Finite Group Databasebeta

← #10804482

Subgroups — #10804482

Order 768.

1194 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5724C211yesmaximal, characteristic
2#79039C211yesmaximal, characteristic, complemented
3#78518C211yesmaximal, characteristic, complemented
4C3×#1966C211yesmaximal, characteristic, complemented
5#71789C211yesmaximal, characteristic, complemented
6#70523C211yesmaximal, characteristic, complemented
7#70114C212yesmaximal
8#70113C212yesmaximal, complemented
9#70099C212yesmaximal, complemented
10#70098C212yesmaximal, complemented
11#69878C211yesmaximal, characteristic, complemented
12#21802—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C22×D3×Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×#947C2211yescharacteristic
17C2×#944C2211yescharacteristic
18#5724—22no—
19#5724—22no—
20#5724—24no—
21C2×#896C414yes—
22C2×#894C414yes—
23C2×#875C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C3×#423C2211yescharacteristic
26C2×C3×#101C2212yes—
27C2×C3×#101C2212yes—
28C3×C4×M16C2211yescharacteristic
29C2×#836C2212yes—
30C2×#836C2212yes—
31C2×#824C2212yes—
32C2×#824C2212yes—
33C2×#823C2212yes—
34C2×#823C2212yes—
35#5126C2212yes—
36#5126C2212yes—
37#5118C2212yes—
38Dic3×M16C2212yes—
39#4924C2211yescharacteristic
40C4×#208C2211yescharacteristic
41#4840—24no—
42#4709—22no—
43#4690—22no—
44C2×#536—31no—
45#2921—31no—
46#2921—31no—
47#1967—31no—
48#1967—32no—
49#1967—31no—
50#1966—32no—
51#1966—31no—
52#1966D311yescharacteristic, complemented
53#1785—32no—
54#1784—32no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C2×#239—22no—
58C2×#239—22no—
59C2×#239—42no—
60C2×#239—42no—
61C2×D3×Q8—22no—
62C2×D3×Q8—22no—
63C2×D3×Q8—42no—
64C2×D3×Q8—42no—
65C22×D12C2×C412yes—
66C22×D12C2311yescharacteristic
67C22×C4×D3C2×C412yes—
68C22×C4×D3C2×C412yes—
69C22×C4×D3C2311yescharacteristic
70C22×Dic6C2×C412yes—
71C22×Dic6C2311yescharacteristic
72C2×C3×M16C2312yes—
73C2×C3×M16C2312yes—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—22no—
77C2×C3×C4⋊C4C2×C414yes—
78C2×C3×C42C2311yescharacteristic
79#947—22no—
80#947—22no—
81#944—22no—
82#944—22no—
83C2×(C2×D3)⋊C4C2×C414yes—
84C2×(C2×D3)⋊C4C2×C414yes—
85C2×(C2×D3)⋊C4C2×C412yes—
86C2×Dic3⋊C4C2×C414yes—
87C2×Dic3⋊C4C2×C412yes—
88C2×C4×Dic3C2311yescharacteristic
89C2×#208C2312yes—
90C2×#208C2312yes—
91#896—28no—
92#896—28no—
93#894—28no—
94#894—28no—
95#875—21no—
96#875—21no—
97#875—22no—
98#872D412yes—
99#872D412yes—
100#872D414yes—
101C3×C4⋊C8—22no—
102C3×#101—42no—
103C3×#101—42no—
104C3×#97—21no—
105C3×C4×C8—21no—
106#836—42no—
107#836—42no—
108#824—42no—
109#824—42no—
110#823—42no—
111#823—42no—
112(C2×D3)⋊C8—24no—
113(C2×D3)⋊C8—24no—
114#815—22no—
115Dic3⋊C8—24no—
116C8×Dic3—22no—
117C4⋊(C3⋊C8)—22no—
118#803—21no—
119C4×C3⋊C8—21no—
120C22×#41—31no—
121C23×Q8—31no—
122#536—64no—
123#536—62no—
124#536—62no—
125C2×C4⋊Q8—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C2×#124—31no—
128C2×#124—31no—
129C2×#123—34no—
130C2×C22⋊Q8—34no—
131C2×#117—31no—
132#433—62no—
133#433—62no—
134#433—64no—
135#423—32no—
136#423—31no—
137#423C2×D311yescharacteristic
138C2×#101—32no—
139C2×#101—32no—
140C2×#101—32no—
141C2×#101—32no—
142C2×#101C2×D312yes—
