Finite Group Databasebeta

← #10690547

Subgroups — #10690547

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C4×#897C211yesmaximal, characteristic
3#70475C211yesmaximal, characteristic
4C2×#4484C211yesmaximal, characteristic, complemented
5C2×#4456C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9077—31nomaximal, Sylow
9C22×ΓL(1,𝔽25)—21no—
10C2×#897C411yescharacteristic
11C2×C42×D3C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C8—21no—
17C2×#815—21no—
18#4987C411yescharacteristic
19#4984C411yescharacteristic
20#4973C411yescharacteristic
21C4×(C2×D3)⋊C4—21no—
22C4×C4⋊Dic3—21no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C4×ΓL(1,𝔽25)—22no—
26C4×ΓL(1,𝔽25)—22no—
27#4668—21no—
28#4653—21no—
29#4550—21no—
30#4484—22no—
31#4456—22no—
32C2×C4×M16—31no—
33C4×C22⋊Q8—31no—
34#1797—31no—
35C2×Q8⋊C8—31no—
36C2×Q8⋊C8—31no—
37#1279—31no—
38#1279D311yescharacteristic, complemented
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3C2×C411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61C2×ΓL(1,𝔽25)—24no—
62C2×ΓL(1,𝔽25)—24no—
63C2×ΓL(1,𝔽25)—24no—
64#897—22no—
65#897—22no—
66C42×D3—21no—
67C42×D3—21no—
68C42×D3—21no—
69C42×D3—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4×C8—21no—
77C3×C4×C8D412yes—
78C3×#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83(C2×D3)⋊C8—42no—
84#815—24no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8D412yes—
89C22×M16—61no—
90C2×C22⋊Q8—31no—
91C2×C4×Q8—31no—
92C22×C42—31no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×C22⋊C8—61no—
96C4×M16—62no—
97C4×M16—62no—
98C2×#97—61no—
99C2×C4×C8—31no—
100C2×C4×C8C2×D311yescharacteristic
101#391—31no—
102#389—31no—
103#387—61no—
104#385—61no—
105#380—31no—
106C4×C4⋊C4—61no—
107C4×C4⋊C4C2×D311yescharacteristic
108C4×C22⋊C4—61no—
109#336—61no—
110Q8⋊C8—62no—
111Q8⋊C8—62no—
112C23×D3C22⋊C411yescharacteristic
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C22×Dic3C22⋊C411yescharacteristic
123C2×C4×D3—22no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—22no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6M1612yescomplemented
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C2×C3×C8—41no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4M1612yescomplemented
155C3×C42—41no—
156C3×C42—41no—
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160C3×C42#4111yescharacteristic
161(C2×D3)⋊C4—24no—
162(C2×D3)⋊C4—42no—
163C4⋊Dic3—24no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—42no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—21no—
172C4×Dic3—21no—
173C4×Dic3—21no—
174C4×Dic3—21no—
175C4×Dic3—41no—
176C4×Dic3—41no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183ΓL(1,𝔽25)—216no—
184C22×Q8—31no—
185C23×C4—31no—
186C23×C4—31no—
187C23×C4—31no—
188C2×M16—64no—
189C2×M16—64no—
190C2×M16—64no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—61no—
194C22×C8—61no—
195C22⋊Q8—62no—
196C22⋊Q8—62no—
197C4×Q8—61no—
198C4×Q8—61no—
199C4×Q8—61no—
200C4×Q8—61no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4C4×D311yescharacteristic
206C2×C4⋊C4C4×D311yescharacteristic
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C42—21no—
210C2×C42—31no—
211C2×C42—31no—
212C2×C42—31no—
213C2×C42—61no—
214C2×C42—61no—
215C2×C42—61no—
216C2×C42—61no—
217C2×C42—61no—
218C2×C42C22×D311yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—61no—
222C4⋊C8—61no—
223C22⋊C8—122no—
224#97—64no—
225C4×C8—21no—
226C4×C8—32no—
227C4×C8—61no—
228C4×C8D1212yes—
229#95—21no—
230#95—61no—
231#95—61no—
232#95—61no—
233#95—61no—
234C23×C3C2×C22⋊C411yescharacteristic
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—21no—
238C22×D3—21no—
239C22×D3—21no—
240C22×D3—21no—
241C22×D3#10412yes—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12311yescharacteristic
256C2×C3×C4C2×M1611yescharacteristic
257C2×C3×C4C2×M1611yescharacteristic
258C2×C3×C4#13111yescharacteristic
259C2×C3×C4#13111yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4#13711yescharacteristic
262C2×C3×C4#13711yescharacteristic
263C2×C3×C4#13711yescharacteristic
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—22no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3#10412yes—
279C4×D3—24no—
