Finite Group Databasebeta

← #11002004

Subgroups — #11002004

Order 768.

2326 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×#5500C211yesmaximal, characteristic, complemented
3#84065C212yesmaximal, complemented
4#84031C212yesmaximal
5#82796C218yesmaximal, complemented
6#82791C218yesmaximal, complemented
7#82183C212yesmaximal, complemented
8#82176C212yesmaximal, complemented
9#82155C211yesmaximal, characteristic, complemented
10#82116C212yesmaximal
11#82113C212yesmaximal
12#36708—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C2×C3×C22⋊Q8—28no—
16C2×C3×C22⋊Q8C2218yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C2×#944C2214yescomplemented
23C2×#926C2212yes—
24C2×#924C2212yes—
25C2×#924C2212yes—
26C2×#924C2214yes—
27#5500—28no—
28C2×#890C2214yescomplemented
29C2×#888C2214yescomplemented
30#5247C2214yes—
31#5230C2218yescomplemented
32#5229C2214yescomplemented
33#5212C2212yes—
34#5212C2214yes—
35#5211C2214yes—
36#5211C2212yes—
37#5210C2214yes—
38#5210C2214yes—
39#5210C2218yes—
40C4×#217C2212yes—
41#4986C2214yescomplemented
42#4986C2214yescomplemented
43#4982C2218yescomplemented
44#4981C2214yescomplemented
45#4980C2218yescomplemented
46#4979C2214yescomplemented
47#4978C2218yescomplemented
48#4953C2214yes—
49#4953C2214yes—
50#4949C2214yes—
51#4947C2214yes—
52#4945C2218yes—
53#4944C2218yes—
54C2×#544—31no—
55C22×C22⋊Q8D311yescharacteristic
56#2848—32no—
57#2690—38no—
58#2677—38no—
59#2627—32no—
60#2561—32no—
61#2484—32no—
62#2482—32no—
63#2434—32no—
64#2313—31no—
65C25×C3C2311yescharacteristic
66C22×C3×Q8—24no—
67C22×C3×Q8—44no—
68C22×C3×Q8C2314yes—
69C23×C3×C4—24no—
70C23×C3×C4—44no—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2314yes—
74C23×C3×C4C2311yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C22×Dic6C2312yes—
79C22×Dic6C2312yes—
80C3×C22⋊Q8—232no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4D418yes—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4C2314yes—
90C2×C3×C22⋊C4C2314yes—
91C2×#217C2318yes—
92C2×#217C2314yes—
93C2×#217C2314yes—
94C2×#217C2318yes—
95#944—48no—
96#926—28no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2314yes—
102#924—28no—
103#924—28no—
104#924—48no—
105C2×Dic3⋊C4C2314yes—
106C2×Dic3⋊C4C2318yes—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2318yes—
109C2×C4×Dic3C2314yes—
110C2×C4×Dic3C2314yes—
111#890—216no—
112#888—216no—
113C3⋊#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2314yes—
118C3⋊#95C2314yes—
119C3⋊#95C2318yes—
120C3⋊#95C2314yes—
121C23×Q8C2×D311yescharacteristic
122C24×C4C2×D311yescharacteristic
123#544—68no—
124C2×C4⋊Q8—34no—
125C2×C4⋊Q8—34no—
126C2×#125—34no—
127C2×C22⋊Q8—28no—
128C2×C22⋊Q8—32no—
129C2×C22⋊Q8—32no—
130C2×C22⋊Q8—34no—
131C2×C22⋊Q8C2×D318yes—
132C2×#117—32no—
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4C2×D311yescharacteristic
137C22×C22⋊C4C2×D311yescharacteristic
138#400—34no—
139#400—34no—
140#399—34no—
141#398—38no—
142#398—34no—
143#397—34no—
144#397—34no—
145#395—38no—
146#395—34no—
147#393—34no—
148#391—34no—
149#388—38no—
150#388—38no—
151#387—38no—
152#387—38no—
153#385—38no—
154#385—34no—
155#385—34no—
156#381—34no—
157#381—32no—
158#380—32no—
159#380—34no—
160#379—34no—
161C4×C22⋊C4—32no—
162C24×C3—22no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—42no—
166C24×C3C2412yes—
167C24×C3C2411yescharacteristic
168C2×C3×Q8—28no—
169C2×C3×Q8—28no—
170C2×C3×Q8—44no—
171C2×C3×Q8—416no—
172C22×C3×C4—22no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—28no—
179C22×C3×C4—28no—
180C22×C3×C4—42no—
181C22×C3×C4—44no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4—416no—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D414yes—
188C22×C3×C4C2412yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2412yes—
191C22×C3×C4C2412yes—
192C22×C3×C4C2412yes—
193C22×C3×C4C2412yes—
194C22×Dic3C2418yes—
195C22×Dic3C2414yes—
196C22×Dic3C2414yes—
197C2×Dic6—24no—
198C2×Dic6—24no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4C2×D4116yes—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—28no—
206#217—28no—
207#217—216no—
208#217—28no—
209#217—48no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—24no—
213C4⋊Dic3—24no—
214C4⋊Dic3—44no—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217Dic3⋊C4—216no—
218Dic3⋊C4—48no—
219C4×Dic3—28no—
220C4×Dic3—44no—
221C25C22×D311yescharacteristic
222C22×Q8—24no—
223C22×Q8—32no—
224C22×Q8—32no—
225C22×Q8—44no—
226C22×Q8C22×D314yes—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D314yes—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C4⋊Q8—616no—
237C4⋊Q8—128no—
238#125—616no—
239C22⋊Q8—232no—
240C22⋊Q8—68no—
