Finite Group Databasebeta

← #78993

Subgroups — #78993

Order 384.

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

normal subgroups only

This group has 762 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
1C2×#1009C211yesmaximal, characteristic, complemented
2#5822C211yesmaximal, characteristic, complemented
3#5748C211yesmaximal, characteristic, complemented
4#5525C211yesmaximal, characteristic, complemented
5C3×#421C211yesmaximal, characteristic, complemented
6C2×#823C211yesmaximal, characteristic, complemented
7#5005C211yesmaximal, characteristic, complemented
8#4825C214yesmaximal, complemented
9#4822C214yesmaximal, complemented
10#3019—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12#1009—22no—
13#1009—22no—
14#1009—22no—
15#1009C2214yes—
16C2×#239—21no—
17C2×D3×D4—22no—
18C2×D3×D4—21no—
19C2×#235—21no—
20C2×#235C2211yescharacteristic
21C2×#235C2211yescharacteristic, complemented
22C22×D12—21no—
23C22×D12C2211yescharacteristic
24C2×C3×M16C2211yescharacteristic, complemented
25C3×#117C2211yescharacteristic, complemented
26#951C2212yescomplemented
27#950—22no—
28#949C2212yescomplemented
29#942—21no—
30#932—21no—
31C2×#208C2211yescharacteristic, complemented
32#909—21no—
33#909C2212yescomplemented
34#908—21no—
35#908C2212yescomplemented
36#904—22no—
37#895—22no—
38#882—22no—
39#875C2212yescomplemented
40#874C2212yescomplemented
41C3×#104C2214yescomplemented
42#823C2214yescomplemented
43#805C2214yescomplemented
44C2×H(5,𝔽2)—31no—
45#575—31no—
46#575—31no—
47#535—31no—
48#455—34no—
49#453—34no—
50#421—31no—
51#421D311yescharacteristic, complemented
52C2×#100—31no—
53C23×D3—22no—
54C23×D3D412yes—
55C3×#41—21no—
56C3×#41—21no—
57C3×#41D412yescomplemented
58C3×#41C2312yes—
59C2×C3×Q8D411yescharacteristic, complemented
60C2×C3×D4—21no—
61C2×C3×D4D411yescharacteristic, complemented
62C22×C3×C4—21no—
63C22×C3×C4C2311yescharacteristic
64C2×#67—22no—
65C2×#67—21no—
66C2×#67—21no—
67#239—24no—
68#239—24no—
69D3×D4—28no—
70D3×D4—24no—
71D3×D4—28no—
72D3×D4—24no—
73#235—24no—
74#235—21no—
75#235—21no—
76#235—21no—
77#235—21no—
78#235—21no—
79#235—21no—
80#235—24no—
81#235D412yescomplemented
82#235C2312yes—
83C2×D12—22no—
84C2×D12—24no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—24no—
90C2×D12—24no—
91C2×D12D412yes—
92C2×D12D411yescharacteristic, complemented
93C2×D12C2312yes—
94C2×D12C2311yescharacteristic, complemented
95C2×C4×D3—22no—
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×Dic6D411yescharacteristic, complemented
99C2×Dic6C2311yescharacteristic, complemented
100C3×M16—21no—
101C3×M16C2312yescomplemented
102C2×C3×C8—21no—
103C3×C4⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C42C2312yescomplemented
106#216—22no—
107#215—22no—
108#214—22no—
109#213—22no—
110(C2×D3)⋊C4—24no—
111#208—21no—
112#208C2312yescomplemented
113C2×C3⋊C8—21no—
114Dic12—22no—
115D24—22no—
116#204—22no—
117#204—22no—
118H(5,𝔽2)—34no—
119H(5,𝔽2)—62no—
120H(5,𝔽2)—62no—
121H(5,𝔽2)—62no—
122C2×#41—31no—
123C2×#41—31no—
124C2×#41—61no—
125C2×#41—61no—
126C2×#41C2×D311yescharacteristic
127C22×D4—31no—
128C22×D4—61no—
129C22×D4—62no—
130C22×D4—61no—
131#137—32no—
132#137—32no—
133#137—61no—
134#137—61no—
135AGL(1,ℤ/8)—32no—
136AGL(1,ℤ/8)—32no—
137AGL(1,ℤ/8)—61no—
138AGL(1,ℤ/8)—61no—
139#135—62no—
140#135—62no—
141C2×M16—31no—
142C2×M16C2×D311yescharacteristic, complemented
143#127—32no—
144#124—32no—
145#121—62no—
146#120—62no—
147#117C2×D311yescharacteristic, complemented
148#104—34no—
149#104C2×D314yescomplemented
150#100—34no—
151C23×C3—21no—
152C23×C3C2×D411yescharacteristic
153C22×D3—22no—
154C22×D3—21no—
155C22×D3—28no—
156C22×D3—22no—
157C22×D3—21no—
158C22×D3—82no—
159C22×D3—84no—
160C22×D3C2×D414yes—
161C3×Q8—21no—
162C3×Q8C2×D412yes—
163C3×D4—22no—
164C3×D4—22no—
165C3×D4—21no—
166C3×D4C2×D412yes—
167C2×C3×C4—21no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4C2×D411yescharacteristic
172C2×C3×C4C2×D411yescharacteristic
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2411yescharacteristic
177#67—28no—
178#67—22no—
179#67—22no—
180#67—41no—
181#67—41no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184D12—28no—
185D12—22no—
186D12—21no—
187D12—22no—
188D12—22no—
189D12—22no—
190D12—21no—
191D12—21no—
192D12—84no—
193D12C2×D412yes—
194C4×D3—28no—
195C4×D3—22no—
196C4×D3—22no—
197C4×D3—41no—
198C4×D3—41no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6C2×D412yes—
