Finite Group Databasebeta

← #10845327

Subgroups — #10845327

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#475C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic
4C2×#4834C211yesmaximal, characteristic
5#71787C211yesmaximal, characteristic, complemented
6C4×#835C211yesmaximal, characteristic, complemented
7#70583C212yesmaximal, complemented
8#70577C212yesmaximal, complemented
9#70454C211yesmaximal, characteristic, complemented
10#70270C212yesmaximal, complemented
11#70243C212yesmaximal, complemented
12#25341—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947—21no—
16C2×#946C2212yes—
17C22×D3×C8C2212yes—
18C2×C4×D12C2211yescharacteristic
19C3×#475—24no—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2212yes—
23#5229—21no—
24C2×#835C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×(C2×D3)⋊C8C2212yes—
27C2×#817C2212yes—
28#4994—22no—
29#4992—22no—
30#4987C2212yes—
31#4987C2212yes—
32#4987C2212yes—
33#4939—21no—
34#4935C2211yescharacteristic
35C4×C4⋊Dic3C2211yescharacteristic
36#4928—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4834—24no—
39#4551C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#475D311yescharacteristic, complemented
45C2×C8×D4—31no—
46#2547—31no—
47#2538—31no—
48#2077—32no—
49#2035—32no—
50#1990—31no—
51#1948—31no—
52#1892—32no—
53#1807—32no—
54C4×#103—31no—
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×D12D411yescharacteristic
58C22×C4×D3C2312yes—
59C22×C3×C8C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70#947—44no—
71#946—24no—
72#946—42no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4D412yes—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C2×D3×C8—24no—
89C2×D3×C8—24no—
90C2×D3×C8—42no—
91C4×D12—22no—
92C4×D12—22no—
93C3×#106—24no—
94C3×C4⋊C8D414yes—
95C3×#103—24no—
96#835—22no—
97#835—22no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102(C2×D3)⋊C8—24no—
103#817—42no—
104C4×C3⋊C8—22no—
105C23×C8—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×#124—61no—
108C2×C22⋊Q8—32no—
109C2×C4×Q8C2×D311yescharacteristic
110C2×C4×D4—31no—
111#475—24no—
112C8×D4—64no—
113C2×#106—32no—
114C2×#106C2×D312yes—
115C2×C4⋊C8C2×D311yescharacteristic
116C2×#103—31no—
117C2×#103—31no—
118C2×#103C2×D312yes—
119C2×C22⋊C8—32no—
120C2×C4×C8—31no—
121#397—62no—
122#396—62no—
123#391—61no—
124#390—61no—
125#389—61no—
126#384—31no—
127#380—32no—
128#380—32no—
129#380—32no—
130C4×C4⋊C4—31no—
131#340—32no—
132#340—31no—
133#340—32no—
134#340—31no—
135#340—32no—
136C23×D3C2×D412yes—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—42no—
143C2×C3×Q8—42no—
144C2×C3×Q8—42no—
145C2×C3×Q8#4112yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—22no—
155C22×Dic3C2×D412yes—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12SD1614yes—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C2×D414yes—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C42—42no—
175C3×C42C2×D412yes—
176C3×C42#4112yes—
177(C2×D3)⋊C4—24no—
178(C2×D3)⋊C4—44no—
179(C2×D3)⋊C4—44no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—82no—
182C4⋊Dic3—22no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3—22no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—21no—
189C4⋊Dic3—42no—
190C4⋊Dic3—81no—
191Dic3⋊C4—44no—
192C4×Dic3—42no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8#4112yes—
199D3×C8—48no—
200C22×Q8—21no—
201C22×Q8C22×D311yescharacteristic
202C22×D4—31no—
203C23×C4—32no—
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—64no—
207C22×C8—64no—
208C22×C8—122no—
209C22×C8C22×D312yes—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212#124—124no—
213C22⋊Q8—64no—
214C22⋊Q8—122no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×D4—62no—
218C4×D4—62no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C42—21no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#106—24no—
239#106—122no—
240C4⋊C8#6714yes—
241#103—24no—
242#103—62no—
243#103—62no—
244C22⋊C8—64no—
245C4×C8—62no—
246#95—62no—
247#95—62no—
248#95—62no—
249#95—61no—
250C23×C3C22×D411yescharacteristic
251C22×D3—22no—
252C22×D3—22no—
253C22×D3—22no—
