Finite Group Databasebeta

← #10876118

Subgroups — #10876118

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2285C211yesmaximal, characteristic, complemented
2C2×#4939C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic, complemented
4#69586C212yesmaximal, complemented
5#69585C212yesmaximal, complemented
6#69521C212yesmaximal
7#69504C212yesmaximal, complemented
8#69462C214yesmaximal, complemented
9#27582—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C2×C3×#117—21no—
12C22×C3×C42C2211yescharacteristic
13C22×(C2×D3)⋊C4—21no—
14C22×(C2×D3)⋊C4—21no—
15C22×C4⋊Dic3C2211yescharacteristic
16C3×#384C2211yescharacteristic, complemented
17C3×#384C2211yescharacteristic, complemented
18C3×#382—21no—
19C3×#380—21no—
20C3×#380C2212yes—
21C3×C4×C4⋊C4—21no—
22C3×C4×C4⋊C4C2212yescomplemented
23C2×#830C2212yes—
24C2×#809C2214yes—
25C2×#809C2214yes—
26C2×#807C2214yescomplemented
27#4939—21no—
28#4939—21no—
29#4939C2212yes—
30#4939C2212yes—
31#4529C2214yescomplemented
32#4524C2212yescomplemented
33#4524C2212yescomplemented
34C2×#390—31no—
35#2285D311yescharacteristic, complemented
36#2093—32no—
37#2071—32no—
38#1925—34no—
39#1852—32no—
40#1817—31no—
41#1786—32no—
42C24×D3—21no—
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—21no—
47C22×D12—21no—
48C22×D12—24no—
49C22×D12—24no—
50C22×D12—42no—
51C22×D12C2312yes—
52C3×#117—44no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—24no—
61C2×C3×C42—22no—
62C2×C3×C42D411yescharacteristic
63C2×C3×C42D411yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—44no—
71C2×(C2×D3)⋊C4—44no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—42no—
74C2×C4⋊Dic3C2312yes—
75C22×C3⋊C8C2314yes—
76C3×#95—22no—
77C3×#95—22no—
78#830—24no—
79#809—28no—
80#809—44no—
81#807—44no—
82C23×D4—31no—
83C2×#117—21no—
84C22×C4⋊C4—31no—
85C22×C22⋊C4—61no—
86C22×C22⋊C4—61no—
87C22×C42C2×D311yescharacteristic
88C2×#107—34no—
89C2×#102—34no—
90C2×#102—34no—
91C2×C22⋊C8—32no—
92#390—32no—
93#390—32no—
94#390—61no—
95#390—61no—
96#384C2×D311yescharacteristic, complemented
97#384C2×D311yescharacteristic, complemented
98#382—21no—
99#380—21no—
100#380C2×D312yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×D312yescomplemented
103#340—34no—
104#336—32no—
105#336—32no—
106C24×C3C2×D411yescharacteristic
107C23×D3—24no—
108C23×D3—22no—
109C23×D3—42no—
110C23×D3—42no—
111C23×D3—42no—
112C23×D3—42no—
113C23×D3—42no—
114C23×D3—42no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—41no—
124C22×C3×C4D814yes—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4114yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—42no—
134C22×Dic3—42no—
135C22×Dic3—42no—
136C2×D12—24no—
137C2×D12—24no—
138C2×D12—24no—
139C2×D12—22no—
140C2×D12—416no—
141C2×D12—42no—
142C2×D12—42no—
143C2×D12#4114yes—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—42no—
149C3×C4⋊C4—42no—
150C3×C22⋊C4—44no—
151C3×C42—22no—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42—22no—
155C3×C42—44no—
156(C2×D3)⋊C4—44no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—44no—
160C4⋊Dic3—24no—
161C4⋊Dic3—42no—
162C2×C3⋊C8—24no—
163C2×C3⋊C8—24no—
164C2×C3⋊C8—24no—
165C25—61no—
166C22×D4—32no—
167C22×D4—64no—
168C22×D4—64no—
169C22×D4—61no—
170C22×D4—122no—
171C23×C4—61no—
172C23×C4#6711yescharacteristic
173C23×C4#6711yescharacteristic
174C23×C4C22×D311yescharacteristic
175C22×C8—34no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—122no—
184C2×C4⋊C4C22×D312yes—
185C2×C4⋊C4C22×D312yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—124no—
192C2×C22⋊C4—124no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42#6711yescharacteristic
196C2×C42#6711yescharacteristic
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199#107—124no—
200#102—68no—
201#102—124no—
202C22⋊C8—64no—
203#95—22no—
204#95—22no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3#12312yes—
210C23×C3C2×D812yes—
211C23×C3#13712yes—
212C23×C3C22×D411yescharacteristic
213C22×D3—24no—
214C22×D3—48no—
215C22×D3—48no—
216C22×D3—48no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—84no—
224C22×D3—84no—
225C22×D3—84no—
226C22×D3—84no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—28no—