241C22⋊Q8—68no—
242C22⋊Q8—128no—
243#117—68no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—38no—
253C2×C4⋊C4—34no—
254C2×C4⋊C4—38no—
255C2×C4⋊C4#6718yes—
256C2×C4⋊C4C22×D314yes—
257C2×C4⋊C4C22×D314yes—
258C2×C4⋊C4C22×D314yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—34no—
262C2×C22⋊C4—34no—
263C2×C22⋊C4—38no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4C22×D314yes—
266C2×C22⋊C4C22×D314yes—
267C2×C42—34no—
268C2×C42—34no—
269#95—34no—
270#95—34no—
271#95—38no—
272#95—34no—
273#95—34no—
274#95—38no—
275#95—38no—
276#95—38no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—28no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—24no—
284C23×C3—416no—
285C23×C3—44no—
286C23×C3—44no—
287C23×C3C22×D412yes—
288C23×C32−1+414yes—
289C23×C32−1+414yes—
290C23×C3C2511yescharacteristic
291C3×Q8—416no—
292C2×C3×C4—28no—
293C2×C3×C4—28no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—416no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4C22×D414yes—
309C2×C3×C4C22×D414yes—
310C2×C3×C4C22×D414yes—
311C2×C3×C4H(5,𝔽2)14yes—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—28no—
315C2×Dic3—28no—
316C2×Dic3—28no—
317C2×Dic3—28no—
318C2×Dic3—28no—
319Dic6—84no—
320Dic6—84no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—42no—
325C24C23×D312yes—
326C24C23×D311yescharacteristic
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—44no—
330C2×Q8—416no—
331C2×Q8—64no—
332C2×Q8—64no—
333C2×Q8—124no—
334C2×Q8—124no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—38no—
344C22×C4—34no—
345C22×C4—34no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—416no—
351C22×C4C2×#6714yes—
352C22×C4C2×#6714yes—
353C22×C4C2×#6714yes—
354C22×C4C23×D312yes—
355C22×C4C23×D312yes—
356C22×C4C23×D312yes—
357C22×C4C23×D312yes—
358C22×C4C23×D312yes—
359C22×C4C23×D312yes—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—616no—
363C4⋊C4—64no—
364C4⋊C4—64no—
365C4⋊C4—64no—
366C4⋊C4—64no—
367C4⋊C4—68no—
368C4⋊C4—68no—
369C4⋊C4—124no—
370C4⋊C4—128no—
371C4⋊C4C2×#67116yes—
372C22⋊C4—28no—
373C22⋊C4—28no—
374C22⋊C4—616no—
375C22⋊C4—68no—
376C22⋊C4—68no—
377C22⋊C4—128no—
378C42—68no—
379C42—124no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—28no—
383C22×C3—24no—
384C22×C3—28no—
385C22×C3—416no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—44no—
389C22×C3#53614yes—
390C22×C3C23×D411yescharacteristic
391C22×C3C2×H(5,𝔽2)12yes—
392C22×C3C2×2−1+412yes—
393C22×C3C2×2−1+412yes—
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—24no—
397C3×C4—24no—
398C3×C4—42no—
399C3×C4—42no—
400Dic3—44no—
401Dic3—44no—
402Dic3—48no—
403C23—22no—
404C23—22no—
405C23—28no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—416no—
411C23—44no—
412C23—44no—
413C23#100714yes—
414C23#101014yes—
415C23C22×#6712yes—
416C23C24×D311yescharacteristic
417Q8—416no—
418Q8—244no—
419Q8—244no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—416no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—68no—
437C2×C4—68no—
438C2×C4—68no—
439C2×C4—68no—
440C2×C4—68no—
441C2×C4—64no—
442C2×C4—64no—
443C2×C4#100414yes—
444C2×C4C22×#6714yes—
445C2×C4C22×#6714yes—
446C2×C4C22×#6714yes—
447C2×C3—24no—
448C2×C3—44no—
449C2×C3C2×#53611yescharacteristic
450C2×C3#348411yescharacteristic
451C2×C3#348611yescharacteristic
452C2×C3#353814yes—
453C22—22no—
454C22—22no—
455C22—28no—
456C22—24no—
457C22—28no—
458C22—416no—
459C22—42no—
460C22—44no—
461C22—42no—
462C22#582514yes—
463C22C2×#100412yes—
464C22C2×#100712yes—
465C22C2×#101012yes—
466C22C23×#6711yescharacteristic
467C4—24no—
468C4—24no—
469C4—24no—
470C4—24no—
471C4—42no—
472C4—42no—
473C4—124no—
474C4—124no—
475C4—128no—
476C3#3670811yescharacteristic, complemented, Sylow
477C2—24no—
478C2—44no—
479C2#8597314yes—
480C2#8723911yescharacteristic
481C2#8725111yescharacteristic
482C2C2×#582511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC22×C3×C22⋊Q8C2
prime radicalC25×C3C23
hypercenterC22×C22⋊Q8D3
hypocenterC3#36708
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#5500C2semidirect
3. #84065C2semidirect
5. #82796C2semidirect
6. #82791C2semidirect
7. #82183C2semidirect
8. #82176C2semidirect
9. #82155C2semidirect
22. C2×#944C22semidirect
28. C2×#890C22semidirect
29. C2×#888C22semidirect
31. #5230C22semidirect
32. #5229C22semidirect
41. #4986C22semidirect
42. #4986C22semidirect
43. #4982C22semidirect
44. #4981C22semidirect
45. #4980C22semidirect
46. #4979C22semidirect
47. #4978C22semidirect
476. C3#36708semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×#55002
#840652
#840312
#827962
#827912
#821832
#821762
#821552
#821162
#821132
#367083