203C3×C8—22no—
204C3⋊C8—22no—
205C24—32no—
206C24—62no—
207#41—21no—
208#41—21no—
209#41—32no—
210#41—32no—
211#41—64no—
212#41—64no—
213#41—64no—
214#41—64no—
215#41—61no—
216#41—61no—
217#41—61no—
218#41—61no—
219#41—61no—
220#41—61no—
221#41D1212yescomplemented
222#41C22×D312yes—
223C2×Q8—31no—
224C2×Q8—31no—
225C2×Q8D1211yescharacteristic, complemented
226C2×D4—21no—
227C2×D4—31no—
228C2×D4—32no—
229C2×D4—32no—
230C2×D4—31no—
231C2×D4—64no—
232C2×D4—62no—
233C2×D4—64no—
234C2×D4—64no—
235C2×D4—68no—
236C2×D4—68no—
237C2×D4—64no—
238C2×D4—62no—
239C2×D4—61no—
240C2×D4—64no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4D1211yescharacteristic, complemented
247C22×C4—21no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4C22×D311yescharacteristic
252Q16—62no—
253Q16—62no—
254SD16—62no—
255SD16—62no—
256SD16—62no—
257SD16—62no—
258D8—62no—
259D8—62no—
260M16—21no—
261M16—32no—
262M16—61no—
263M16C22×D312yescomplemented
264C2×C8—21no—
265C2×C8—61no—
266C4⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—64no—
269C42C22×D312yescomplemented
270C22×C3—22no—
271C22×C3—41no—
272C22×C3—42no—
273C22×C3#12011yescharacteristic
274C22×C3#12011yescharacteristic
275C22×C3C22×D411yescharacteristic
276C2×D3—24no—
277C2×D3—22no—
278C2×D3—44no—
279C2×D3—42no—
280C2×D3—44no—
281C2×D3—42no—
282C2×D3—41no—
283C2×D3—82no—
284C2×D3—88no—
285C3×C4—22no—
286C3×C4—42no—
287C3×C4#12011yescharacteristic
288C3×C4#12011yescharacteristic
289C3×C4C22×D411yescharacteristic
290C3×C4C22×D411yescharacteristic
291Dic3—22no—
292Dic3—41no—
293C23—21no—
294C23—34no—
295C23—68no—
296C23—61no—
297C23—62no—
298C23—62no—
299C23—61no—
300C23—242no—
301C23—244no—
302C23D3×D411yescharacteristic
303Q8—21no—
304Q8—32no—
305Q8—61no—
306Q8—61no—
307Q8—61no—
308Q8C2×D1212yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—32no—
313D4—68no—
314D4—68no—
315D4—62no—
316D4—62no—
317D4—61no—
318D4—62no—
319D4—62no—
320D4—62no—
321D4—61no—
322D4—62no—
323D4—61no—
324D4—121no—
325D4—121no—
326D4—244no—
327D4C2×D1212yes—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—68no—
333C2×C4—61no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4C2×D1211yescharacteristic
340C2×C4C2×D1211yescharacteristic
341C2×C4D3×D411yescharacteristic
342C2×C4D3×D411yescharacteristic
343C2×C4D3×D411yescharacteristic
344C2×C4C23×D311yescharacteristic
345C8—22no—
346C8—62no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—42no—
351C2×C3C2×#12011yescharacteristic
352D3—44no—
353D3—42no—
354D3—81no—
355C22—22no—
356C22—41no—
357C22—42no—
358C22—62no—
359C22—64no—
360C22—124no—
361C22—124no—
362C22—122no—
363C22—122no—
364C22—121no—
365C22—242no—
366C22—248no—
367C22#88211yescharacteristic
368C22#88211yescharacteristic
369C22C2×D3×D411yescharacteristic
370C4—22no—
371C4—42no—
372C4—62no—
373C4—121no—
374C4#88211yescharacteristic
375C4#88211yescharacteristic
376C4C22×D1211yescharacteristic
377C4C2×D3×D411yescharacteristic
378C3#301911yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—124no—
384C2—122no—
385C2—241no—
386C2C2×#88211yescharacteristic
subgroupG/H
centerC2C2×#882
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#421C2
socleC2×C3C2×#120
hypercenter#421D3
hypocenterC3#3019
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. C2×#1009C2semidirect
2. #5822C2semidirect
3. #5748C2semidirect
4. #5525C2semidirect
5. C3×#421C2semidirect
6. C2×#823C2semidirect
7. #5005C2semidirect
8. #4825C2semidirect
9. #4822C2semidirect
21. C2×#235C22semidirect
24. C2×C3×M16C22semidirect
25. C3×#117C22semidirect
26. #951C22semidirect
28. #949C22semidirect
31. C2×#208C22semidirect
33. #909C22semidirect
35. #908C22semidirect
39. #875C22semidirect
40. #874C22semidirect
41. C3×#104C22semidirect
42. #823C22semidirect
43. #805C22semidirect
51. #421D3semidirect
57. C3×#41D4semidirect
59. C2×C3×Q8D4semidirect
61. C2×C3×D4D4semidirect
81. #235D4semidirect
92. C2×D12D4semidirect
94. C2×D12C23semidirect
98. C2×Dic6D4semidirect
99. C2×Dic6C23semidirect
101. C3×M16C23semidirect
105. C3×C42C23semidirect
112. #208C23semidirect
142. C2×M16C2×D3semidirect
147. #117C2×D3semidirect
149. #104C2×D3semidirect
221. #41D12semidirect
225. C2×Q8D12semidirect
246. C2×D4D12semidirect
263. M16C22×D3semidirect
269. C42C22×D3semidirect
378. C3#3019semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
#58222
#57482
#55252
C3×#4212
C2×#8232
#50052
#48252
#48222
#30193