Finite Group Databasebeta

← #78994

Subgroups — #78994

Order 384.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#5964C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C2×#908C211yesmaximal, characteristic, complemented
4#5525C211yesmaximal, characteristic, complemented
5C3×#421C211yesmaximal, characteristic, complemented
6#5132C211yesmaximal, characteristic, complemented
7#5005C211yesmaximal, characteristic, complemented
8#4827C214yesmaximal, complemented
9#4822C214yesmaximal, complemented
10#3021—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12#1010—21no—
13#1010C2212yescomplemented
14#1009—21no—
15#1009C2212yescomplemented
16D3×#41—22no—
17D3×#41—22no—
18#1007—21no—
19#1004—21no—
20#1004—22no—
21C2×#235—22no—
22C2×#235C2211yescharacteristic
23C2×#235C2211yescharacteristic
24C22×D12C2211yescharacteristic, complemented
25C2×C3×M16C2211yescharacteristic, complemented
26C3×#117C2211yescharacteristic, complemented
27#949—22no—
28#949C2214yescomplemented
29C2×#215—21no—
30C2×#213—21no—
31C2×#208C2211yescharacteristic, complemented
32#908—22no—
33#908C2214yescomplemented
34C2×D24—21no—
35C2×#204—21no—
36#895—22no—
37#882—22no—
38#875C2212yescomplemented
39#874C2212yescomplemented
40C3×#104C2214yescomplemented
41#824C2212yescomplemented
42#823C2212yescomplemented
43#805C2214yescomplemented
44#585—31no—
45C2×AGL(1,ℤ/8)—31no—
46C2×AGL(1,ℤ/8)—31no—
47#535—31no—
48#454—34no—
49#453—34no—
50#421—31no—
51#421D311yescharacteristic, complemented
52#413—31no—
53C23×D3—21no—
54C3×#41—21no—
55C3×#41—21no—
56C3×#41D412yescomplemented
57C3×#41C2312yes—
58C2×C3×Q8D411yescharacteristic, complemented
59C2×C3×D4—21no—
60C2×C3×D4D411yescharacteristic, complemented
61C22×C3×C4—21no—
62C22×C3×C4C2311yescharacteristic
63C2×#67—24no—
64C2×#67—21no—
65C2×#67D412yes—
66#239—22no—
67#239—22no—
68D3×Q8—22no—
69D3×Q8—22no—
70#237—24no—
71#237—22no—
72#237—22no—
73#237—24no—
74D3×D4—24no—
75D3×D4—22no—
76D3×D4—24no—
77D3×D4—22no—
78#235—24no—
79#235—24no—
80#235—21no—
81#235—21no—
82#235—21no—
83#235—21no—
84#235—21no—
85#235—21no—
86#235—22no—
87#235—22no—
88#235D412yescomplemented
89#235C2312yes—
90C2×D12—22no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12—24no—
97C2×D12D411yescharacteristic, complemented
98C2×D12C2312yes—
99C2×D12C2311yescharacteristic
100C2×C4×D3—24no—
101C2×C4×D3—21no—
102C2×C4×D3D412yes—
103C2×Dic6—22no—
104C2×Dic6D411yescharacteristic, complemented
105C2×Dic6C2311yescharacteristic
106C3×M16—21no—
107C3×M16C2312yes—
108C2×C3×C8—21no—
109C3×C4⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C42C2312yes—
112#215—24no—
113#213—24no—
114(C2×D3)⋊C4—24no—
115#208—21no—
116#208C2312yes—
117C2×C3⋊C8—21no—
118D24—24no—
119#204—24no—
1202−1+4—32no—
1212−1+4—61no—
1222−1+4—61no—
123H(5,𝔽2)—32no—
124H(5,𝔽2)—62no—
125H(5,𝔽2)—61no—
126H(5,𝔽2)—61no—
127C2×#41—31no—
128C2×#41—31no—
129C2×#41—62no—
130C2×#41—62no—
131C2×#41—62no—
132C2×#41C2×D311yescharacteristic
133C22×D4—31no—
134AGL(1,ℤ/8)—34no—
135AGL(1,ℤ/8)—34no—
136AGL(1,ℤ/8)—62no—
137AGL(1,ℤ/8)—62no—
138C2×SD16—61no—
139C2×SD16—61no—
140C2×D8—61no—
141C2×D8—61no—
142C2×M16—31no—
143C2×M16C2×D311yescharacteristic, complemented
144#127—32no—
145#124—32no—
146#121—62no—
147#120—62no—
148#117C2×D311yescharacteristic, complemented
149#104—34no—
150#104C2×D314yescomplemented
151#101—32no—
152#100—32no—
153C23×C3—21no—
154C23×C3C2×D411yescharacteristic
155C22×D3—24no—
156C22×D3—21no—
157C22×D3—22no—
158C22×D3—41no—
159C22×D3—41no—
160C22×D3—82no—
161C22×D3C2×D412yes—
162C3×Q8—21no—
163C3×Q8C2×D412yes—
164C3×D4—22no—
165C3×D4—22no—
166C3×D4—21no—
167C3×D4C2×D412yes—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4C2×D411yescharacteristic
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178#67—24no—
179#67—24no—
180#67—22no—
181#67—22no—
182#67—22no—
183#67—22no—
184C2×Dic3—24no—
185C2×Dic3—21no—
186C2×Dic3C2×D412yes—
187D12—24no—
188D12—21no—
189D12—21no—
190D12—21no—
191D12—22no—
192D12—22no—
193D12—41no—
194D12—41no—
195D12—84no—
196D12C2×D412yes—
197C4×D3—24no—
198C4×D3—24no—
199C4×D3—22no—
200C4×D3—22no—
201C4×D3—22no—
202C4×D3—22no—
203Dic6—24no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6C2×D412yes—
208C3×C8—22no—
209C3⋊C8—22no—
210C24—61no—
211#41—21no—
212#41—21no—
213#41—32no—
