Finite Group Databasebeta

← #10690550

Subgroups — #10690550

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
1C4×#938C211yesmaximal, characteristic
2C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
3C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
4C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic, complemented
6#67704C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9078—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938C411yescharacteristic
11C2×C4×#67—21no—
12C22×ΓL(1,𝔽25)—21no—
13C2×C42×D3C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16#5221C411yescharacteristic
17C4×#217—21no—
18C2×(C2×D3)⋊C8—21no—
19C2×#815—21no—
20#4987C411yescharacteristic
21#4972C411yescharacteristic
22#4956—21no—
23#4944—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27C4×ΓL(1,𝔽25)—22no—
28C4×ΓL(1,𝔽25)—22no—
29C3×D4⋊C8—22no—
30#4550—21no—
31D4⋊(C3⋊C8)—22no—
32C2×C4×M16—31no—
33C4×#121—31no—
34#1797—31no—
35C2×D4⋊C8—31no—
36C2×D4⋊C8D311yescharacteristic, complemented
37#1279—31no—
38#1279—31no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—21no—
42C22×C4×D3C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57#938—22no—
58#938—22no—
59C4×#67—42no—
60C4×#67—42no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3C2×C411yescharacteristic
67C2×C4×Dic3D411yescharacteristic
68C2×C4×Dic3D411yescharacteristic
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
71C2×ΓL(1,𝔽25)—24no—
72C2×ΓL(1,𝔽25)—24no—
73C2×ΓL(1,𝔽25)—24no—
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—61no—
94C2×C4×D4C2×D311yescharacteristic
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⋊C4—31no—
110C4×C22⋊C4—61no—
111#336—61no—
112D4⋊C8—22no—
113D4⋊C8—62no—
114C24×C3—21no—
115C23×D3C22⋊C411yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4M1612yescomplemented
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C2×#67—24no—
133C2×#67—42no—
134C2×#67—42no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3C22⋊C411yescharacteristic
138C22×Dic3C22⋊C411yescharacteristic
139C22×Dic3C22⋊C411yescharacteristic
140C2×C4×D3—22no—
141C2×C4×D3—21no—
142C2×C4×D3—21no—
143C2×C4×D3—22no—
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—21no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—21no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
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×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4111yescharacteristic
169C3×C42#4111yescharacteristic
170#217—24no—
171#217—42no—
172#217—42no—
173(C2×D3)⋊C4—42no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3M1612yescomplemented
178Dic3⋊C4—42no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C4×Dic3—41no—
186C4×Dic3—41no—
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—61no—
195C22×D4C4×D311yescharacteristic
196C23×C4—21no—
197C23×C4—31no—
198C23×C4—31no—
199C23×C4—31no—
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—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—122no—
214C4×D4—122no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4C4×D311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—61no—
224C2×C42—31no—
225C2×C42—31no—
226C2×C42—31no—
227C2×C42—61no—
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×C8#6712yescomplemented
243#95—61no—
244#95—61no—
245#95—61no—
246C23×C3—22no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3C2×C22⋊C411yescharacteristic
254C22×D3—21no—
255C22×D3—21no—
256C22×D3—21no—
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—21no—
260C22×D3#10412yes—
261C3×D4—22no—
262C3×D4—22no—
263C3×D4—41no—
264C3×D4—41no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—81no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4C4×D411yescharacteristic
275C2×C3×C4C4×D411yescharacteristic
276C2×C3×C4#12011yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4#12311yescharacteristic
280C2×C3×C4C2×M1611yescharacteristic
281C2×C3×C4C2×M1611yescharacteristic
282C2×C3×C4#13111yescharacteristic
283C2×C3×C4#13111yescharacteristic
284C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
285C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
286C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
287C2×C3×C4#13711yescharacteristic
288#67—48no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—22no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3#10412yes—
304C4×D3—24no—
305C4×D3—24no—
306C4×D3—41no—
307C4×D3—41no—
308C4×D3—41no—
309C4×D3—41no—
310C4×D3—41no—
311C4×D3—41no—
312C4×D3—41no—
313C4×D3—41no—
314C3×C8—24no—
315C3×C8—81no—
316C3⋊C8—24no—
