Finite Group Databasebeta

← #10945159

Subgroups — #10945159

Order 768.

1098 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C42×D4C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic
3C2×#5008C211yesmaximal, characteristic, complemented
4#73313C211yesmaximal, characteristic, complemented
5#71613C214yesmaximal, complemented
6#69489C211yesmaximal, characteristic, complemented
7#69487C211yesmaximal, characteristic, complemented
8#69463C214yesmaximal, complemented
9#69392C211yesmaximal, characteristic, complemented
10#33030—31nomaximal, Sylow
11C2×C3×C4×D4C2212yes—
12C2×C3×C4×D4C2211yescharacteristic
13C22×C3×C42—21no—
14C22×#213C2212yes—
15C2×#875—22no—
16C2×#874C2211yescharacteristic
17C2×#869C2211yescharacteristic
18C3×C4×C4⋊C4C2212yes—
19C3×C4×C4⋊C4C2211yescharacteristic
20C3×C4×C22⋊C4—21no—
21C3×C4×C22⋊C4—22no—
22C3×C43C2211yescharacteristic
23C2×#832C2212yes—
24#5015—24no—
25#5008—24no—
26C2×#809C2212yes—
27C2×#809C2214yes—
28C2×#808C2212yes—
29#4939—21no—
30#4939—21no—
31#4939C2214yes—
32C2×C4⋊(C3⋊C8)C2211yescharacteristic
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4529C2214yes—
35#4524C2214yes—
36C2×#476—31no—
37C2×#459—31no—
38#2858—31no—
39C42×D4D311yescharacteristic, complemented
40#1987—31no—
41#1982—31no—
42#1926—34no—
43#1879—34no—
44#1831—31no—
45C22×C3×D4C2311yescharacteristic
46C23×C3×C4—22no—
47C23×C3×C4—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12C2312yes—
51C22×Dic6—21no—
52C3×C4×D4—24no—
53C3×C4×D4—24no—
54C3×C4×D4—22no—
55C3×C4×D4D414yes—
56C2×C3×C4⋊C4C2311yescharacteristic
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—22no—
61C2×C3×C42—21no—
62C2×C3×C42—44no—
63C2×C3×C42D411yescharacteristic
64C2×C3×C42D411yescharacteristic
65C2×C3×C42D412yes—
66C2×C3×C42C2312yes—
67C2×C3×C42C2311yescharacteristic
68C2×#213—24no—
69C2×#213—28no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—24no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3C2312yes—
74C22×C3⋊C8C2312yes—
75C22×C3⋊C8C2312yes—
76#875—48no—
77#874—22no—
78#874—22no—
79#869—22no—
80#869—22no—
81C3×#95—22no—
82C3×#95—21no—
83#832—24no—
84#809—24no—
85#809—28no—
86#808—24no—
87C4⋊(C3⋊C8)—22no—
88C4⋊(C3⋊C8)—22no—
89C22×D8—32no—
90C2×C4⋊Q8—31no—
91C2×#127—31no—
92C2×#124—62no—
93C2×C4×D4C2×D312yes—
94C2×C4×D4C2×D311yescharacteristic
95C22×C42—21no—
96#476—64no—
97#459—64no—
98C2×#106—32no—
99C2×C4⋊C8—31no—
100C2×C4⋊C8—31no—
101C2×#102—32no—
102C2×#102—34no—
103C2×#102—32no—
104#390—34no—
105#390—61no—
106#390—61no—
107C4×C4⋊C4C2×D312yes—
108C4×C4⋊C4C2×D311yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C43C2×D311yescharacteristic
112#340—34no—
113#336—34no—
114C24×C3—21no—
115C23×D3—22no—
116C2×C3×D4—21no—
117C2×C3×D4—22no—
118C2×C3×D4C2×D412yes—
119C2×C3×D4#4114yes—
120C22×C3×C4—28no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—24no—
124C22×C3×C4—44no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4—44no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2411yescharacteristic
133C22×Dic3—22no—
134C2×D12—22no—
135C2×D12—24no—
136C2×D12—22no—
137C2×D12—24no—
138C2×D12—44no—
139C2×D12—42no—
140C2×D12—42no—
141C2×D12—42no—
142C2×D12—42no—
143C2×Dic6—44no—
144C2×Dic6—42no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4C2×D412yes—
148C3×C4⋊C4#4114yes—
149C3×C22⋊C4—28no—
150C3×C22⋊C4—24no—
151C3×C42—22no—
152C3×C42—24no—
153C3×C42—22no—
154C3×C42—22no—
155C3×C42—22no—
156C3×C42—44no—
157C3×C42—42no—
158C3×C42C2×D412yes—
159C3×C42#4112yes—
160#213—48no—
161(C2×D3)⋊C4—48no—
162(C2×D3)⋊C4—48no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—24no—
169C2×C3⋊C8—24no—
170C22×Q8—61no—
171C22×D4—32no—
172C22×D4—61no—
173C22×D4—61no—
174C22×D4C22×D311yescharacteristic
175C23×C4—22no—
176C23×C4—21no—
177C2×D8—64no—
178C2×D8—68no—
179C22×C8—32no—
180C22×C8—32no—
181C4⋊Q8—62no—
182C4⋊Q8—62no—
183#127—62no—
184#127—62no—
185#124—128no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4D1214yes—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4C22×D311yescharacteristic
193C2×C4⋊C4C22×D312yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—64no—
197C2×C22⋊C4—64no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—44no—
201C2×C42#6712yes—
202C2×C42#6711yescharacteristic
203C2×C42#6711yescharacteristic
204C2×C42C22×D312yes—
205C2×C42C22×D311yescharacteristic
206#106—64no—
207C4⋊C8—62no—
208C4⋊C8—62no—
209#102—64no—
210#102—64no—
