Finite Group Databasebeta

← #10751713

Subgroups — #10751713

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic
2C2×C3×#387C211yesmaximal, characteristic
3C2×#4979C212yesmaximal
4#71565C212yesmaximal
5#71534C211yesmaximal, characteristic
6#69424C212yesmaximal
7#69420C212yesmaximal
8#69420C212yesmaximal
9#69415C212yesmaximal
10#16603—31nomaximal, Sylow
11C23×Dic6C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×C4⋊Dic3C2212yes—
17C22×Dic3⋊C4C2214yes—
18C2×#869—28no—
19C2×#869—24no—
20C3×#387—24no—
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2212yes—
31#4979—28no—
32C22×C4⋊Q8—31no—
33C2×#395—32no—
34C2×#387D311yescharacteristic
35#1446—32no—
36#1444—32no—
37#1444—32no—
38#1442—32no—
39#1441—32no—
40#1434—31no—
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2314yes—
49C23×Dic3C2314yes—
50C22×Dic6—24no—
51C22×Dic6—24no—
52C22×Dic6—44no—
53C22×Dic6—44no—
54C22×Dic6D418yes—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4Q814yescomplemented
60C2×C3×C42—24no—
61C2×C3×C42D414yes—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—28no—
65C2×C4⋊Dic3—28no—
66C2×Dic3⋊C4—28no—
67C2×Dic3⋊C4—44no—
68C2×Dic3⋊C4—44no—
69#869—232no—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—22no—
73C3×#95—22no—
74C3⋊#95—24no—
75C3⋊#95—24no—
76C3⋊#95—28no—
77C3⋊#95—44no—
78C3⋊#95—44no—
79C3⋊#95Q818yescomplemented
80C23×Q8—32no—
81C2×C4⋊Q8—68no—
82C2×C4⋊Q8—64no—
83C22×C4⋊C4—32no—
84C22×C4⋊C4—32no—
85C22×C4⋊C4—34no—
86C22×C4⋊C4C2×D311yescharacteristic
87C22×C4⋊C4C2×D311yescharacteristic
88C22×C42C2×D311yescharacteristic
89#395—68no—
90#387—24no—
91C2×#95—32no—
92C2×#95—32no—
93C2×#95—34no—
94C2×#95—32no—
95C2×#95—34no—
96C2×#95—34no—
97C2×#95C2×D311yescharacteristic
98C2×#95C2×D311yescharacteristic
99C2×#95C2×D311yescharacteristic
100C2×#95C2×D311yescharacteristic
101C24×C3C2411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4C2×D414yes—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×Q812yes—
129C22×C3×C4C2×Q812yes—
130C22×C3×C4C2×Q812yes—
131C22×C3×C4C2×Q814yes—
132C22×C3×C4#4112yes—
133C22×Dic3—24no—
134C22×Dic3—24no—
135C22×Dic3—24no—
136C22×Dic3—24no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—24no—
140C22×Dic3—24no—
141C22×Dic3—24no—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—24no—
145C22×Dic3C2×D418yes—
146C22×Dic3C2×Q818yes—
147C2×Dic6—216no—
148C2×Dic6—416no—
149C2×Dic6—416no—
150C2×Dic6—48no—
151C2×Dic6—48no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—44no—
154C3×C42—28no—
155C4⋊Dic3—216no—
156C4⋊Dic3—216no—
157Dic3⋊C4—416no—
158C22×Q8—38no—
159C22×Q8—64no—
160C22×Q8—64no—
161C22×Q8—124no—
162C22×Q8—124no—
163C23×C4—34no—
164C23×C4—34no—
165C23×C4C22×D311yescharacteristic
166C23×C4C22×D311yescharacteristic
167C23×C4C22×D311yescharacteristic
168C23×C4C22×D311yescharacteristic
169C23×C4C22×D311yescharacteristic
170C23×C4C22×D311yescharacteristic
171C23×C4C22×D311yescharacteristic
172C4⋊Q8—632no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—68no—
178C2×C4⋊C4—68no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4—68no—
182C2×C4⋊C4—124no—
183C2×C4⋊C4—124no—
184C2×C4⋊C4Dic614yescomplemented
185C2×C42—24no—
186C2×C42#6714yes—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—38no—
192#95—68no—
193#95—64no—
194#95—64no—
195#95—124no—
196#95—124no—
197C23×C3C22×D412yes—
198C23×C3C22×D411yescharacteristic
199C23×C3C22×Q811yescharacteristic
200C23×C3C22×Q812yes—
201C23×C3C2×#4111yescharacteristic
202C23×C3H(5,𝔽2)12yes—
203C23×C3H(5,𝔽2)12yes—
204C23×C32−1+412yes—
205C23×C32−1+412yes—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—28no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—24no—
