Finite Group Databasebeta

← #10944179

Subgroups — #10944179

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#5242C212yesmaximal
3#82533C212yesmaximal, complemented
4#82382C211yesmaximal, characteristic, complemented
5#71666C214yesmaximal, complemented
6#71649C212yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71617C211yesmaximal, characteristic, complemented
9#32922—31nomaximal, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C3×#520—22no—
12C3×#520C2212yes—
13C3×#518—22no—
14C3×#518—22no—
15C3×#518C2211yescharacteristic
16C3×C4×#41—22no—
17C3×C4×#41C2212yes—
18C2×C3×C4×D4—22no—
19C2×C3×C4×D4—22no—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#951C2214yes—
22C2×#936C2212yes—
23C2×#869C2211yescharacteristic, complemented
24#5242—28no—
25C2×#837C2214yes—
26C2×#833C2212yes—
27#5212—22no—
28#5031—22no—
29#5031—22no—
30#5019—22no—
31#5019—22no—
32#5004C2214yescomplemented
33#4973C2212yescomplemented
34#4953—22no—
35C4×C4⋊Dic3C2211yescharacteristic, complemented
36#4924C2212yescomplemented
37#4532C2214yescomplemented
38#4530C2212yescomplemented
39C4×H(5,𝔽2)D311yescharacteristic
40#3039—32no—
41C2×#455—32no—
42#2547—31no—
43#1885—32no—
44#1882—34no—
45#1806—32no—
46#1800—31no—
47C3×H(5,𝔽2)—24no—
48C3×H(5,𝔽2)—22no—
49C3×H(5,𝔽2)D414yes—
50C2×C3×#41—22no—
51C2×C3×#41C2312yes—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4D412yes—
57C22×Dic6C2312yes—
58C3×C4×Q8—22no—
59C3×C4×Q8D412yes—
60C3×C4×Q8D412yes—
61C3×C4×D4—24no—
62C3×C4×D4—24no—
63C3×C4×D4—24no—
64C3×C4×D4—24no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—24no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4D412yes—
74C3×C4×D4D412yes—
75C3×#117—24no—
76C3×#117—22no—
77C3×#117—22no—
78C3×#117C2312yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D412yes—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—24no—
90#951—28no—
91#951—28no—
92C2×#216—24no—
93#936—24no—
94#936—24no—
95C2×#214—24no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4×Dic3C2312yes—
100C2×#208C2314yes—
101#869—22no—
102#869—22no—
103#837—28no—
104#833—24no—
105C3⋊#95—21no—
106C3⋊#95—24no—
107#810—24no—
108#810—24no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)—22no—
111C2×H(5,𝔽2)C2×D311yescharacteristic
112C2×#137—34no—
113C2×C4⋊Q8—31no—
114C2×#124—32no—
115#520—22no—
116#520C2×D312yes—
117#518—22no—
118#518—22no—
119#518C2×D311yescharacteristic
120C4×#41—22no—
121C4×#41C2×D312yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2×D311yescharacteristic
125#462—62no—
126#462—62no—
127#461—62no—
128#461—62no—
129#455—68no—
130#423—32no—
131C2×#104—34no—
132#419—34no—
133C2×#100—32no—
134#397—62no—
135#391—32no—
136#380—62no—
137C4×C4⋊C4—31no—
138#343—32no—
139#339—34no—
140C24×C3—22no—
141C24×C3C2×D412yes—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—22no—
145C3×#41—22no—
146C3×#41C2×D414yes—
147C3×#41#4114yes—
148C2×C3×Q8C2×D412yes—
149C2×C3×D4—24no—
150C2×C3×D4—28no—
151C2×C3×D4—28no—
152C2×C3×D4—28no—
153C2×C3×D4—22no—
154C2×C3×D4—28no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4C2×D412yes—
158C2×C3×D4C2×D412yes—
159C2×C3×D4#4112yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—44no—
170C22×C3×C4—84no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3—22no—
177C2×Dic6—24no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×D412yes—
186C3×C4⋊C4C2×D412yes—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—24no—
193C3×C42—24no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—48no—
200#216—48no—
201#214—48no—
202C4⋊Dic3—42no—
203C4⋊Dic3—42no—
204C4⋊Dic3—41no—
205C4⋊Dic3—41no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208#208—24no—
209#208—24no—
210C2×C3⋊C8—24no—
211H(5,𝔽2)—24no—
212H(5,𝔽2)—22no—
213H(5,𝔽2)D1214yes—
214C2×#41—22no—
215C2×#41C22×D312yes—
216C22×Q8—32no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4C22×D311yescharacteristic
220C23×C4—22no—
221C23×C4#6712yes—
222#137—68no—
223#137—68no—
224C2×Q16—64no—
225C2×SD16—64no—
226C2×M16—34no—
227C4⋊Q8—62no—
228C4⋊Q8—62no—
229#124—64no—
230#124—64no—
231C4×Q8—22no—
232C4×Q8#6712yes—
233C4×Q8#6712yes—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4#6712yes—
247C4×D4#6712yes—
248#117—24no—
249#117—22no—
250#117—22no—
251#117C22×D312yes—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4#6712yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—32no—
