Finite Group Databasebeta

← #10945158

Subgroups — #10945158

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
1#83184C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic, complemented
4C4×#893C211yesmaximal, characteristic, complemented
5C3×#1982C211yesmaximal, characteristic, complemented
6#70443C211yesmaximal, characteristic, complemented
7#70208C214yesmaximal, complemented
8#70156C214yesmaximal, complemented
9#69489C211yesmaximal, characteristic, complemented
10#33030—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×#936—22no—
13C22×#213C2212yes—
14C2×#893C2212yes—
15C2×C4×D12C2211yescharacteristic
16C2×C42×D3—21no—
17C2×#869C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#102C2214yes—
20C3×C4×C4⋊C4C2211yescharacteristic
21#5221—21no—
22#5221—21no—
23#5221C2214yes—
24C2×#832C2212yes—
25C2×#821C2212yes—
26C2×#817C2212yes—
27#5072—24no—
28Dic3×C4⋊C4C2212yes—
29C4×(C2×D3)⋊C4—21no—
30C42×Dic3C2211yescharacteristic
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32#4842—24no—
33#4666—22no—
34#4550C2214yes—
35#4529C2214yes—
36C2×#476—31no—
37C2×#459—31no—
38#2858—31no—
39C42×D4—31no—
40#1987—31no—
41#1982D311yescharacteristic, complemented
42#1926—34no—
43#1879—34no—
44#1831—31no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4C2312yes—
48C22×D12C2311yescharacteristic
49C22×C4×D3—21no—
50C22×C4×D3—22no—
51C22×Dic6—21no—
52C22×C3×C8C2312yes—
53C3×#127—22no—
54C3×#127—22no—
55C2×C3×C4⋊C4C2312yes—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#217—24no—
59C2×#217—24no—
60#936—48no—
61C2×#213—24no—
62C2×#213—28no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2312yes—
67C2×C4⋊Dic3C2311yescharacteristic
68C2×C4×Dic3—22no—
69C2×C4×Dic3D412yes—
70C2×C4×Dic3D411yescharacteristic
71C2×C4×Dic3D411yescharacteristic
72C2×C4×Dic3C2312yes—
73C22×C3⋊C8C2312yes—
74#893—24no—
75#893—24no—
76C4×D12—22no—
77C4×D12D414yes—
78C42×D3—44no—
79#869—22no—
80#869—22no—
81C3×C4⋊C8—22no—
82C3×#102—28no—
83C3×#95—21no—
84#832—24no—
85C3⋊#95—22no—
86#821—24no—
87#817—24no—
88C4⋊(C3⋊C8)—22no—
89C22×D8—32no—
90C2×C4⋊Q8—31no—
91C2×#127C2×D311yescharacteristic
92C2×#124—62no—
93C2×C4×D4—32no—
94C2×C4×D4—31no—
95C22×C42—61no—
96#476—64no—
97#459—64no—
98C2×#106—32no—
99C2×C4⋊C8—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#102—32no—
102C2×#102—32no—
103C2×#102C2×D314yes—
104#390—34no—
105#390—61no—
106#390—61no—
107C4×C4⋊C4—32no—
108C4×C4⋊C4C2×D311yescharacteristic
109C4×C22⋊C4—62no—
110C4×C22⋊C4—61no—
111C43—31no—
112#340—34no—
113#336—34no—
114C24×C3—22no—
115C23×D3—21no—
116C2×C3×D4—22no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—22no—
120C2×C3×D4—44no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C22×C3×C4—22no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2411yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3C2×D412yes—
132C22×Dic3C2×D412yes—
133C2×D12—21no—
134C2×D12—22no—
135C2×D12C2×D412yes—
136C2×D12#4114yes—
137C2×C4×D3—28no—
138C2×C4×D3—24no—
139C2×C4×D3—44no—
140C2×C4×D3—42no—
141C2×C4×D3—42no—
142C2×C4×D3—44no—
143C2×Dic6—44no—
144C2×Dic6—42no—
145C2×C3×C8—22no—
146C2×C3×C8—24no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4#4114yes—
149C3×C42—42no—
150C3×C42C2×D412yes—
151C3×C42#4112yes—
152#217—48no—
153#217—48no—
154#213—48no—
155(C2×D3)⋊C4—28no—
156(C2×D3)⋊C4—24no—
157C4⋊Dic3—21no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—42no—
161C4⋊Dic3C2×D412yes—
162C4×Dic3—22no—
163C4×Dic3—22no—
164C4×Dic3—22no—
165C4×Dic3—22no—
166C4×Dic3—24no—
167C4×Dic3—44no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—24no—
170C22×Q8—61no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—31no—
174C22×D4C22×D312yes—
175C23×C4—62no—
176C23×C4—61no—
177C2×D8—68no—
178C2×D8—64no—
179C22×C8—32no—
180C22×C8C22×D312yes—
181C4⋊Q8—62no—
182C4⋊Q8—62no—
183#127—22no—
184#127—22no—
185#124—128no—
186C4×D4—34no—
187C4×D4—64no—
188C4×D4—64no—
189C4×D4—62no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4C22×D312yes—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—64no—
197C2×C22⋊C4—64no—
198C2×C42—21no—
199C2×C42—32no—
200C2×C42—31no—
201C2×C42—31no—
202C2×C42—32no—
203C2×C42—62no—
204C2×C42—124no—
205C2×C42C22×D311yescharacteristic
206#106—64no—
207C4⋊C8—22no—
208C4⋊C8—62no—
209#102—28no—
