Finite Group Databasebeta

← #10845331

Subgroups — #10845331

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
1#82500C211yesmaximal, characteristic
2C2×#5020C211yesmaximal, characteristic, complemented
3C2×C8×D12C211yesmaximal, characteristic
4#73802C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70274C212yesmaximal, complemented
7#70270C212yesmaximal, complemented
8#69777C211yesmaximal, characteristic, complemented
9#69467C211yesmaximal, characteristic, complemented
10#68868C212yesmaximal, complemented
11#68833C212yesmaximal, complemented
12#25341—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2212yes—
15C22×D3×C8C2212yes—
16C2×#875—21no—
17C2×C4×D12C2211yescharacteristic
18C2×#869C2211yescharacteristic
19C2×C3×#103C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#835C2212yes—
24C2×(C2×D3)⋊C8C2212yes—
25#5020—24no—
26#4994—22no—
27#4987C2212yes—
28#4987C2212yes—
29#4987C2212yes—
30C2×#808C2212yes—
31C2×#808C2212yes—
32#4939—21no—
33#4935C2211yescharacteristic
34#4929—21no—
35#4928—21no—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37C8×D12—24no—
38#4674—22no—
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4529C2212yes—
44C2×#475—31no—
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×#103D311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×D12D411yescharacteristic
57C22×C4×D3C2312yes—
58C22×Dic6—21no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70#946—24no—
71#946—42no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4D412yes—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C22×C3⋊C8C2312yes—
85C2×D3×C8—24no—
86C2×D3×C8—24no—
87C2×D3×C8—42no—
88#875—44no—
89C4×D12—22no—
90C4×D12—22no—
91#869—22no—
92#869—22no—
93C3×#103—22no—
94C3×#103—22no—
95C3×C4×C8—22no—
96C3×#95—21no—
97#835—24no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101(C2×D3)⋊C8—24no—
102#808—24no—
103#808—42no—
104C4⋊(C3⋊C8)D414yes—
105C23×C8—32no—
106C2×C4⋊Q8—31no—
107C2×#124—61no—
108C2×C22⋊Q8—32no—
109C2×C4×Q8C2×D311yescharacteristic
110C2×C4×D4—31no—
111#475—64no—
112C8×D4—64no—
113C2×#106—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#103—32no—
117C2×#103C2×D311yescharacteristic
118C2×#103C2×D311yescharacteristic
119C2×C22⋊C8—32no—
120C2×C4×C8C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
131#340—32no—
132#340—32no—
133#340—32no—
134#340C2×D311yescharacteristic
135#340C2×D311yescharacteristic
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#4112yes—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3C2×D412yes—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12SD1614yes—
156C2×C4×D3—22no—
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C2×Dic6—42no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8#4112yes—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—42no—
178C3×C42—42no—
179C3×C42—42no—
180C3×C42C2×D412yes—
181C3×C42#4112yes—
182(C2×D3)⋊C4—24no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—82no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3—81no—
195Dic3⋊C4—44no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8C2×D414yes—
199D3×C8—48no—
200C22×Q8—61no—
201C22×Q8C22×D311yescharacteristic
202C22×D4—31no—
203C23×C4—32no—
204C22×C8—32no—
205C22×C8—64no—
206C22×C8—64no—
207C22×C8—122no—
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C4⋊Q8—62no—
211C4⋊Q8—62no—
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—31no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4C22×D311yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D311yescharacteristic
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—21no—
237C2×C42C22×D311yescharacteristic
238#106—64no—
239#106—122no—
240C4⋊C8—34no—
241#103—22no—
242#103—22no—
243#103—64no—
244C22⋊C8—64no—
245C4×C8—22no—
246#95—21no—
247#95—62no—
248#95—62no—
249#95—62no—
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—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4#12112yes—
277C2×C3×C4#12112yes—
278C2×C3×C4C2×SD1612yes—
279C2×C3×C4#13512yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4H(5,𝔽2)11yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295D12—28no—
296C4×D3—42no—
297C4×D3—42no—
298C4×D3—42no—
299C4×D3—42no—
300Dic6—84no—
301C3×C8—22no—
302C3×C8—41no—
303C3⋊C8—24no—
304C24—32no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—122no—
311C2×Q8—122no—
312C2×Q8—122no—
313C2×Q8#23512yes—
314C2×D4—34no—
315C2×D4—62no—
316C2×D4—62no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—32no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4C23×D311yescharacteristic
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—34no—
339C2×C8—62no—
340C2×C8—62no—
341C2×C8—128no—
342C2×C8#23512yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—42no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—62no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—124no—
356C4⋊C4—122no—
357C4⋊C4—122no—
358C4⋊C4—241no—
359C22⋊C4—64no—
360C22⋊C4—124no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—242no—
364C42—42no—
365C42—42no—
366C42#23512yes—
367C42D3×D412yes—
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—81no—
385C3×C4—81no—
386C3×C4—81no—
387C3×C4#46514yes—
388C3×C4#52911yescharacteristic
389C3×C4#54211yescharacteristic
390C3×C4#54611yescharacteristic
391C3×C4#54811yescharacteristic
392Dic3—42no—
393Dic3—82no—
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—244no—
408D4—68no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4#88412yes—
437C2×C4#88412yes—
438C2×C4D3×SD1612yes—
439C2×C4#91912yes—
440C2×C4C2×#23511yescharacteristic
441C2×C4C2×#23511yescharacteristic
442C2×C4C2×#23511yescharacteristic
443C2×C4C2×D3×D411yescharacteristic
444C2×C4#100911yescharacteristic
445C2×C4#101011yescharacteristic
446C8—22no—
447C8—41no—
448C8—64no—
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—
464C22C2×#88411yescharacteristic
465C22#555011yescharacteristic
466C22#555111yescharacteristic
467C22C2×D3×SD1611yescharacteristic
468C22C2×#91911yescharacteristic
469C22#577011yescharacteristic
470C22#577111yescharacteristic
471C4—22no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—122no—
477C4—242no—
478C4—242no—
479C4#480714yes—
480C4#547911yescharacteristic
481C4#553411yescharacteristic
482C4#555011yescharacteristic
483C4#555211yescharacteristic
484C3#2534111yescharacteristic, complemented, Sylow
485C2—124no—
486C2#7348611yescharacteristic
487C2C2×#480711yescharacteristic
488C2#7884111yescharacteristic
489C2#7967811yescharacteristic
490C2#7968811yescharacteristic
491C2#7973711yescharacteristic
492C2#7974311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenterC4×#103D3
hypocenterC3#25341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
5. C3×C4×#103C2semidirect
6. #70274C2semidirect
7. #70270C2semidirect
8. #69777C2semidirect
9. #69467C2semidirect
10. #68868C2semidirect
11. #68833C2semidirect
54. C4×#103D3semidirect
484. C3#25341semidirect

Maximal subgroups

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

subgroupindex
#825002
C2×#50202
C2×C8×D122
#738022
C3×C4×#1032
#702742
#702702
#697772
#694672
#688682
#688332
#253413