Finite Group Databasebeta

← #10697532

Subgroups — #10697532

Order 768.

628 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#477C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic
4#71785C211yesmaximal, characteristic, complemented
5C2×#4535C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67702C211yesmaximal, characteristic, complemented
8#10236—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#946—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#477—22no—
14C3×#477—22no—
15C2×C3×#107—21no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18C2×#835—22no—
19#5084—22no—
20#5084—22no—
21#4994—22no—
22#4987—21no—
23C2×#809—21no—
24C2×#808—21no—
25#4939—21no—
26#4935C2211yescharacteristic
27#4928—21no—
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4535—22no—
31#4483—22no—
32C2×#477D311yescharacteristic, complemented
33C2×#476—31no—
34#2538—31no—
35#1956—31no—
36C2×Q8⋊C8—31no—
37C2×D4⋊C8—31no—
38#1296—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58#946—42no—
59#946—42no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—22no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D412yes—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C3×#107—24no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×#103—24no—
78#835—44no—
79C3⋊#95—22no—
80#809—24no—
81#808—24no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×C22⋊Q8—61no—
88C2×C4×Q8C2×D311yescharacteristic
89C2×C4×D4—31no—
90#477—22no—
91#477—22no—
92#476—62no—
93#476—62no—
94C2×#107—21no—
95C2×#106—61no—
96C2×C4⋊C8—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#103—21no—
99C2×#103—62no—
100C2×#102—61no—
101C2×C4×C8—31no—
102#396—62no—
103#390—61no—
104#389—61no—
105#384—31no—
106#380—61no—
107Q8⋊C8—62no—
108D4⋊C8—62no—
109C23×D3—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—42no—
117C2×C3×Q8—42no—
118C2×C3×Q8—41no—
119C2×C3×Q8SD1612yes—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—22no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12Q1612yes—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—82no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—41no—
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167C4⋊Dic3—42no—
168Dic3⋊C4—44no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C22×Q8—21no—
176C22×Q8#6711yescharacteristic
177C22×D4—31no—
178C23×C4—61no—
179C22×C8—21no—
180C22×C8—61no—
181C22×C8—61no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8#6712yes—
186C22⋊Q8—122no—
187C22⋊Q8—122no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—62no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—62no—
208C2×C42—21no—
209C2×C42C22×D311yescharacteristic
210#107—24no—
211#106—64no—
212C4⋊C8—21no—
213C4⋊C8—32no—
214C4⋊C8—61no—
215C4⋊C8#6712yes—
216#103—24no—
217#103—124no—
218#102—64no—
219C4×C8—61no—
220C4×C8—61no—
221#95—62no—
222C23×C3#12011yescharacteristic
223C22×D3—22no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C3×Q8—22no—
231C3×Q8—22no—
232C3×Q8—44no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—82no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×C3×C4C2×SD1611yescharacteristic
254C2×C3×C4C2×Q1611yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4#13711yescharacteristic
259C2×C3×C4#13711yescharacteristic
260C2×C3×C4#13711yescharacteristic
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267D12—22no—
268D12—22no—
269D12—41no—
270D12—41no—
271C4×D3—81no—
272C4×D3—81no—
273C4×D3—81no—
274C4×D3—81no—
275C3×C8—42no—
276C3⋊C8—42no—
277C3⋊C8—42no—
278C24—61no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8#21512yes—
289C2×D4—32no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—61no—
299C22×C4—62no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4C2×#6711yescharacteristic
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—122no—
329C4⋊C4—124no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C4⋊C4—121no—
334C22⋊C4—124no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—242no—
338C42—41no—
339C42—41no—
340C42D3×D411yescharacteristic
341C42#23711yescharacteristic
342C42C2×#6711yescharacteristic
343C42C2×#6711yescharacteristic
344C22×C3#39211yescharacteristic
345C22×C3#45011yescharacteristic
346C22×C3#45111yescharacteristic
347C22×C3#45211yescharacteristic
348C22×C3#45211yescharacteristic
349C22×C3#45311yescharacteristic
350C22×C3#45411yescharacteristic
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—42no—
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—42no—
361C3×C4—81no—
362C3×C4—82no—
363C3×C4#44811yescharacteristic
364C3×C4#44911yescharacteristic
365C3×C4#45111yescharacteristic
366C3×C4#45211yescharacteristic
367C3×C4#46511yescharacteristic
368C3×C4#46711yescharacteristic
369C3×C4#46811yescharacteristic
370C3×C4#47011yescharacteristic
371Dic3—82no—
372Dic3—81no—
373C23—62no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—121no—
378C23—121no—
379C23—121no—
380C23#93711yescharacteristic
381Q8—22no—
382Q8—22no—
383Q8—44no—
384Q8—41no—
385Q8—41no—
386Q8—82no—
387D4—62no—
388D4—62no—
389D4—121no—
390D4—121no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—241no—
409C2×C4—241no—
410C2×C4—241no—
411C2×C4—241no—
412C2×C4D3×Q1611yescharacteristic
413C2×C4#91811yescharacteristic
414C2×C4#91811yescharacteristic
415C2×C4#91911yescharacteristic
416C2×C4#93711yescharacteristic
417C2×C4#93811yescharacteristic
418C2×C4#93911yescharacteristic
419C2×C4#93911yescharacteristic
420C2×C4#94011yescharacteristic
421C2×C4C2×#21511yescharacteristic
422C2×C4#94211yescharacteristic
423C2×C4#94911yescharacteristic
424C2×C4#95011yescharacteristic
425C2×C4#95311yescharacteristic
426C8—42no—
427C8—122no—
428C8—122no—
429C2×C3#163111yescharacteristic
430C2×C3#163211yescharacteristic
431C2×C3#163711yescharacteristic
432C2×C3#163811yescharacteristic
433C2×C3#200011yescharacteristic
434C2×C3#200111yescharacteristic
435C2×C3#200611yescharacteristic
436D3—82no—
437C22—122no—
438C22—122no—
439C22—122no—
440C22—241no—
441C22—241no—
442C22—241no—
443C22—241no—
444C22#504611yescharacteristic
445C22#504711yescharacteristic
446C22#518611yescharacteristic
447C22#518711yescharacteristic
448C22#519811yescharacteristic
449C22#520111yescharacteristic
450C22#522311yescharacteristic
451C4—21no—
452C4—21no—
453C4—42no—
454C4—81no—
455C4—82no—
456C4—241no—
457C4—242no—
458C4#504811yescharacteristic
459C4#504911yescharacteristic
460C4#518611yescharacteristic
461C4#518711yescharacteristic
462C4#518811yescharacteristic
463C4#518911yescharacteristic
464C4#522711yescharacteristic
465C4#523811yescharacteristic
466C3#1023611yescharacteristic, complemented, Sylow
467C2—242no—
468C2#7115611yescharacteristic
469C2#7115711yescharacteristic
470C2#7116011yescharacteristic
471C2#7116311yescharacteristic
472C2#7169911yescharacteristic
473C2#7207111yescharacteristic
474C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#477C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#477D3
hypocenterC3#10236
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×C3×#477C2semidirect
4. #71785C2semidirect
5. C2×#4535C2semidirect
7. #67702C2semidirect
32. C2×#477D3semidirect
466. C3#10236semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4772
C2×#50842
#825002
#717852
C2×#45352
C2×#44832
#677022
#102363