Finite Group Databasebeta

← #78747

Subgroups — #78747

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×#932C211yesmaximal, characteristic, complemented
3C2×#904C211yesmaximal, characteristic, complemented
4#5510C211yesmaximal, characteristic, complemented
5C2×C3×#102C211yesmaximal, characteristic, complemented
6#5110C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic, complemented
8#4783C212yesmaximal, complemented
9#4781C212yesmaximal, complemented
10#4776C212yesmaximal, complemented
11#4773C212yesmaximal, complemented
12#3012—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—22no—
15#1004—24no—
16#1004—24no—
17C2×#237—21no—
18C2×#237C2212yes—
19C2×D3×D4—21no—
20C2×D3×D4C2212yes—
21C2×#235C2211yescharacteristic
22C2×#235C2211yescharacteristic
23C2×#235C2211yescharacteristic, complemented
24C22×C3×C8C2211yescharacteristic, complemented
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×#214C2212yescomplemented
27#932—24no—
28C2×#213C2212yescomplemented
29#930—22no—
30C2×#208C2211yescharacteristic, complemented
31C2×Dic12C2211yescharacteristic, complemented
32#904—24no—
33C2×D24C2211yescharacteristic, complemented
34C2×#204C2211yescharacteristic, complemented
35C2×#204C2211yescharacteristic, complemented
36#897—21no—
37#897C2212yescomplemented
38#895—21no—
39#895C2212yescomplemented
40C3×#102C2212yescomplemented
41C3×#102C2212yescomplemented
42(C2×D3)⋊C8C2212yescomplemented
43(C2×D3)⋊C8C2212yescomplemented
44#810C2212yescomplemented
45#809C2212yescomplemented
46C2×H(5,𝔽2)—31no—
47C2×AGL(1,ℤ/8)—31no—
48C2×#135—31no—
49#537—31no—
50#452—32no—
51#450—32no—
52#449—32no—
53#447—32no—
54#417—31no—
55C2×#102D311yescharacteristic, complemented
56#408—31no—
57C24×C3—21no—
58C23×D3—22no—
59C2×C3×D4—21no—
60C2×C3×D4—22no—
61C2×C3×D4D414yescomplemented
62C2×C3×D4C2312yes—
63C22×C3×C4—21no—
64C22×C3×C4C2311yescharacteristic
65C2×#67—216no—
66C2×#67—21no—
67C2×#67—22no—
68C2×#67—21no—
69C2×#67—22no—
70C2×#67D412yes—
71C22×Dic3—22no—
72#237—28no—
73#237—24no—
74#237—24no—
75D3×D4—28no—
76D3×D4—24no—
77D3×D4—24no—
78#235—22no—
79#235—22no—
80#235—22no—
81#235—22no—
82#235—22no—
83#235D414yescomplemented
84C2×D12—22no—
85C2×D12D411yescharacteristic
86C2×D12C2311yescharacteristic
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3C2312yes—
91C2×Dic6D411yescharacteristic
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C2×C3×C8—22no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2312yes—
99#214—24no—
100#213—24no—
101(C2×D3)⋊C4—22no—
102(C2×D3)⋊C4—22no—
103C4⋊Dic3—22no—
104#208—22no—
105C2×C3⋊C8C2312yes—
106Dic12—22no—
107D24—22no—
108#204—22no—
109#204—22no—
110H(5,𝔽2)—64no—
111H(5,𝔽2)—64no—
112C2×#41—31no—
113C2×#41—32no—
114C2×#41—31no—
115C2×#41—31no—
116C2×#41—61no—
117C22×D4—32no—
118C22×D4—61no—
119C22×D4—62no—
120C22×D4C2×D311yescharacteristic
121AGL(1,ℤ/8)—64no—
122#135—64no—
123C2×Q16—31no—
124C2×SD16—32no—
125C2×SD16—31no—
126C2×SD16—31no—
127C2×D8—32no—
128C2×D8—31no—
129C2×M16—31no—
130C22×C8C2×D311yescharacteristic, complemented
131C22⋊Q8—32no—
132C22⋊Q8—61no—
133#121—32no—
134#121—61no—
135#121—62no—
136C2×C4⋊C4C2×D311yescharacteristic, complemented
137#103—32no—
138#102—32no—
139#102C2×D312yescomplemented
140#102C2×D312yescomplemented
141C22⋊C8—32no—
142C22⋊C8—32no—
143C23×C3—22no—
144C23×C3—24no—
145C23×C3—81no—
146C23×C3—81no—
147C23×C3C2×D411yescharacteristic
148C22×D3—21no—
149C22×D3—28no—
150C22×D3—21no—
151C22×D3—82no—
152C22×D3—82no—
153C22×D3C2×D412yes—
154C3×D4—24no—
155C3×D4—22no—
156C3×D4C2×D414yes—
157C2×C3×C4—22no—
158C2×C3×C4—81no—
159C2×C3×C4C2×D412yes—
160C2×C3×C4C2×D411yescharacteristic
161C2×C3×C4C2×D412yes—
162C2×C3×C4C2411yescharacteristic
163#67—28no—
164#67—28no—
165#67—24no—
166#67—24no—
167#67—42no—
168C2×Dic3—28no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3C2×D412yes—
172D12—22no—
173D12—21no—
174D12—21no—
175D12—21no—
176D12—82no—
177D12C2×D412yes—
178C4×D3—24no—
179C4×D3—22no—
180C4×D3—22no—
181C4×D3—42no—
182Dic6—22no—
183Dic6—21no—
