Finite Group Databasebeta

← #10828481

Subgroups — #10828481

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic
3C2×#4971C212yesmaximal
4#71559C212yesmaximal
5#71531C211yesmaximal, characteristic
6#69633C212yesmaximal, complemented
7#69632C214yesmaximal, complemented
8#69605C212yesmaximal, complemented
9#24063—31nomaximal, Sylow
10C22×C3×C4⋊C4—21no—
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C2×#926—21no—
15C2×#924—21no—
16C2×#924C2212yes—
17#5470—28no—
18C2×#870—21no—
19C2×#869C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6—21no—
22C2×C4×Dic6C2212yes—
23C3×#389—21no—
24C3×#389—21no—
25C3×#389C2212yes—
26C3×#389C2212yes—
27C2×C3×#95—21no—
28C2×#830C2212yes—
29#4971—28no—
30C2×#810C2214yes—
31C2×#810C2214yes—
32C2×#808C2214yes—
33#4529C2214yes—
34#4524C2212yes—
35#4524C2212yes—
36C2×#533—31no—
37C2×#483—32no—
38C2×#389D311yescharacteristic
39#2058—32no—
40#2043—32no—
41#1872—32no—
42#1865—34no—
43#1817—31no—
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×Dic6—21no—
50C22×Dic6C2312yes—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—44no—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C42—24no—
61C2×C3×C42D411yescharacteristic
62C2×C3×C42D411yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×#217—22no—
66#926—28no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3C2312yes—
71#924—28no—
72#924—24no—
73#924D418yes—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×C4×Dic3—22no—
77C22×C3⋊C8C2314yes—
78#870—28no—
79#869—24no—
80#869—24no—
81C4×Dic6—28no—
82C4×Dic6—24no—
83C4×Dic6—24no—
84C3×#95—22no—
85C3×#95—22no—
86#830—24no—
87#810—28no—
88#810—28no—
89#808—28no—
90#533—68no—
91C2×C4⋊Q8—31no—
92C2×C4⋊Q8—31no—
93C2×#125—61no—
94C2×C22⋊Q8—32no—
95C2×C22⋊Q8—61no—
96C2×C4×Q8—32no—
97C2×C4×Q8—61no—
98C2×#117—61no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2×D311yescharacteristic
101C22×C4⋊C4C2×D311yescharacteristic
102C22×C42C2×D311yescharacteristic
103#483—68no—
104C2×#106—34no—
105C2×#103—34no—
106C2×#103—34no—
107C2×C22⋊C8—32no—
108#389—21no—
109#389—21no—
110#389C2×D312yes—
111#389C2×D312yes—
112C2×#95—21no—
113#340—34no—
114#336—32no—
115#336—32no—
116C24×C3C2×D411yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—41no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4SD1614yes—
133C22×C3×C4C2×D414yes—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—22no—
142C2×Dic6—24no—
143C2×Dic6—24no—
144C2×Dic6—24no—
145C2×Dic6—22no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6C2×D414yes—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—44no—
155C3×C4⋊C4#4114yes—
156C3×C42—22no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—22no—
160#217—28no—
161C4⋊Dic3—24no—
162C4⋊Dic3—24no—
163C4⋊Dic3—24no—
164C4⋊Dic3—22no—
165C4⋊Dic3C2×D414yes—
166Dic3⋊C4—28no—
167Dic3⋊C4—28no—
168C4×Dic3—28no—
169C2×C3⋊C8—24no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—24no—
172C22×Q8—32no—
173C22×Q8—61no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8—34no—
180C4⋊Q8—64no—
181C4⋊Q8—64no—
182#125—68no—
183C22⋊Q8—38no—
184C22⋊Q8—68no—
185C22⋊Q8—64no—
186C4×Q8—64no—
187C4×Q8—64no—
188C4×Q8—68no—
189#117—68no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4C22×D312yes—
204C2×C4⋊C4C22×D312yes—
205C2×C22⋊C4—62no—
206C2×C42—24no—
207C2×C42—62no—
208C2×C42#6711yescharacteristic
209C2×C42#6711yescharacteristic
210C2×C42C22×D311yescharacteristic
211C2×C42C22×D311yescharacteristic
212#106—68no—
213#103—68no—
214#103—68no—
215C22⋊C8—64no—
216#95—22no—
217#95—22no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3#12012yes—
