Finite Group Databasebeta

← #10691371

Subgroups — #10691371

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×#5072C211yesmaximal, characteristic
2C2×C8×D12C211yesmaximal, characteristic, complemented
3C3×#1957C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#69584C211yesmaximal, characteristic, complemented
6#67654C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#9221—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C22×#213—21no—
11C22×D3×C8—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×#102—21no—
14C2×C3×#102—21no—
15C2×C3×C4×C8C2211yescharacteristic
16C3×#384C2211yescharacteristic
17C2×#832—21no—
18C2×(C2×D3)⋊C8—21no—
19#5072—22no—
20#5072C414yescomplemented
21C2×#809—21no—
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C8×D12—22no—
26C8×D12—22no—
27#4544—22no—
28#4529—21no—
29C2×#459—31no—
30C2×C8×D4—31no—
31#1957D311yescharacteristic, complemented
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×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×C3×C8D411yescharacteristic
42C22×C3×C8D411yescharacteristic
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127C2×C412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#213—24no—
52C2×#213—42no—
53C2×#213—42no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C2×D3×C8—24no—
60C2×D3×C8—42no—
61C2×D3×C8—42no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12C2×C412yes—
65C4×D12D412yes—
66C3×#102—42no—
67C3×#102—42no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70#832—24no—
71(C2×D3)⋊C8—24no—
72#809—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4⋊(C3⋊C8)D412yes—
79C22×D8—61no—
80C23×C8—61no—
81C2×#127C2×D311yescharacteristic
82C2×C4×D4—31no—
83#459—34no—
84#459—62no—
85C8×D4—62no—
86C8×D4—62no—
87C2×C4⋊C8—31no—
88C2×C4⋊C8—31no—
89C2×C4⋊C8—31no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—61no—
93C2×#102—61no—
94C2×C22⋊C8—61no—
95C2×C4×C8C2×D311yescharacteristic
96#384C2×D311yescharacteristic
97#340—61no—
98#331—62no—
99C24×C3—41no—
100C23×D3—21no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—22no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
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—81no—
116C2×C3×D4—81no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12C22⋊C412yes—
126C2×D12D812yes—
127C2×D12SD1612yes—
128C2×C4×D3—22no—
129C2×C4×D3—22no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8D812yescomplemented
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—81no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#213—48no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152C4⋊Dic3—21no—
153C4⋊Dic3C22⋊C412yes—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163D3×C8—48no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—31no—
168C23×C4—61no—
169C2×D8—64no—
170C2×D8—122no—
171C2×D8—122no—
172C22×C8—64no—
173C22×C8—61no—
174C22×C8—61no—
175C22×C8—61no—
176C22×C8—122no—
177C22×C8—122no—
178C22×C8#6711yescharacteristic
179C22×C8#6711yescharacteristic
180#127—21no—
181#127—21no—
182#127—21no—
183#127C4×D312yes—
184C4×D4—32no—
185C4×D4—32no—
186C4×D4—61no—
187C4×D4—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C22⋊C4—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—32no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#102—42no—
201#102—42no—
202#102—64no—
203#102—122no—
204C22⋊C8—64no—
205C4×C8—21no—
206C4×C8—21no—
207C23×C3—42no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#12111yescharacteristic
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—82no—
227C3×D4—82no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—81no—
231C3×D4—81no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×C3×C4C2×D811yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×Dic3—22no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254D12—22no—
255D12—22no—
256D12—22no—
257D12#10214yescomplemented
258C4×D3—42no—
259C4×D3—42no—
260C4×D3—42no—
261C4×D3—42no—
262C3×C8—22no—
263C3×C8—41no—
264C3⋊C8—42no—
265C3⋊C8—42no—
266C3⋊C8—81no—
267C24—41no—
268C24—61no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—32no—
273C2×D4—32no—
274C2×D4—32no—
