Finite Group Databasebeta

← #10814916

Subgroups — #10814916

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2355C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3#73638C211yesmaximal, characteristic
4#71626C211yesmaximal, characteristic, complemented
5#69676C212yesmaximal
6#69666C212yesmaximal, complemented
7#69584C211yesmaximal, characteristic
8#69576C211yesmaximal, characteristic, complemented
9C4×#808C212yesmaximal, complemented
10C4⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
11#69463C212yesmaximal
12#22805—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#215C2211yescharacteristic
15C2×#892—21no—
16C2×C4×D12C2211yescharacteristic
17C3×#391C2212yes—
18C3×#387—22no—
19C3×#384C2211yescharacteristic
20C3×#382—21no—
21C3×C4×C4⋊C4—21no—
22C3×C4×C4⋊C4C2212yes—
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#835C2212yes—
25C2×#835C2211yescharacteristic
26C4×#215C418yescomplemented
27#4987—21no—
28C2×#809C2212yes—
29C2×#809C2211yescharacteristic
30C2×#808C2212yes—
31C2×#808C2212yes—
32C2×#808C2211yescharacteristic
33#4939C2212yes—
34#4935C2211yescharacteristic
35C4×C4⋊Dic3C2212yes—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37C2×C4×C3⋊C8C2212yes—
38C2×C4×C3⋊C8C2211yescharacteristic
39#4670—22no—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43C2×C4×SD16—31no—
44#2355D311yescharacteristic, complemented
45#2354—31no—
46#1968—32no—
47#1927—32no—
48#1926—32no—
49#1844—31no—
50#1803—31no—
51#1802—31no—
52C4×#106—32no—
53C8×C4⋊C4—31no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C3×C4×Q8—22no—
58C3×C4×Q8C2×C414yes—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42—22no—
68C2×C3×C42C2312yes—
69C2×C3×C42C2311yescharacteristic
70C2×#215—22no—
71C2×#215—22no—
72C2×#215C2×C414yes—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4×Dic3—22no—
81C22×C3⋊C8C2312yes—
82C22×C3⋊C8C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84#892—44no—
85C4×D12—22no—
86C4×D12C2×C414yes—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95D412yes—
91#835—42no—
92#835C2×C414yes—
93C3⋊#95—22no—
94#809—42no—
95#809C2×C414yes—
96#808—24no—
97#808—42no—
98#808C2×C414yes—
99C4⋊(C3⋊C8)—22no—
100C4×C3⋊C8—24no—
101C4×C3⋊C8C2×C414yes—
102C22×SD16—31no—
103C2×C4×Q8C2×D311yescharacteristic
104C2×C4×D4—31no—
105C2×#117—61no—
106C4×SD16—38no—
107C2×#106—31no—
108C2×#106—32no—
109C2×#106—32no—
110C2×C4⋊C8—31no—
111C2×#103—31no—
112C2×#103—32no—
113C2×#102—31no—
114C2×#102—32no—
115C2×C4×C8—31no—
116C2×C4×C8—32no—
117#391C2×D312yes—
118#390—32no—
119#388—62no—
120#387—22no—
121#384—31no—
122#384C2×D311yescharacteristic
123#382—21no—
124#380—61no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4C2×D312yes—
128C4×C4⋊C4C2×D311yescharacteristic
129#340—32no—
130#340—32no—
131#336—32no—
132C23×D3—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8C22×C412yes—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—21no—
148C22×Dic3—22no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12C22×C412yes—
155C2×C4×D3—24no—
156C2×C4×D3—42no—
157C2×C4×D3—42no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—81no—
165C3×C4⋊C4SD1614yescomplemented
166C3×C4⋊C4C22×C412yes—
167C3×C42—24no—
168C3×C42—22no—
169C3×C42—22no—
170C3×C42—82no—
171C3×C42C22×C412yes—
172C3×C42#4112yes—
173#215—28no—
174(C2×D3)⋊C4—24no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—82no—
177C4⋊Dic3—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—42no—
181C4⋊Dic3—81no—
182C4⋊Dic3C22×C412yes—
183C4×Dic3—44no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8C22×C412yes—
190C2×C3⋊C8C22×C412yes—
191C2×C3⋊C8#4112yes—
192C2×C3⋊C8#4112yes—
193C22×Q8C22×D311yescharacteristic
194C22×D4—31no—
195C23×C4—61no—
196C2×SD16—34no—
197C2×SD16—62no—
198C2×SD16—62no—
199C22×C8—31no—
200C22×C8—31no—
201C22×C8—32no—
202C4×Q8—22no—
203C4×Q8C4×D314yes—
204C4×D4—34no—
205C4×D4—62no—
206#117—124no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—31no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4#6711yescharacteristic
215C2×C4⋊C4#6711yescharacteristic
216C2×C4⋊C4C22×D312yes—
217C2×C4⋊C4C22×D311yescharacteristic
218C2×C22⋊C4—61no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—62no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—62no—
224C2×C42C22×D312yes—
225C2×C42C22×D311yescharacteristic
226#106—34no—
227#106—64no—
228#106—122no—
229C4⋊C8—62no—
230#103—34no—
231#103—122no—
232#102—34no—
233#102—122no—
234C4×C8—34no—
235C4×C8—64no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—62no—
240#95#6712yes—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C3×Q8—22no—
250C3×Q8—22no—
