidirect
25. C2×C3×SD16C22semidirect
26. C2×C3×D8C22semidirect
27. C2×C3×M16C22semidirect
29. #946C22semidirect
30. C2×#216C22semidirect
31. C2×#215C22semidirect
33. #938C22semidirect
34. C2×#214C22semidirect
35. C2×#213C22semidirect
37. C2×C4⋊Dic3C22semidirect
39. C22×C3⋊C8C22semidirect
40. #832C22semidirect
41. #832C22semidirect
42. #821C22semidirect
43. (C2×D3)⋊C8C22semidirect
44. (C2×D3)⋊C8C22semidirect
45. #816C22semidirect
47. C2×AGL(1,ℤ/8)D3semidirect
65. C2×C3×D4D4semidirect
87. #235D4semidirect
116. C2×#41C2×D3semidirect
126. C2×SD16C2×D3semidirect
128. C2×D8C2×D3semidirect
129. C2×M16C2×D3semidirect
218. C2×D4#67semidirect
348. C3#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C2×C3×AGL(1,ℤ/8)2
#58332
C2×#9502
C2×#8322
#51732
#51692
#51582
#51562
#51332
#51292
#30123