Finite Group Databasebeta

← #10828458

Subgroups — #10828458

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5034C212yesmaximal, complemented
3C2×#4928C211yesmaximal, characteristic, complemented
4#71783C212yesmaximal, complemented
5#71762C212yesmaximal, complemented
6#71677C212yesmaximal, complemented
7#71607C214yesmaximal, complemented
8#71531C211yesmaximal, characteristic, complemented
9#24061—31nomaximal, Sylow
10C22×C3×#41—21no—
11C3×#532—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×C3×#124—21no—
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4—21no—
23C2×#832C2214yes—
24C2×#832C2214yes—
25C2×C3⋊#95—21no—
26C2×#830C2212yes—
27#5034—28no—
28C2×#808C2214yes—
29#4928—21no—
30#4928—21no—
31#4928C2212yes—
32#4928C2212yes—
33#4529C2214yes—
34#4524C2212yes—
35#4524C2212yes—
36C2×#532D311yescharacteristic
37C2×#481—32no—
38C2×#389—31no—
39#2058—32no—
40#2041—32no—
41#1874—32no—
42#1864—34no—
43#1817—31no—
44C2×C3×#41—28no—
45C2×C3×#41—44no—
46C2×C3×#41—44no—
47C22×C3×Q8—21no—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C23×C3×C4—22no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—21no—
57C23×Dic3—21no—
58C3×C4⋊Q8—24no—
59C3×#127—24no—
60C3×#124—28no—
61C3×#121—28no—
62C3×#121—24no—
63C3×#121D418yes—
64C3×C4×D4—28no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42—24no—
72C2×C3×C42D411yescharacteristic
73C2×C3×C42D411yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2312yes—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—44no—
85C22×C3⋊C8C2314yes—
86#832—28no—
87#832—28no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90#830—24no—
91#808—28no—
92C22×#41—21no—
93#532—28no—
94C2×C4⋊Q8C2×D311yescharacteristic
95C2×#127C2×D311yescharacteristic
96C2×#124—21no—
97C2×#121—21no—
98C2×#121C2×D312yes—
99C2×C4×D4—21no—
100C2×C4×D4C2×D312yes—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—61no—
104C22×C42C2×D311yescharacteristic
105#481—68no—
106C2×#106—34no—
107C2×#102—34no—
108C2×#102—34no—
109C2×C22⋊C8—32no—
110#389—32no—
111#389—32no—
112#389—61no—
113#389—61no—
114C2×#95—61no—
115#340—34no—
116#336—32no—
117#336—32no—
118C24×C3—22no—
119C24×C3C2×D411yescharacteristic
120C3×#41—432no—
121C2×C3×Q8—24no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—28no—
127C2×C3×D4—22no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—42no—
135C2×C3×D4—42no—
136C2×C3×D4C2×D414yes—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—28no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—41no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4SD1614yes—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—24no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—42no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4C2×D414yes—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—28no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—22no—
173C4⋊Dic3—24no—
174C4⋊Dic3—22no—
175C4⋊Dic3—24no—
176C4⋊Dic3—42no—
177C4⋊Dic3#4114yescomplemented
178Dic3⋊C4—44no—
179Dic3⋊C4—44no—
180C2×C3⋊C8—24no—
181C2×C3⋊C8—24no—
182C2×C3⋊C8—24no—
183C2×#41—28no—
184C2×#41—44no—
185C2×#41—44no—
186C22×Q8—21no—
187C22×D4—22no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C22×D312yes—
191C23×C4—22no—
192C23×C4—61no—
193C23×C4—61no—
194C23×C4#6711yescharacteristic
195C23×C4#6711yescharacteristic
196C23×C4C22×D311yescharacteristic
197C22×C8—34no—
198C4⋊Q8—24no—
199#127—24no—
200#124—28no—
201#121—28no—
202#121—24no—
203#121#6718yes—
204C4×D4—28no—
205C4×D4—24no—
206C4×D4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—64no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—64no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4—124no—
217C2×C4⋊C4C22×D312yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—24no—
221C2×C42#6711yescharacteristic
222C2×C42#6711yescharacteristic
223C2×C42C22×D311yescharacteristic
224C2×C42C22×D311yescharacteristic
225#106—68no—
226#102—68no—
227#102—68no—