266C2×C42C22×D311yescharacteristic
267C4⋊C8—62no—
268C4⋊C8—62no—
269#104—68no—
270#103—64no—
271#103—64no—
272#100—64no—
273#95—61no—
274#95—64no—
275C23×C3—28no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—82no—
282C23×C3—82no—
283C23×C3#12114yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—22no—
286C3×Q8#12114yes—
287C3×D4—216no—
288C3×D4—24no—
289C3×D4—24no—
290C3×D4—24no—
291C3×D4—22no—
292C3×D4—24no—
293C3×D4#12114yes—
294C2×C3×C4—28no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—41no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—42no—
306C2×C3×C4—82no—
307C2×C3×C4—82no—
308C2×C3×C4—82no—
309C2×C3×C4—84no—
310C2×C3×C4—82no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D412yes—
316C2×C3×C4C2×#4112yes—
317C2×C3×C4C2×#4111yescharacteristic
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324Dic6—44no—
325Dic6—44no—
326C3⋊C8—44no—
327C24—22no—
328C24C2×#6712yes—
329#41—28no—
330#41—28no—
331#41—22no—
332#41—22no—
333#41C2×D1214yes—
334#41#23514yes—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—64no—
338C2×Q8—64no—
339C2×Q8C2×#6712yes—
340C2×D4—24no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—22no—
345C2×D4—28no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4C2×D1212yes—
349C2×D4#23512yes—
350C2×D4C2×#6712yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—84no—
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6712yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C23×D311yescharacteristic
368Q16—128no—
369SD16—128no—
370M16—64no—
371M16—64no—
372C2×C8—64no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—122no—
378C4⋊C4—122no—
379C4⋊C4—121no—
380C4⋊C4—121no—
381C4⋊C4C2×#6712yes—
382C4⋊C4C2×#6712yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—128no—
390C42—24no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—62no—
395C42—62no—
396C42C2×#6712yes—
397C42C2×#6712yes—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—44no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—82no—
408C22×C3—88no—
409C22×C3C2×#12011yescharacteristic
410C22×C3C2×#12111yescharacteristic
411C22×C3#52511yescharacteristic
412C3×C4—24no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4#45512yescomplemented
419C3×C4C2×#12112yes—
420C3×C4#52512yes—
421Dic3—82no—
422Dic3—82no—
423C23—28no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—82no—
430C23—82no—
431C23#93014yes—
432C23C22×#6711yescharacteristic
433Q8—22no—
434Q8—124no—
435Q8—124no—
436Q8#93014yes—
437D4—216no—
438D4—24no—
439D4—24no—
440D4—24no—
441D4—24no—
442D4—22no—
443D4#93014yes—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4#95312yes—
468C2×C4#95311yescharacteristic
469C2×C4#95311yescharacteristic
470C2×C4C22×D1211yescharacteristic
471C2×C4C2×#23512yes—
472C2×C4C2×#23511yescharacteristic
473C2×C4C22×#6712yes—
474C8—124no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—44no—
478C2×C3—44no—
479C2×C3#284711yescharacteristic
480C2×C3C2×#45511yescharacteristic
481C2×C3#302211yescharacteristic
482C22—21no—
483C22—21no—
484C22—24no—
485C22—24no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—88no—
493C22C2×#93011yescharacteristic
494C22#579111yescharacteristic
495C22C2×#95311yescharacteristic
496C4—24no—
497C4—21no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—242no—
503C4—242no—
504C4#524212yescomplemented
505C4C2×#93012yes—
506C4#579112yes—
507C3#3292211yescharacteristic, complemented, Sylow
508C2—21no—
509C2—21no—
510C2—44no—
511C2—44no—
512C2#8403211yescharacteristic
513C2C2×#524211yescharacteristic
514C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×H(5,𝔽2)C2
socleC22×C3C2×#120
prime radicalC2×C3×H(5,𝔽2)C22
hypercenterC4×H(5,𝔽2)D3
hypocenterC3#32922
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
3. #82533C2semidirect
4. #82382C2semidirect
5. #71666C2semidirect
6. #71649C2semidirect
7. #71646C2semidirect
8. #71617C2semidirect
23. C2×#869C22semidirect
32. #5004C22semidirect
33. #4973C22semidirect
35. C4×C4⋊Dic3C22semidirect
36. #4924C22semidirect
37. #4532C22semidirect
38. #4530C22semidirect
418. C3×C4#455semidirect
504. C4#5242semidirect
507. C3#32922semidirect

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C2×#52422
#825332
#823822
#716662
#716492
#716462
#716172
#329223