Finite Group Databasebeta

← #10848838

Subgroups — #10848838

Order 768.

1094 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×#541C211yesmaximal, characteristic, complemented
2C2×#5086C211yesmaximal, characteristic, complemented
3C2×#5068C211yesmaximal, characteristic, complemented
4#82287C211yesmaximal, characteristic
5#71785C211yesmaximal, characteristic, complemented
6#71741C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#71699C212yesmaximal
9#71633C212yesmaximal, complemented
10#71632C212yesmaximal, complemented
11#71582C211yesmaximal, characteristic, complemented
12#25612—31nomaximal, Sylow
13C3×#541—24no—
14C3×#541—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#125—22no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117C2211yescharacteristic
22C22×#216C2212yes—
23C22×#215C2212yes—
24C2×#930C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C2×#875C2211yescharacteristic
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#830C2212yes—
30#5086—24no—
31#5068—24no—
32#4994—22no—
33C2×#810C2212yes—
34C2×#809C2212yes—
35#4961—22no—
36#4939C2211yescharacteristic
37#4935C2211yescharacteristic
38#4929C2211yescharacteristic
39C2×#803C2211yescharacteristic
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C2×#541D311yescharacteristic, complemented
45C2×#497—31no—
46C2×#488—31no—
47#2620—31no—
48#2001—32no—
49#2000—32no—
50#1992—31no—
51#1956—31no—
52#1880—32no—
53#1878—32no—
54#1750—31no—
55C22×C3×Q8C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C22×D12C2311yescharacteristic
59C22×Dic6C2311yescharacteristic
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8D414yescomplemented
62C3×#126—216no—
63C3×#125—216no—
64C3×#125—22no—
65C3×#125D414yescomplemented
66C3×C22⋊Q8—24no—
67C3×C22⋊Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D412yes—
80C2×C3×C42—22no—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84C2×#216—24no—
85C2×#216—24no—
86C2×#216—42no—
87C2×#215—24no—
88C2×#215—24no—
89C2×#215—42no—
90#930—42no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97#924—42no—
98C2×Dic3⋊C4—21no—
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2312yes—
101#875—22no—
102#875—22no—
103#835—42no—
104#835—42no—
105C3⋊#95—22no—
106#830—24no—
107#810—24no—
108#809—24no—
109#803—22no—
110C4×C3⋊C8—22no—
111C22×Q16—32no—
112C22×SD16—32no—
113#541—24no—
114#541—24no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#126—22no—
117C2×#125—22no—
118C2×#125C2×D311yescharacteristic
119C2×#124—31no—
120C2×C22⋊Q8—31no—
121C2×C22⋊Q8C2×D312yes—
122C2×#121—31no—
123C2×C4×Q8C2×D312yes—
124C2×#117C2×D311yescharacteristic
125#497—64no—
126#488—64no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#102—32no—
131C2×C22⋊C8—32no—
132C2×#97—31no—
133C2×C4×C8—31no—
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384—31no—
139#340—32no—
140#340—32no—
141#336—32no—
142C24×C3C2×D411yescharacteristic
143C23×D3—21no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8C2×D414yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2411yescharacteristic
160C2×#67—42no—
161C2×#67—82no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—22no—
168C2×D12—22no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C3×C4⋊C4—28no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4C2×D414yes—
180C3×C22⋊C4—28no—
181C3×C22⋊C4—24no—
182C3×C42—28no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42C2×D412yes—
186C3×C42C2×D412yes—
187#217—42no—
188#216—48no—
189#215—48no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—82no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—82no—
196Dic3⋊C4—42no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8#4114yes—
203C22×Q8—31no—
204C22×Q8C22×D312yes—
205C22×D4—31no—
206C22×D4—61no—
207C23×C4C22×D311yescharacteristic
208C2×Q16—64no—
209C2×Q16—64no—
210C2×Q16—122no—
211C2×SD16—64no—
212C2×SD16—64no—
213C2×SD16—122no—
214C22×C8—32no—
215C22×C8—32no—
216C4⋊Q8—22no—
217C4⋊Q8#6714yescomplemented
218#126—216no—
