Finite Group Databasebeta

← #10944175

Subgroups — #10944175

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
1C4×#1004C211yesmaximal, characteristic
2C3×#2547C211yesmaximal, characteristic, complemented
3#83209C212yesmaximal, complemented
4C2×#5061C212yesmaximal
5#71723C212yesmaximal, complemented
6#71617C211yesmaximal, characteristic, complemented
7#69693C214yesmaximal, complemented
8#69494C212yesmaximal, complemented
9#32922—31nomaximal, Sylow
10C2×#1004C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic, complemented
12C2×C3×#124C2212yes—
13C2×C3×C4×D4C2211yescharacteristic
14C2×#942C2214yes—
15#5798C2211yescharacteristic
16C2×C4×#67—22no—
17#5546—22no—
18#5545—22no—
19C4×D3×D4—22no—
20C4×#237—22no—
21#5504C2212yes—
22#5486—22no—
23C4×#235C2212yes—
24C3×#397—22no—
25C3×#391C2212yescomplemented
26C3×#380—22no—
27C3×C4×C4⋊C4C2211yescharacteristic, complemented
28#5225C2214yescomplemented
29C2×#833C2212yes—
30#5090—22no—
31#5090—22no—
32#5083—22no—
33#5083—22no—
34#5061—28no—
35C2×#805C2214yes—
36#4924C2212yescomplemented
37#4530C2212yescomplemented
38#4523C2214yescomplemented
39C4×H(5,𝔽2)—31no—
40#3039—32no—
41C2×#455—32no—
42#2547D311yescharacteristic, complemented
43#1885—32no—
44#1882—34no—
45#1806—32no—
46#1800—31no—
47C22×C3×Q8C2312yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4D412yes—
50C22×#67—22no—
51#1004—22no—
52#1004—24no—
53#1004D414yes—
54C2×#237—22no—
55C2×D3×D4—22no—
56C2×#235C2312yes—
57C22×C4×D3—22no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×#124—24no—
61C3×#124—24no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#216—24no—
74#942—28no—
75#942—28no—
76C2×#215—24no—
77D4×Dic3—22no—
78D4×Dic3—22no—
79C4×#67—24no—
80C4×#67—24no—
81C4×#67—22no—
82C4×#67—22no—
83C4×#67—24no—
84C2×(C2×D3)⋊C4—22no—
85#926C2312yes—
86C2×Dic3⋊C4—22no—
87C2×C4×Dic3—22no—
88C2×#208C2314yes—
89#893—22no—
90#892—22no—
91D3×C4⋊C4—22no—
92#887—22no—
93#67⋊C4—24no—
94#67⋊C4—24no—
95D3×C22⋊C4—24no—
96#877—24no—
97C4×D12D412yes—
98C4×D12D412yes—
99#872—22no—
100C42×D3—22no—
101C4×Dic6D412yes—
102C4×Dic6D412yes—
103C3×#95—21no—
104C3×#95—24no—
105#835—24no—
106#835—24no—
107#833—24no—
108#805—28no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)—22no—
111C2×H(5,𝔽2)—31no—
112C2×#137—34no—
113C2×C4⋊Q8C2×D311yescharacteristic, complemented
114C2×#124C2×D312yes—
115#520—32no—
116#520—62no—
117#518—31no—
118#518—62no—
119#518—62no—
120C4×#41—32no—
121C4×#41—62no—
122C2×C4×D4—62no—
123C2×C4×D4—62no—
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—22no—
135#391C2×D312yescomplemented
136#380—22no—
137C4×C4⋊C4C2×D311yescharacteristic, complemented
138#343—32no—
139#339—34no—
140C24×C3C2×D412yes—
141C23×D3—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—22no—
144C2×C3×Q8—24no—
145C2×C3×Q8—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4C2×D412yes—
149C2×C3×D4#4112yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—44no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—28no—
161C2×#67—28no—
162C2×#67—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166#237—28no—
167#237—28no—
168D3×D4—28no—
169D3×D4—28no—
170#235—22no—
171#235—22no—
172#235C2×D414yes—
173#235#4114yes—
174C2×D12C2×D412yes—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—24no—
178C2×C4×D3—22no—
179C2×C4×D3—84no—
180C2×Dic6C2×D412yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—41no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—48no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192#217—24no—
193#217—22no—
194#216—48no—
195#215—48no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—22no—
198(C2×D3)⋊C4—22no—
199C4⋊Dic3C2×D412yes—
200C4⋊Dic3C2×D412yes—
201Dic3⋊C4—24no—
202Dic3⋊C4—22no—
203Dic3⋊C4—22no—
204C4×Dic3—24no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208#208—24no—
209#208—24no—
210C2×C3⋊C8—24no—
211H(5,𝔽2)—34no—
212H(5,𝔽2)—64no—
213H(5,𝔽2)—62no—
214C2×#41—32no—
215C2×#41—62no—
216C22×Q8C22×D312yes—
217C22×D4—62no—
218C22×D4—62no—
219C22×D4C22×D311yescharacteristic
220C23×C4—62no—
221C23×C4#6712yes—
222#137—68no—
223#137—68no—
224C2×Q16—64no—
225C2×SD16—64no—
226C2×M16—34no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229#124—24no—
230#124—24no—
231C4×Q8—32no—
232C4×Q8—32no—
233C4×Q8—62no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—32no—
237C4×D4—32no—
238C4×D4—64no—
239C4×D4—64no—
240C4×D4—62no—
241C4×D4—62no—
242C4×D4—62no—
243C4×D4—64no—
244C4×D4—64no—
245C4×D4—62no—
246C4×D4—64no—
247C4×D4—62no—
248#117—32no—
249#117—64no—
250#117—62no—
251#117—62no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—24no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4#6712yes—
263C2×C42—62no—
264C2×C42—62no—
