Finite Group Databasebeta

← #10802089

Subgroups — #10802089

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
1C3×#2875C211yesmaximal, characteristic, complemented
2C2×#5142C211yesmaximal, characteristic
3C4×#949C212yesmaximal, complemented
4C2×#4711C211yesmaximal, characteristic, complemented
5#72018C212yesmaximal, complemented
6#72015C211yesmaximal, characteristic, complemented
7#69903C211yesmaximal, characteristic, complemented
8#69893C211yesmaximal, characteristic, complemented
9#69892C212yesmaximal, complemented
10C4×#823C212yesmaximal, complemented
11#69840C211yesmaximal, characteristic, complemented
12#21555—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16C2×#949C2212yes—
17C22×D24C2211yescharacteristic
18C2×C4×D12C2212yes—
19C3×#405—21no—
20C3×#405—21no—
21C3×#405C2211yescharacteristic
22C3×#405C2211yescharacteristic
23C3×C4×M16—21no—
24C3×C4×M16C2211yescharacteristic
25C2×C3×#97—21no—
26C2×C3×#97C2211yescharacteristic
27#5232C2212yes—
28#5142C418yescomplemented
29C2×#823C2212yes—
30C2×#823C2212yes—
31C2×#821C2212yes—
32C2×#819C2211yescharacteristic
33C2×#819C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#5026—22no—
36C4×#215—22no—
37#5014—22no—
38C4×#213—22no—
39#5001C2212yes—
40#4995C2212yes—
41C4×#208C2212yes—
42#4711—21no—
43#4711—21no—
44#4711—21no—
45#4711—21no—
46#4554C2212yes—
47#4554C2212yes—
48#4552C2212yes—
49C2×#472—31no—
50C4×AGL(1,ℤ/8)—32no—
51C2×#442—31no—
52#2875D311yescharacteristic, complemented
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×D12C2312yes—
62C22×C4×D3—22no—
63C3×#131—22no—
64C3×#131—22no—
65C3×#131—22no—
66C3×#131C2×C414yescomplemented
67C2×C3×M16—21no—
68C2×C3×M16—21no—
69C2×C3×M16C2311yescharacteristic
70C2×C3×M16C2311yescharacteristic
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C3×C4×Q8D412yes—
76C3×C4×D4—22no—
77C3×C4×D4D412yes—
78C3×#117—21no—
79C3×#117C2311yescharacteristic
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82#949—24no—
83#949C2×C418yescomplemented
84C2×#215—22no—
85C2×#213—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×C4⋊Dic3C2312yes—
88C2×#208C2312yes—
89C2×#208C2312yes—
90C2×D24—21no—
91C2×D24—21no—
92C2×D24—22no—
93C2×D24C2×C414yescomplemented
94C4×D12—22no—
95C4×D12—22no—
96C4×D12—22no—
97C4×D12—22no—
98C3×#97—21no—
99C3×#97—21no—
100C3×#97—21no—
101C3×#97D411yescharacteristic
102C3×#97D411yescharacteristic
103C3×#97D411yescharacteristic
104C3×#97D411yescharacteristic
105C3×C4×C8—21no—
106C3×C4×C8—21no—
107C3×C4×C8—21no—
108#835—22no—
109#832—22no—
110#823—24no—
111#823C2×C418yescomplemented
112#821—22no—
113#821—22no—
114#819—22no—
115#819C2×C414yescomplemented
116#817—21no—
117#817—21no—
118#809—22no—
119#809—22no—
120#808—22no—
121#807—22no—
122#803—22no—
123C4×C3⋊C8—22no—
124C2×AGL(1,ℤ/8)—32no—
125C22×D8—31no—
126C2×#131C2×D311yescharacteristic
127C2×#131C2×D311yescharacteristic
128C4×#41C2×D311yescharacteristic
129C2×C4×D4—32no—
130#472—38no—
131#442—62no—
132#442—61no—
133#442—61no—
134#442—61no—
135#442—61no—
136#440—62no—
137C4×SD16—62no—
138C4×D8—62no—
139C2×#108—31no—
140C2×#108—31no—
141#428—32no—
142C2×#106—31no—
143#418—32no—
144#417—32no—
145C2×#102—32no—
146C2×#100—32no—
147C2×#100—32no—
148#405—21no—
149#405—21no—
150#405C2×D311yescharacteristic
151#405C2×D311yescharacteristic
152C4×M16—21no—
153C4×M16—32no—
154C4×M16C2×D311yescharacteristic
155C2×#97—21no—
156C2×#97C2×D311yescharacteristic
157#343—32no—
158#343—32no—