214#41—32no—
215#41—62no—
216#41—62no—
217#41—64no—
218#41—64no—
219#41—64no—
220#41—64no—
221#41—62no—
222#41—62no—
223#41—62no—
224#41—62no—
225#41—61no—
226#41—61no—
227#41—61no—
228#41—61no—
229#41—61no—
230#41—61no—
231#41D1212yescomplemented
232#41C22×D312yes—
233C2×Q8—31no—
234C2×Q8—31no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8D1211yescharacteristic, complemented
239C2×D4—21no—
240C2×D4—32no—
241C2×D4—31no—
242C2×D4—31no—
243C2×D4—32no—
244C2×D4—62no—
245C2×D4—64no—
246C2×D4—64no—
247C2×D4—64no—
248C2×D4—62no—
249C2×D4—64no—
250C2×D4—61no—
251C2×D4—62no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4D1211yescharacteristic, complemented
258C22×C4—21no—
259C22×C4—32no—
260C22×C4—64no—
261C22×C4—61no—
262C22×C4C22×D311yescharacteristic
263SD16—64no—
264SD16—64no—
265D8—64no—
266D8—64no—
267M16—21no—
268M16—32no—
269M16—61no—
270M16C22×D312yes—
271C2×C8—21no—
272C2×C8—61no—
273C4⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—64no—
276C42C22×D312yes—
277C22×C3—22no—
278C22×C3—41no—
279C22×C3—81no—
280C22×C3#12011yescharacteristic
281C22×C3#12011yescharacteristic
282C22×C3C22×D411yescharacteristic
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—42no—
286C2×D3—44no—
287C2×D3—42no—
288C2×D3—81no—
289C2×D3—84no—
290C2×D3—82no—
291C2×D3—82no—
292C3×C4—22no—
293C3×C4—42no—
294C3×C4#12011yescharacteristic
295C3×C4#12011yescharacteristic
296C3×C4C22×D411yescharacteristic
297C3×C4C22×D411yescharacteristic
298Dic3—22no—
299Dic3—22no—
300C23—21no—
301C23—32no—
302C23—64no—
303C23—61no—
304C23—62no—
305C23—121no—
306C23—121no—
307C23—242no—
308C23D3×D411yescharacteristic
309Q8—21no—
310Q8—32no—
311Q8—64no—
312Q8—61no—
313Q8—61no—
314Q8—61no—
315Q8C2×D1212yes—
316D4—22no—
317D4—22no—
318D4—21no—
319D4—32no—
320D4—64no—
321D4—64no—
322D4—64no—
323D4—62no—
324D4—62no—
325D4—62no—
326D4—62no—
327D4—61no—
328D4—61no—
329D4—62no—
330D4—61no—
331D4—62no—
332D4—121no—
333D4—121no—
334D4—244no—
335D4C2×D1212yes—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—32no—
341C2×C4—64no—
342C2×C4—64no—
343C2×C4—64no—
344C2×C4—61no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4C2×D1211yescharacteristic
350C2×C4C2×D1211yescharacteristic
351C2×C4D3×D411yescharacteristic
352C2×C4D3×D411yescharacteristic
353C2×C4D3×D411yescharacteristic
354C2×C4C23×D311yescharacteristic
355C8—22no—
356C8—62no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—42no—
361C2×C3C2×#12011yescharacteristic
362D3—42no—
363D3—42no—
364D3—82no—
365C22—22no—
366C22—41no—
367C22—62no—
368C22—62no—
369C22—81no—
370C22—122no—
371C22—124no—
372C22—122no—
373C22—241no—
374C22—242no—
375C22—242no—
376C22—244no—
377C22#88211yescharacteristic
378C22#88211yescharacteristic
379C22C2×D3×D411yescharacteristic
380C4—22no—
381C4—42no—
382C4—62no—
383C4—62no—
384C4#88211yescharacteristic
385C4#88211yescharacteristic
386C4C22×D1211yescharacteristic
387C4C2×D3×D411yescharacteristic
388C3#302111yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—122no—
394C2—122no—
395C2—242no—
396C2C2×#88211yescharacteristic
subgroupG/H
centerC2C2×#882
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#421C2
socleC2×C3C2×#120
hypercenter#421D3
hypocenterC3#3021
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C2×#949C2semidirect
3. C2×#908C2semidirect
4. #5525C2semidirect
5. C3×#421C2semidirect
6. #5132C2semidirect
7. #5005C2semidirect
8. #4827C2semidirect
9. #4822C2semidirect
13. #1010C22semidirect
15. #1009C22semidirect
24. C22×D12C22semidirect
25. C2×C3×M16C22semidirect
26. C3×#117C22semidirect
28. #949C22semidirect
31. C2×#208C22semidirect
33. #908C22semidirect
38. #875C22semidirect
39. #874C22semidirect
40. C3×#104C22semidirect
41. #824C22semidirect
42. #823C22semidirect
43. #805C22semidirect
51. #421D3semidirect
56. C3×#41D4semidirect
58. C2×C3×Q8D4semidirect
60. C2×C3×D4D4semidirect
88. #235D4semidirect
97. C2×D12D4semidirect
104. C2×Dic6D4semidirect
143. C2×M16C2×D3semidirect
148. #117C2×D3semidirect
150. #104C2×D3semidirect
231. #41D12semidirect
238. C2×Q8D12semidirect
257. C2×D4D12semidirect
388. C3#3021semidirect

Maximal subgroups

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

subgroupindex
#59642
C2×#9492
C2×#9082
#55252
C3×#4212
#51322
#50052
#48272
#48222
#30213