Finite Group Databasebeta

← #10828483

Subgroups — #10828483

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×#4928C211yesmaximal, characteristic
3C2×#4721C212yesmaximal
4C3×#1817C211yesmaximal, characteristic
5#69930C212yesmaximal
6#68684C214yesmaximal, complemented
7#68633C212yesmaximal, complemented
8#68631C212yesmaximal, complemented
9#24063—31nomaximal, Sylow
10C22×C3×C42C2211yescharacteristic
11C2×#926—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C2×#924—21no—
15C2×#924C2212yes—
16C22×Dic3⋊C4—21no—
17#5470—28no—
18C2×#870—21no—
19C2×#869C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6—21no—
22C2×C4×Dic6C2212yes—
23C2×C3×C22⋊C8C2212yes—
24C2×C3⋊#95—21no—
25C2×#817C2214yes—
26C2×#816C2214yes—
27C2×#816C2214yes—
28#4928—21no—
29#4928—21no—
30#4928C2212yes—
31#4928C2212yes—
32#4721—28no—
33C3×#336C2212yes—
34C3×#336C2212yes—
35#4551C2214yes—
36C2×#533—31no—
37C2×#483—32no—
38C2×#389—31no—
39#2058—32no—
40#2043—32no—
41#1872—32no—
42#1865—34no—
43#1817D311yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3—21no—
48C23×Dic3—21no—
49C22×Dic6—21no—
50C22×Dic6C2312yes—
51C22×C3×C8C2314yes—
52C2×C3×C42—24no—
53C2×C3×C42D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#217—22no—
58#926—28no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—24no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2312yes—
67C2×C4⋊Dic3C2312yes—
68C2×C4⋊Dic3C2312yes—
69#924—28no—
70#924—24no—
71#924D418yes—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—44no—
77C2×C4×Dic3—22no—
78#870—28no—
79#869—24no—
80#869—24no—
81C4×Dic6—28no—
82C4×Dic6—24no—
83C4×Dic6—24no—
84C3×C22⋊C8—24no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#817—28no—
88#816—28no—
89#816—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—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—61no—
102C22×C42C2×D311yescharacteristic
103#483—68no—
104C2×#106—34no—
105C2×#103—34no—
106C2×#103—34no—
107C2×C22⋊C8C2×D312yes—
108#389—32no—
109#389—32no—
110#389—61no—
111#389—61no—
112C2×#95—61no—
113#340—34no—
114#336C2×D312yes—
115#336C2×D312yes—
116C24×C3C2×D411yescharacteristic
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×C4SD1614yes—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D414yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—24no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—42no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C22×Dic3—42no—
141C22×Dic3—42no—
142C2×Dic6—24no—
143C2×Dic6—24no—
144C2×Dic6—24no—
145C2×Dic6—22no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6C2×D414yes—
149C2×C3×C8—24no—
150C2×C3×C8—24no—
151C2×C3×C8—24no—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42—22no—
155C3×C42—22no—
156#217—28no—
157C4⋊Dic3—24no—
158C4⋊Dic3—24no—
159C4⋊Dic3—24no—
160C4⋊Dic3—24no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—24no—
164C4⋊Dic3—42no—
165C4⋊Dic3C2×D414yes—
166C4⋊Dic3#4114yes—
167Dic3⋊C4—28no—
168Dic3⋊C4—28no—
169Dic3⋊C4—44no—
170Dic3⋊C4—44no—
171C4×Dic3—28no—
172C22×Q8—32no—
173C22×Q8—61no—
174C23×C4—61no—
175C23×C4—61no—
176C23×C4D1211yescharacteristic
177C23×C4D1211yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8C22×D314yes—
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—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—124no—
205C2×C22⋊C4—62no—
206C2×C42—24no—
207C2×C42—62no—
208C2×C42D1211yescharacteristic
209C2×C42D1211yescharacteristic
210C2×C42C22×D311yescharacteristic
211C2×C42C22×D311yescharacteristic
212#106—68no—
213#103—68no—
214#103—68no—
215C22⋊C8—24no—
216#95—62no—
217#95—62no—
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—21no—
238C2×C3×C4—21no—
239C2×C3×C4—44no—
240C2×C3×C4#12012yes—
241C2×C3×C4C2×SD1612yes—
242C2×C3×C4C2×SD1612yes—
243C2×C3×C4#13712yes—
244C2×C3×C4C22×D412yes—
245C2×C3×C4C2×#4112yes—
246C2×C3×C42−1+412yes—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—48no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—84no—
261Dic6—28no—
262Dic6—48no—
263Dic6—44no—
264C3×C8—44no—
265C24C2×D1211yescharacteristic
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—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—64no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4#20414yes—
291C22×C4C2×D1211yescharacteristic
292C22×C4C2×D1211yescharacteristic
293C22×C4C2×D1211yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4C2×D1211yescharacteristic
296C22×C4D3×D414yes—
297C22×C4C23×D311yescharacteristic
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—64no—
304C4⋊C4—64no—
305C4⋊C4—64no—
306C4⋊C4—68no—
307C4⋊C4—64no—
308C4⋊C4—68no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—64no—
312C4⋊C4—122no—
313C4⋊C4—124no—
314C4⋊C4—124no—
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—41no—
340C3×C4#54714yes—
341Dic3—44no—
342Dic3—44no—
343Dic3—44no—
344Dic3—82no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23#88212yes—
350C23C2×#20412yes—
351C23#90912yes—
352C23C22×D1211yescharacteristic
353Q8—68no—
354Q8—128no—
355Q8—124no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—28no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—44no—
370C2×C4—64no—
371C2×C4—64no—
372C2×C4—64no—
373C2×C4—128no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—244no—
384C2×C4#88212yes—
385C2×C4C2×#20412yes—
386C2×C4C2×#20412yes—
387C2×C4#90912yes—
388C2×C4C2×D3×D412yes—
389C2×C4D3×#4112yes—
390C2×C4#101012yes—
391C8—44no—
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#473814yes—
407C22C2×#88211yescharacteristic
408C22#552411yescharacteristic
409C22#552611yescharacteristic
410C22C22×#20411yescharacteristic
411C22C2×#90911yescharacteristic
412C22C2×#90911yescharacteristic
413C22#574811yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—124no—
420C4—124no—
421C4—124no—
422C4—242no—
423C4#565514yes—
424C3#2406311yescharacteristic, complemented, Sylow
425C2—24no—
426C2#7406311yescharacteristic
427C2C2×#473811yescharacteristic
428C2#7864011yescharacteristic
429C2#7927111yescharacteristic
430C2#7929811yescharacteristic
431C2#7932011yescharacteristic
432C2#7935611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1817D3
hypocenterC3#24063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
6. #68684C2semidirect
7. #68633C2semidirect
8. #68631C2semidirect
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×#49282
C2×#47212
C3×#18172
#699302
#686842
#686332
#686312
#240633