143C2×#101C2×D312yes—
144C2×#100—32no—
145C2×#100—32no—
146#407—32no—
147#407—32no—
148C4×M16—32no—
149C4×M16—31no—
150C4×M16C2×D311yescharacteristic
151C23×D3C22×C412yes—
152C23×D3C22×C411yescharacteristic
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C22×C3×C4C22×C412yes—
162C22×C3×C4C22×C412yes—
163C22×C3×C4C22×C412yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3C22×C412yes—
166C22×Dic3C22×C411yescharacteristic
167C22×Dic3C22×C412yes—
168#239—44no—
169#239—44no—
170#239—48no—
171D3×Q8—44no—
172D3×Q8—44no—
173D3×Q8—48no—
174C2×D12—24no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—44no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—24no—
183C2×C4×D3—24no—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3C2×D412yes—
187C2×C4×D3C2×D412yes—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—24no—
193C2×Dic6—44no—
194C3×M16—22no—
195C3×M16—22no—
196C3×M16—22no—
197C3×M16—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D412yes—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4C22⋊C414yes—
209(C2×D3)⋊C4C22⋊C414yes—
210Dic3⋊C4—28no—
211Dic3⋊C4C22⋊C414yes—
212Dic3⋊C4C22⋊C414yes—
213C4×Dic3C2×D412yes—
214C4×Dic3C2×D412yes—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×#41—62no—
222C2×#41—62no—
223C2×#41—122no—
224C2×#41—122no—
225C22×Q8—32no—
226C22×Q8—31no—
227C22×Q8—62no—
228C22×Q8—62no—
229C22×Q8—122no—
230C22×Q8—122no—
231C22×Q8C22×D311yescharacteristic
232C22×Q8C22×D311yescharacteristic
233C22×D4—32no—
234C22×D4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—32no—
238C2×M16—32no—
239C2×M16—32no—
240C2×M16C22×D312yes—
241C2×M16C22×D312yes—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—22no—
245C4⋊Q8—62no—
246C4⋊Q8—62no—
247#124—62no—
248#124—62no—
249#124—61no—
250#124—61no—
251#124—62no—
252#123—68no—
253#123—68no—
254C22⋊Q8—68no—
255C22⋊Q8—68no—
256#117—34no—
257#117—32no—
258#117—32no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4C4×D314yes—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—34no—
264C2×C22⋊C4—34no—
265C2×C42—31no—
266C2×C42C22×D311yescharacteristic
267C4⋊C8—22no—
268C4⋊C8—64no—
269C4⋊C8—62no—
270#101—42no—
271#101—42no—
272#101—122no—
273#101—122no—
274#101—122no—
275#101—122no—
276#100—122no—
277#100—122no—
278C22⋊C8—64no—
279C22⋊C8—64no—
280#97—21no—
281#97—62no—
282#97—61no—
283C4×C8—21no—
284C4×C8—62no—
285C4×C8—61no—
286C23×C3C23×C411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—24no—
290C22×D3—42no—
291C22×D3—44no—
292C22×D3C2×C22⋊C412yes—
293C3×Q8—41no—
294C3×Q8—41no—
295C3×Q8—42no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—42no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—42no—
304C2×C3×C4C2×C22⋊C412yes—
305C2×C3×C4C2×C22⋊C412yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4H(5,𝔽2)12yes—
309C2×C3×C4H(5,𝔽2)12yes—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—42no—
314C2×Dic3C2×C22⋊C414yes—
315C2×Dic3C2×C22⋊C412yes—
316D12—48no—
317D12—41no—
318D12—41no—
319D12—42no—
320C4×D3—22no—
321C4×D3—22no—
322C4×D3—24no—
323C4×D3—48no—
324C4×D3—48no—
325Dic6—48no—
326Dic6—41no—
327Dic6—41no—
328Dic6—42no—
329C3×C8—42no—
330C3×C8—42no—
331C3⋊C8—42no—
332C3⋊C8—42no—
333C24—31no—