265C2×C42C22×D312yes—
266C2×C42C22×D311yescharacteristic
267C4⋊C8—62no—
268C4⋊C8—62no—
269#104—68no—
270#103—64no—
271#103—64no—
272#100—64no—
273#95—21no—
274#95—24no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3#12114yes—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—28no—
283C22×D3—82no—
284C22×D3—82no—
285C3×Q8—44no—
286C3×Q8—44no—
287C3×D4—24no—
288C3×D4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—41no—
292C2×C3×C4—44no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—82no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12011yescharacteristic
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D412yes—
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4112yes—
308#67—216no—
309#67—24no—
310#67—24no—
311C2×Dic3—28no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—82no—
316C2×Dic3—82no—
317D12—22no—
318D12#12114yes—
319C4×D3—24no—
320C4×D3—24no—
321C4×D3—44no—
322C4×D3—82no—
323C4×D3—84no—
324Dic6—22no—
325Dic6#12114yes—
326C3⋊C8—44no—
327C24—62no—
328C24C2×#6712yes—
329#41—34no—
330#41—34no—
331#41—68no—
332#41—68no—
333#41—62no—
334#41—62no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—32no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—32no—
343C2×D4—68no—
344C2×D4—68no—
345C2×D4—64no—
346C2×D4—68no—
347C2×D4—68no—
348C2×D4—62no—
349C2×D4D3×D412yes—
350C2×D4#23712yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4—64no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—244no—
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—22no—
374C4⋊C4—32no—
375C4⋊C4—32no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—64no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C22⋊C4—24no—
384C22⋊C4—48no—
385C22⋊C4—64no—
386C22⋊C4—64no—
387C22⋊C4—62no—
388C22⋊C4—62no—
389C22⋊C4—62no—
390C42—22no—
391C42—22no—
392C42—64no—
393C42—62no—
394C42—62no—
395C42—62no—
396C42D3×D412yes—
397C42D3×D412yes—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—24no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—82no—
405C22×C3C2×#12011yescharacteristic
406C22×C3C2×#12111yescharacteristic
407C22×C3#52511yescharacteristic
408C2×D3—24no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—88no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4—82no—
417C3×C4#45512yescomplemented
418C3×C4C2×#12112yes—
419C3×C4#52512yes—
420Dic3—24no—
421Dic3—42no—
422Dic3—42no—
423C23—22no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—68no—
428C23—62no—
429C23—242no—
430C23—242no—
431C23#93814yes—
432C23C22×#6711yescharacteristic
433Q8—34no—
434Q8—44no—
435Q8—44no—
436Q8—62no—
437D4—24no—
438D4—24no—
439D4—34no—
440D4—616no—
441D4—64no—
442D4—64no—
443D4—62no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—68no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—82no—
462C2×C4—124no—
463C2×C4—242no—
464C2×C4—244no—
465C2×C4—242no—
466C2×C4—242no—
467C2×C4#93712yes—
468C2×C4#93711yescharacteristic
469C2×C4#93711yescharacteristic
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×D3×D412yes—
472C2×C4C2×#23711yescharacteristic
473C2×C4D3×#4112yes—
474C8—124no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—44no—
478C2×C3#284711yescharacteristic
479C2×C3C2×#45511yescharacteristic
480C2×C3#302211yescharacteristic
481D3—44no—
482C22—21no—
483C22—21no—
484C22—24no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—64no—
489C22—82no—
490C22—122no—
491C22—122no—
492C22—248no—
493C22C2×#93711yescharacteristic
494C22C2×#93811yescharacteristic
495C22#582811yescharacteristic
496C4—21no—
497C4—22no—
498C4—42no—
499C4—64no—
500C4—82no—
501C4—82no—
502C4—122no—
503C4—122no—
504C4#506112yescomplemented
505C4D3×#12112yes—
506C4#560612yes—
507C3#3292211yescharacteristic, complemented, Sylow
508C2—21no—
509C2—21no—
510C2—44no—
511C2—124no—
512C2#8272311yescharacteristic
513C2C2×#506111yescharacteristic
514C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2547C2
socleC22×C3C2×#120
prime radicalC2×#1004C22
hypercenter#2547D3
hypocenterC3#32922
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. C3×#2547C2semidirect
3. #83209C2semidirect
5. #71723C2semidirect
6. #71617C2semidirect
7. #69693C2semidirect
8. #69494C2semidirect
11. C2×C3×C4⋊Q8C22semidirect
25. C3×#391C22semidirect
27. C3×C4×C4⋊C4C22semidirect
28. #5225C22semidirect
36. #4924C22semidirect
37. #4530C22semidirect
38. #4523C22semidirect
42. #2547D3semidirect
113. C2×C4⋊Q8C2×D3semidirect
135. #391C2×D3semidirect
137. C4×C4⋊C4C2×D3semidirect
417. C3×C4#455semidirect
504. C4#5061semidirect
507. C3#32922semidirect

Maximal subgroups

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

subgroupindex
C4×#10042
C3×#25472
#832092
C2×#50612
#717232
#716172
#696932
#694942
#329223