211#102—68no—
212#95—22no—
213#95—21no—
214C23×C3—22no—
215C23×C3—24no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—42no—
219C23×C3C22×D411yescharacteristic
220C22×D3—42no—
221C22×D3—44no—
222C22×D3—42no—
223C22×D3—44no—
224C22×D3—42no—
225C3×D4—24no—
226C3×D4—22no—
227C3×D4#12414yescomplemented
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—416no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—48no—
242C2×C3×C4#12112yes—
243C2×C3×C4#12112yes—
244C2×C3×C4#13512yes—
245C2×C3×C4AGL(1,ℤ/8)12yes—
246C2×C3×C4AGL(1,ℤ/8)14yes—
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4C2×#4112yes—
251C2×Dic3—42no—
252C2×Dic3—44no—
253D12—42no—
254D12—44no—
255D12—42no—
256D12—84no—
257D12—84no—
258Dic6—84no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C24—21no—
262C24—62no—
263C2×Q8—124no—
264C2×Q8—122no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—62no—
268C2×D4—64no—
269C2×D4—62no—
270C2×D4—64no—
271C2×D4—124no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—122no—
275C2×D4—122no—
276C2×D4C2×D1212yes—
277C2×D4#23514yes—
278C22×C4—28no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—62no—
287C22×C4C2×#6712yes—
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6712yes—
290C22×C4C2×#6711yescharacteristic
291C22×C4C23×D311yescharacteristic
292D8—128no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—64no—
296C2×C8—64no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—122no—
302C4⋊C4C2×D1212yes—
303C4⋊C4#23514yes—
304C22⋊C4—28no—
305C22⋊C4—24no—
306C22⋊C4—128no—
307C22⋊C4—128no—
308C42—22no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—44no—
314C42—42no—
315C42C2×D1212yes—
316C42#23512yes—
317C22×C3—24no—
318C22×C3—44no—
319C22×C3—44no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3C2×#12111yescharacteristic
323C22×C3#52512yes—
324C22×C3C2×#13511yescharacteristic
325C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
326C22×C3C2×AGL(1,ℤ/8)12yes—
327C2×D3—82no—
328C2×D3—84no—
329C2×D3—84no—
330C2×D3—82no—
331C2×D3—82no—
332C3×C4—24no—
333C3×C4—22no—
334C3×C4—42no—
335C3×C4—44no—
336C3×C4—42no—
337C3×C4#46314yes—
338C3×C4C2×#12411yescharacteristic
339C3×C4#52912yes—
340C3×C4#53211yescharacteristic
341Dic3—82no—
342C23—22no—
343C23—24no—
344C23—41no—
345C23—42no—
346C23—41no—
347C23—122no—
348C23—124no—
349C23—122no—
350C23—124no—
351C23—122no—
352C23C22×#6711yescharacteristic
353Q8—244no—
354D4—22no—
355D4—24no—
356D4—122no—
357D4—122no—
358D4—124no—
359D4—244no—
360D4—244no—
361D4#87514yescomplemented
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—416no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—48no—
376C2×C4—122no—
377C2×C4—124no—
378C2×C4#93012yes—
379C2×C4#93012yes—
380C2×C4#93212yes—
381C2×C4#94914yes—
382C2×C4#95012yes—
383C2×C4C22×D1211yescharacteristic
384C2×C4C2×#23512yes—
385C2×C4C2×#23512yes—
386C2×C4C2×#23511yescharacteristic
387C8—122no—
388C8—122no—
389C2×C3—44no—
390C2×C3#285011yescharacteristic
391C2×C3C2×#46311yescharacteristic
392C2×C3#303811yescharacteristic
393C2×C3#310612yes—
394C2×C3#310912yes—
395D3—162no—
396C22—24no—
397C22—42no—
398C22—44no—
399C22—42no—
400C22—44no—
401C22—242no—
402C22—244no—
403C22—242no—
404C22—244no—
405C22—242no—
406C22C2×#93011yescharacteristic
407C22#579112yes—
408C22C2×#93211yescharacteristic
409C22C2×#94912yes—
410C22C2×#95011yescharacteristic
411C4—22no—
412C4—24no—
413C4—42no—
414C4—42no—
415C4—44no—
416C4—242no—
417C4#501614yes—
418C4C2×#87511yescharacteristic
419C4#547711yescharacteristic
420C4#547912yes—
421C3#3303011yescharacteristic, complemented, Sylow
422C2—44no—
423C2—482no—
424C2#8207711yescharacteristic
425C2C2×#501611yescharacteristic
426C2#8231012yes—
427C2#8252611yescharacteristic
428C2#8252912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C42×D4C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenterC42×D4D3
hypocenterC3#33030
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
1. C3×C42×D4C2semidirect
3. C2×#5008C2semidirect
4. #73313C2semidirect
5. #71613C2semidirect
6. #69489C2semidirect
7. #69487C2semidirect
8. #69463C2semidirect
9. #69392C2semidirect
39. C42×D4D3semidirect
227. C3×D4#124semidirect
361. D4#875semidirect
421. C3#33030semidirect

Maximal subgroups

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

subgroupindex
C3×C42×D42
C2×#50152
C2×#50082
#733132
#716132
#694892
#694872
#694632
#693922
#330303