334C24—32no—
335#41—124no—
336#41—124no—
337#41—128no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—64no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—124no—
352C2×Q8—124no—
353C2×Q8—124no—
354C2×Q8—128no—
355C2×D4—64no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—124no—
361C22×C4—31no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—64no—
367C22×C4—64no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—124no—
371C22×C4—124no—
372C22×C4C2×C4×D312yes—
373C22×C4C2×C4×D312yes—
374C22×C4C2×C4×D312yes—
375C22×C4C23×D311yescharacteristic
376M16—22no—
377M16—22no—
378M16—22no—
379M16—22no—
380M16—62no—
381M16—62no—
382M16—62no—
383M16—62no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—34no—
391C4⋊C4—34no—
392C4⋊C4—68no—
393C22⋊C4—34no—
394C22⋊C4—34no—
395C22⋊C4—68no—
396C22⋊C4—64no—
397C22⋊C4—64no—
398C42—32no—
399C42—32no—
400C42D3×D411yescharacteristic
401C42D3×D411yescharacteristic
402C42D3×D412yes—
403C22×C3—22no—
404C22×C3C22×C22⋊C411yescharacteristic
405C22×C3#51812yes—
406C2×D3—22no—
407C2×D3—24no—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—44no—
411C2×D3—84no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4#53511yescharacteristic
417C3×C4#53511yescharacteristic
418C3×C4#53512yes—
419Dic3—22no—
420Dic3—24no—
421Dic3—42no—
422C23—32no—
423C23—62no—
424C23—62no—
425C23—64no—
426C23—122no—
427C23—124no—
428C23C22×C4×D311yescharacteristic
429Q8—41no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434Q8—42no—
435Q8—128no—
436Q8—121no—
437Q8—121no—
438Q8—122no—
439D4—128no—
440D4—121no—
441D4—121no—
442D4—122no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—34no—
448C2×C4—32no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—64no—
456C2×C4—128no—
457C2×C4—128no—
458C2×C4—122no—
459C2×C4D3×C22⋊C412yes—
460C2×C4D3×C22⋊C412yes—
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4#100412yes—
464C2×C4#100912yes—
465C8—42no—
466C8—42no—
467C8—122no—
468C8—122no—
469C2×C3—22no—
470C2×C3#230911yescharacteristic
471C2×C3#288912yes—
472D3—42no—
473D3—82no—
474C22—22no—
475C22—64no—
476C22—62no—
477C22—124no—
478C22—124no—
479C22—122no—
480C22—244no—
481C22C2×D3×C22⋊C411yescharacteristic
482C22#554512yes—
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—64no—
488C4—62no—
489C4—122no—
490C4#560712yes—
491C4#567411yescharacteristic
492C4#571511yescharacteristic
493C3#2180211yescharacteristic, complemented, Sylow
494C2—22no—
495C2—122no—
496C2—242no—
497C2#7642811yescharacteristic
498C2#7870612yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1966C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#239C22
cosocleC22×C3×C4C24
hypercenter#1966D3
hypocenterC3#21802
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
2. #79039C2semidirect
3. #78518C2semidirect
4. C3×#1966C2semidirect
5. #71789C2semidirect
6. #70523C2semidirect
8. #70113C2semidirect
9. #70099C2semidirect
10. #70098C2semidirect
11. #69878C2semidirect
52. #1966D3semidirect
493. C3#21802semidirect

Maximal subgroups

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

subgroupindex
C2×#57242
#790392
#785182
C3×#19662
#717892
#705232
#701142
#701132
#700992
#700982
#698782
#218023