280C4×D3—24no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—41no—
288C4×D3—41no—
289Dic6—22no—
290Dic6—22no—
291Dic6—41no—
292Dic6—41no—
293C3×C8—24no—
294C3×C8—81no—
295C3⋊C8—24no—
296C3⋊C8—81no—
297C24—31no—
298C2×Q8—32no—
299C2×Q8—61no—
300C2×Q8—61no—
301C2×Q8—61no—
302C2×Q8—61no—
303C2×Q8—61no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4C2×C4×D311yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330M16—616no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—62no—
338C2×C8—62no—
339C2×C8—62no—
340C2×C8—121no—
341C2×C8—121no—
342C2×C8—121no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—64no—
347C4⋊C4—61no—
348C4⋊C4—61no—
349C4⋊C4—122no—
350C4⋊C4—121no—
351C4⋊C4—121no—
352C4⋊C4ΓL(1,𝔽25)12yescomplemented
353C22⋊C4—64no—
354C22⋊C4—122no—
355C42—41no—
356C42—41no—
357C42—61no—
358C42—61no—
359C42—61no—
360C42—61no—
361C42—61no—
362C42—61no—
363C42—121no—
364C42—121no—
365C42C2×D1211yescharacteristic
366C42#23511yescharacteristic
367C42D3×D411yescharacteristic
368C42#23911yescharacteristic
369C22×C3#38611yescharacteristic
370C22×C3#40711yescharacteristic
371C22×C3#40811yescharacteristic
372C22×C3#41711yescharacteristic
373C22×C3#41911yescharacteristic
374C22×C3C2×#10411yescharacteristic
375C22×C3C2×#10411yescharacteristic
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—22no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—42no—
386C3×C4—81no—
387C3×C4#43511yescharacteristic
388C3×C4#43511yescharacteristic
389C3×C4#43611yescharacteristic
390C3×C4#43611yescharacteristic
391C3×C4#46811yescharacteristic
392C3×C4#46811yescharacteristic
393C3×C4#47011yescharacteristic
394C3×C4#47011yescharacteristic
395Dic3—22no—
396Dic3—42no—
397Dic3—41no—
398Dic3—41no—
399Dic3—81no—
400C23—32no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23D3×C22⋊C411yescharacteristic
408Q8—62no—
409Q8—62no—
410Q8—121no—
411Q8—121no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—32no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—62no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4C4×D1211yescharacteristic
446C2×C4#88211yescharacteristic
447C2×C4#88411yescharacteristic
448C2×C4#89311yescharacteristic
449C2×C4#89411yescharacteristic
450C2×C4#89511yescharacteristic
451C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
452C2×C4#90111yescharacteristic
453C2×C4D3×M1611yescharacteristic
454C2×C4#90711yescharacteristic
455C2×C4#91111yescharacteristic
456C2×C4#91511yescharacteristic
457C2×C4#91511yescharacteristic
458C2×C4#91811yescharacteristic
459C8—24no—
460C8—64no—
461C8—81no—
462C8—241no—
463C2×C3#148311yescharacteristic
464C2×C3#148511yescharacteristic
465C2×C3#148911yescharacteristic
466C2×C3#149111yescharacteristic
467C2×C3#187011yescharacteristic
468C2×C3#189411yescharacteristic
469C2×C3#189611yescharacteristic
470D3—42no—
471C22—62no—
472C22—62no—
473C22—62no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22#466511yescharacteristic
479C22#472611yescharacteristic
480C22#472711yescharacteristic
481C22#477111yescharacteristic
482C22#480211yescharacteristic
483C22D3×#10411yescharacteristic
484C22D3×#10411yescharacteristic
485C4—21no—
486C4—21no—
487C4—42no—
488C4—62no—
489C4—81no—
490C4—121no—
491C4—121no—
492C4—122no—
493C4—241no—
494C4#468811yescharacteristic
495C4#470611yescharacteristic
496C4#478011yescharacteristic
497C4#481611yescharacteristic
498C4#483611yescharacteristic
499C4#483811yescharacteristic
500C4#486711yescharacteristic
501C4#488611yescharacteristic
502C3#907711yescharacteristic, complemented, Sylow
503C2—122no—
504C2#6817011yescharacteristic
505C2#6817411yescharacteristic
506C2#6823511yescharacteristic
507C2#6823911yescharacteristic
508C2#6866911yescharacteristic
509C2#6875611yescharacteristic
510C2#6888411yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9077
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
4. C2×#4484C2semidirect
5. C2×#4456C2semidirect
6. C3×#1279C2semidirect
7. #67619C2semidirect
38. #1279D3semidirect
144. C2×Dic6M16semidirect
154. C3×C4⋊C4M16semidirect
352. C4⋊C4ΓL(1,𝔽25)semidirect
502. C3#9077semidirect

Maximal subgroups

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

subgroupindex
C2×C4×ΓL(1,𝔽25)2
C4×#8972
#704752
C2×#44842
C2×#44562
C3×#12792
#676192
#90773