210#102—64no—
211#102—64no—
212#95—21no—
213#95—62no—
214C23×C3—42no—
215C23×C3—44no—
216C23×C3—42no—
217C23×C3—44no—
218C23×C3—42no—
219C23×C3C22×D411yescharacteristic
220C22×D3—22no—
221C22×D3—24no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—42no—
225C3×D4—42no—
226C3×D4—44no—
227C3×D4—42no—
228C3×D4—84no—
229C3×D4—84no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4#12112yes—
235C2×C3×C4#12112yes—
236C2×C3×C4#13512yes—
237C2×C3×C4AGL(1,ℤ/8)12yes—
238C2×C3×C4C22×D411yescharacteristic
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4C2×#4112yes—
241C2×C3×C4C2×#4112yes—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—44no—
252C2×Dic3AGL(1,ℤ/8)14yes—
253D12—24no—
254D12—22no—
255D12#12414yescomplemented
256C4×D3—416no—
257C4×D3—48no—
258Dic6—84no—
259C3×C8—42no—
260C3⋊C8—42no—
261C24—22no—
262C24—61no—
263C2×Q8—124no—
264C2×Q8—122no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—32no—
270C2×D4—34no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—61no—
277C2×D4—62no—
278C22×C4—22no—
279C22×C4—32no—
280C22×C4—32no—
281C22×C4—68no—
282C22×C4—61no—
283C22×C4—64no—
284C22×C4—62no—
285C22×C4—124no—
286C22×C4—124no—
287C22×C4—122no—
288C22×C4—122no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C23×D311yescharacteristic
292D8—128no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—62no—
296C2×C8—64no—
297C4⋊C4—22no—
298C4⋊C4—32no—
299C4⋊C4—61no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—122no—
303C4⋊C4#23714yes—
304C22⋊C4—68no—
305C22⋊C4—64no—
306C22⋊C4—128no—
307C22⋊C4—128no—
308C42—42no—
309C42—64no—
310C42—62no—
311C42—62no—
312C42—62no—
313C42—62no—
314C42—124no—
315C42#23512yes—
316C42D3×D412yes—
317C22×C3—82no—
318C22×C3—84no—
319C22×C3—84no—
320C22×C3—82no—
321C22×C3—82no—
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—24no—
328C2×D3—44no—
329C2×D3—44no—
330C2×D3—42no—
331C2×D3—42no—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4#46314yes—
335C3×C4C2×#12411yescharacteristic
336C3×C4#52912yes—
337C3×C4#53211yescharacteristic
338Dic3—24no—
339Dic3—42no—
340Dic3—42no—
341Dic3—82no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—44no—
346C23—44no—
347C23—62no—
348C23—64no—
349C23—121no—
350C23—121no—
351C23—122no—
352C23C2×D3×D411yescharacteristic
353Q8—244no—
354D4—34no—
355D4—42no—
356D4—42no—
357D4—44no—
358D4—62no—
359D4—64no—
360D4—84no—
361D4—84no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—34no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—1216no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—128no—
377C2×C4—122no—
378C2×C4—124no—
379C2×C4#88412yes—
380C2×C4#88412yes—
381C2×C4#91112yes—
382C2×C4#91612yes—
383C2×C4C2×#23511yescharacteristic
384C2×C4C2×D3×D411yescharacteristic
385C2×C4C2×#23712yes—
386C2×C4D3×#4112yes—
387C8—42no—
388C8—122no—
389C2×C3—162no—
390C2×C3#285011yescharacteristic
391C2×C3C2×#46311yescharacteristic
392C2×C3#303811yescharacteristic
393C2×C3#310612yes—
394C2×C3#310912yes—
395D3—44no—
396C22—64no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—84no—
401C22—84no—
402C22—122no—
403C22—122no—
404C22—124no—
405C22—124no—
406C22C2×#88411yescharacteristic
407C22#560612yes—
408C22C2×#91111yescharacteristic
409C22C2×#91611yescharacteristic
410C22D3×AGL(1,ℤ/8)12yes—
411C4—22no—
412C4—44no—
413C4—64no—
414C4—122no—
415C4—122no—
416C4—242no—
417C4#478714yes—
418C4#555712yes—
419C4D3×#12411yescharacteristic
420C4#571611yescharacteristic
421C3#3303011yescharacteristic, complemented, Sylow
422C2—124no—
423C2—162no—
424C2#7647411yescharacteristic
425C2C2×#478711yescharacteristic
426C2#7894211yescharacteristic
427C2#8031412yes—
428C2#8036112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1982C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenter#1982D3
hypocenterC3#33030
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
1. #83184C2semidirect
3. C2×#4842C2semidirect
4. C4×#893C2semidirect
5. C3×#1982C2semidirect
6. #70443C2semidirect
7. #70208C2semidirect
8. #70156C2semidirect
9. #69489C2semidirect
41. #1982D3semidirect
255. D12#124semidirect
421. C3#33030semidirect

Maximal subgroups

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

subgroupindex
#831842
C2×#50722
C2×#48422
C4×#8932
C3×#19822
#704432
#702082
#701562
#694892
#330303