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—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—81no—
290C2×D4—81no—
291C22×C4—41no—
292C22×C4—61no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302D8—128no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—128no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C2×C8#21312yescomplemented
319C4⋊C4—32no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—81no—
326C22⋊C4—62no—
327C22⋊C4—62no—
328C42C2×C4×D311yescharacteristic
329C42#23511yescharacteristic
330C42D3×D411yescharacteristic
331C42D3×D411yescharacteristic
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3—161no—
336C22×C3—161no—
337C22×C3—161no—
338C22×C3—161no—
339C22×C3#38611yescharacteristic
340C22×C3#45911yescharacteristic
341C22×C3#46311yescharacteristic
342C22×C3#46511yescharacteristic
343C22×C3#46611yescharacteristic
344C22×C3#47111yescharacteristic
345C22×C3#47211yescharacteristic
346C2×D3—24no—
347C2×D3—42no—
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—42no—
351C2×D3—42no—
352C2×D3—42no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—161no—
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—42no—
365C23—42no—
366C23—62no—
367C23—62no—
368C23—62no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—121no—
376C23—121no—
377C23—121no—
378C23—121no—
379C23#93811yescharacteristic
380D4—34no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—62no—
386D4—62no—
387D4—62no—
388D4—82no—
389D4—82no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—81no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—62no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4D3×C22⋊C411yescharacteristic
407C2×C4#67⋊C411yescharacteristic
408C2×C4#88311yescharacteristic
409C2×C4#88411yescharacteristic
410C2×C4D3×D811yescharacteristic
411C2×C4#91211yescharacteristic
412C2×C4D3×SD1611yescharacteristic
413C2×C4#91611yescharacteristic
414C2×C4C4×#6711yescharacteristic
415C2×C4C2×#21311yescharacteristic
416C2×C4#93211yescharacteristic
417C2×C4#93711yescharacteristic
418C2×C4#94911yescharacteristic
419C2×C4#95011yescharacteristic
420C8—22no—
421C8—41no—
422C8—122no—
423C8—122no—
424C8—241no—
425C2×C3—162no—
426C2×C3#166511yescharacteristic
427C2×C3#166711yescharacteristic
428C2×C3#167711yescharacteristic
429C2×C3#168111yescharacteristic
430C2×C3#187911yescharacteristic
431C2×C3#189311yescharacteristic
432C2×C3#190711yescharacteristic
433D3—44no—
434C22—64no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—122no—
439C22—122no—
440C22—122no—
441C22—122no—
442C22—122no—
443C22—122no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22#495611yescharacteristic
449C22#505711yescharacteristic
450C22#507211yescharacteristic
451C22#515711yescharacteristic
452C22#516011yescharacteristic
453C22#517111yescharacteristic
454C22#517711yescharacteristic
455C4—21no—
456C4—21no—
457C4—122no—
458C4—161no—
459C4D3×#10211yescharacteristic
460C4#477211yescharacteristic
461C4#477511yescharacteristic
462C4#477811yescharacteristic
463C4C4×#21311yescharacteristic
464C4#501411yescharacteristic
465C4#503611yescharacteristic
466C4#503711yescharacteristic
467C3#922111yescharacteristic, complemented, Sylow
468C2—124no—
469C2—162no—
470C2#6965711yescharacteristic
471C2#7020211yescharacteristic
472C2#7021811yescharacteristic
473C2#7120011yescharacteristic
474C2#7120111yescharacteristic
475C2#7126711yescharacteristic
476C2#7128211yescharacteristic
subgroupG/H
centerC23#938
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1957C2
socleC23×C3#121
prime radicalC2×#5072C2
cosocleC2×C3×C42C23
hypercenter#1957D3
hypocenterC3#9221
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×C8×D12C2semidirect
3. C3×#1957C2semidirect
4. C2×#4544C2semidirect
5. #69584C2semidirect
6. #67654C2semidirect
7. #67634C2semidirect
20. #5072C4semidirect
31. #1957D3semidirect
139. C2×C3×C8D8semidirect
257. D12#102semidirect
318. C2×C8#213semidirect
467. C3#9221semidirect

Maximal subgroups

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

subgroupindex
C2×#50722
C2×C8×D122
C3×#19572
C2×#45442
#695842
#676542
#676342
#92213