159#341—32no—
160C23×D3—22no—
161C3×#41—21no—
162C3×#41—21no—
163C3×#41C22×C412yes—
164C3×#41#4112yes—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4C2×D411yescharacteristic
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12C22×C414yes—
179C2×C4×D3—42no—
180C2×C4×D3—42no—
181C2×C4×D3—84no—
182C3×M16—24no—
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—24no—
186C3×M16C22×C412yes—
187C3×M16#4112yes—
188C2×C3×C8—24no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—24no—
194C2×C3×C8C22×C412yes—
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8#4112yes—
200C3×C4⋊C4—22no—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—21no—
203C3×C22⋊C4—22no—
204C3×C42—22no—
205C3×C42—21no—
206C3×C42C2×D412yes—
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209#215—28no—
210#213—28no—
211(C2×D3)⋊C4—42no—
212(C2×D3)⋊C4—42no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208C22×C414yes—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221D24—24no—
222D24—24no—
223C2×#41C22×D311yescharacteristic
224C22×D4—32no—
225C23×C4—62no—
226AGL(1,ℤ/8)—38no—
227AGL(1,ℤ/8)—64no—
228C2×SD16—62no—
229C2×D8—34no—
230C2×D8—62no—
231C2×D8—62no—
232C2×D8—61no—
233C2×D8—61no—
234#131—22no—
235#131—22no—
236#131—22no—
237#131C4×D314yescomplemented
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—32no—
241C2×M16—32no—
242C2×M16C22×D311yescharacteristic
243C2×M16C22×D311yescharacteristic
244C22×C8—21no—
245C22×C8—21no—
246C22×C8C22×D311yescharacteristic
247C22×C8C22×D311yescharacteristic
248C4×Q8D1212yes—
249C4×D4—22no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4D1212yes—
255#117—21no—
256#117C22×D311yescharacteristic
257C2×C4⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C42—21no—
260C2×C42C22×D311yescharacteristic
261#108—34no—
262#108—62no—
263#107—62no—
264#106—61no—
265#106—61no—
266#106—62no—
267#103—62no—
268#102—62no—
269#102—62no—
270#102—62no—
271#102—62no—
272#102—62no—
273#100—38no—
274#100—64no—
275#97—21no—
276#97—21no—
277#97—21no—
278#97—62no—
279#97#6711yescharacteristic
280#97#6711yescharacteristic
281#97#6711yescharacteristic
282#97#6711yescharacteristic
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—62no—
287C23×C3—21no—
288C23×C3C2×#4111yescharacteristic
289C22×D3—24no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—82no—
293C22×D3—82no—
294C3×Q8—21no—
295C3×Q8C4×D412yes—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—21no—
299C3×D4C4×D412yes—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—42no—
305C2×C3×C4—82no—
306C2×C3×C4#12112yes—
307C2×C3×C4#12112yes—
308C2×C3×C4C23×C411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×Dic3—42no—
315C2×Dic3—82no—
316D12—24no—
317D12—24no—
318D12—42no—
319D12—42no—
320C4×D3—84no—
321C4×D3—84no—
322C3×C8—22no—
323C3×C8—21no—
324C3×C8—22no—
325C3×C8—21no—
326C3×C8C4×D412yes—
327C3×C8C4×D412yes—
328C3⋊C8—24no—
329C3⋊C8—42no—
330C24—62no—
331#41—21no—
332#41—21no—
333#41C2×C4×D312yes—
334#41#23512yes—
335C2×Q8C2×D1211yescharacteristic
336C2×D4—21no—
337C2×D4—34no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4C2×D1211yescharacteristic
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—62no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4—244no—
350C22×C4#23512yes—
351C22×C4C23×D311yescharacteristic
352SD16—68no—
353D8—68no—
354D8—64no—
355D8—64no—
356M16—24no—
357M16—21no—
358M16—21no—
359M16—24no—
360M16—34no—
361M16—62no—
362M16—62no—
