Finite Group Databasebeta

← #10813398

Subgroups — #10813398

Order 768.

1022 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C212yesmaximal, complemented
2C2×#4688C211yesmaximal, characteristic
3#74360C211yesmaximal, characteristic
4#74324C211yesmaximal, characteristic
5C3×#1844C211yesmaximal, characteristic, complemented
6#70580C212yesmaximal, complemented
7#70579C212yesmaximal, complemented
8#70578C212yesmaximal, complemented
9C4×#818C211yesmaximal, characteristic, complemented
10#69568C211yesmaximal, characteristic, complemented
11#69567C211yesmaximal, characteristic, complemented
12#22663—31nomaximal, Sylow
13C22×ΓL(1,𝔽25)C2212yes—
14C2×#893—21no—
15C2×#892C2212yes—
16C2×D3×C4⋊C4C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×C3×#106C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C3×#384C2211yescharacteristic
23C2×(C2×D3)⋊C8C2212yes—
24C2×#818C2211yescharacteristic
25C2×#818C2211yescharacteristic
26#4990—22no—
27#4989—22no—
28#4988—21no—
29#4987C2212yes—
30#4987C2212yes—
31Dic3×C4⋊C4—21no—
32#4972—21no—
33C2×#808C2212yes—
34C2×#807C2212yes—
35C4×C4⋊Dic3C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38#4861—28no—
39#4688C418yescomplemented
40#4551C2211yescharacteristic
41#4551C2211yescharacteristic
42#4529C2212yes—
43#4529C2212yes—
44C2×#436—31no—
45C2×#428—32no—
46#2354—31no—
47D4×C4⋊C4—31no—
48#1948—31no—
49#1947—31no—
50#1854—32no—
51#1853—32no—
52#1844D311yescharacteristic, complemented
53#1809—32no—
54C4×#107—31no—
55C22×D12D411yescharacteristic
56C22×C4×D3—22no—
57C22×C4×D3C2312yes—
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4D412yes—
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×C4⋊Dic3C2311yescharacteristic
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—21no—
79C2×C4×Dic3—22no—
80C22×C3⋊C8C2312yes—
81C2×ΓL(1,𝔽25)—24no—
82C2×ΓL(1,𝔽25)—24no—
83C2×ΓL(1,𝔽25)C2×C418yescomplemented
84#893—44no—
85#892—24no—
86#892—42no—
87D3×C4⋊C4—24no—
88D3×C4⋊C4—42no—
89C4×D12C2×C414yes—
90C4×D12Q814yes—
91C3×#106—24no—
92C3×#106—42no—
93C3×C4×C8C2×C414yescomplemented
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98(C2×D3)⋊C8C2×C418yescomplemented
99#818—22no—
100#818D414yescomplemented
101#808—24no—
102#807—24no—
103C4⋊(C3⋊C8)C2×C414yescomplemented
104C22×M16—32no—
105C2×C4×D4—31no—
106C2×C4×D4—61no—
107C2×#117—32no—
108C22×C4⋊C4—32no—
109#436—38no—
110#428—68no—
111C2×#107—31no—
112C2×#107—31no—
113C2×#107—32no—
114C2×#106—32no—
115C2×#106C2×D312yes—
116C2×#106C2×D312yes—
117C2×C4⋊C8—31no—
118C2×C22⋊C8—32no—
119C2×C4×C8C2×D311yescharacteristic
120#390—61no—
121#389—61no—
122#388—62no—
123#385—62no—
124#384C2×D311yescharacteristic
125#384C2×D311yescharacteristic
126#380—32no—
127#380—32no—
128C4×C4⋊C4—31no—
129C4×C4⋊C4—31no—
130C4×C4⋊C4—61no—
131#340—31no—
132#340—31no—
133#340—32no—
134#340—32no—
135C23×D3C2×D412yes—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3C2×D412yes—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12C4⋊C414yes—
149C2×C4×D3—22no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—44no—
154C2×C4×D3—44no—
155C2×C4×D3—82no—
156C2×C4×D3C22×C414yes—
