Finite Group Databasebeta

← #80277

Subgroups — #80277

Order 384.

724 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#5966C211yesmaximal, characteristic, complemented
2#5773C212yesmaximal, complemented
3#5716C211yesmaximal, characteristic, complemented
4C3×#453C211yesmaximal, characteristic, complemented
5#5242C211yesmaximal, characteristic, complemented
6#5201C212yesmaximal, complemented
7#5077C211yesmaximal, characteristic, complemented
8#4826C212yesmaximal, complemented
9D3×#104C212yesmaximal, complemented
10#4746C211yesmaximal, characteristic, complemented
11#4745C211yesmaximal, characteristic, complemented
12#3018—31nomaximal, Sylow
13C3×H(5,𝔽2)C2211yescharacteristic
14#1010—22no—
15#1010C2212yes—
16D3×#41—22no—
17D3×#41C2212yes—
18#1004—22no—
19#1004—22no—
20#1004C2211yescharacteristic
21C2×#237—22no—
22C2×#237—22no—
23C2×#237C2211yescharacteristic
24C2×#237C2211yescharacteristic, complemented
25C3×AGL(1,ℤ/8)C2212yescomplemented
26C3×#127C2211yescharacteristic, complemented
27#951C2212yescomplemented
28#938—22no—
29#936C2211yescharacteristic, complemented
30#936C2211yescharacteristic, complemented
31D4×Dic3—22no—
32#932C2212yescomplemented
33#916—22no—
34#915—22no—
35#912—22no—
36#911—22no—
37#909C2212yescomplemented
38D3×M16C2212yescomplemented
39C42×D3C2211yescharacteristic, complemented
40#869C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#100C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#833C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#824C2212yescomplemented
47#805C2212yescomplemented
48#585—31no—
49#575—32no—
50#532—31no—
51#456—32no—
52#455—31no—
53#455—31no—
54#454—32no—
55#453—31no—
56#453D311yescharacteristic, complemented
57C2×#104—32no—
58#413—31no—
59C3×#41—22no—
60C3×#41C2312yes—
61C2×C3×D4—22no—
62C2×C3×D4—22no—
63C2×C3×D4—22no—
64C2×C3×D4C2311yescharacteristic
65C2×C3×D4C2311yescharacteristic
66C2×#67—24no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—21no—
70C2×#67D412yes—
71C22×Dic3—22no—
72C22×Dic3—21no—
73C22×Dic3D412yes—
74#239D412yescomplemented
75D3×Q8—22no—
76D3×Q8D412yescomplemented
77#237—24no—
78#237—24no—
79#237—24no—
80#237—22no—
81#237—22no—
82#237—24no—
83#237—22no—
84#237—22no—
85#237—22no—
86#237—22no—
87#237—22no—
88#237D412yescomplemented
89D3×D4—24no—
90D3×D4—22no—
91D3×D4—22no—
92D3×D4D412yescomplemented
93#235—24no—
94#235—22no—
95#235—22no—
96#235C2312yes—
97C2×C4×D3—22no—
98C2×C4×D3—22no—
99C2×C4×D3—21no—
100C2×C4×D3C2311yescharacteristic
101C2×Dic6—22no—
102C2×Dic6—22no—
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×SD16—22no—
106C3×D8—22no—
107C3×M16C2312yes—
108C3×C42C2311yescharacteristic
109#217—22no—
110#217—22no—
111#216—22no—
112#214—22no—
113#214—22no—
114#213—22no—
115C4⋊Dic3—22no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118#208C2312yes—
119Dic12—22no—
120#204—22no—
121ΓL(1,𝔽25)—22no—
122D3×C8—22no—
1232−1+4—32no—
1242−1+4—62no—
125H(5,𝔽2)—31no—
126H(5,𝔽2)—62no—
127H(5,𝔽2)—62no—
128H(5,𝔽2)C2×D311yescharacteristic
129C2×#41—31no—
130C2×#41—32no—
131C2×#41—31no—
132C2×#41—62no—
133C2×#41—62no—
134C2×#41—62no—
135#137—32no—
136#137—32no—
137#137—62no—
138AGL(1,ℤ/8)—32no—
139AGL(1,ℤ/8)—62no—
140AGL(1,ℤ/8)C2×D312yescomplemented
141#135—62no—
142#135—62no—
143C2×M16—32no—
144C4⋊Q8—31no—
145#127C2×D311yescharacteristic, complemented
146#124—31no—
147#124—31no—
148#121—62no—
149C4×D4—62no—
150C2×C42—31no—
151#104—32no—
152#104—32no—
153#104—32no—
154#104C2×D312yescomplemented
155#101—32no—
156#100—31no—
157#100C2×D311yescharacteristic, complemented
158C23×C3—22no—
159C23×C3—21no—
160C23×C3C2×D412yes—
161C22×D3—22no—
162C22×D3—22no—
163C22×D3C2×D411yescharacteristic
164C3×Q8C2×D412yes—
165C3×D4—24no—
166C3×D4—22no—
167C3×D4—22no—
168C3×D4—22no—
169C3×D4—44no—
170C3×D4C2×D412yes—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—22no—
174C2×C3×C4C2411yescharacteristic
175#67—24no—
176#67—24no—
177#67—22no—
178#67—24no—
179#67—22no—
180#67—22no—
181#67—42no—
182C2×Dic3—24no—
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—22no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3C2×D412yes—
