Finite Group Databasebeta

← #10801893

Subgroups — #10801893

Order 768.

1170 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5159C211yesmaximal, characteristic
2C2×#5152C211yesmaximal, characteristic, complemented
3C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4#76291C212yesmaximal, complemented
5C3×#1970C211yesmaximal, characteristic, complemented
6#70631C211yesmaximal, characteristic, complemented
7#70605C211yesmaximal, characteristic, complemented
8#70203C212yesmaximal, complemented
9#70152C212yesmaximal, complemented
10#69769C211yesmaximal, characteristic, complemented
11#69644C212yesmaximal, complemented
12#21538—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×#938C2212yes—
15C2×D4×Dic3C2212yes—
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#67⋊C4—22no—
19C2×C42×D3C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#102C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#390C2212yes—
24C2×#832C2212yes—
25C2×#832C2212yes—
26#5159C418yescomplemented
27#5152—24no—
28C2×#817C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×#815C2211yescharacteristic
32#4987C2212yes—
33Dic3×C4⋊C4C2212yes—
34#4956—22no—
35#4945—22no—
36Dic3×C22⋊C4—22no—
37#4928C2212yes—
38C2×C4×C3⋊C8C2211yescharacteristic
39C4×ΓL(1,𝔽25)—24no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#469—31no—
44C2×#442—31no—
45C2×C4×M16—31no—
46C4×#121—32no—
47#1970D311yescharacteristic, complemented
48#1941—31no—
49#1934—32no—
50#1895—32no—
51#1809—31no—
52#1776—31no—
53#1762—32no—
54C22×C3×D4C2312yes—
55C22×#67—22no—
56C23×Dic3—22no—
57C22×C4×D3D412yes—
58C22×C4×D3C2311yescharacteristic
59C2×C3×D8—22no—
60C2×C3×D8—22no—
61C2×C3×D8C2×C414yescomplemented
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#217—22no—
70#938—24no—
71#938C2×C418yescomplemented
72D4×Dic3—24no—
73D4×Dic3—24no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D412yes—
80C2×C4×Dic3C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C2×ΓL(1,𝔽25)—22no—
84C2×ΓL(1,𝔽25)—22no—
85C2×ΓL(1,𝔽25)—22no—
86C2×ΓL(1,𝔽25)—22no—
87C2×ΓL(1,𝔽25)—22no—
88C2×ΓL(1,𝔽25)C2×C414yescomplemented
89#67⋊C4—48no—
90C42×D3—22no—
91C42×D3—22no—
92C3×#107—22no—
93C3×#102—24no—
94C3×C4×C8—22no—
95#832—24no—
96#832C2×C418yescomplemented
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#817—22no—
101#817C2×C414yescomplemented
102#815—22no—
103#815D414yescomplemented
104C4×C3⋊C8—22no—
105C22×D8C2×D311yescharacteristic
106C22×M16—31no—
107C22×M16—31no—
108C2×#121—32no—
109C2×C4×D4—32no—
110C2×C4×D4—62no—
111C22×C42—31no—
112#469—38no—
113#442—64no—
114C2×#107C2×D311yescharacteristic
115C2×#106—31no—
116C2×#106—31no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102C2×D312yes—
120C4×M16—64no—
121C2×#97—31no—
122C2×#97—31no—
123C2×C4×C8—31no—
124C2×C4×C8C2×D311yescharacteristic
125#390C2×D312yes—
126#389—32no—
127#388—62no—
128#386—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131C4×C22⋊C4—62no—
132#340—32no—
133#340—32no—
134#340—32no—
135C24×C3—22no—
136C23×D3C2×D411yescharacteristic
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4C22×C414yes—
143C22×C3×C4—22no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—28no—
147C2×#67—44no—
148C2×#67—44no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—44no—
152C22×Dic3—44no—
153C22×Dic3—44no—
154C22×Dic3C2×D411yescharacteristic
155C22×Dic3C2×D412yes—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3—21no—
160C2×C4×D3—24no—
161C2×C4×D3—24no—
162C2×C4×D3—42no—
163C2×C4×D3C22×C412yes—
164C2×C4×D3#4112yes—
165C3×D8—28no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8C22×C412yes—
169C2×C3×C8C2×D412yes—
170C2×C3×C8#4112yes—
171C2×C3×C8#4112yes—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—44no—
176C3×C42—21no—
177C3×C42—21no—
178#217—28no—
179#217—44no—
180(C2×D3)⋊C4—44no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3C22×C414yes—
185Dic3⋊C4—44no—
186C4×Dic3—24no—
187C4×Dic3—44no—
188C4×Dic3C2×D412yes—
189C4×Dic3#4112yes—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8C22×C412yes—
195C2×C3⋊C8C2×D412yes—
196ΓL(1,𝔽25)—28no—
197ΓL(1,𝔽25)—28no—
198C22×D4—62no—
199C22×D4C22×D312yes—
200C23×C4—31no—
201C23×C4—32no—
202C23×C4—62no—
203C2×D8—22no—
204C2×D8—22no—
205C2×D8C4×D314yescomplemented
206C2×M16—34no—
207C2×M16—62no—
208C2×M16—62no—
209C2×M16—62no—
210C2×M16—62no—
211C2×M16—62no—
212C22×C8—31no—
213C22×C8—31no—
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D311yescharacteristic
216#121—38no—
217#121—64no—
218C4×D4—64no—
219C4×D4—64no—
220C4×D4—128no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—62no—
230C2×C42—31no—
231C2×C42—32no—
232C2×C42—62no—
233C2×C42—62no—
234C2×C42—62no—
235C2×C42C22×D311yescharacteristic
236#107—22no—
237#106—34no—
238#106—62no—
239#102—24no—
240#102—38no—
241#102—64no—
242#97—34no—
243#97—62no—
244C4×C8—22no—
245C4×C8—62no—
246#95—62no—
247#95—62no—