219#125—216no—
220#125—22no—
221#125#6714yescomplemented
222#124—62no—
223#124—62no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—122no—
227#121—122no—
228C4×Q8—24no—
229C4×Q8—24no—
230#117—22no—
231#117—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4#6712yes—
241C2×C4⋊C4C22×D312yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4#6712yes—
248C2×C42—22no—
249C2×C42C22×D311yescharacteristic
250C2×C42C22×D311yescharacteristic
251#103—64no—
252#103—122no—
253#103—122no—
254#102—64no—
255C22⋊C8—64no—
256#97—62no—
257C4×C8—62no—
258#95—62no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3C22×D411yescharacteristic
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—42no—
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4#12014yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4H(5,𝔽2)12yes—
295#67—84no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303D12—42no—
304D12—42no—
305Dic6—42no—
306Dic6—42no—
307C3⋊C8—24no—
308C3⋊C8—42no—
309C24—61no—
310C24C2×#6711yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—62no—
317C2×Q8—62no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8C2×#6714yes—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—61no—
324C2×D4—61no—
325C2×D4—122no—
326C2×D4—242no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6712yes—
339C22×C4C23×D311yescharacteristic
340Q16—128no—
341SD16—128no—
342C2×C8—34no—
343C2×C8—62no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C4⋊C4—242no—
358C4⋊C4C2×#6714yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C22⋊C4—124no—
364C22⋊C4—242no—
365C42—28no—
366C42—21no—
367C42—21no—
368C42C2×#6712yes—
369C42C2×#6712yes—
370C22×C3—42no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3C2×#12011yescharacteristic
376C22×C3#53211yescharacteristic
377C22×C3#53511yescharacteristic
378C22×C3#57712yes—
379C22×C3#57812yes—
380C2×D3—82no—
381C2×D3—81no—
382C2×D3—81no—
383C2×D3—81no—
384C2×D3—82no—
385C3×C4—24no—
386C3×C4—41no—
387C3×C4—44no—
388C3×C4—42no—
389C3×C4—44no—
390C3×C4#54612yes—
391C3×C4#55012yes—
392Dic3—81no—
393Dic3—81no—
394Dic3—81no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—121no—
401C23—121no—
402C23—122no—
403C23—122no—
404C23—121no—
405C23C22×#6711yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—122no—
411Q8—122no—
412D4—122no—
413D4—122no—
414D4—244no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—121no—
438C2×C4#95314yes—
439C2×C4D3×#4112yes—
440C2×C4#100912yes—
441C2×C4C22×#6711yescharacteristic
442C2×C4C22×#6711yescharacteristic
443C8—64no—
444C8—122no—
445C2×C3—81no—
446C2×C3#243111yescharacteristic
447C2×C3#301312yes—
448C2×C3#317812yes—
449C2×C3#322211yescharacteristic
450C2×C3#322311yescharacteristic
451D3—161no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—241no—
458C22—242no—
459C22—242no—
460C22—241no—
461C22—241no—
462C22#583011yescharacteristic
463C22#583211yescharacteristic
464C22#583712yes—
465C22#583812yes—
466C22C2×#95311yescharacteristic
467C4—24no—
468C4—41no—
469C4—44no—
470C4—42no—
471C4—44no—
472C4—241no—
473C4—241no—
474C4—241no—
475C4#582912yes—
476C4#583412yes—
477C3#2561211yescharacteristic, complemented, Sylow
478C2—81no—
479C2—481no—
480C2#8344512yes—
481C2#8344811yescharacteristic
482C2#8345211yescharacteristic
483C2#8409211yescharacteristic
484C2#8411412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#541C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#541D3
hypocenterC3#25612
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
2. C2×#5086C2semidirect
3. C2×#5068C2semidirect
5. #71785C2semidirect
6. #71741C2semidirect
7. #71700C2semidirect
9. #71633C2semidirect
10. #71632C2semidirect
11. #71582C2semidirect
44. C2×#541D3semidirect
61. C3×C4⋊Q8D4semidirect
65. C3×#125D4semidirect
217. C4⋊Q8#67semidirect
221. #125#67semidirect
477. C3#25612semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#50862
C2×#50682
#822872
#717852
#717412
#717002
#716992
#716332
#716322
#715822
#256123