Finite Group Databasebeta

← #10945157

Subgroups — #10945157

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
1C2×C3×#459C211yesmaximal, characteristic
2#83184C211yesmaximal, characteristic, complemented
3C4×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×#5008C211yesmaximal, characteristic, complemented
5#72037C214yesmaximal, complemented
6#71761C211yesmaximal, characteristic, complemented
7#70459C211yesmaximal, characteristic, complemented
8#70443C211yesmaximal, characteristic, complemented
9#70174C214yesmaximal, complemented
10#33030—31nomaximal, Sylow
11C22×C3×D8C2212yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936—22no—
15C2×D4×Dic3C2212yes—
16C22×C4×Dic3—21no—
17C2×#869C2211yescharacteristic
18C3×#459—24no—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#102C2212yes—
21#5221—21no—
22#5221—21no—
23#5221C2214yes—
24C2×#832C2214yes—
25C2×#832C2212yes—
26C4×#217—21no—
27#5008—24no—
28Dic3×C4⋊C4C2212yes—
29C2×#808C2212yes—
30Dic3×C22⋊C4—22no—
31C4×C4⋊Dic3C2211yescharacteristic
32C42×Dic3C2211yescharacteristic
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4551C2214yes—
35#4550C2214yes—
36C2×#476—31no—
37C2×#459D311yescharacteristic
38#2858—31no—
39C42×D4—31no—
40#1987—31no—
41#1982—31no—
42#1926—34no—
43#1879—34no—
44#1831—31no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4C2312yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4—21no—
50C23×Dic3—22no—
51C22×Dic6—21no—
52C2×C3×D8—24no—
53C2×C3×D8—28no—
54C22×C3×C8C2312yes—
55C3×#127—22no—
56C3×#127—22no—
57C3×C4×D4—22no—
58C3×C4×D4D414yes—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×#217—24no—
64C2×#217—24no—
65#936—48no—
66D4×Dic3—24no—
67D4×Dic3—24no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3—44no—
74C2×C4×Dic3D412yes—
75C2×C4×Dic3D411yescharacteristic
76C2×C4×Dic3D411yescharacteristic
77C2×C4×Dic3C2312yes—
78C22×C3⋊C8C2312yes—
79#869—22no—
80#869—22no—
81C3×C4⋊C8—22no—
82C3×#102—24no—
83#832—24no—
84#832—28no—
85C3⋊#95—21no—
86C3⋊#95—22no—
87#808—24no—
88C4⋊(C3⋊C8)—22no—
89C22×D8C2×D312yes—
90C2×C4⋊Q8—31no—
91C2×#127C2×D311yescharacteristic
92C2×#124—62no—
93C2×C4×D4—32no—
94C2×C4×D4C2×D311yescharacteristic
95C22×C42—61no—
96#476—64no—
97#459—24no—
98C2×#106—32no—
99C2×C4⋊C8—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#102—32no—
102C2×#102—34no—
103C2×#102C2×D312yes—
104#390—34no—
105#390—61no—
106#390—61no—
107C4×C4⋊C4—32no—
108C4×C4⋊C4—31no—
109C4×C22⋊C4—62no—
110C4×C22⋊C4—61no—
111C43—31no—
112#340—34no—
113#336—34no—
114C24×C3—21no—
115C24×C3—22no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—21no—
119C2×C3×D4—24no—
120C2×C3×D4—22no—
121C2×C3×D4—24no—
122C2×C3×D4—44no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4C2×D412yes—
128C2×C3×D4#4114yes—
129C22×C3×C4—21no—
130C22×C3×C4—24no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—28no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—44no—
140C22×Dic3—44no—
141C22×Dic3C2×D412yes—
142C22×Dic3C2×D412yes—
143C2×Dic6—44no—
144C2×Dic6—42no—
145C3×D8—48no—
146C2×C3×C8—22no—
147C2×C3×C8—24no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4C2×D412yes—
150C3×C22⋊C4—24no—
151C3×C42C2×D412yes—
152C3×C42#4112yes—
153#217—28no—
154#217—48no—
155#217—48no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—22no—
159C4⋊Dic3—42no—
160C4⋊Dic3#4114yes—
161C4×Dic3—22no—
162C4×Dic3—22no—
163C4×Dic3—22no—
164C4×Dic3—22no—
165C4×Dic3—24no—
166C4×Dic3—44no—
167C4×Dic3—42no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—24no—
170C22×Q8—61no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22×D312yes—
174C22×D4C22×D311yescharacteristic
175C23×C4—21no—
176C23×C4—62no—
177C2×D8—28no—
178C2×D8—24no—
179C22×C8—32no—
180C22×C8C22×D312yes—
181C4⋊Q8—62no—
182C4⋊Q8—62no—
183#127—22no—
184#127—22no—
185#124—128no—
186C4×D4—22no—
187C4×D4—64no—
188C4×D4—64no—
189C4×D4#6714yes—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—64no—
197C2×C22⋊C4—64no—
198C2×C42—32no—
199C2×C42—31no—
200C2×C42—31no—
201C2×C42—32no—
202C2×C42—62no—
203C2×C42—61no—
204C2×C42—124no—
205C2×C42C22×D311yescharacteristic
206#106—64no—
207C4⋊C8—22no—
208C4⋊C8—62no—
209#102—24no—
210#102—68no—
211#102—64no—