214C2×C3×C4—28no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—44no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—44no—
234C2×C3×C4#12014yes—
235C2×C3×C4C22⋊Q814yes—
236C2×C3×C4C4⋊Q814yes—
237C2×Dic3—28no—
238C2×Dic3—28no—
239C2×Dic3—28no—
240C2×Dic3—28no—
241C2×Dic3—28no—
242C2×Dic3—28no—
243C2×Dic3—44no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—44no—
247C2×Dic3—44no—
248C2×Dic3—44no—
249C2×Dic3—44no—
250C2×Dic3—44no—
251Dic6—432no—
252C24C23×D311yescharacteristic
253C2×Q8—616no—
254C2×Q8—1216no—
255C2×Q8—1216no—
256C2×Q8—128no—
257C2×Q8—128no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—38no—
281C22×C4—38no—
282C22×C4—64no—
283C22×C4—64no—
284C22×C4—64no—
285C22×C4—64no—
286C22×C4—64no—
287C22×C4—64no—
288C22×C4—64no—
289C22×C4—64no—
290C22×C4—64no—
291C22×C4—64no—
292C22×C4—64no—
293C22×C4—64no—
294C22×C4C2×Dic612yes—
295C22×C4C2×Dic612yes—
296C22×C4C2×Dic612yes—
297C22×C4#23512yes—
298C22×C4D3×D414yes—
299C22×C4D3×Q814yes—
300C22×C4C2×#6712yes—
301C22×C4C2×#6712yes—
302C22×C4C2×#6712yes—
303C4⋊C4—28no—
304C4⋊C4—44no—
305C4⋊C4—616no—
306C4⋊C4—616no—
307C4⋊C4—1216no—
308C42—28no—
309C22×C3C2×#12011yescharacteristic
310C22×C3C2×C22⋊Q811yescharacteristic
311C22×C3C2×C4⋊Q811yescharacteristic
312C22×C3#53414yes—
313C22×C3#53511yescharacteristic
314C22×C3#53511yescharacteristic
315C22×C3#53611yescharacteristic
316C22×C3#53611yescharacteristic
317C22×C3#54212yes—
318C22×C3#54414yes—
319C22×C3#54412yes—
320C22×C3#54812yes—
321C22×C3D4×Q814yes—
322C22×C3D4×Q812yes—
323C22×C3#55412yes—
324C22×C3Q8212yes—
325C22×C3#56314yes—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—24no—
329C3×C4—42no—
330C3×C4—42no—
331C3×C4—81no—
332C3×C4—81no—
333Dic3—48no—
334Dic3—48no—
335C23C22×Dic611yescharacteristic
336C23C2×#23511yescharacteristic
337C23C2×D3×D412yes—
338C23C2×D3×Q812yes—
339C23#100912yes—
340C23#100912yes—
341C23#101012yes—
342C23#101012yes—
343C23C22×#6711yescharacteristic
344Q8—1232no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—68no—
374C2×C4—68no—
375C2×C4—68no—
376C2×C4—68no—
377C2×C4—68no—
378C2×C4—68no—
379C2×C4—124no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—124no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4#92414yes—
388C2×C4#93714yes—
389C2×C4#94414yes—
390C2×C3#240911yescharacteristic
391C2×C3#243411yescharacteristic
392C2×C3#256711yescharacteristic
393C2×C3#261311yescharacteristic
394C2×C3#262811yescharacteristic
395C2×C3#266011yescharacteristic
396C2×C3#272914yes—
397C2×C3#273414yes—
398C2×C3#282111yescharacteristic
399C22#551912yes—
400C22#553412yes—
401C22D4×Dic612yes—
402C22#555212yes—
403C22Q8×Dic612yes—
404C22#557912yes—
405C22#563214yes—
406C22#565314yes—
407C22#568914yes—
408C22#571214yes—
409C22C2×#92411yescharacteristic
410C22C2×#93711yescharacteristic
411C22C2×#94411yescharacteristic
412C22#582511yescharacteristic
413C22#582511yescharacteristic
414C22#583011yescharacteristic
415C22#583011yescharacteristic
416C4—24no—
417C4—24no—
418C4—24no—
419C4—42no—
420C4—42no—
421C4—81no—
422C4—81no—
423C4—128no—
424C4—128no—
425C3#1660311yescharacteristic, complemented, Sylow
426C2#7517714yes—
427C2#7517914yes—
428C2#8226411yescharacteristic
429C2#8234911yescharacteristic
430C2#8247811yescharacteristic
431C2#8276011yescharacteristic
432C2#8290311yescharacteristic
433C2#8304411yescharacteristic
434C2#8310311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#387C2
hypercenterC2×#387D3
hypocenterC3#16603
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
59. C2×C3×C4⋊C4Q8semidirect
79. C3⋊#95Q8semidirect
184. C2×C4⋊C4Dic6semidirect
425. C3#16603semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
C2×C3×#3872
C2×#49792
#715652
#715342
#694242
#694202
#694152
#166033