Finite Group Databasebeta

← #10691370

Subgroups — #10691370

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
3#71764C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#71604C211yesmaximal, characteristic, complemented
6#67703C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9221—31nomaximal, Sylow
9C22×C3×D8—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C23×C3⋊C8—21no—
13C3×#459—22no—
14C3×#459C414yescomplemented
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×#102—21no—
17C2×#832—21no—
18C2×#832—21no—
19C2×#832—21no—
20C2×#830—21no—
21D4×C3⋊C8—22no—
22D4×C3⋊C8—22no—
23#4935C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4544—22no—
28#4529—21no—
29C2×#459D311yescharacteristic
30C2×C8×D4—31no—
31#1957—31no—
32#1802—31no—
33C2×#331—31no—
34#1296—31no—
35#1277—31no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C2×C3×D8—24no—
42C2×C3×D8—42no—
43C2×C3×D8—42no—
44C22×C3×C8—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127C2×C412yes—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4C2×C412yes—
52C3×C4×D4D412yes—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C22×C3⋊C8—24no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8—42no—
63C22×C3⋊C8—42no—
64C22×C3⋊C8D411yescharacteristic
65C22×C3⋊C8D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×C4⋊C8C2×C412yes—
68C3×#102—24no—
69#832—42no—
70#832—42no—
71#832—42no—
72#830—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C22×D8—21no—
80C23×C8—61no—
81C2×#127C2×D311yescharacteristic
82C2×C4×D4C2×D311yescharacteristic
83#459—22no—
84#459Dic314yescomplemented
85C8×D4—62no—
86C8×D4—62no—
87C2×C4⋊C8—31no—
88C2×C4⋊C8—31no—
89C2×C4⋊C8C2×D311yescharacteristic
90C2×#102—21no—
91C2×#102—61no—
92C2×#102—61no—
93C2×#102—61no—
94C2×C22⋊C8—61no—
95C2×C4×C8—31no—
96#384—31no—
97#340—61no—
98#331—62no—
99C24×C3—21no—
100C24×C3—41no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—22no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—81no—
119C2×C3×D4—81no—
120C2×C3×D4C22⋊C412yes—
121C2×C3×D4D812yes—
122C2×C3×D4SD1612yes—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—41no—
134C3×D8—48no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4C22⋊C412yes—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—81no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—48no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8D812yescomplemented
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4#6711yescharacteristic
168C23×C4—21no—
169C2×D8—24no—
170C2×D8—42no—
171C2×D8—42no—
172C22×C8—21no—
173C22×C8—31no—
174C22×C8—31no—
175C22×C8—64no—
176C22×C8—61no—
177C22×C8—61no—
178C22×C8—122no—
179C22×C8—122no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C2×Dic312yes—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C2×Dic312yes—
187C4×D4#6712yes—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8C2×Dic312yes—
200#102—24no—
201#102—122no—
202#102—122no—
203#102—122no—
204C22⋊C8—64no—
205C4×C8—61no—
206C4×C8—61no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3—81no—
216C23×C3—81no—
217C23×C3—81no—
218C23×C3—81no—
219C23×C3—81no—
220C23×C3—81no—
221C23×C3#12111yescharacteristic
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—42no—
228C3×D4—42no—
229C3×D4—82no—
230C3×D4—82no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4#10214yescomplemented
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×C3×C4C2×D811yescharacteristic
252C2×C3×C4C2×D811yescharacteristic
253C2×C3×C4C2×SD1611yescharacteristic
254C2×C3×C4#13511yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×Dic3—81no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262C3×C8—42no—
263C3⋊C8—22no—
264C3⋊C8—42no—
265C3⋊C8—41no—
266C3⋊C8—81no—
267C24—21no—
268C24—41no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—81no—
287C2×D4—81no—
288C2×D4#21312yes—
289C2×D4#21412yes—
