Finite Group Databasebeta

← #10944140

Subgroups — #10944140

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×#5812C211yesmaximal, characteristic, complemented
2C2×C3×#451C211yesmaximal, characteristic, complemented
3C2×#5227C211yesmaximal, characteristic
4C2×#5185C212yesmaximal, complemented
5#82821C211yesmaximal, characteristic
6#72065C212yesmaximal, complemented
7#71693C211yesmaximal, characteristic, complemented
8#70240C212yesmaximal, complemented
9#70221C212yesmaximal, complemented
10#69933C211yesmaximal, characteristic
11#69909C211yesmaximal, characteristic, complemented
12#32918—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#947C2211yescharacteristic
17#5812—28no—
18C2×Q8×Dic3—22no—
19C2×Q8×Dic3C2212yes—
20C2×Q8×Dic3C2212yes—
21C22×#215C2212yes—
22C2×#930C2211yescharacteristic
23C2×#926—22no—
24C2×#926C2211yescharacteristic
25C2×#926C2211yescharacteristic
26C3×#451—24no—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29#5229C2211yescharacteristic
30#5227—24no—
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#830C2211yescharacteristic
34#5185—28no—
35C2×#821C2212yes—
36C2×Dic3⋊C8C2212yes—
37#4994—22no—
38#4988C2211yescharacteristic
39#4975C2211yescharacteristic
40#4959—22no—
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#519—31no—
45C2×#464—32no—
46C2×#451D311yescharacteristic, complemented
47C2×#448—31no—
48#2620—31no—
49#1880—32no—
50#1878—32no—
51#1875—31no—
52#1873—31no—
53#1803—32no—
54#1798—31no—
55C22×C3×Q8—24no—
56C22×C3×Q8—24no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8D414yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2312yes—
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×D12C2311yescharacteristic
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C22×C3×C8C2312yes—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×#217—21no—
76C2×#217D412yes—
77#947—24no—
78Q8×Dic3—216no—
79Q8×Dic3—24no—
80Q8×Dic3—24no—
81Q8×Dic3—24no—
82Q8×Dic3D418yescomplemented
83C2×#215—24no—
84C2×#215—24no—
85C2×#215—24no—
86#930—22no—
87#930—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91#926—216no—
92#926—24no—
93#926—24no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3D412yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×Dic3⋊C4—22no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C3×#103—24no—
107C3×C22⋊C8—22no—
108#835—24no—
109#835—24no—
110C3⋊#95—22no—
111#830—22no—
112#821—24no—
113Dic3⋊C8—24no—
114C23×Q8C2×D311yescharacteristic
115C22×Q16C2×D312yes—
116C22×SD16—32no—
117C2×#124—31no—
118C2×C22⋊Q8C2×D311yescharacteristic
119C2×#121—31no—
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—31no—
126C2×#117—62no—
127#464—68no—
128#451—24no—
129#448—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103C2×D312yes—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3C2×D411yescharacteristic
146C23×D3—21no—
147C2×C3×Q8—24no—
148C2×C3×Q8—216no—
149C2×C3×Q8—24no—
150C2×C3×Q8—216no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8C2×D414yes—
161C2×C3×Q8#4114yes—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C22×C3×C4C2411yescharacteristic
174C2×#67—42no—
175C2×#67—82no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3C2×D412yes—
179C22×Dic3C2×D412yes—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—22no—
184C3×Q16—48no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—42no—
191C3×C22⋊C4—42no—
192#217—24no—
193#217—24no—
194#215—48no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—82no—
198C4⋊Dic3—28no—
199C4⋊Dic3—22no—
200C4⋊Dic3—28no—
201C4⋊Dic3—24no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3C2×D414yes—
205Dic3⋊C4—82no—
206C4×Dic3—28no—
207C4×Dic3—24no—
208C4×Dic3—22no—
209C4×Dic3—28no—
210C4×Dic3C2×D414yes—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8#6714yes—
219C22×Q8C22×D311yescharacteristic
220C22×Q8C22×D312yes—
221C22×D4—31no—
222C22×D4—61no—
223C23×C4—21no—
224C23×C4C22×D311yescharacteristic
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—24no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C22×C8—32no—
232C22×C8C22×D312yes—
233#124—64no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—62no—