236C2×C3×C4—28no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—44no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4#12312yes—
248C2×C3×C4C2×D812yes—
249C2×C3×C4C2×D812yes—
250C2×C3×C4AGL(1,ℤ/8)12yes—
251C2×C3×C4C2×#4112yes—
252C2×C3×C4C2×#4112yes—
253C2×C3×C4H(5,𝔽2)12yes—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—84no—
258D12—28no—
259D12—48no—
260D12—44no—
261C3⋊C8—44no—
262C24—64no—
263C24—62no—
264C24—122no—
265C24—122no—
266C24—122no—
267C24—122no—
268C24—122no—
269C24—122no—
270C24C2×#6711yescharacteristic
271C2×D4—34no—
272C2×D4—64no—
273C2×D4—64no—
274C2×D4—62no—
275C2×D4—64no—
276C2×D4—1216no—
277C2×D4—122no—
278C2×D4—122no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—62no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4—122no—
292C22×C4#21314yes—
293C22×C4#23514yes—
294C22×C4C2×#6711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—64no—
301C2×C8—64no—
302C2×C8—64no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—64no—
310C4⋊C4—122no—
311C22⋊C4—44no—
312C22⋊C4—124no—
313C22⋊C4—124no—
314C22⋊C4—124no—
315C22⋊C4—124no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—44no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3#48014yes—
328C22×C3C2×#12311yescharacteristic
329C22×C3#52511yescharacteristic
330C22×C3#53511yescharacteristic
331C22×C3C22×D811yescharacteristic
332C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
333C22×C3C2×#13711yescharacteristic
334C22×C3#57511yescharacteristic
335C2×D3—44no—
336C2×D3—44no—
337C2×D3—44no—
338C2×D3—816no—
339C2×D3—82no—
340C2×D3—82no—
341C2×D3—82no—
342C2×D3—82no—
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—41no—
348C3×C4—82no—
349C3×C4—82no—
350C3×C4#55114yes—
351Dic3—82no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—64no—
357C23—128no—
358C23—128no—
359C23—128no—
360C23—122no—
361C23—122no—
362C23—122no—
363C23—122no—
364C23—122no—
365C23—122no—
366C23—244no—
367C23—244no—
368C23—244no—
369C23—244no—
370C23#92912yes—
371C23C2×#21312yes—
372C23#94212yes—
373C23C22×#6711yescharacteristic
374D4—68no—
375D4—128no—
376D4—124no—
377C2×C4—28no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—28no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—244no—
401C2×C4#92912yes—
402C2×C4C2×#21312yes—
403C2×C4C2×#21312yes—
404C2×C4#94912yes—
405C2×C4C2×#23512yes—
406C2×C4D3×#4112yes—
407C2×C4#100912yes—
408C8—124no—
409C2×C3—24no—
410C2×C3#248211yescharacteristic
411C2×C3C2×#48011yescharacteristic
412C2×C3#309411yescharacteristic
413C2×C3#310011yescharacteristic
414C2×C3#311011yescharacteristic
415C2×C3#320511yescharacteristic
416C2×C3#321511yescharacteristic
417D3—84no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—124no—
425C22—124no—
426C22—124no—
427C22—2416no—
428C22—242no—
429C22—242no—
430C22—242no—
431C22—242no—
432C22#496714yes—
433C22C2×#92911yescharacteristic
434C22C22×#21311yescharacteristic
435C22C2×#94211yescharacteristic
436C22C2×#94911yescharacteristic
437C22#582211yescharacteristic
438C22#582811yescharacteristic
439C22#583011yescharacteristic
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—82no—
446C4—82no—
447C4—242no—
448C4#552714yes—
449C3#2758211yescharacteristic, complemented, Sylow
450C2—24no—
451C2—244no—
452C2C2×#496711yescharacteristic
453C2#8221711yescharacteristic
454C2#8228211yescharacteristic
455C2#8329811yescharacteristic
456C2#8330211yescharacteristic
457C2#8335311yescharacteristic
458C2#8335711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2285C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#2285D3
hypocenterC3#27582
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C3×#2285C2semidirect
3. #71531C2semidirect
4. #69586C2semidirect
5. #69585C2semidirect
7. #69504C2semidirect
8. #69462C2semidirect
16. C3×#384C22semidirect
17. C3×#384C22semidirect
22. C3×C4×C4⋊C4C22semidirect
26. C2×#807C22semidirect
31. #4529C22semidirect
32. #4524C22semidirect
33. #4524C22semidirect
35. #2285D3semidirect
96. #384C2×D3semidirect
97. #384C2×D3semidirect
102. C4×C4⋊C4C2×D3semidirect
449. C3#27582semidirect

Maximal subgroups

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

subgroupindex
C3×#22852
C2×#49392
#715312
#695862
#695852
#695212
#695042
#694622
#275823