Finite Group Databasebeta

← #10845298

Subgroups — #10845298

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82417C211yesmaximal, characteristic
2C2×#5007C211yesmaximal, characteristic, complemented
3C2×C8×D12C211yesmaximal, characteristic, complemented
4#73802C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic, complemented
6#70206C212yesmaximal, complemented
7#70202C212yesmaximal, complemented
8#69777C211yesmaximal, characteristic, complemented
9#69466C211yesmaximal, characteristic, complemented
10#68889C212yesmaximal, complemented
11#68869C212yesmaximal, complemented
12#25338—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#938C2212yes—
15C2×#930—21no—
16C22×D3×C8C2212yes—
17C2×#875—21no—
18C2×C4×D12C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#832C2212yes—
25#5212—21no—
26C2×(C2×D3)⋊C8C2212yes—
27#5007—24no—
28#4987C2212yes—
29#4987C2212yes—
30#4987C2212yes—
31C2×#807C2212yes—
32C2×#807C2212yes—
33#4961—22no—
34#4935C2211yescharacteristic
35#4929—21no—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37C8×D12—24no—
38#4674—22no—
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4529C2212yes—
44C2×#474—31no—
45C2×C8×D4—31no—
46#2547—31no—
47#2537—31no—
48#2075—32no—
49#2033—32no—
50#1990—31no—
51#1947—31no—
52#1893—32no—
53#1808—32no—
54C4×#102D311yescharacteristic, complemented
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×#67—22no—
58C22×D12D411yescharacteristic
59C22×C4×D3C2312yes—
60C22×Dic6—21no—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72#938—24no—
73#938—42no—
74#930—44no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4D412yes—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C22×C3⋊C8C2312yes—
86C2×D3×C8—24no—
87C2×D3×C8—24no—
88C2×D3×C8—42no—
89#875—44no—
90C4×D12—22no—
91C4×D12—22no—
92#869—22no—
93#869—22no—
94C3×#102—22no—
95C3×#102—22no—
96C3×C4×C8—22no—
97C3×#95—21no—
98#832—24no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102(C2×D3)⋊C8—24no—
103#807—24no—
104#807—42no—
105C4⋊(C3⋊C8)D414yescomplemented
106C23×C8—32no—
107C2×C4⋊Q8—31no—
108C2×#124—61no—
109C2×#121—32no—
110C2×#121—61no—
111C2×C4×D4—31no—
112C2×C4×D4C2×D311yescharacteristic
113#474—64no—
114C8×D4—64no—
115C2×#107—32no—
116C2×#107—32no—
117C2×C4⋊C8—31no—
118C2×#102—32no—
119C2×#102C2×D311yescharacteristic
120C2×#102C2×D311yescharacteristic
121C2×C22⋊C8—32no—
122C2×C4×C8C2×D311yescharacteristic
123#397—62no—
124#394—62no—
125#391—61no—
126#384—31no—
127#380—32no—
128#380—32no—
129#380—32no—
130#380—61no—
131C4×C4⋊C4C2×D311yescharacteristic
132#340—32no—
133#340—32no—
134#340—32no—
135#340C2×D311yescharacteristic
136#340C2×D311yescharacteristic
137C24×C3—21no—
138C23×D3C2×D412yes—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4#4112yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C2×#67—44no—
155C2×#67—44no—
156C2×#67—82no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3C2×D412yes—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12D814yes—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×Dic6—42no—
170C2×Dic6—42no—
171C2×Dic6—42no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8#4112yes—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C22⋊C4—42no—
185C3×C42—42no—
186C3×C42C2×D412yes—
187C3×C42#4112yes—
188#217—44no—
189(C2×D3)⋊C4—24no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—42no—
199C4⋊Dic3—42no—
200C4⋊Dic3—81no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8C2×D414yes—
204D3×C8—48no—
205C22×Q8—61no—
206C22×D4—31no—
207C22×D4—62no—
208C22×D4C22×D311yescharacteristic
209C23×C4—21no—
210C23×C4—32no—
211C22×C8—32no—
212C22×C8—64no—
213C22×C8—64no—
214C22×C8—122no—
215C22×C8C22×D311yescharacteristic
216C22×C8C22×D311yescharacteristic
217C4⋊Q8—62no—
218C4⋊Q8—62no—
219#124—124no—
220#121—64no—
221#121—122no—
222#121—124no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—62no—
226C4×D4—62no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—32no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C42—21no—
243C2×C42C22×D311yescharacteristic
244#107—64no—
245#107—122no—
246C4⋊C8—34no—
247#102—22no—
248#102—22no—
249#102—64no—
250C22⋊C8—64no—
251C4×C8—22no—
252#95—21no—
253#95—62no—
254#95—62no—
255#95—62no—
256C23×C3—22no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C22×D411yescharacteristic
264C22×D3—22no—
265C22×D3—22no—
