Finite Group Databasebeta

← #10802090

Subgroups — #10802090

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#472C211yesmaximal, characteristic
2Dic3×AGL(1,ℤ/8)C212yesmaximal, complemented
3C2×#5152C211yesmaximal, characteristic, complemented
4#83782C211yesmaximal, characteristic, complemented
5#72091C212yesmaximal, complemented
6#72041C211yesmaximal, characteristic, complemented
7#72015C211yesmaximal, characteristic, complemented
8#70700C211yesmaximal, characteristic, complemented
9#70692C211yesmaximal, characteristic, complemented
10#70072C212yesmaximal, complemented
11Dic3×#100C212yesmaximal, complemented
12#21555—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16Dic3×#41C2211yescharacteristic
17C2×#948C2211yescharacteristic
18C2×D4×Dic3C2212yes—
19C3×#472C418yescomplemented
20C2×C3×#108C2211yescharacteristic
21C2×C3×#100C2212yes—
22#5232C2212yes—
23C2×#833C2212yes—
24#5215C2212yes—
25C2×#832C2212yes—
26#5164—22no—
27Dic3×SD16—22no—
28#5152—22no—
29#5152—21no—
30#5152—21no—
31#5152—21no—
32#5152—21no—
33Dic3×D8—22no—
34#5122—21no—
35#5122C2211yescharacteristic
36#5121C2212yes—
37Dic3×M16—21no—
38Dic3×M16C2212yes—
39Dic3×M16C2211yescharacteristic
40C2×#819C2211yescharacteristic
41C2×#817C2211yescharacteristic
42#5101—21no—
43#5101C2211yescharacteristic
44C2×#815—21no—
45C2×#815C2211yescharacteristic
46#4558C2212yes—
47#4554C2212yes—
48#4554C2212yes—
49C2×#472D311yescharacteristic
50C4×AGL(1,ℤ/8)—32no—
51C2×#442—31no—
52#2875—31no—
53#1978—31no—
54#1951—31no—
55#1931—32no—
56#1907—32no—
57#1815—31no—
58#1778—31no—
59C4×#100—32no—
60C2×C3×#41C2311yescharacteristic
61C22×C3×D4C2312yes—
62C23×Dic3—22no—
63C3×AGL(1,ℤ/8)—24no—
64C3×AGL(1,ℤ/8)C2×C418yescomplemented
65C2×C3×SD16—22no—
66C2×C3×D8—22no—
67C2×C3×D8—21no—
68C2×C3×D8—21no—
69C2×C3×D8—22no—
70C2×C3×D8C2×C414yescomplemented
71C3×#131—22no—
72C3×#131C2×C414yescomplemented
73C2×C3×M16—21no—
74C2×C3×M16C2312yes—
75C2×C3×M16C2311yescharacteristic
76C22×C3×C8—21no—
77C22×C3×C8C2311yescharacteristic
78C2×#217—22no—
79#948—22no—
80#948—22no—
81Q8×Dic3D412yes—
82D4×Dic3—22no—
83D4×Dic3—22no—
84D4×Dic3—22no—
85D4×Dic3—22no—
86D4×Dic3—22no—
87D4×Dic3D412yes—
88#926—21no—
89#926C2311yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3—21no—
92C2×C4×Dic3C2311yescharacteristic
93C2×#208—21no—
94C2×#208C2312yes—
95C2×#208C2311yescharacteristic
96C22×C3⋊C8—21no—
97C22×C3⋊C8C2311yescharacteristic
98C3×#108C2×C414yescomplemented
99C3×#100C2×C418yescomplemented
100#835—22no—
101#833—24no—
102#832—22no—
103#832—22no—
104#832—22no—
105#832—22no—
106#832—22no—
107#819—22no—
108#818—22no—
109#817—21no—
110#817—21no—
111#817—22no—
112#815—21no—
113#815—22no—
114#815—21no—
115#815—21no—
116#815D411yescharacteristic
117#815D411yescharacteristic
118#815D411yescharacteristic
119#815D411yescharacteristic
120C8×Dic3—21no—
121C8×Dic3—21no—
122C8×Dic3—22no—
123C8×Dic3—21no—
124C2×AGL(1,ℤ/8)C2×D312yes—
125C22×D8C2×D311yescharacteristic
126C2×#131—31no—
127C2×#131C2×D311yescharacteristic
128C4×#41—31no—
129C2×C4×D4—32no—
130#472Dic318yescomplemented
131#442—61no—
132#442—61no—
133#442—61no—
134#442—61no—
135#442—62no—
136#440—62no—
137C4×SD16—62no—
138C4×D8—62no—
139C2×#108—31no—
140C2×#108C2×D311yescharacteristic
141#428—32no—
142C2×#106—31no—
143#418—32no—
144#417—32no—
145C2×#102—32no—
146C2×#100—32no—
147C2×#100C2×D312yes—
148#405—31no—
149#405—31no—
150#405—61no—
151#405—61no—
