Finite Group Databasebeta

← #10818114

Subgroups — #10818114

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4897C211yesmaximal, characteristic
2#79178C211yesmaximal, characteristic, complemented
3#79167C211yesmaximal, characteristic, complemented
4#78560C212yesmaximal, complemented
5C3×#1773C211yesmaximal, characteristic, complemented
6#70700C211yesmaximal, characteristic, complemented
7#69892C212yesmaximal, complemented
8#69839C211yesmaximal, characteristic, complemented
9#69825C211yesmaximal, characteristic, complemented
10#68776C212yesmaximal, complemented
11#68758C212yesmaximal, complemented
12#23048—31nomaximal, Sylow
13C2×#908C2212yes—
14C22×D24C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5523C2212yes—
17#5521C2212yes—
18C2×C3×#108C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#405C2212yes—
21C2×C3×#97C2211yescharacteristic
22C2×#823C2212yes—
23#5121C2212yes—
24#5113C2211yescharacteristic
25C2×#821C2212yes—
26C2×#817C2211yescharacteristic
27#5101C2211yescharacteristic
28#5001C2212yes—
29#4995C2212yes—
30#4939C2212yes—
31C2×#805C2211yescharacteristic
32#4897C418yescomplemented
33#4881—22no—
34#4872—22no—
35#4869—21no—
36#4869—21no—
37#4860—21no—
38D3×#106—21no—
39#4711—22no—
40#4710—22no—
41C4×D24—22no—
42C4×#204—22no—
43C3×#339C2212yes—
44#4554C2212yes—
45#4552C2211yescharacteristic
46#4523C2211yescharacteristic
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2912—31no—
51#1972—31no—
52#1951—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1773D311yescharacteristic, complemented
57#1765—31no—
58C2×#235C2311yescharacteristic
59C22×D12C2312yes—
60C2×C3×M16C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#117C2312yes—
64C2×C3×C42C2311yescharacteristic
65C2×(C2×D3)⋊C4—42no—
66#926C2311yescharacteristic
67C2×C4⋊Dic3C2312yes—
68C2×#208C2311yescharacteristic
69C2×#208C2311yescharacteristic
70#908—24no—
71#908C2×C418yescomplemented
72C2×D24—22no—
73C2×D24—21no—
74C2×D24—21no—
75C2×D24—22no—
76C2×D24C2×C414yescomplemented
77C2×#204—22no—
78#901—22no—
79#901C2×C414yescomplemented
80C2×ΓL(1,𝔽25)—21no—
81C2×D3×C8—21no—
82#893—22no—
83#893—22no—
84#892—22no—
85D3×C4⋊C4—22no—
86#67⋊C4—42no—
87D3×C22⋊C4—42no—
88C4×D12—22no—
89C4×D12—22no—
90C4×D12—22no—
91C4×Dic6—22no—
92C3×#108C2×C414yescomplemented
93C3×#107D412yes—
94C3×#106D411yescharacteristic
95C3×#106D411yescharacteristic
96C3×#97—22no—
97C3×#97—22no—
98C3×C4×C8—22no—
99#823C2×C418yescomplemented
100#821—22no—
101#821—22no—
102#821D412yes—
103#818—22no—
104#817—21no—
105#817—21no—
106#817—22no—
107#816D412yes—
108#815—21no—
109C8×Dic3—21no—
110#809—22no—
111#809—22no—
112#808—22no—
113#807—22no—
114#805—22no—
115C2×AGL(1,ℤ/8)—32no—
116C22×D8—31no—
117C2×#131—31no—
118#520—32no—
119#518—32no—
120#472—38no—
121#442—62no—
122#442—61no—
123#442—61no—
124#440—62no—
125C4×SD16—62no—
126C4×D8—62no—
127C4×D8—62no—
128C2×#108C2×D311yescharacteristic
129#428—32no—
130#428—32no—
131#428—62no—
132#427—61no—
133#427C2×D311yescharacteristic
134C2×#106—31no—
135C2×#106—61no—
136C2×#104—31no—
137#418—32no—
138#416—31no—
139C2×#102—32no—
140C2×#100—32no—
141#405—31no—
142#405C2×D312yes—
143C2×#97C2×D311yescharacteristic
144#390—32no—
145#343—32no—
