Finite Group Databasebeta

← #10690552

Subgroups — #10690552

Order 768.

754 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
2C4×#895C211yesmaximal, characteristic
3#70475C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9078—31nomaximal, Sylow
9C22×ΓL(1,𝔽25)—21no—
10C2×#895C411yescharacteristic
11C2×C4×D12—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C42×D3C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17C2×(C2×D3)⋊C8—21no—
18C2×#815—21no—
19#4988C411yescharacteristic
20#4987C411yescharacteristic
21#4984C411yescharacteristic
22C4×(C2×D3)⋊C4—21no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C4×ΓL(1,𝔽25)—22no—
26C4×ΓL(1,𝔽25)—22no—
27#4670—21no—
28#4665—21no—
29#4550—21no—
30#4483—22no—
31#4459—22no—
32C2×C4×M16—31no—
33C4×#121—31no—
34#1797—31no—
35C2×D4⋊C8—31no—
36C2×D4⋊C8—31no—
37#1279—31no—
38#1279D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C22×C4×D3C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×C4×Dic3C2×C411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63C2×ΓL(1,𝔽25)—24no—
64C2×ΓL(1,𝔽25)—24no—
65C2×ΓL(1,𝔽25)—24no—
66#895—22no—
67#895—22no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—42no—
73C4×D12—42no—
74C42×D3—21no—
75C42×D3—21no—
76C42×D3—21no—
77C42×D3—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C4×C8—21no—
81C3×C4×C8D412yescomplemented
82C3×#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85(C2×D3)⋊C8—42no—
86#815—24no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8D412yescomplemented
91C22×M16—61no—
92C2×#121—31no—
93C2×C4×D4—31no—
94C2×C4×D4—61no—
95C22×C42—31no—
96C2×C4⋊C8—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×C22⋊C8—61no—
99C4×M16—62no—
100C4×M16—62no—
101C2×#97—61no—
102C2×C4×C8—31no—
103C2×C4×C8C2×D311yescharacteristic
104#390—31no—
105#389—31no—
106#388—61no—
107#386—61no—
108#380—31no—
109C4×C4⋊C4C2×D311yescharacteristic
110C4×C22⋊C4—61no—
111#336—61no—
112D4⋊C8—62no—
113D4⋊C8—62no—
114C23×D3—21no—
115C23×D3C22⋊C411yescharacteristic
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3C22⋊C411yescharacteristic
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—24no—
130C2×D12—21no—
131C2×D12—42no—
132C2×D12—42no—
133C2×D12M1612yescomplemented
134C2×C4×D3—22no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—22no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—21no—
145C2×C4×D3—21no—
146C2×C4×D3—21no—
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—21no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—41no—
160C2×C3×C8—41no—
161C2×C3×C8—41no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4M1612yescomplemented
166C3×C42—41no—
167C3×C42—41no—
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42#4111yescharacteristic
171C3×C42#4111yescharacteristic
172(C2×D3)⋊C4—24no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—42no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—42no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—21no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—41no—
191C2×C3⋊C8—41no—
192C2×C3⋊C8—41no—
193ΓL(1,𝔽25)—216no—
194C22×D4—31no—
195C22×D4—61no—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4—31no—
199C23×C4—61no—
200C2×M16—64no—
201C2×M16—64no—
202C2×M16—64no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—61no—
206C22×C8—61no—
207#121—62no—
208#121—62no—
209C4×D4—61no—
210C4×D4—61no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—122no—
214C4×D4—122no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4C4×D311yescharacteristic
218C2×C4⋊C4C4×D311yescharacteristic
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—61no—
224C2×C42—21no—
225C2×C42—31no—
226C2×C42—31no—
227C2×C42—31no—
228C2×C42—61no—
229C2×C42—61no—
230C2×C42—61no—
231C2×C42—61no—
232C2×C42C22×D311yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—61no—
236C4⋊C8—61no—
237C22⋊C8—122no—
238#97—64no—
239C4×C8—21no—
240C4×C8—32no—
241C4×C8—61no—
242C4×C8D1212yescomplemented
243#95—21no—
244#95—61no—
245#95—61no—
246C23×C3C2×C22⋊C411yescharacteristic
247C22×D3—21no—
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—22no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3#10412yes—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4C4×D411yescharacteristic
270C2×C3×C4C4×D411yescharacteristic
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12311yescharacteristic
275C2×C3×C4C2×M1611yescharacteristic
276C2×C3×C4C2×M1611yescharacteristic
277C2×C3×C4#13111yescharacteristic
278C2×C3×C4#13111yescharacteristic
279C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
280C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
281C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
282C2×C3×C4#13711yescharacteristic
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3#10412yes—
295D12—22no—