212#95—62no—
213#95—61no—
214C23×C3—22no—
215C23×C3—24no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—44no—
221C23×C3—42no—
222C23×C3—44no—
223C23×C3—42no—
224C23×C3C22×D411yescharacteristic
225C3×D4—24no—
226C3×D4—22no—
227C3×D4—42no—
228C3×D4—44no—
229C3×D4—42no—
230C3×D4—84no—
231C3×D4—84no—
232C3×D4#12414yescomplemented
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—48no—
238C2×C3×C4#12112yes—
239C2×C3×C4#12112yes—
240C2×C3×C4#13512yes—
241C2×C3×C4AGL(1,ℤ/8)12yes—
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C2×#4112yes—
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4112yes—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—24no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—416no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—44no—
257C2×Dic3AGL(1,ℤ/8)14yes—
258Dic6—84no—
259C3×C8—42no—
260C3⋊C8—42no—
261C24—21no—
262C24—22no—
263C2×Q8—124no—
264C2×Q8—122no—
265C2×D4—21no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4#23714yes—
277C2×D4C2×#6712yes—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—68no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—124no—
288C22×C4—124no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C23×D311yescharacteristic
292D8—48no—
293C2×C8—24no—
294C2×C8—22no—
295C2×C8—64no—
296C2×C8—62no—
297C4⋊C4—21no—
298C4⋊C4—34no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—122no—
303C4⋊C4C2×#6712yes—
304C22⋊C4—24no—
305C22⋊C4—68no—
306C22⋊C4—128no—
307C22⋊C4—128no—
308C42—64no—
309C42—62no—
310C42—62no—
311C42—62no—
312C42—62no—
313C42—124no—
314C42—122no—
315C42#23712yes—
316C42C2×#6712yes—
317C22×C3—24no—
318C22×C3—44no—
319C22×C3—44no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—82no—
323C22×C3—84no—
324C22×C3—84no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3C2×#12111yescharacteristic
328C22×C3#52512yes—
329C22×C3C2×#13511yescharacteristic
330C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
331C22×C3C2×AGL(1,ℤ/8)12yes—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4#46314yes—
335C3×C4C2×#12411yescharacteristic
336C3×C4#52912yes—
337C3×C4#53211yescharacteristic
338Dic3—24no—
339Dic3—42no—
340Dic3—44no—
341Dic3—82no—
342C23—22no—
343C23—24no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—42no—
351C23—44no—
352C23C2×D3×D411yescharacteristic
353Q8—244no—
354D4—22no—
355D4—24no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—84no—
360D4—84no—
361D4#93614yescomplemented
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—34no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—64no—
374C2×C4—1216no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—124no—
378C2×C4—122no—
379C2×C4#91112yes—
380C2×C4#91212yes—
381C2×C4#93912yes—
382C2×C4#93912yes—
383C2×C4C2×#23712yes—
384C2×C4C2×#23711yescharacteristic
385C2×C4D3×#4112yes—
386C2×C4C22×#6711yescharacteristic
387C8—42no—
388C8—122no—
389C2×C3—44no—
390C2×C3—162no—
391C2×C3#285011yescharacteristic
392C2×C3C2×#46311yescharacteristic
393C2×C3#303811yescharacteristic
394C2×C3#310612yes—
395C2×C3#310912yes—
396C22—24no—
397C22—42no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—82no—
403C22—84no—
404C22—84no—
405C22—82no—
406C22#560612yes—
407C22C2×#91111yescharacteristic
408C22C2×#91211yescharacteristic
409C22D3×AGL(1,ℤ/8)12yes—
410C22C2×#93911yescharacteristic
411C4—22no—
412C4—42no—
413C4—64no—
414C4—122no—
415C4—124no—
416C4—242no—
417C4#515314yes—
418C4#560012yes—
419C4C2×#93611yescharacteristic
420C4#583211yescharacteristic
421C3#3303011yescharacteristic, complemented, Sylow
422C2—44no—
423C2—162no—
424C2#8031312yes—
425C2#8031412yes—
426C2#8269811yescharacteristic
427C2C2×#515311yescharacteristic
428C2#8393211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#33030
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. #83184C2semidirect
3. C4×D4×Dic3C2semidirect
4. C2×#5008C2semidirect
5. #72037C2semidirect
6. #71761C2semidirect
7. #70459C2semidirect
8. #70443C2semidirect
9. #70174C2semidirect
232. C3×D4#124semidirect
361. D4#936semidirect
421. C3#33030semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
#831842
C4×D4×Dic32
C2×#50082
#720372
#717612
#704592
#704432
#701742
#330303