251C3×Q8—41no—
252C3×Q8—41no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4C4×D412yes—
270C2×C3×C4C4×D412yes—
271C2×C3×C4C2×SD1614yes—
272C2×C3×C4C2×SD1612yes—
273C2×C3×C4#13512yes—
274C2×C3×C4C23×C411yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4H(5,𝔽2)11yescharacteristic
279C2×C3×C42−1+411yescharacteristic
280C2×Dic3—22no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286D12—22no—
287D12—22no—
288D12—41no—
289D12—41no—
290C4×D3—48no—
291C3⋊C8—22no—
292C3⋊C8—42no—
293C3⋊C8#11714yescomplemented
294C24—61no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8C2×C4×D312yes—
301C2×D4—32no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—61no—
311C22×C4—64no—
312C22×C4—62no—
313C22×C4—122no—
314C22×C4—122no—
315C22×C4C2×#6712yes—
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6712yes—
318C22×C4C2×#6711yescharacteristic
319C22×C4C23×D311yescharacteristic
320SD16—68no—
321C2×C8—32no—
322C2×C8—32no—
323C2×C8—32no—
324C2×C8—32no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—62no—
328C2×C8—62no—
329C2×C8—62no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—32no—
336C4⋊C4—42no—
337C4⋊C4—61no—
338C4⋊C4—62no—
339C4⋊C4—62no—
340C4⋊C4—81no—
341C4⋊C4—122no—
342C4⋊C4—241no—
343C4⋊C4#21514yescomplemented
344C4⋊C4C2×C4×D312yes—
345C22⋊C4—64no—
346C22⋊C4—124no—
347C22⋊C4—242no—
348C42—24no—
349C42—22no—
350C42—22no—
351C42—82no—
352C42—124no—
353C42C2×C4×D312yes—
354C42#23512yes—
355C22×C3C2×C4×D411yescharacteristic
356C22×C3#54711yescharacteristic
357C22×C3#55011yescharacteristic
358C22×C3C22×SD1611yescharacteristic
359C22×C3C2×#13511yescharacteristic
360C22×C3#57711yescharacteristic
361C22×C3#57711yescharacteristic
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—81no—
369C3×C4—24no—
370C3×C4—22no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—82no—
375C3×C4C4×SD1614yes—
376C3×C4C2×#11711yescharacteristic
377C3×C4#52011yescharacteristic
378C3×C4#55511yescharacteristic
379C3×C4#55611yescharacteristic
380Dic3—42no—
381Dic3—82no—
382C23—62no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23C22×#6711yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—41no—
393Q8—41no—
394D4—62no—
395D4—62no—
396D4—121no—
397D4—121no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—62no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—128no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4C4×#6712yes—
422C2×C4C4×#6712yes—
423C2×C4C2×#21514yes—
424C2×C4C2×#21512yes—
425C2×C4#95012yes—
426C2×C4C22×C4×D311yescharacteristic
427C2×C4C2×#23511yescharacteristic
428C2×C4D3×#4111yescharacteristic
429C2×C4D3×#4111yescharacteristic
430C2×C4#100911yescharacteristic
431C2×C4#101011yescharacteristic
432C8—34no—
433C8—62no—
434C8—122no—
435C2×C3#235411yescharacteristic
436C2×C3C2×C4×SD1611yescharacteristic
437C2×C3#295211yescharacteristic
438C2×C3#333511yescharacteristic
439C2×C3#334011yescharacteristic
440C2×C3#339211yescharacteristic
441C2×C3#339411yescharacteristic
442D3—82no—
443C22—122no—
444C22—122no—
445C22—122no—
446C22—241no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22C2×C4×#6711yescharacteristic
451C22C22×#21511yescharacteristic
452C22C2×#95011yescharacteristic
453C22#582411yescharacteristic
454C22#583411yescharacteristic
455C22#583611yescharacteristic
456C22#583711yescharacteristic
457C4—22no—
458C4—24no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—82no—
463C4—122no—
464C4—242no—
465C4C4×#21514yes—
466C4D3×#11711yescharacteristic
467C4#552311yescharacteristic
468C4#553211yescharacteristic
469C4#553311yescharacteristic
470C3#2280511yescharacteristic, complemented, Sylow
471C2—242no—
472C2#8224111yescharacteristic
473C2C2×C4×#21511yescharacteristic
474C2#8255811yescharacteristic
475C2#8362211yescharacteristic
476C2#8365811yescharacteristic
477C2#8368511yescharacteristic
478C2#8371011yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2355C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#2355D3
hypocenterC3#22805
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2355

Complements

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

NH ≅ G/Ntype
1. C3×#2355C2semidirect
4. #71626C2semidirect
6. #69666C2semidirect
8. #69576C2semidirect
9. C4×#808C2semidirect
10. C4⋊C4×C3⋊C8C2semidirect
26. C4×#215C4semidirect
44. #2355D3semidirect
165. C3×C4⋊C4SD16semidirect
293. C3⋊C8#117semidirect
343. C4⋊C4#215semidirect
470. C3#22805semidirect

Maximal subgroups

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

subgroupindex
C3×#23552
C2×C4×#2152
#736382
#716262
#696762
#696662
#695842
#695762
C4×#8082
C4⋊C4×C3⋊C82
#694632
#228053