296D12—22no—
297D12—48no—
298D12—41no—
299D12—41no—
300C4×D3—24no—
301C4×D3—24no—
302C4×D3—41no—
303C4×D3—41no—
304C4×D3—41no—
305C4×D3—41no—
306C4×D3—41no—
307C4×D3—41no—
308C4×D3—41no—
309C4×D3—41no—
310C4×D3—81no—
311C4×D3—81no—
312C4×D3—81no—
313C4×D3—81no—
314C3×C8—24no—
315C3×C8—81no—
316C3⋊C8—24no—
317C3⋊C8—81no—
318C24—31no—
319C24—61no—
320C2×D4—32no—
321C2×D4—64no—
322C2×D4—61no—
323C2×D4—61no—
324C2×D4—61no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—122no—
328C2×D4—122no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—121no—
352C22×C4—121no—
353C22×C4—121no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4C2×C4×D311yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4D3×D411yescharacteristic
360M16—616no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—121no—
371C2×C8—121no—
372C2×C8—121no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—122no—
379C4⋊C4ΓL(1,𝔽25)12yescomplemented
380C22⋊C4—64no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—121no—
385C22⋊C4—121no—
386C42—41no—
387C42—41no—
388C42—61no—
389C42—61no—
390C42—61no—
391C42—61no—
392C42—61no—
393C42—61no—
394C42C2×D1211yescharacteristic
395C42#23511yescharacteristic
396C42D3×D411yescharacteristic
397C42#23911yescharacteristic
398C22×C3#38611yescharacteristic
399C22×C3#40711yescharacteristic
400C22×C3#40811yescharacteristic
401C22×C3#41711yescharacteristic
402C22×C3#41811yescharacteristic
403C22×C3C2×#10411yescharacteristic
404C22×C3C2×#10411yescharacteristic
405C2×D3—22no—
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—41no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—81no—
416C2×D3—81no—
417C2×D3—81no—
418C2×D3—81no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—42no—
422C3×C4—81no—
423C3×C4#43511yescharacteristic
424C3×C4#43511yescharacteristic
425C3×C4#43611yescharacteristic
426C3×C4#43611yescharacteristic
427C3×C4#46811yescharacteristic
428C3×C4#46811yescharacteristic
429C3×C4#46911yescharacteristic
430C3×C4#46911yescharacteristic
431Dic3—22no—
432Dic3—41no—
433Dic3—41no—
434Dic3—81no—
435C23—32no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23—62no—
443C23—121no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23—121no—
449C23D3×C22⋊C411yescharacteristic
450D4—62no—
451D4—62no—
452D4—128no—
453D4—121no—
454D4—121no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—32no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4C4×D1211yescharacteristic
490C2×C4#88211yescharacteristic
491C2×C4#88411yescharacteristic
492C2×C4#89311yescharacteristic
493C2×C4#89411yescharacteristic
494C2×C4#89511yescharacteristic
495C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
496C2×C4#90111yescharacteristic
497C2×C4D3×M1611yescharacteristic
498C2×C4#90711yescharacteristic
499C2×C4#91111yescharacteristic
500C2×C4#91411yescharacteristic
501C2×C4#91411yescharacteristic
502C2×C4#91811yescharacteristic
503C8—24no—
504C8—64no—
505C8—81no—
506C8—241no—
507C2×C3#148311yescharacteristic
508C2×C3#148411yescharacteristic
509C2×C3#148911yescharacteristic
510C2×C3#149011yescharacteristic
511C2×C3#187011yescharacteristic
512C2×C3#189511yescharacteristic
513C2×C3#189611yescharacteristic
514D3—42no—
515D3—82no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—241no—
527C22—241no—
528C22—241no—
529C22—241no—
530C22#466511yescharacteristic
531C22#472611yescharacteristic
532C22#472711yescharacteristic
533C22#477011yescharacteristic
534C22#480311yescharacteristic
535C22D3×#10411yescharacteristic
536C22D3×#10411yescharacteristic
537C4—21no—
538C4—21no—
539C4—42no—
540C4—62no—
541C4—81no—
542C4—121no—
543C4—121no—
544C4—241no—
545C4#468811yescharacteristic
546C4#470611yescharacteristic
547C4#478211yescharacteristic
548C4#481211yescharacteristic
549C4#483611yescharacteristic
550C4#483811yescharacteristic
551C4#486511yescharacteristic
552C4#488611yescharacteristic
553C3#907811yescharacteristic, complemented, Sylow
554C2—122no—
555C2—242no—
556C2#6817111yescharacteristic
557C2#6817511yescharacteristic
558C2#6823411yescharacteristic
559C2#6823811yescharacteristic
560C2#6866911yescharacteristic
561C2#6875611yescharacteristic
562C2#6888511yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#895C4
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9078
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
1. C2×C4×ΓL(1,𝔽25)C2semidirect
3. #70475C2semidirect
4. C2×#4483C2semidirect
5. C2×#4459C2semidirect
6. C3×#1279C2semidirect
7. #67619C2semidirect
38. #1279D3semidirect
81. C3×C4×C8D4semidirect
90. C4×C3⋊C8D4semidirect
133. C2×D12M16semidirect
165. C3×C4⋊C4M16semidirect
242. C4×C8D12semidirect
379. C4⋊C4ΓL(1,𝔽25)semidirect
553. C3#9078semidirect

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×#8952
#704752
C2×#44832
C2×#44592
C3×#12792
#676192
#90783