Finite Group Databasebeta

← #79506

Subgroups — #79506

Order 384.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5563C211yesmaximal, characteristic, complemented
2#5550C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5134C211yesmaximal, characteristic, complemented
5#5131C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7#5113C211yesmaximal, characteristic, complemented
8#4846C211yesmaximal, characteristic, complemented
9#4844C211yesmaximal, characteristic, complemented
10#4737C211yesmaximal, characteristic, complemented
11#4734C211yesmaximal, characteristic, complemented
12#4732C211yesmaximal, characteristic, complemented
13#4723C211yesmaximal, characteristic, complemented
14#4714C211yesmaximal, characteristic, complemented
15#4711C211yesmaximal, characteristic, complemented
16#3319—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic, complemented
21C3×C4×D4C2211yescharacteristic
22#938—21no—
23#937—21no—
24#930C2211yescharacteristic
25#930C2211yescharacteristic
26#930C2211yescharacteristic, complemented
27#926C2211yescharacteristic
28#926C2211yescharacteristic, complemented
29#924C2211yescharacteristic
30C2×D24C2211yescharacteristic
31C2×D24C2211yescharacteristic
32C2×#204C2211yescharacteristic, complemented
33#894—21no—
34#892—21no—
35#885—21no—
36#885—21no—
37#884—21no—
38D3×C22⋊C4—21no—
39#875C2211yescharacteristic
40C4×D12C2211yescharacteristic
41C4×D12C2211yescharacteristic, complemented
42#869C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic, complemented
46C3×#97C2211yescharacteristic, complemented
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#821C2211yescharacteristic, complemented
54#818C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#817C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#555—31no—
61#546—31no—
62#489—31no—
63#485—31no—
64#482—31no—
65#474—31no—
66#468—31no—
67#466—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71#442—31no—
72#435D311yescharacteristic, complemented
73#418—31no—
74#416—31no—
75C23×D3—21no—
76C2×C3×D4D411yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83#235—22no—
84#235—41no—
85C2×D12—22no—
86C2×D12—21no—
87C2×D12—41no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×Dic6—21no—
95C2×Dic6C2311yescharacteristic
96C3×M16—41no—
97C2×C3×C8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4D411yescharacteristic
103C3×C22⋊C4D411yescharacteristic
104C3×C22⋊C4D411yescharacteristic
105C3×C42C2311yescharacteristic
106#217—21no—
107#217—21no—
108#217—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C4×Dic3—21no—
124D24—22no—
125D24—41no—
126#204—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×SD16—31no—
130C2×D8—31no—
131C2×D8—31no—
132C2×M16C2×D311yescharacteristic
133C22×C8C2×D311yescharacteristic, complemented
134C4⋊Q8—31no—
135#124—31no—
136#124—61no—
137#124—61no—
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121—61no—
145#120—61no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4C2×D311yescharacteristic
149#117—31no—
150#117—31no—
151#117—61no—
152C2×C22⋊C4—61no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8C2×D311yescharacteristic, complemented
157#103—31no—
158#103—31no—
159#103—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8C2×D311yescharacteristic
168C22⋊C8C2×D311yescharacteristic, complemented
169#97C2×D311yescharacteristic, complemented
170C23×C3C2×D411yescharacteristic
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—81no—
178C3×D4—41no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—42no—
189#67—81no—
190#67—81no—
191#67—81no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197D12—21no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—21no—
202D12—42no—
203D12#4112yes—
204C4×D3—42no—
205C4×D3—42no—
206C4×D3—81no—
207Dic6—21no—
208Dic6—21no—
209Dic6—81no—
210C3×C8—21no—
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—21no—
214C24—61no—
215#41—62no—
216#41—121no—
217C2×Q8—31no—
218C2×Q8—61no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—62no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—121no—
229C2×D4D1211yescharacteristic
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235SD16—121no—
236D8—62no—
237D8—121no—
238M16—41no—
239C2×C8—41no—
240C2×C8C22×D311yescharacteristic
241C2×C8C22×D311yescharacteristic
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4D1211yescharacteristic
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4D1211yescharacteristic
262C22⋊C4D1211yescharacteristic
263C42—61no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3C22×D411yescharacteristic
273C2×D3—22no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—84no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4C2×#4111yescharacteristic
288C3×C4H(5,𝔽2)11yescharacteristic
289Dic3—41no—
290Dic3—41no—
291Dic3—41no—
292Dic3—41no—
293Dic3—41no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23—122no—
298C23—122no—
299C23—241no—
300C23C2×D1211yescharacteristic
301C23C2×D1211yescharacteristic
302Q8—61no—
303Q8—61no—
304Q8—241no—
305D4—32no—
306D4—41no—
307D4—61no—
308D4—61no—
309D4—61no—
310D4—61no—
311D4—61no—
312D4—122no—
313D4—122no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—241no—
329C2×C4C2×D1211yescharacteristic
330C2×C4C2×D1211yescharacteristic
331C2×C4C2×D1211yescharacteristic
332C2×C4C2×D1211yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3#54611yescharacteristic
341C2×C3#57511yescharacteristic
342C2×C3#57611yescharacteristic
343D3—42no—
344D3—81no—
345D3—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—62no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—244no—
362C22C22×D1211yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4#100411yescharacteristic
373C4D3×#4111yescharacteristic
374C3#331911yescharacteristic, complemented, Sylow
375C2—41no—
376C2—41no—
377C2—122no—
378C2—241no—
379C2—241no—
380C2#555411yescharacteristic
381C2#574811yescharacteristic
382C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3319
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #5563C2semidirect
2. #5550C2semidirect
3. C3×#435C2semidirect
4. #5134C2semidirect
5. #5131C2semidirect
6. #5115C2semidirect
7. #5113C2semidirect
8. #4846C2semidirect
9. #4844C2semidirect
10. #4737C2semidirect
11. #4734C2semidirect
12. #4732C2semidirect
13. #4723C2semidirect
14. #4714C2semidirect
15. #4711C2semidirect
20. C22×C3×C8C22semidirect
26. #930C22semidirect
28. #926C22semidirect
32. C2×#204C22semidirect
41. C4×D12C22semidirect
42. #869C22semidirect
43. C3×C4⋊C8C22semidirect
45. C3×C22⋊C8C22semidirect
46. C3×#97C22semidirect
53. #821C22semidirect
54. #818C22semidirect
55. #818C22semidirect
56. #817C22semidirect
57. #816C22semidirect
58. #816C22semidirect
59. #816C22semidirect
72. #435D3semidirect
133. C22×C8C2×D3semidirect
156. C4⋊C8C2×D3semidirect
168. C22⋊C8C2×D3semidirect
169. #97C2×D3semidirect
374. C3#3319semidirect

Maximal subgroups

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

subgroupindex
#55632
#55502
C3×#4352
#51342
#51312
#51152
#51132
#48462
#48442
#47372
#47342
#47322
#47232
#47142
#47112
#33193