192C2×Dic3C2×D411yescharacteristic
193D12—22no—
194D12C2×D412yes—
195C4×D3—24no—
196C4×D3—22no—
197C4×D3—22no—
198C4×D3—22no—
199C4×D3—42no—
200C4×D3C2×D412yes—
201C4×D3C2×D412yes—
202Dic6—24no—
203Dic6—22no—
204Dic6—22no—
205Dic6—22no—
206Dic6C2×D412yes—
207C3×C8—22no—
208C3⋊C8—22no—
209#41—22no—
210#41—32no—
211#41—32no—
212#41—32no—
213#41—62no—
214#41—62no—
215#41—64no—
216#41—62no—
217#41—62no—
218#41—62no—
219#41—64no—
220#41—62no—
221#41—62no—
222#41—62no—
223#41—64no—
224#41—64no—
225#41—64no—
226#41—62no—
227#41C22×D312yes—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8—31no—
231C2×Q8—62no—
232C2×Q8—62no—
233C2×Q8—62no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—32no—
238C2×D4—32no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—64no—
242C2×D4—62no—
243C2×D4—64no—
244C2×D4—62no—
245C2×D4—61no—
246C2×D4C22×D311yescharacteristic
247C2×D4C22×D311yescharacteristic
248C22×C4—31no—
249C22×C4—32no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—62no—
255Q16—62no—
256Q16—62no—
257SD16—22no—
258SD16—62no—
259SD16—62no—
260SD16—62no—
261D8—22no—
262D8—62no—
263M16—32no—
264M16—62no—
265M16C22×D312yes—
266C2×C8—62no—
267C4⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C42—31no—
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—82no—
279C22×C3C22×D411yescharacteristic
280C2×D3—22no—
281C2×D3—41no—
282C2×D3—82no—
283C2×D3—82no—
284C2×D3#12012yes—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4C22×D412yes—
288Dic3—22no—
289Dic3—22no—
290Dic3—41no—
291Dic3—41no—
292Dic3#12011yescharacteristic
293Dic3#12011yescharacteristic
294C23—22no—
295C23—21no—
296C23—31no—
297C23—62no—
298C23—62no—
299C23D3×D412yes—
300Q8—32no—
301Q8—62no—
302Q8—64no—
303Q8—62no—
304Q8—62no—
305Q8D3×D412yes—
306D4—24no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—32no—
311D4—44no—
312D4—62no—
313D4—62no—
314D4—64no—
315D4—62no—
316D4—64no—
317D4—64no—
318D4—62no—
319D4—122no—
320D4D3×D412yes—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—31no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—64no—
331C2×C4—62no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—64no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4C23×D311yescharacteristic
343C8—22no—
344C8—62no—
345C2×C3—21no—
346C2×C3—42no—
347C2×C3—42no—
348C2×C3—81no—
349C2×C3C2×#12011yescharacteristic
350D3—22no—
351D3—42no—
352C22—22no—
353C22—22no—
354C22—32no—
355C22—42no—
356C22—42no—
357C22—41no—
358C22—62no—
359C22—82no—
360C22—121no—
361C22—242no—
362C22—242no—
363C22C2×D3×D411yescharacteristic
364C4—22no—
365C4—22no—
366C4—31no—
367C4—31no—
368C4—62no—
369C4—62no—
370C4—121no—
371C4—121no—
372C4C2×D3×D412yes—
373C3#301811yescharacteristic, complemented, Sylow
374C2—21no—
375C2—42no—
376C2—42no—
377C2—62no—
378C2—81no—
379C2—122no—
380C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#453C2
socleC2×C3C2×#120
hypercenter#453D3
hypocenterC3#3018
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #5966C2semidirect
2. #5773C2semidirect
3. #5716C2semidirect
4. C3×#453C2semidirect
5. #5242C2semidirect
6. #5201C2semidirect
7. #5077C2semidirect
8. #4826C2semidirect
9. D3×#104C2semidirect
10. #4746C2semidirect
11. #4745C2semidirect
24. C2×#237C22semidirect
25. C3×AGL(1,ℤ/8)C22semidirect
26. C3×#127C22semidirect
27. #951C22semidirect
29. #936C22semidirect
30. #936C22semidirect
32. #932C22semidirect
37. #909C22semidirect
38. D3×M16C22semidirect
39. C42×D3C22semidirect
40. #869C22semidirect
41. C3×#104C22semidirect
42. C3×#100C22semidirect
43. #837C22semidirect
44. #833C22semidirect
45. #825C22semidirect
46. #824C22semidirect
47. #805C22semidirect
56. #453D3semidirect
74. #239D4semidirect
76. D3×Q8D4semidirect
88. #237D4semidirect
92. D3×D4D4semidirect
140. AGL(1,ℤ/8)C2×D3semidirect
145. #127C2×D3semidirect
154. #104C2×D3semidirect
157. #100C2×D3semidirect
373. C3#3018semidirect

Maximal subgroups

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

subgroupindex
#59662
#57732
#57162
C3×#4532
#52422
#52012
#50772
#48262
D3×#1042
#47462
#47452
#30183