152C4×M16—31no—
153C4×M16—32no—
154C4×M16—61no—
155C2×#97—31no—
156C2×#97—61no—
157#343—32no—
158#343—32no—
159#341—32no—
160C24×C3—22no—
161C3×#41—21no—
162C3×#41—21no—
163C3×#41C22×C412yes—
164C3×#41#4112yes—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4C22×C414yes—
173C2×C3×D4C2×D411yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—42no—
179C22×Dic3—42no—
180C22×Dic3—84no—
181C22×Dic3#4112yes—
182C3×SD16—28no—
183C3×D8—28no—
184C3×D8—24no—
185C3×D8—24no—
186C3×M16—21no—
187C3×M16—22no—
188C3×M16—24no—
189C3×M16C22×C412yes—
190C3×M16C22×C414yes—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—24no—
194C2×C3×C8C22×C412yes—
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8#4112yes—
198#217—21no—
199#217—22no—
200#217—42no—
201#217—42no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3C2×D412yes—
206C4×Dic3—21no—
207C4×Dic3—22no—
208C4×Dic3C2×D412yes—
209C4×Dic3C2×D411yescharacteristic
210C4×Dic3C2×D411yescharacteristic
211#208—24no—
212#208—21no—
213#208—22no—
214#208—22no—
215#208#4112yes—
216C2×C3⋊C8—24no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223C2×#41C22×D311yescharacteristic
224C22×D4C22×D312yes—
225C23×C4—62no—
226AGL(1,ℤ/8)—24no—
227AGL(1,ℤ/8)C2×Dic318yescomplemented
228C2×SD16—22no—
229C2×D8—22no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—22no—
233C2×D8C2×Dic314yescomplemented
234#131—22no—
235#131—62no—
236#131—62no—
237#131C2×Dic314yescomplemented
238C2×M16—21no—
239C2×M16—31no—
240C2×M16—32no—
241C2×M16—61no—
242C2×M16C22×D311yescharacteristic
243C2×M16C22×D312yes—
244C22×C8—21no—
245C22×C8—31no—
246C22×C8—61no—
247C22×C8C22×D311yescharacteristic
248C4×Q8—32no—
249C4×D4—32no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4—62no—
255#117—31no—
256#117—61no—
257C2×C4⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C42—31no—
260C2×C42—61no—
261#108—62no—
262#108C2×Dic314yescomplemented
263#107—62no—
264#106—62no—
265#106—61no—
266#106—61no—
267#103—62no—
268#102—62no—
269#102—62no—
270#102—62no—
271#102—62no—
272#102—62no—
273#100—64no—
274#100C2×Dic318yescomplemented
275#97—31no—
276#97—31no—
277#97—31no—
278#97—31no—
279#97—61no—
280#97—62no—
281#97—61no—
282#97—61no—
283C4×C8—61no—
284C4×C8—62no—
285C4×C8—61no—
286C4×C8—61no—
287C23×C3—21no—
288C23×C3—24no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—82no—
292C23×C3—82no—
293C23×C3C2×#4111yescharacteristic
294C3×Q8—21no—
295C3×Q8C4×D412yes—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—24no—
299C3×D4—24no—
300C3×D4—21no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4C4×D412yes—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4C23×C411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×Dic3—22no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—82no—
317C2×Dic3—84no—
318C2×Dic3—84no—
319C2×Dic3—82no—
320C2×Dic3#12112yes—
321C2×Dic3#12112yes—
322C3×C8—22no—
323C3×C8—24no—
324C3×C8C4×D412yes—
325C3×C8C4×D412yes—
326C3⋊C8—21no—
327C3⋊C8—22no—
328C3⋊C8—21no—
329C3⋊C8—42no—
330C24—22no—
331#41—21no—
332#41—21no—
333#41#23712yes—
334#41C22×Dic312yes—
335C2×Q8D3×D411yescharacteristic
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4D3×D411yescharacteristic
343C2×D4C22×Dic314yes—
344C22×C4—21no—
345C22×C4—32no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4—244no—
351C22×C4C23×D311yescharacteristic
352SD16—28no—
353D8—28no—
354D8—24no—
355D8—24no—
356M16—21no—