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—22no—
257C22×D3C2×SD1614yes—
258C3×Q8—22no—
259C3×Q8—22no—
260C3×Q8—41no—
261C3×Q8—41no—
262C3×Q8—84no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4#12112yes—
275C2×C3×C4#12112yes—
276C2×C3×C4C2×SD1612yes—
277C2×C3×C4#13512yes—
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4H(5,𝔽2)11yescharacteristic
283C2×C3×C42−1+411yescharacteristic
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296D12—28no—
297C4×D3—42no—
298C4×D3—42no—
299C4×D3—42no—
300C4×D3—42no—
301C3×C8—24no—
302C3⋊C8—22no—
303C3⋊C8—41no—
304C24—32no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8#23912yes—
314C2×D4—34no—
315C2×D4—62no—
316C2×D4—62no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—32no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4C23×D311yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—32no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—128no—
342C2×C8C2×#6714yes—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—62no—
350C4⋊C4—61no—
351C4⋊C4—62no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—62no—
356C4⋊C4—124no—
357C4⋊C4—122no—
358C4⋊C4—241no—
359C22⋊C4—64no—
360C22⋊C4—124no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—242no—
364C42—42no—
365C42—122no—
366C42#23712yes—
367C42C2×#6712yes—
368C22×C3C2×#12111yescharacteristic
369C22×C3#54611yescharacteristic
370C22×C3#54711yescharacteristic
371C22×C3C22×SD1611yescharacteristic
372C22×C3C2×#13511yescharacteristic
373C22×C3#57711yescharacteristic
374C22×C3#57811yescharacteristic
375C2×D3—24no—
376C2×D3—24no—
377C2×D3—24no—
378C2×D3—42no—
379C2×D3—42no—
380C2×D3—42no—
381C2×D3—42no—
382C3×C4—22no—
383C3×C4—42no—
384C3×C4—82no—
385C3×C4—81no—
386C3×C4#46514yes—
387C3×C4#52911yescharacteristic
388C3×C4#54211yescharacteristic
389C3×C4#54611yescharacteristic
390C3×C4#54811yescharacteristic
391Dic3—42no—
392Dic3—81no—
393Dic3—81no—
394Dic3—82no—
395C23—34no—
396C23—62no—
397C23—62no—
398C23—62no—
399C23—62no—
400C23—62no—
401C23—62no—
402C23C2×D3×D411yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407Q8—84no—
408D4—68no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4D3×SD1612yes—
437C2×C4#91612yes—
438C2×C4#93812yes—
439C2×C4#93812yes—
440C2×C4C2×#23711yescharacteristic
441C2×C4C2×#23911yescharacteristic
442C2×C4#100711yescharacteristic
443C2×C4D3×#4111yescharacteristic
444C2×C4#100911yescharacteristic
445C2×C4C22×#6711yescharacteristic
446C8—24no—
447C8—62no—
448C8—121no—
449C2×C3#242711yescharacteristic
450C2×C3C2×#46511yescharacteristic
451C2×C3#306311yescharacteristic
452C2×C3#329811yescharacteristic
453C2×C3#330411yescharacteristic
454C2×C3#333511yescharacteristic
455C2×C3#334211yescharacteristic
456D3—44no—
457C22—64no—
458C22—64no—
459C22—64no—
460C22—122no—
461C22—122no—
462C22—122no—
463C22—122no—
464C22#563011yescharacteristic
465C22#563111yescharacteristic
466C22C2×D3×SD1611yescharacteristic
467C22C2×#91611yescharacteristic
468C22#577811yescharacteristic
469C22#577911yescharacteristic
470C22C2×#93811yescharacteristic
471C4—22no—
472C4—42no—
473C4—81no—
474C4—82no—
475C4—122no—
476C4—241no—
477C4—242no—
478C4—241no—
479C4#517114yes—
480C4#562411yescharacteristic
481C4#564311yescharacteristic
482C4#581711yescharacteristic
483C4#582911yescharacteristic
484C3#2534111yescharacteristic, complemented, Sylow
485C2—124no—
486C2#8102211yescharacteristic
487C2#8102311yescharacteristic
488C2#8102811yescharacteristic
489C2#8102911yescharacteristic
490C2#8275611yescharacteristic
491C2C2×#517111yescharacteristic
492C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#475C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenterC2×#475D3
hypocenterC3#25341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
5. #71787C2semidirect
6. C4×#835C2semidirect
7. #70583C2semidirect
8. #70577C2semidirect
9. #70454C2semidirect
10. #70270C2semidirect
11. #70243C2semidirect
44. C2×#475D3semidirect
484. C3#25341semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4752
#832402
#825002
C2×#48342
#717872
C4×#8352
#705832
#705772
#704542
#702702
#702432
#253413