237#121—62no—
238C4×Q8—38no—
239C4×Q8—64no—
240C4×Q8—64no—
241C4×Q8—616no—
242C4×Q8—64no—
243#117—64no—
244#117—64no—
245#117—616no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—32no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C42—31no—
261C2×C42—31no—
262C2×C42—62no—
263C2×C42—62no—
264C2×C42—61no—
265C4⋊C8—64no—
266#103—24no—
267#103—64no—
268#103—64no—
269#102—64no—
270C22⋊C8—22no—
271C22⋊C8—62no—
272#95—62no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3#12112yes—
278C23×C3C22×D411yescharacteristic
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—216no—
285C3×Q8—24no—
286C3×Q8—24no—
287C3×Q8—24no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×Q8#12118yes—
291C2×C3×C4—216no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4#12112yes—
305C2×C3×C4C22×D412yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4C2×#4111yescharacteristic
309#67—84no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3#12014yes—
321D12—42no—
322D12—42no—
323C3×C8—42no—
324C3⋊C8—42no—
325C24—61no—
326C24D3×D411yescharacteristic
327C2×Q8—24no—
328C2×Q8—216no—
329C2×Q8—24no—
330C2×Q8—216no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8D3×D414yes—
340C2×Q8#23714yes—
341C2×Q8C2×#6714yes—
342C2×D4—62no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—61no—
346C2×D4—122no—
347C2×D4—242no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4D3×D411yescharacteristic
361C22×C4#23712yes—
362C22×C4C2×#6712yes—
363C22×C4C23×D311yescharacteristic
364Q16—48no—
365SD16—128no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—34no—
375C4⋊C4—42no—
376C4⋊C4—64no—
377C4⋊C4—68no—
378C4⋊C4—68no—
379C4⋊C4—62no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—242no—
383C22⋊C4—42no—
384C22⋊C4—64no—
385C22⋊C4—64no—
386C22⋊C4—124no—
387C22⋊C4—122no—
388C22⋊C4—242no—
389C42—34no—
390C42—68no—
391C42—68no—
392C42—64no—
393C42—62no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3C2×#12011yescharacteristic
400C22×C3C2×#12111yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#57712yes—
403C22×C3#57812yes—
404C2×D3—81no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—28no—
410C3×C4—22no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4C2×#12112yes—
414C3×C4#52512yes—
415Dic3—24no—
416Dic3—42no—
417Dic3—44no—
418Dic3—81no—
419C23—22no—
420C23—22no—
421C23—21no—
422C23—21no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23#93912yes—
429C23C2×D3×D411yescharacteristic
430Q8—216no—
431Q8—24no—
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8#93918yes—
437D4—122no—
438D4—122no—
439D4—244no—
440C2×C4—216no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—34no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4#93912yes—
465C2×C4C2×D3×D412yes—
466C2×C4C2×#23712yes—
467C2×C4C2×#23711yescharacteristic
468C2×C4C22×#6711yescharacteristic
469C8—42no—
470C8—122no—
471C2×C3—42no—
472C2×C3#284711yescharacteristic
473C2×C3#301312yes—
474C2×C3#304311yescharacteristic
475C2×C3#304411yescharacteristic
476C2×C3#312112yes—
477D3—161no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—241no—
484C22—242no—
485C22—242no—
486C22—241no—
487C22—241no—
488C22#558611yescharacteristic
489C22D3×#12011yescharacteristic
490C22#576612yes—
491C22#577812yes—
492C22C2×#93911yescharacteristic
493C4—28no—
494C4—22no—
495C4—64no—
496C4—81no—
497C4—81no—
498C4—122no—
499C4—124no—
500C4—241no—
501C4#558612yes—
502C4C2×#93912yes—
503C3#3291811yescharacteristic, complemented, Sylow
504C2—42no—
505C2—481no—
506C2#8017612yes—
507C2#8018612yes—
508C2#8260111yescharacteristic
509C2#8387711yescharacteristic
510C2#8396911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#451D3
hypocenterC3#32918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
1. C2×#5812C2semidirect
2. C2×C3×#451C2semidirect
4. C2×#5185C2semidirect
6. #72065C2semidirect
7. #71693C2semidirect
8. #70240C2semidirect
9. #70221C2semidirect
11. #69909C2semidirect
46. C2×#451D3semidirect
82. Q8×Dic3D4semidirect
503. C3#32918semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
C2×C3×#4512
C2×#52272
C2×#51852
#828212
#720652
#716932
#702402
#702212
#699332
#699092
#329183