146#341—31no—
147#339—31no—
148#339C2×D312yes—
149C23×D3—22no—
150C22×C3×C4—21no—
151C22×C3×C4C2411yescharacteristic
152C2×#67—21no—
153C22×Dic3—22no—
154#235—21no—
155#235—21no—
156#235C22×C412yes—
157#235#4112yes—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12C22×C414yes—
164C2×D12C2×D411yescharacteristic
165C2×C4×D3—21no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×Dic6C2×D411yescharacteristic
170C3×M16—22no—
171C3×M16C22×C414yes—
172C2×C3×C8—22no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8C22×C412yes—
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D412yes—
179C2×C3×C8#4112yes—
180C3×C4⋊C4C2×D412yes—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—22no—
183C3×C42—22no—
184C3×C42—21no—
185C3×C42—21no—
186#217—21no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—84no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3C2×D412yes—
194Dic3⋊C4—42no—
195C4×Dic3—21no—
196C4×Dic3—42no—
197#208—21no—
198#208—21no—
199#208—21no—
200#208C22×C412yes—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203D24—28no—
204D24—24no—
205D24—24no—
206#204—28no—
207ΓL(1,𝔽25)—24no—
208D3×C8—24no—
209C2×#41—31no—
210C22×D4—32no—
211AGL(1,ℤ/8)—38no—
212AGL(1,ℤ/8)—64no—
213C2×SD16—62no—
214C2×D8—34no—
215C2×D8—62no—
216C2×D8—61no—
217C2×D8—61no—
218C2×D8—62no—
219#131—34no—
220#131—62no—
221C2×M16—31no—
222C2×M16—31no—
223C2×M16—61no—
224C2×M16C22×D312yes—
225C22×C8—61no—
226C22×C8C22×D311yescharacteristic
227C22×C8C22×D311yescharacteristic
228C4×Q8—62no—
229C4×D4—62no—
230C4×D4—62no—
231C4×D4—62no—
232C4×D4—62no—
233C4×D4—62no—
234C4×D4—122no—
235#117—31no—
236#117—62no—
237#117C22×D312yes—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—62no—
240C2×C22⋊C4—122no—
241C2×C22⋊C4—122no—
242C2×C42C22×D311yescharacteristic
243#108C4×D314yescomplemented
244#107—62no—
245#107—62no—
246#107D1212yes—
247#106—62no—
248#106—61no—
249#106—61no—
250#106—62no—
251#106D1211yescharacteristic
252#106D1211yescharacteristic
253#104—62no—
254#103—32no—
255#102—32no—
256#102—62no—
257#102—62no—
258#102—62no—
259#102—62no—
260#100—38no—
261#97—22no—
262#97—22no—
263#97—61no—
264C4×C8—22no—
265C4×C8—61no—
266C23×C3H(5,𝔽2)11yescharacteristic
267C22×D3—21no—
268C22×D3—24no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—82no—
272C22×D3—82no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4C23×C411yescharacteristic
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C42−1+411yescharacteristic
283#67—22no—
284#67—22no—
285C2×Dic3—21no—
286C2×Dic3—22no—
287C2×Dic3—42no—
288C2×Dic3—82no—
289D12—24no—
290D12—24no—
291D12—21no—
292D12—42no—
293D12—42no—
294D12C4×D412yes—
295C4×D3—22no—
296C4×D3—22no—
297C4×D3—162no—
298C4×D3—162no—
299C4×D3—162no—
300Dic6—21no—
301Dic6C4×D412yes—
302C3×C8—24no—
303C3×C8—22no—
304C3×C8C4×D412yes—
305C3×C8C4×D412yes—
306C3⋊C8—22no—
307C3⋊C8—41no—
308C24—62no—
309#41—32no—
310#41—32no—
311#41—61no—
312#41—61no—
313C2×Q8—31no—
314C2×D4—34no—
315C2×D4—31no—
316C2×D4—61no—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C22×C4—21no—
323C22×C4—61no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C23×D311yescharacteristic
329SD16—68no—
330D8—68no—
331D8—64no—
332D8—64no—
