Finite Group Databasebeta

← #10944133

Subgroups — #10944133

Order 768.

1372 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic, complemented
2C3×#2620C211yesmaximal, characteristic
3C2×#5054C212yesmaximal, complemented
4C2×#5050C211yesmaximal, characteristic, complemented
5C2×#5040C211yesmaximal, characteristic
6#71693C211yesmaximal, characteristic, complemented
7#71678C211yesmaximal, characteristic
8#71583C211yesmaximal, characteristic, complemented
9#69690C212yesmaximal, complemented
10#69660C212yesmaximal, complemented
11#69588C212yesmaximal, complemented
12#32918—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×#216C2212yes—
19C22×#214C2212yes—
20C2×#926C2211yescharacteristic
21#5516—28no—
22C2×#887—22no—
23C2×#877—22no—
24C2×C4×Dic6C2212yes—
25C2×C4×Dic6C2212yes—
26C3×#396—22no—
27C3×#394—22no—
28C3×#391C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5054—28no—
36#5050—24no—
37#5040—24no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×C4⋊(C3⋊C8)C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C2×#519—31no—
45C2×#464—32no—
46C2×#451—31no—
47C2×#448—31no—
48#2620D311yescharacteristic
49#1880—32no—
50#1878—32no—
51#1875—31no—
52#1873—31no—
53#1803—32no—
54#1798—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C22×Dic6—21no—
61C22×Dic6—22no—
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6D414yes—
65C22×Dic6C2312yes—
66C3×#124—24no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×#121—22no—
70C3×#121—22no—
71C3×#117—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4D412yes—
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—24no—
88C2×#214—24no—
89C2×#214—24no—
90C2×#214—24no—
91#926—24no—
92C2×C4⋊Dic3C2312yes—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—21no—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2312yes—
100#887—216no—
101#877—216no—
102C4×Dic6—24no—
103C4×Dic6—24no—
104C4×Dic6—24no—
105C4×Dic6D418yescomplemented
106C3×#95—22no—
107#835—24no—
108#832—24no—
109#830—22no—
110#830—22no—
111#810—24no—
112#810—24no—
113C4⋊(C3⋊C8)—24no—
114C23×Q8—31no—
115C22×Q16—32no—
116C22×SD16—32no—
117C2×#124C2×D311yescharacteristic
118C2×C22⋊Q8C2×D311yescharacteristic
119C2×#121C2×D311yescharacteristic
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—62no—
126C2×#117C2×D311yescharacteristic
127#464—68no—
128#451—64no—
129#448—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8—31no—
137#396—22no—
138#394—22no—
139#391C2×D311yescharacteristic
140#390C2×D311yescharacteristic
141#384C2×D311yescharacteristic
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3—21no—
146C24×C3C2×D411yescharacteristic
147C2×C3×Q8—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—21no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—42no—
156C2×C3×D4—82no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C2×Dic6—24no—
174C2×Dic6—24no—
175C2×Dic6—216no—
176C2×Dic6—24no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—216no—
181C2×Dic6C2×D414yes—
182C2×Dic6C2×D414yes—
183C2×Dic6#4114yes—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—82no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—82no—
196C3×C42—22no—
197C3×C42C2×D414yes—
198#217—24no—
199#216—48no—
200#214—48no—
201C4⋊Dic3—22no—
202C4⋊Dic3C2×D414yes—
203Dic3⋊C4—28no—
204Dic3⋊C4—28no—
205C4×Dic3—28no—
206C4×Dic3—24no—
207C4×Dic3—28no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C22×Q8—34no—
215C22×Q8—32no—
216C22×Q8—64no—
217C22×Q8—64no—
218C22×Q8—62no—
219C22×Q8—61no—
220C22×Q8C22×D311yescharacteristic
221C22×D4—21no—
222C22×D4C22×D311yescharacteristic
223C23×C4—61no—
224C23×C4C22×D311yescharacteristic
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—64no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C22×C8—32no—
232C22×C8—32no—
233#124—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—22no—
237#121—22no—
238C4×Q8—38no—
239C4×Q8—64no—
240C4×Q8—64no—
241C4×Q8—616no—
242C4×Q8—64no—
243#117—24no—
244#117—616no—
245#117—64no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4#6712yes—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4#6712yes—