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—22no—
269C22×D3—22no—
270C22×D3C2×D814yes—
271C3×D4—22no—
272C3×D4—22no—
273C3×D4—41no—
274C3×D4—41no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—84no—
287C2×C3×C4#12112yes—
288C2×C3×C4#12112yes—
289C2×C3×C4C2×D812yes—
290C2×C3×C4#13512yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4H(5,𝔽2)11yescharacteristic
296C2×C3×C42−1+411yescharacteristic
297#67—88no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307D12—28no—
308C4×D3—42no—
309C4×D3—42no—
310C4×D3—42no—
311C4×D3—42no—
312Dic6—84no—
313C3×C8—22no—
314C3×C8—41no—
315C3⋊C8—24no—
316C24—21no—
317C24—32no—
318C2×Q8—122no—
319C2×Q8—122no—
320C2×Q8—122no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—34no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—242no—
332C2×D4#23512yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—32no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4C23×D311yescharacteristic
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—34no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—128no—
360C2×C8#23512yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C4⋊C4—241no—
375C22⋊C4—42no—
376C22⋊C4—64no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—124no—
381C42—42no—
382C42#23512yes—
383C42D3×D412yes—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3C2×#12111yescharacteristic
392C22×C3#54611yescharacteristic
393C22×C3#54711yescharacteristic
394C22×C3C22×D811yescharacteristic
395C22×C3C2×#13511yescharacteristic
396C22×C3#57711yescharacteristic
397C22×C3#57811yescharacteristic
398C2×D3—24no—
399C2×D3—24no—
400C2×D3—24no—
401C2×D3—42no—
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—42no—
405C3×C4—22no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4#46614yes—
410C3×C4#52911yescharacteristic
411C3×C4#54211yescharacteristic
412C3×C4#54611yescharacteristic
413C3×C4#54811yescharacteristic
414Dic3—42no—
415Dic3—82no—
416Dic3—82no—
417C23—22no—
418C23—34no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23—62no—
430C23—62no—
431C23C2×D3×D411yescharacteristic
432Q8—244no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437D4—68no—
438D4—248no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—84no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4#88412yes—
465C2×C4#88412yes—
466C2×C4D3×D812yes—
467C2×C4#91612yes—
468C2×C4C2×#23511yescharacteristic
469C2×C4C2×#23511yescharacteristic
470C2×C4C2×#23511yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4#100911yescharacteristic
473C2×C4#101011yescharacteristic
474C8—22no—
475C8—41no—
476C8—64no—
477C2×C3—82no—
478C2×C3#242711yescharacteristic
479C2×C3C2×#46611yescharacteristic
480C2×C3#306311yescharacteristic
481C2×C3#329411yescharacteristic
482C2×C3#330211yescharacteristic
483C2×C3#333411yescharacteristic
484C2×C3#334011yescharacteristic
485D3—44no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—64no—
490C22—64no—
491C22—64no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22C2×#88411yescharacteristic
501C22#555011yescharacteristic
502C22#555111yescharacteristic
503C22C2×D3×D811yescharacteristic
504C22C2×#91611yescharacteristic
505C22#577011yescharacteristic
506C22#577111yescharacteristic
507C4—22no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—122no—
512C4—242no—
513C4—242no—
514C4#477514yes—
515C4#547911yescharacteristic
516C4#553411yescharacteristic
517C4#555011yescharacteristic
518C4#555211yescharacteristic
519C3#2533811yescharacteristic, complemented, Sylow
520C2—82no—
521C2—124no—
522C2#7348611yescharacteristic
523C2C2×#477511yescharacteristic
524C2#7884111yescharacteristic
525C2#7960811yescharacteristic
526C2#7961411yescharacteristic
527C2#7968011yescharacteristic
528C2#7969011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC4×#102D3
hypocenterC3#25338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5007C2semidirect
3. C2×C8×D12C2semidirect
4. #73802C2semidirect
5. C3×C4×#102C2semidirect
6. #70206C2semidirect
7. #70202C2semidirect
8. #69777C2semidirect
9. #69466C2semidirect
10. #68889C2semidirect
11. #68869C2semidirect
54. C4×#102D3semidirect
105. C4⋊(C3⋊C8)D4semidirect
519. C3#25338semidirect

Maximal subgroups

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

subgroupindex
#824172
C2×#50072
C2×C8×D122
#738022
C3×C4×#1022
#702062
#702022
#697772
#694662
#688892
#688692
#253383