248#95—62no—
249C23×C3—24no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3C22×D411yescharacteristic
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—22no—
260C22×D3—22no—
261C22×D3C4×D412yes—
262C3×D4—24no—
263C3×D4—24no—
264C3×D4—42no—
265C3×D4—42no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4AGL(1,ℤ/8)14yes—
272C2×C3×C4C23×C411yescharacteristic
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278#67—416no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—24no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—88no—
290C2×Dic3C4×D412yes—
291C2×Dic3#12114yes—
292C4×D3—22no—
293C4×D3—22no—
294C4×D3—22no—
295C4×D3—22no—
296C4×D3—48no—
297C3×C8—22no—
298C3×C8C4×D414yescomplemented
299C3⋊C8—22no—
300C3⋊C8—22no—
301C24—22no—
302C24—31no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—68no—
309C2×D4—124no—
310C2×D4—124no—
311C2×D4C2×C4×D314yes—
312C22×C4—22no—
313C22×C4—31no—
314C22×C4—32no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—64no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—64no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4—124no—
327C22×C4—124no—
328C22×C4—124no—
329C22×C4D3×D412yes—
330C22×C4C23×D311yescharacteristic
331D8—28no—
332M16—68no—
333M16—68no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—32no—
337C2×C8—32no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8C2×C4×D312yes—
343C2×C8D3×D412yes—
344C2×C8#23712yes—
345C2×C8#23712yes—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—34no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—124no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—68no—
356C22⋊C4—124no—
357C22⋊C4—124no—
358C42—21no—
359C42—21no—
360C42—32no—
361C42—32no—
362C42—64no—
363C42—124no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3—44no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—82no—
371C22×C3C2×C4×D411yescharacteristic
372C22×C3C2×#12111yescharacteristic
373C22×C3#53211yescharacteristic
374C22×C3C2×AGL(1,ℤ/8)12yes—
375C22×C3#57512yes—
376C2×D3—24no—
377C2×D3—22no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—42no—
381C3×C4—24no—
382C3×C4—82no—
383C3×C4C2×C4×D411yescharacteristic
384C3×C4C4×#4111yescharacteristic
385C3×C4#52511yescharacteristic
386C3×C4#52911yescharacteristic
387Dic3—24no—
388Dic3—22no—
389Dic3—44no—
390Dic3—82no—
391C23—24no—
392C23—32no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—61no—
400C23—62no—
401C23—62no—
402C23—61no—
403C23C2×D3×D411yescharacteristic
404D4—24no—
405D4—24no—
406D4—42no—
407D4—42no—
408D4—1216no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—32no—
412C2×C4—34no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—64no—
425C2×C4—128no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—248no—
432C2×C4#91114yes—
433C2×C4C22×C4×D311yescharacteristic
434C2×C4C2×D3×D411yescharacteristic
435C2×C4C2×#23711yescharacteristic
436C2×C4C2×#23711yescharacteristic
437C2×C4D3×#4111yescharacteristic
438C2×C4D3×#4111yescharacteristic
439C8—22no—
440C8—62no—
441C8—62no—
442C8#67⋊C414yescomplemented
443C2×C3—84no—
444C2×C3C4×#12111yescharacteristic
445C2×C3C4×AGL(1,ℤ/8)12yes—
446C2×C3#305512yes—
447C2×C3#318311yescharacteristic
448C2×C3#318411yescharacteristic
449D3—42no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—62no—
454C22—64no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—122no—
460C22—121no—
461C22—121no—
462C22C4×D3×D411yescharacteristic
463C22D3×#12111yescharacteristic
464C22#571611yescharacteristic
465C22C2×#91112yes—
466C22#576812yes—
467C4—24no—
468C4—62no—
469C4—64no—
470C4—82no—
471C4—124no—
472C4—242no—
473C4C2×#67⋊C411yescharacteristic
474C4C4×#23711yescharacteristic
475C4#560611yescharacteristic
476C4#568011yescharacteristic
477C3#2153811yescharacteristic, complemented, Sylow
478C2—84no—
479C2—122no—
480C2#7629111yescharacteristic
481C2C4×#91112yes—
482C2#8071512yes—
483C2#8177711yescharacteristic
484C2#8181211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1970C2
socleC23×C3C22×D4
prime radicalC2×#5159C2
cosocleC22×C3×C4C24
hypercenter#1970D3
hypocenterC3#21538
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×#5152C2semidirect
3. C2×C4×ΓL(1,𝔽25)C2semidirect
4. #76291C2semidirect
5. C3×#1970C2semidirect
6. #70631C2semidirect
7. #70605C2semidirect
8. #70203C2semidirect
9. #70152C2semidirect
10. #69769C2semidirect
11. #69644C2semidirect
26. #5159C4semidirect
47. #1970D3semidirect
61. C2×C3×D8C2×C4semidirect
71. #938C2×C4semidirect
88. C2×ΓL(1,𝔽25)C2×C4semidirect
96. #832C2×C4semidirect
101. #817C2×C4semidirect
103. #815D4semidirect
205. C2×D8C4×D3semidirect
298. C3×C8C4×D4semidirect
442. C8#67⋊C4semidirect
477. C3#21538semidirect

Maximal subgroups

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

subgroupindex
C2×#51592
C2×#51522
C2×C4×ΓL(1,𝔽25)2
#762912
C3×#19702
#706312
#706052
#702032
#701522
#697692
#696442
#215383