290C2×D4#21712yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—121no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302D8—48no—
303C2×C8—22no—
304C2×C8—32no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—61no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—62no—
313C2×C8—128no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C2×C8—121no—
319C4⋊C4—21no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C4⋊C4—121no—
323C4⋊C4—121no—
324C4⋊C4—241no—
325C4⋊C4#21712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C42D3×D411yescharacteristic
329C42#23711yescharacteristic
330C42C22×Dic311yescharacteristic
331C42C2×#6711yescharacteristic
332C22×C3—24no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—161no—
343C22×C3—161no—
344C22×C3—161no—
345C22×C3—161no—
346C22×C3#38611yescharacteristic
347C22×C3#45911yescharacteristic
348C22×C3#46311yescharacteristic
349C22×C3#46511yescharacteristic
350C22×C3#46611yescharacteristic
351C22×C3#47111yescharacteristic
352C22×C3#47211yescharacteristic
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—42no—
356C3×C4C2×#10211yescharacteristic
357C3×C4#41611yescharacteristic
358C3×C4C4×D811yescharacteristic
359C3×C4#44211yescharacteristic
360C3×C4#44711yescharacteristic
361C3×C4#44911yescharacteristic
362C3×C4#46511yescharacteristic
363C3×C4#46611yescharacteristic
364Dic3—161no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23#93911yescharacteristic
380D4—22no—
381D4—22no—
382D4—22no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—82no—
388D4—82no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4#83214yescomplemented
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4—241no—
406C2×C4D3×D811yescharacteristic
407C2×C4#91111yescharacteristic
408C2×C4#91111yescharacteristic
409C2×C4#91211yescharacteristic
410C2×C4C2×#21311yescharacteristic
411C2×C4C2×#21411yescharacteristic
412C2×C4D4×Dic311yescharacteristic
413C2×C4D4×Dic311yescharacteristic
414C2×C4#93511yescharacteristic
415C2×C4#93711yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#95011yescharacteristic
418C2×C4#95011yescharacteristic
419C2×C4C2×#21711yescharacteristic
420C8—42no—
421C8—62no—
422C8—122no—
423C8—121no—
424C8—241no—
425C2×C3—44no—
426C2×C3—162no—
427C2×C3#166511yescharacteristic
428C2×C3#166711yescharacteristic
429C2×C3#167711yescharacteristic
430C2×C3#168111yescharacteristic
431C2×C3#187911yescharacteristic
432C2×C3#189311yescharacteristic
433C2×C3#190711yescharacteristic
434C22—24no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22#503811yescharacteristic
449C22#504111yescharacteristic
450C22#515011yescharacteristic
451C22#515311yescharacteristic
452C22#519911yescharacteristic
453C22#520011yescharacteristic
454C22#522011yescharacteristic
455C4—21no—
456C4—21no—
457C4—42no—
458C4—481no—
459C4#503811yescharacteristic
460C4#504111yescharacteristic
461C4Dic3×D811yescharacteristic
462C4#515211yescharacteristic
463C4#515611yescharacteristic
464C4#515811yescharacteristic
465C4C2×#83211yescharacteristic
466C4#523311yescharacteristic
467C3#922111yescharacteristic, complemented, Sylow
468C2—44no—
469C2—162no—
470C2#7100411yescharacteristic
471C2#7100511yescharacteristic
472C2#7100611yescharacteristic
473C2#7100711yescharacteristic
474C2#7168111yescharacteristic
475C2#7203711yescharacteristic
476C2#7209111yescharacteristic
subgroupG/H
centerC23#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3#121
cosocleC2×C3×C42C23
hypercenterC2×#459D3
hypocenterC3#9221
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. C2×D4×C3⋊C8C2semidirect
3. #71764C2semidirect
4. C2×#4544C2semidirect
5. #71604C2semidirect
6. #67703C2semidirect
7. #67702C2semidirect
14. C3×#459C4semidirect
84. #459Dic3semidirect
163. C2×C3⋊C8D8semidirect
235. C3×D4#102semidirect
393. D4#832semidirect
467. C3#9221semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×D4×C3⋊C82
#717642
C2×#45442
#716042
#677032
#677022
#92213