363M16—62no—
364M16C2×C4×D312yes—
365M16#23512yes—
366C2×C8—24no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—24no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8C2×C4×D312yes—
375C2×C8#23512yes—
376C2×C8C2×#6711yescharacteristic
377C2×C8C2×#6711yescharacteristic
378C2×C8C2×#6711yescharacteristic
379C2×C8C2×#6711yescharacteristic
380C4⋊C4—22no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4C2×D1212yes—
384C22⋊C4—21no—
385C22⋊C4—22no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C42—22no—
389C42—21no—
390C42C2×D1212yes—
391C42C2×#6711yescharacteristic
392C42C2×#6711yescharacteristic
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—42no—
396C22×C3C4×#4111yescharacteristic
397C22×C3C2×#12111yescharacteristic
398C22×C3#52911yescharacteristic
399C2×D3—44no—
400C2×D3—84no—
401C2×D3—82no—
402C2×D3—82no—
403C2×D3—84no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—42no—
408C3×C4C2×C4×D411yescharacteristic
409C3×C4C2×C4×D411yescharacteristic
410C3×C4#52511yescharacteristic
411C3×C4#53211yescharacteristic
412Dic3—82no—
413C23—21no—
414C23—64no—
415C23—122no—
416C23—122no—
417C23—242no—
418C23—242no—
419C23C2×#23511yescharacteristic
420Q8—21no—
421Q8C4×D1212yes—
422D4—22no—
423D4—22no—
424D4—21no—
425D4—64no—
426D4—64no—
427D4—122no—
428D4—122no—
429D4C4×D1212yes—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—122no—
437C2×C4—244no—
438C2×C4—244no—
439C2×C4—242no—
440C2×C4#93012yes—
441C2×C4#93012yes—
442C2×C4C22×C4×D311yescharacteristic
443C2×C4C22×D1211yescharacteristic
444C2×C4C2×#23511yescharacteristic
445C2×C4C2×#23511yescharacteristic
446C2×C4C2×#23511yescharacteristic
447C2×C4C22×#6711yescharacteristic
448C8—22no—
449C8—21no—
450C8—22no—
451C8—21no—
452C8—64no—
453C8—122no—
454C8C4×#6712yes—
455C8C4×#6712yes—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—42no—
459C2×C3C4×#12111yescharacteristic
460C2×C3#306812yes—
461D3—84no—
462C22—22no—
463C22—22no—
464C22—42no—
465C22—124no—
466C22—244no—
467C22—244no—
468C22—242no—
469C22—242no—
470C22C4×#23511yescharacteristic
471C22#547911yescharacteristic
472C22C2×#93011yescharacteristic
473C4—22no—
474C4—22no—
475C4—22no—
476C4—42no—
477C4—242no—
478C4C2×C4×D1211yescharacteristic
479C4#547711yescharacteristic
480C4C2×C4×#6711yescharacteristic
481C4#579111yescharacteristic
482C3#2155511yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—42no—
486C2—244no—
487C2C4×#93011yescharacteristic
488C2#8381412yes—
subgroupG/H
centerC22C2×#930
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2875C2
socleC22×C3C2×#121
prime radicalC2×#5142C2
cosocleC22×C3×C4C24
hypercenter#2875D3
hypocenterC3#21555
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2875

Complements

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

NH ≅ G/Ntype
1. C3×#2875C2semidirect
3. C4×#949C2semidirect
4. C2×#4711C2semidirect
5. #72018C2semidirect
6. #72015C2semidirect
7. #69903C2semidirect
8. #69893C2semidirect
9. #69892C2semidirect
10. C4×#823C2semidirect
11. #69840C2semidirect
28. #5142C4semidirect
52. #2875D3semidirect
66. C3×#131C2×C4semidirect
83. #949C2×C4semidirect
93. C2×D24C2×C4semidirect
111. #823C2×C4semidirect
115. #819C2×C4semidirect
237. #131C4×D3semidirect
482. C3#21555semidirect

Maximal subgroups

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

subgroupindex
C3×#28752
C2×#51422
C4×#9492
C2×#47112
#720182
#720152
#699032
#698932
#698922
C4×#8232
#698402
#215553