157C2×C4×D3C2×Q814yes—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8C22×C412yes—
161C2×C3×C8C22×C412yes—
162C2×C3×C8C2×D412yes—
163C2×C3×C8#4112yes—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—81no—
171C3×C4⋊C4—81no—
172C3×C42C22×C412yes—
173C3×C42C2×Q812yes—
174(C2×D3)⋊C4—44no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—82no—
177(C2×D3)⋊C4C4⋊C418yes—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3C4⋊C414yes—
185C4⋊Dic3C2×D412yes—
186C4⋊Dic3C2×D412yes—
187Dic3⋊C4—44no—
188C4×Dic3—44no—
189C4×Dic3—42no—
190C4×Dic3—42no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8C22×C414yes—
194ΓL(1,𝔽25)—216no—
195C22×D4—31no—
196C23×C4—32no—
197C23×C4—62no—
198C2×M16—38no—
199C2×M16—64no—
200C2×M16—64no—
201C22×C8—32no—
202C22×C8C22×D311yescharacteristic
203C22×C8C22×D311yescharacteristic
204C4×D4—34no—
205C4×D4—34no—
206C4×D4—124no—
207#117—64no—
208#117—122no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—122no—
221C2×C4⋊C4C22×D312yes—
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—32no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C42—62no—
228C2×C42—61no—
229C2×C42—61no—
230C2×C42C22×D311yescharacteristic
231#107—34no—
232#107—64no—
233#107—62no—
234#106—24no—
235#106—42no—
236#106—64no—
237C4⋊C8—34no—
238C22⋊C8—38no—
239C4×C8C4×D314yescomplemented
240#95—62no—
241#95—62no—
242#95—61no—
243#95—61no—
244C23×C3C22×D411yescharacteristic
245C22×D3—22no—
246C22×D3—22no—
247C22×D3—22no—
248C22×D3—22no—
249C22×D3—22no—
250C22×D3—22no—
251C22×D3C2×C4⋊C414yes—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4C2×C4⋊C412yes—
261C2×C3×C4C2×C4⋊C412yes—
262C2×C3×C4AGL(1,ℤ/8)12yes—
263C2×C3×C4#13712yes—
264C2×C3×C4C23×C411yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C22×Q811yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268C2×C3×C4H(5,𝔽2)11yescharacteristic
269C2×C3×C42−1+411yescharacteristic
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3C2×C4⋊C414yes—
285D12—28no—
286C4×D3—24no—
287C4×D3—24no—
288C4×D3—24no—
289C4×D3—24no—
290C4×D3—88no—
291C3×C8—22no—
292C3×C8C4×D414yescomplemented
293C3⋊C8—24no—
294C24—32no—
295C2×D4—34no—
296C2×D4—62no—
297C2×D4—62no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—32no—
301C22×C4—34no—
302C22×C4—34no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—124no—
312C22×C4—124no—
313C22×C4—242no—
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4C23×D311yescharacteristic
317M16—616no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—34no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8C2×C4×D312yes—
324C2×C8C2×C4×D312yes—
325C2×C8D3×D412yes—
326C2×C8#23912yes—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—34no—
332C4⋊C4—32no—
333C4⋊C4—32no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4—61no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—61no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—124no—
345C22⋊C4—38no—
346C22⋊C4—124no—
347C22⋊C4—124no—
348C22⋊C4—242no—
349C42—124no—
350C42—122no—
351C42—122no—
352C42C2×C4×D312yes—
353C42D3×Q812yes—
354C22×C3C22×C4⋊C411yescharacteristic
355C22×C3D4211yescharacteristic
356C22×C3#54711yescharacteristic