260C2×C42—61no—
261C2×C42—62no—
262C2×C42—62no—
263C2×C42C22×D311yescharacteristic
264C2×C42C22×D311yescharacteristic
265C4⋊C8—64no—
266#103—64no—
267#103—64no—
268#103—64no—
269#102—64no—
270C22⋊C8—62no—
271C22⋊C8—62no—
272#95—22no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3#12112yes—
283C23×C3C22×D411yescharacteristic
284C3×Q8—42no—
285C3×Q8—42no—
286C3×D4—42no—
287C3×D4—42no—
288C3×D4—84no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4#12014yes—
306C2×C3×C4#12112yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D412yes—
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4112yes—
311C2×Dic3—216no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318Dic6—216no—
319Dic6—24no—
320Dic6—24no—
321Dic6—24no—
322Dic6#12118yes—
323C3⋊C8—42no—
324C3⋊C8—42no—
325C24—21no—
326C24C2×#6711yescharacteristic
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—34no—
332C2×Q8—34no—
333C2×Q8—34no—
334C2×Q8—616no—
335C2×Q8—64no—
336C2×Q8—64no—
337C2×Q8—616no—
338C2×Q8—64no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—42no—
347C2×D4—82no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—64no—
354C22×C4—64no—
355C22×C4—64no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4D3×D412yes—
359C22×C4#23712yes—
360C22×C4C2×#6712yes—
361C22×C4C2×#6712yes—
362C22×C4C2×#6711yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—128no—
365SD16—128no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C4⋊C4—24no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—34no—
378C4⋊C4—42no—
379C4⋊C4—68no—
380C4⋊C4—68no—
381C4⋊C4—62no—
382C4⋊C4—82no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—64no—
388C22⋊C4—82no—
389C42—22no—
390C42—64no—
391C42—68no—
392C42—68no—
393C42D3×D414yes—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—81no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3C2×#12011yescharacteristic
405C22×C3C2×#12111yescharacteristic
406C22×C3#52511yescharacteristic
407C22×C3#57712yes—
408C22×C3#57812yes—
409C3×C4—22no—
410C3×C4—24no—
411C3×C4—42no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4—81no—
415C3×C4C2×#12112yes—
416C3×C4#52512yes—
417Dic3—28no—
418Dic3—44no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23#93812yes—
429C23C22×#6711yescharacteristic
430Q8—38no—
431Q8—42no—
432Q8—42no—
433Q8—616no—
434Q8—64no—
435Q8—64no—
436Q8—64no—
437D4—42no—
438D4—42no—
439D4—84no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—616no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4#93714yes—
464C2×C4#93812yes—
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×D3×D412yes—
467C2×C4C2×#23711yescharacteristic
468C2×C4D3×#4112yes—
469C8—122no—
470C8—122no—
471C2×C3—42no—
472C2×C3—161no—
473C2×C3#284711yescharacteristic
474C2×C3#301312yes—
475C2×C3#304311yescharacteristic
476C2×C3#304411yescharacteristic
477C2×C3#312112yes—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—82no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22C2×#93711yescharacteristic
489C22C2×#93811yescharacteristic
490C22#582811yescharacteristic
491C22#583612yes—
492C22#583812yes—
493C4—22no—
494C4—24no—
495C4—42no—
496C4—68no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—124no—
501C4D3×#12112yes—
502C4#560612yes—
503C3#3291811yescharacteristic, complemented, Sylow
504C2—42no—
505C2—161no—
506C2#8272311yescharacteristic
507C2#8339111yescharacteristic
508C2#8341211yescharacteristic
509C2#8342712yes—
510C2#8346712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2620C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2620D3
hypocenterC3#32918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. C2×#5516C2semidirect
3. C2×#5054C2semidirect
4. C2×#5050C2semidirect
6. #71693C2semidirect
8. #71583C2semidirect
9. #69690C2semidirect
10. #69660C2semidirect
11. #69588C2semidirect
105. C4×Dic6D4semidirect
503. C3#32918semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
C3×#26202
C2×#50542
C2×#50502
C2×#50402
#716932
#716782
#715832
#696902
#696602
#695882
#329183