223C23×C3C2×SD1612yes—
224C23×C3#13712yes—
225C23×C3C22×D411yescharacteristic
226C2×C3×C4—21no—
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—28no—
235C2×C3×C4—28no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—44no—
241C2×C3×C4—48no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—84no—
248C2×C3×C4#12012yes—
249C2×C3×C4C2×SD1612yes—
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4#13712yes—
252C2×C3×C4C22×D412yes—
253C2×C3×C4C2×#4112yes—
254C2×C3×C42−1+412yes—
255C2×Dic3—24no—
256C2×Dic3—24no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261Dic6—28no—
262Dic6—48no—
263Dic6—44no—
264C3⋊C8—44no—
265C24C2×#6711yescharacteristic
266C2×Q8—34no—
267C2×Q8—64no—
268C2×Q8—62no—
269C2×Q8—64no—
270C2×Q8—64no—
271C2×Q8—122no—
272C2×Q8—122no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4#21414yes—
291C22×C4D3×D414yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C22×C4C23×D311yescharacteristic
298C2×C8—64no—
299C2×C8—64no—
300C2×C8—64no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—34no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—64no—
309C4⋊C4—64no—
310C4⋊C4—68no—
311C4⋊C4—64no—
312C4⋊C4—68no—
313C4⋊C4—62no—
314C4⋊C4#23514yes—
315C22⋊C4—68no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—68no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3#44814yes—
328C22×C3C2×#12011yescharacteristic
329C22×C3#52511yescharacteristic
330C22×C3#53611yescharacteristic
331C22×C3C22×SD1611yescharacteristic
332C22×C3C2×#13711yescharacteristic
333C22×C3C2×#13711yescharacteristic
334C22×C3#57511yescharacteristic
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—44no—
340C3×C4—41no—
341C3×C4—82no—
342C3×C4#54714yes—
343Dic3—44no—
344Dic3—44no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23C2×#21412yes—
350C23#93712yes—
351C23#95112yes—
352C23C22×#6711yescharacteristic
353Q8—68no—
354Q8—128no—
355Q8—124no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—28no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—64no—
378C2×C4—64no—
379C2×C4—84no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4C2×#21412yes—
385C2×C4C2×#21412yes—
386C2×C4#93712yes—
387C2×C4#94212yes—
388C2×C4C2×#23512yes—
389C2×C4C2×D3×D412yes—
390C2×C4#100712yes—
391C8—124no—
392C2×C3—24no—
393C2×C3#240611yescharacteristic
394C2×C3C2×#44811yescharacteristic
395C2×C3#300711yescharacteristic
396C2×C3#310111yescharacteristic
397C2×C3#311311yescharacteristic
398C2×C3#322911yescharacteristic
399C2×C3#323611yescharacteristic
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22#504014yes—
407C22#579111yescharacteristic
408C22C22×#21411yescharacteristic
409C22C2×#93711yescharacteristic
410C22C2×#94211yescharacteristic
411C22#581611yescharacteristic
412C22#582211yescharacteristic
413C22C2×#95111yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—44no—
420C4—82no—
421C4—124no—
422C4—124no—
423C4#550714yes—
424C3#2406311yescharacteristic, complemented, Sylow
425C2—24no—
426C2#8219211yescharacteristic
427C2#8231411yescharacteristic
428C2#8244811yescharacteristic
429C2C2×#504011yescharacteristic
430C2#8273511yescharacteristic
431C2#8302111yescharacteristic
432C2#8322311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#389C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#389D3
hypocenterC3#24063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
6. #69633C2semidirect
7. #69632C2semidirect
8. #69605C2semidirect
424. C3#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C2×C3×#3892
C2×#49712
#715592
#715312
#696332
#696322
#696052
#240633