357C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C3C2×#13711yescharacteristic
359C22×C3#57711yescharacteristic
360C22×C3#57711yescharacteristic
361C2×D3—24no—
362C2×D3—24no—
363C2×D3—24no—
364C2×D3—42no—
365C2×D3—42no—
366C2×D3—42no—
367C2×D3—42no—
368C3×C4—22no—
369C3×C4—82no—
370C3×C4—82no—
371C3×C4#42814yescomplemented
372C3×C4C2×C4×D411yescharacteristic
373C3×C4#52011yescharacteristic
374C3×C4D4×Q811yescharacteristic
375C3×C4#55411yescharacteristic
376Dic3—24no—
377Dic3—42no—
378Dic3—42no—
379Dic3—81no—
380Dic3—81no—
381C23—34no—
382C23—62no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—62no—
387C23—62no—
388C23C2×D3×D411yescharacteristic
389D4—68no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—34no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—64no—
404C2×C4—64no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—248no—
418C2×C4D3×C4⋊C412yes—
419C2×C4D3×C4⋊C412yes—
420C2×C4#91412yes—
421C2×C4#91512yes—
422C2×C4C22×C4×D311yescharacteristic
423C2×C4C2×D3×D411yescharacteristic
424C2×C4#100411yescharacteristic
425C2×C4C2×D3×Q811yescharacteristic
426C2×C4C2×#23911yescharacteristic
427C2×C4#100711yescharacteristic
428C8—22no—
429C8—64no—
430C8#89314yescomplemented
431C2×C3D4×C4⋊C411yescharacteristic
432C2×C3C2×#42811yescharacteristic
433C2×C3#291211yescharacteristic
434C2×C3#327211yescharacteristic
435C2×C3#327811yescharacteristic
436C2×C3#335411yescharacteristic
437C2×C3#335611yescharacteristic
438D3—44no—
439C22—64no—
440C22—64no—
441C22—64no—
442C22—122no—
443C22—122no—
444C22—122no—
445C22—122no—
446C22C2×D3×C4⋊C411yescharacteristic
447C22#571711yescharacteristic
448C22#572611yescharacteristic
449C22C2×#91411yescharacteristic
450C22C2×#91511yescharacteristic
451C22#576211yescharacteristic
452C22#576211yescharacteristic
453C4—22no—
454C4—64no—
455C4—82no—
456C4—82no—
457C4—122no—
458C4—122no—
459C4—241no—
460C4—241no—
461C4#486114yescomplemented
462C4C2×#89311yescharacteristic
463C4#550411yescharacteristic
464C4#572711yescharacteristic
465C4#573011yescharacteristic
466C3#2266311yescharacteristic, complemented, Sylow
467C2—124no—
468C2#7436011yescharacteristic
469C2C2×#486111yescharacteristic
470C2#7911611yescharacteristic
471C2#8177011yescharacteristic
472C2#8177311yescharacteristic
473C2#8190111yescharacteristic
474C2#8190211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1844C2
socleC23×C3C22×D4
prime radicalC22×D12D4
cosocleC22×C3×C4C24
hypercenter#1844D3
hypocenterC3#22663
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
1. C2×#4861C2semidirect
5. C3×#1844C2semidirect
6. #70580C2semidirect
7. #70579C2semidirect
8. #70578C2semidirect
9. C4×#818C2semidirect
10. #69568C2semidirect
11. #69567C2semidirect
39. #4688C4semidirect
52. #1844D3semidirect
83. C2×ΓL(1,𝔽25)C2×C4semidirect
93. C3×C4×C8C2×C4semidirect
98. (C2×D3)⋊C8C2×C4semidirect
100. #818D4semidirect
103. C4⋊(C3⋊C8)C2×C4semidirect
239. C4×C8C4×D3semidirect
292. C3×C8C4×D4semidirect
371. C3×C4#428semidirect
430. C8#893semidirect
461. C4#4861semidirect
466. C3#22663semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
C2×#46882
#743602
#743242
C3×#18442
#705802
#705792
#705782
C4×#8182
#695682
#695672
#226633