228C22⋊C8—64no—
229#95—62no—
230#95—62no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—24no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3#12012yes—
243C23×C3C2×SD1612yes—
244C23×C3AGL(1,ℤ/8)12yes—
245C23×C3C22×D411yescharacteristic
246C3×Q8—48no—
247C3×D4—28no—
248C3×D4—48no—
249C3×D4—48no—
250C3×D4—416no—
251C3×D4—44no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—44no—
267C2×C3×C4—416no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4#12012yes—
271C2×C3×C4C2×SD1612yes—
272C2×C3×C4C2×SD1612yes—
273C2×C3×C4AGL(1,ℤ/8)12yes—
274C2×C3×C4C22×D412yes—
275C2×C3×C4C2×#4112yes—
276C2×C3×C42−1+412yes—
277C2×Dic3—24no—
278C2×Dic3—48no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—84no—
285C3⋊C8—44no—
286C24—22no—
287C24C2×#6711yescharacteristic
288#41—432no—
289C2×Q8—24no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—28no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4C2×#6714yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—64no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—122no—
320C22×C4—122no—
321C22×C4—122no—
322C22×C4—122no—
323C22×C4—122no—
324C22×C4#21414yes—
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C2×#6714yes—
331C22×C4C23×D311yescharacteristic
332C2×C8—64no—
333C2×C8—64no—
334C2×C8—64no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—34no—
338C4⋊C4—64no—
339C4⋊C4—62no—
340C4⋊C4—64no—
341C4⋊C4—124no—
342C4⋊C4—124no—
343C4⋊C4—122no—
344C4⋊C4C2×#6714yes—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3#45014yes—
365C22×C3C2×#12011yescharacteristic
366C22×C3#52511yescharacteristic
367C22×C3#53611yescharacteristic
368C22×C3C22×SD1611yescharacteristic
369C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C3#57511yescharacteristic
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—41no—
378C3×C4#54714yes—
379Dic3—44no—
380Dic3—82no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—24no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23#93212yes—
393C23C2×#21412yes—
394C23#95312yes—
395C23C22×#6711yescharacteristic
396Q8—48no—
397D4—28no—
398D4—48no—
399D4—48no—
400D4—416no—
401D4—44no—
402C2×C4—28no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—28no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—44no—
417C2×C4—416no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—64no—
421C2×C4—128no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—244no—
428C2×C4C2×#21412yes—
429C2×C4C2×#21412yes—
430C2×C4#94912yes—
431C2×C4#95312yes—
432C2×C4D3×#4112yes—
433C2×C4#101012yes—
434C2×C4C22×#6712yes—
435C8—124no—
436C2×C3—24no—
437C2×C3—84no—
438C2×C3#240611yescharacteristic
439C2×C3C2×#45011yescharacteristic
440C2×C3#300611yescharacteristic
441C2×C3#310311yescharacteristic
442C2×C3#310911yescharacteristic
443C2×C3#322711yescharacteristic
444C2×C3#323411yescharacteristic
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22#521814yes—
459C22C2×#93211yescharacteristic
460C22C22×#21411yescharacteristic
461C22C2×#94911yescharacteristic
462C22#582211yescharacteristic
463C22#582511yescharacteristic
464C22#582811yescharacteristic
465C22C2×#95311yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—44no—
471C4—41no—
472C4—124no—
473C4—242no—
474C4#582414yes—
475C3#2406111yescharacteristic, complemented, Sylow
476C2—24no—
477C2—84no—
478C2#8327111yescharacteristic
479C2#8327311yescharacteristic
480C2#8330011yescharacteristic
481C2#8331511yescharacteristic
482C2C2×#521811yescharacteristic
483C2#8408211yescharacteristic
484C2#8409011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#24061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5034C2semidirect
3. C2×#4928C2semidirect
4. #71783C2semidirect
5. #71762C2semidirect
6. #71677C2semidirect
7. #71607C2semidirect
8. #71531C2semidirect
177. C4⋊Dic3#41semidirect
475. C3#24061semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50342
C2×#49282
#717832
#717622
#716772
#716072
#715312
#240613