357M16—22no—
358M16—24no—
359M16—32no—
360M16—64no—
361M16—61no—
362M16—62no—
363M16—62no—
364M16C22×Dic312yes—
365M16C22×Dic314yes—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—31no—
370C2×C8—31no—
371C2×C8—64no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—62no—
376C2×C8D3×D411yescharacteristic
377C2×C8D3×D411yescharacteristic
378C2×C8#23712yes—
379C2×C8C22×Dic312yes—
380C4⋊C4—32no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C22⋊C4—62no—
385C22⋊C4—61no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C42—31no—
389C42—32no—
390C42—31no—
391C42—62no—
392C42—61no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—84no—
398C22×C3—84no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3C4×#4111yescharacteristic
402C22×C3C2×#12111yescharacteristic
403C22×C3#52911yescharacteristic
404C3×C4—22no—
405C3×C4C2×C4×D411yescharacteristic
406C3×C4C2×C4×D411yescharacteristic
407C3×C4#52511yescharacteristic
408C3×C4#53211yescharacteristic
409Dic3—22no—
410Dic3—22no—
411Dic3—42no—
412Dic3—82no—
413C23—21no—
414C23—24no—
415C23—42no—
416C23—42no—
417C23—82no—
418C23—82no—
419C23D3×#4111yescharacteristic
420Q8—21no—
421Q8D4×Dic312yes—
422D4—22no—
423D4—22no—
424D4—24no—
425D4—24no—
426D4—21no—
427D4—42no—
428D4—42no—
429D4D4×Dic312yes—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—62no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—242no—
439C2×C4—244no—
440C2×C4—244no—
441C2×C4—242no—
442C2×C4C2×D3×D411yescharacteristic
443C2×C4C2×D3×D411yescharacteristic
444C2×C4C2×#23711yescharacteristic
445C2×C4C2×#23711yescharacteristic
446C2×C4D3×#4111yescharacteristic
447C2×C4C23×Dic311yescharacteristic
448C8—22no—
449C8—24no—
450C8—61no—
451C8—61no—
452C8—62no—
453C8—122no—
454C8D4×Dic312yes—
455C8D4×Dic312yes—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—42no—
459C2×C3—84no—
460C2×C3C4×#12111yescharacteristic
461C2×C3#306812yes—
462C22—22no—
463C22—22no—
464C22—42no—
465C22—44no—
466C22—84no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22#560011yescharacteristic
471C22D3×#12111yescharacteristic
472C22Dic3×#4111yescharacteristic
473C4—22no—
474C4—62no—
475C4—62no—
476C4—122no—
477C4—242no—
478C4#559611yescharacteristic
479C4#560611yescharacteristic
480C4C2×D4×Dic311yescharacteristic
481C4C2×D4×Dic311yescharacteristic
482C3#2155511yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—42no—
486C2—84no—
487C2#8093812yes—
488C2Dic3×#12111yescharacteristic
subgroupG/H
centerC22D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#472C2
socleC22×C3C2×#121
cosocleC22×C3×C4C24
hypercenterC2×#472D3
hypocenterC3#21555
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
2. Dic3×AGL(1,ℤ/8)C2semidirect
3. C2×#5152C2semidirect
4. #83782C2semidirect
5. #72091C2semidirect
6. #72041C2semidirect
7. #72015C2semidirect
8. #70700C2semidirect
9. #70692C2semidirect
10. #70072C2semidirect
11. Dic3×#100C2semidirect
19. C3×#472C4semidirect
64. C3×AGL(1,ℤ/8)C2×C4semidirect
70. C2×C3×D8C2×C4semidirect
72. C3×#131C2×C4semidirect
98. C3×#108C2×C4semidirect
99. C3×#100C2×C4semidirect
130. #472Dic3semidirect
227. AGL(1,ℤ/8)C2×Dic3semidirect
233. C2×D8C2×Dic3semidirect
237. #131C2×Dic3semidirect
262. #108C2×Dic3semidirect
274. #100C2×Dic3semidirect
482. C3#21555semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4722
Dic3×AGL(1,ℤ/8)2
C2×#51522
#837822
#720912
#720412
#720152
#707002
#706922
#700722
Dic3×#1002
#215553