333M16—22no—
334M16—32no—
335M16—61no—
336M16—64no—
337M16—61no—
338M16—61no—
339M16C2×C4×D314yes—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—61no—
344C2×C8—64no—
345C2×C8—61no—
346C2×C8C2×C4×D312yes—
347C2×C8C2×D1211yescharacteristic
348C2×C8C2×D1211yescharacteristic
349C2×C8#23512yes—
350C2×C8D3×D412yes—
351C4⋊C4—32no—
352C4⋊C4—62no—
353C4⋊C4—62no—
354C4⋊C4—122no—
355C4⋊C4C2×D1212yes—
356C4⋊C4C2×D1212yes—
357C22⋊C4—22no—
358C22⋊C4—61no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—244no—
363C42—22no—
364C42—21no—
365C42—21no—
366C42—61no—
367C42—122no—
368C22×C3—22no—
369C22×C3#52011yescharacteristic
370C22×C3D4211yescharacteristic
371C22×C3#55311yescharacteristic
372C2×D3—22no—
373C2×D3—42no—
374C2×D3—44no—
375C2×D3—82no—
376C2×D3—82no—
377C2×D3—84no—
378C2×D3—162no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4C2×C4×D411yescharacteristic
383C3×C4C2×C4×D411yescharacteristic
384C3×C4#54711yescharacteristic
385C3×C4#54811yescharacteristic
386Dic3—22no—
387Dic3—42no—
388Dic3—82no—
389C23—61no—
390C23—64no—
391C23—122no—
392C23—122no—
393C23—242no—
394C23—242no—
395C23#100911yescharacteristic
396Q8—32no—
397Q8—61no—
398D4—32no—
399D4—62no—
400D4—62no—
401D4—64no—
402D4—61no—
403D4—64no—
404D4—122no—
405D4—122no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—61no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—122no—
415C2×C4—242no—
416C2×C4—482no—
417C2×C4—482no—
418C2×C4—482no—
419C2×C4C22×C4×D311yescharacteristic
420C2×C4C22×D1211yescharacteristic
421C2×C4C2×#23511yescharacteristic
422C2×C4C2×D3×D411yescharacteristic
423C2×C4#100711yescharacteristic
424C2×C4D3×#4111yescharacteristic
425C8—22no—
426C8—24no—
427C8—62no—
428C8—121no—
429C8C4×D1212yes—
430C8C4×D1212yes—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3#236111yescharacteristic
434C2×C3#328812yes—
435D3—42no—
436D3—84no—
437C22—22no—
438C22—62no—
439C22—122no—
440C22—124no—
441C22—244no—
442C22—242no—
443C22—242no—
444C22—482no—
445C22#553011yescharacteristic
446C22D4×D1211yescharacteristic
447C22#557411yescharacteristic
448C4—42no—
449C4—42no—
450C4—62no—
451C4—81no—
452C4—122no—
453C4—242no—
454C4C2×C4×D1211yescharacteristic
455C4#550711yescharacteristic
456C4C4×D3×D411yescharacteristic
457C4#557611yescharacteristic
458C3#2304811yescharacteristic, complemented, Sylow
459C2—21no—
460C2—21no—
461C2—122no—
462C2—244no—
463C2#7377411yescharacteristic
464C2#7990012yes—
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC22×C3D42
prime radicalC2×#4897C2
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #79178C2semidirect
3. #79167C2semidirect
4. #78560C2semidirect
5. C3×#1773C2semidirect
6. #70700C2semidirect
7. #69892C2semidirect
8. #69839C2semidirect
9. #69825C2semidirect
10. #68776C2semidirect
11. #68758C2semidirect
32. #4897C4semidirect
56. #1773D3semidirect
71. #908C2×C4semidirect
76. C2×D24C2×C4semidirect
79. #901C2×C4semidirect
92. C3×#108C2×C4semidirect
99. #823C2×C4semidirect
243. #108C4×D3semidirect
458. C3#23048semidirect

Maximal subgroups

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

subgroupindex
C2×#48972
#791782
#791672
#785602
C3×#17732
#707002
#698922
#698392
#698252
#687762
#687582
#230483