317C3⋊C8—81no—
318C24—21no—
319C24—31no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—64no—
326C2×D4—122no—
327C2×D4—122no—
328C2×D4ΓL(1,𝔽25)12yescomplemented
329C22×C4—21no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—62no—
340C22×C4—62no—
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—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
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—32no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—122no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—64no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C42—61no—
387C42—61no—
388C42—61no—
389C42—61no—
390C42—61no—
391C42—61no—
392C42—121no—
393C42—121no—
394C42#23511yescharacteristic
395C42D3×D411yescharacteristic
396C42#23711yescharacteristic
397C42C2×#6711yescharacteristic
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3#38611yescharacteristic
406C22×C3#40711yescharacteristic
407C22×C3#40811yescharacteristic
408C22×C3#41711yescharacteristic
409C22×C3#41811yescharacteristic
410C22×C3C2×#10411yescharacteristic
411C22×C3C2×#10411yescharacteristic
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—41no—
416C2×D3—41no—
417C2×D3—41no—
418C2×D3—41no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—81no—
422C3×C4#43511yescharacteristic
423C3×C4#43511yescharacteristic
424C3×C4#43611yescharacteristic
425C3×C4#43611yescharacteristic
426C3×C4#46811yescharacteristic
427C3×C4#46811yescharacteristic
428C3×C4#46911yescharacteristic
429C3×C4#46911yescharacteristic
430Dic3—22no—
431Dic3—41no—
432Dic3—41no—
433Dic3—42no—
434Dic3—81no—
435C23—22no—
436C23—32no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—61no—
444C23—61no—
445C23—61no—
446C23—61no—
447C23—61no—
448C23—61no—
449C23D3×C22⋊C411yescharacteristic
450D4—22no—
451D4—22no—
452D4—41no—
453D4—41no—
454D4—128no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—32no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—62no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
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—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4#67⋊C411yescharacteristic
490C2×C4#88311yescharacteristic
491C2×C4#88411yescharacteristic
492C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
493C2×C4#90111yescharacteristic
494C2×C4D3×M1611yescharacteristic
495C2×C4#90711yescharacteristic
496C2×C4#91111yescharacteristic
497C2×C4#91111yescharacteristic
498C2×C4#91411yescharacteristic
499C2×C4#91511yescharacteristic
500C2×C4C4×#6711yescharacteristic
501C2×C4#93711yescharacteristic
502C2×C4#93811yescharacteristic
503C8—24no—
504C8—64no—
505C8—81no—
506C8—241no—
507C2×C3—82no—
508C2×C3#148311yescharacteristic
509C2×C3#148411yescharacteristic
510C2×C3#148911yescharacteristic
511C2×C3#149011yescharacteristic
512C2×C3#187011yescharacteristic
513C2×C3#189511yescharacteristic
514C2×C3#189611yescharacteristic
515D3—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—62no—
520C22—62no—
521C22—62no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—121no—
527C22—121no—
528C22—121no—
529C22—121no—
530C22#472611yescharacteristic
531C22#472711yescharacteristic
532C22#477011yescharacteristic
533C22#477111yescharacteristic
534C22D3×#10411yescharacteristic
535C22D3×#10411yescharacteristic
536C22#495611yescharacteristic
537C4—21no—
538C4—21no—
539C4—62no—
540C4—81no—
541C4—121no—
542C4—121no—
543C4—122no—
544C4—241no—
545C4#472911yescharacteristic
546C4#473011yescharacteristic
547C4#478011yescharacteristic
548C4#478211yescharacteristic
549C4#511111yescharacteristic
550C4#512811yescharacteristic
551C4#515911yescharacteristic
552C4#517411yescharacteristic
553C3#907811yescharacteristic, complemented, Sylow
554C2—82no—
555C2—122no—
556C2#6817011yescharacteristic
557C2#6817111yescharacteristic
558C2#6817411yescharacteristic
559C2#6817511yescharacteristic
560C2#6995811yescharacteristic
561C2#7020311yescharacteristic
562C2#7038211yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#938C4
cosocleC2×C3×C42C23
hypercenterC2×D4⋊C8D3
hypocenterC3#9078
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C4×ΓL(1,𝔽25)C2semidirect
3. C2×C3×D4⋊C8C2semidirect
4. C2×D4⋊(C3⋊C8)C2semidirect
5. #70475C2semidirect
6. #67704C2semidirect
7. #67655C2semidirect
36. C2×D4⋊C8D3semidirect
81. C3×C4×C8D4semidirect
90. C4×C3⋊C8D4semidirect
121. C2×C3×D4M16semidirect
177. C4⋊Dic3M16semidirect
242. C4×C8#67semidirect
328. C2×D4ΓL(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
C4×#9382
C2×C4×ΓL(1,𝔽25)2
C2×C3×D4⋊C82
C2×D4⋊(C3⋊C8)2
#704752
#677042
#676552
#90783