184Dic6—21no—
185Dic6—21no—
186Dic6C2×D412yes—
187C3×C8—21no—
188C3×C8—21no—
189C3⋊C8—22no—
190C24—21no—
191C24—62no—
192#41—34no—
193#41—64no—
194#41—68no—
195#41—64no—
196#41—62no—
197#41—62no—
198#41—62no—
199#41—62no—
200#41—62no—
201C2×Q8—31no—
202C2×Q8—31no—
203C2×Q8—31no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—32no—
207C2×D4—31no—
208C2×D4—31no—
209C2×D4—31no—
210C2×D4—62no—
211C2×D4—62no—
212C2×D4—64no—
213C2×D4—68no—
214C2×D4—62no—
215C2×D4—616no—
216C2×D4—61no—
217C2×D4—64no—
218C2×D4—61no—
219C2×D4D1214yescomplemented
220C2×D4C22×D312yes—
221C22×C4—21no—
222C22×C4—32no—
223C22×C4—62no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4C22×D311yescharacteristic
227Q16—62no—
228SD16—62no—
229SD16—62no—
230SD16—64no—
231D8—62no—
232D8—64no—
233M16—62no—
234C2×C8—22no—
235C2×C8—32no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—62no—
240C4⋊C4C22×D312yes—
241C22⋊C4—62no—
242C22⋊C4—62no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—24no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—84no—
249C22×C3#12012yes—
250C22×C3C22×D411yescharacteristic
251C2×D3—22no—
252C2×D3—22no—
253C2×D3—22no—
254C2×D3—22no—
255C2×D3—41no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—88no—
261C3×C4—42no—
262C3×C4#12012yes—
263C3×C4C22×D411yescharacteristic
264C3×C4C22×D411yescharacteristic
265Dic3—22no—
266Dic3—22no—
267Dic3—41no—
268C23—22no—
269C23—24no—
270C23—32no—
271C23—68no—
272C23—61no—
273C23—61no—
274C23—81no—
275C23—81no—
276C23—242no—
277C23—242no—
278C23D3×D411yescharacteristic
279Q8—32no—
280Q8—62no—
281Q8—61no—
282Q8—61no—
283Q8—61no—
284D4—24no—
285D4—22no—
286D4—32no—
287D4—68no—
288D4—68no—
289D4—64no—
290D4—64no—
291D4—62no—
292D4—61no—
293D4—61no—
294D4—61no—
295D4—122no—
296D4—242no—
297D4C2×D1214yes—
298C2×C4—22no—
299C2×C4—32no—
300C2×C4—68no—
301C2×C4—61no—
302C2×C4—64no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—61no—
306C2×C4—81no—
307C2×C4—122no—
308C2×C4C2×D1212yes—
309C2×C4D3×D411yescharacteristic
310C2×C4D3×D412yes—
311C2×C4C23×D311yescharacteristic
312C8—21no—
313C8—21no—
314C8—62no—
315C2×C3—22no—
316C2×C3—44no—
317C2×C3C2×#12011yescharacteristic
318C2×C3#57611yescharacteristic
319C2×C3#57711yescharacteristic
320D3—42no—
321D3—42no—
322D3—81no—
323C22—21no—
324C22—21no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22—62no—
329C22—62no—
330C22—62no—
331C22—62no—
332C22—84no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22—248no—
339C22#88212yes—
340C22C2×D3×D411yescharacteristic
341C4—42no—
342C4—62no—
343C4—62no—
344C4—121no—
345C4#88212yes—
346C4C22×D1211yescharacteristic
347C4C2×D3×D411yescharacteristic
348C3#301211yescharacteristic, complemented, Sylow
349C2—22no—
350C2—44no—
351C2—122no—
352C2—122no—
353C2—241no—
354C2C2×#88211yescharacteristic
355C2#575711yescharacteristic
356C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#102C2
socleC22×C3C22×D4
hypercenterC2×#102D3
hypocenterC3#3012
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C2×#932C2semidirect
3. C2×#904C2semidirect
4. #5510C2semidirect
5. C2×C3×#102C2semidirect
6. #5110C2semidirect
7. #4966C2semidirect
8. #4783C2semidirect
9. #4781C2semidirect
10. #4776C2semidirect
11. #4773C2semidirect
23. C2×#235C22semidirect
24. C22×C3×C8C22semidirect
25. C2×C3×C4⋊C4C22semidirect
26. C2×#214C22semidirect
28. C2×#213C22semidirect
30. C2×#208C22semidirect
31. C2×Dic12C22semidirect
33. C2×D24C22semidirect
34. C2×#204C22semidirect
35. C2×#204C22semidirect
37. #897C22semidirect
39. #895C22semidirect
40. C3×#102C22semidirect
41. C3×#102C22semidirect
42. (C2×D3)⋊C8C22semidirect
43. (C2×D3)⋊C8C22semidirect
44. #810C22semidirect
45. #809C22semidirect
55. C2×#102D3semidirect
61. C2×C3×D4D4semidirect
83. #235D4semidirect
130. C22×C8C2×D3semidirect
136. C2×C4⋊C4C2×D3semidirect
139. #102C2×D3semidirect
140. #102C2×D3semidirect
219. C2×D4D12semidirect
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×#9322
C2×#9042
#55102
C2×C3×#1022
#51102
#49662
#47832
#47812
#47762
#47732
#30123