Finite Group Databasebeta

← #10690729

Subgroups — #10690729

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82096C211yesmaximal, characteristic
2#78602C211yesmaximal, characteristic, complemented
3C2×C3×#323C211yesmaximal, characteristic, complemented
4C2×#4525C211yesmaximal, characteristic, complemented
5#69919C211yesmaximal, characteristic, complemented
6#67678C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9109—31nomaximal, Sylow
9C22×C3×C22⋊C4C2211yescharacteristic
10C2×#953C411yescharacteristic
11C2×C4×#67C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5247C411yescharacteristic
15#5213C411yescharacteristic
16#5210C411yescharacteristic
17C4×#217C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20#5111—22no—
21#4957—21no—
22#4956—21no—
23#4945—21no—
24#4944—21no—
25#4730—22no—
26#4726—21no—
27#4719—21no—
28C3×#323—22no—
29#4525—22no—
30#4469—21no—
31#4468—21no—
32#2973—31no—
33#2321—31no—
34#1459—31no—
35C2×#323—31no—
36C2×#323D311yescharacteristic, complemented
37#1282—31no—
38#1282—31no—
39C25×C3C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67C2×C411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53#953—22no—
54#953—22no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217C2×C411yescharacteristic
59C2×#217C2×C411yescharacteristic
60C2×#217C2×C411yescharacteristic
61C4×#67—21no—
62C4×#67—21no—
63C4×#67—21no—
64C4×#67—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4D411yescharacteristic
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4D411yescharacteristic
69C2×C4×Dic3—21no—
70C2×C4×Dic3D411yescharacteristic
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C2×ΓL(1,𝔽25)—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yescomplemented
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yescomplemented
85(C2×D3)⋊C8—22no—
86#815—22no—
87Dic3⋊C8—22no—
88C2×#120—31no—
89C2×C4×D4—31no—
90C22×C22⋊C4C2×D311yescharacteristic
91#435—62no—
92#435—62no—
93#432—61no—
94#407—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98C2×C22⋊C8C2×D311yescharacteristic
99#390—61no—
100#388—61no—
101#388—61no—
102#386—31no—
103#386—61no—
104#385—31no—
105#379—31no—
106C4×C22⋊C4—31no—
107#324—61no—
108#323—22no—
109#323—61no—
110#323—62no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—81no—
118C24×C3—81no—
119C24×C3M1612yescomplemented
120C24×C3C22×C411yescharacteristic
121C23×D3C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—81no—
126C22×C3×C4—81no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×#67—21no—
134C2×#67—21no—
135C2×#67—21no—
136C2×#67—21no—
137C2×#67—21no—
138C2×#67—21no—
139C2×#67—24no—
140C2×#67—81no—
141C2×#67—81no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3C22⋊C411yescharacteristic
145C22×Dic3C22⋊C411yescharacteristic
146C22×Dic3C22⋊C411yescharacteristic
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—81no—
150C2×C3×C8—22no—
151C2×C3×C8—41no—
152C2×C3×C8—81no—
153C2×C3×C8—81no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158#217—24no—
159#217—21no—
160#217—21no—
161#217—21no—
162#217—21no—
163#217—21no—
164#217—81no—
165#217—81no—
166#217M1612yescomplemented
167(C2×D3)⋊C4—21no—
168(C2×D3)⋊C4—21no—
169(C2×D3)⋊C4—81no—
170Dic3⋊C4—21no—
171Dic3⋊C4—21no—
172Dic3⋊C4—81no—
173C4×Dic3—21no—
174C4×Dic3—21no—
175C4×Dic3—81no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—81no—
179C2×C3⋊C8—81no—
180ΓL(1,𝔽25)—82no—
181C25C4×D311yescharacteristic
182C22×D4—31no—
183C22×D4—61no—
184C23×C4—31no—
185C23×C4C4×D311yescharacteristic
186C23×C4C22×D311yescharacteristic
187C2×M16—62no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—61no—
191C22×C8—61no—
192#120—62no—
193#120—62no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—31no—
207C2×C22⋊C4—31no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C42—31no—
215C2×C42—61no—
216C4⋊C8—62no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—32no—
221C22⋊C8—62no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8#6712yescomplemented
226#97—62no—
227#95—61no—
228#95—61no—
229#95—61no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—81no—
244C23×C3—81no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3—82no—
248C23×C3—82no—
249C23×C3—82no—
250C23×C3—82no—
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3C4×D411yescharacteristic
253C23×C3C4×D411yescharacteristic
254C23×C3C2×M1611yescharacteristic
255C23×C3C2×M1611yescharacteristic
256C23×C3#13111yescharacteristic
257C23×C3#13111yescharacteristic
258C22×D3—21no—
259C22×D3—21no—
260C22×D3—21no—
261C22×D3—41no—
262C22×D3—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4#12011yescharacteristic
275C2×C3×C4#12111yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12311yescharacteristic
278#67—41no—
279#67—41no—
280#67—41no—
281#67—41no—
282#67—84no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—22no—
288C2×Dic3—21no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C4×D3—42no—
297C4×D3—81no—
298C3×C8—42no—
299C3×C8—81no—
300C3⋊C8—42no—
301C3⋊C8—81no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—31no—
309C24—81no—
310C24—81no—
311C24ΓL(1,𝔽25)12yescomplemented
312C24C2×C4×D311yescharacteristic
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—64no—
320C2×D4—241no—
321C2×D4—241no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—241no—
335C22×C4#23511yescharacteristic
336C22×C4D3×D411yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4#23711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341M16—242no—
342C2×C8—22no—
343C2×C8—41no—
344C2×C8—62no—
345C2×C8—81no—
346C2×C8—81no—
347C2×C8—121no—
348C2×C8—241no—
349C2×C8—241no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—241no—
353C22⋊C4—32no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—64no—
359C22⋊C4—61no—
360C22⋊C4—61no—
361C22⋊C4—61no—
362C22⋊C4—61no—
363C22⋊C4—61no—
364C22⋊C4—61no—
365C22⋊C4—61no—
366C22⋊C4—241no—
367C22⋊C4—241no—
368C22⋊C4—241no—
369C42—61no—
370C42—61no—
371C42—241no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3#38611yescharacteristic
394C22×C3#40711yescharacteristic
395C22×C3#40811yescharacteristic
396C22×C3#43511yescharacteristic
397C22×C3#43511yescharacteristic
398C22×C3#43611yescharacteristic
399C22×C3#43611yescharacteristic
400C2×D3—42no—
401C2×D3—41no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—81no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—81no—
410Dic3—42no—
411Dic3—41no—
412Dic3—41no—
413Dic3—41no—
414Dic3—81no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—121no—
440C23—121no—
441C23D3×C22⋊C411yescharacteristic
442C23#67⋊C411yescharacteristic
443C23C2×ΓL(1,𝔽25)11yescharacteristic
444C23#90111yescharacteristic
445C23D3×M1611yescharacteristic
446C23#90711yescharacteristic
447C23C4×#6711yescharacteristic
448D4—121no—
449D4—121no—
450D4—121no—
451D4—121no—
452D4—244no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—241no—
479C2×C4#88311yescharacteristic
480C2×C4#88411yescharacteristic
481C2×C4#93711yescharacteristic
482C2×C4#93811yescharacteristic
483C8—42no—
484C8—81no—
485C8—122no—
486C8—241no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—82no—
494C2×C3#187011yescharacteristic
495C2×C3#191311yescharacteristic
496C2×C3#191411yescharacteristic
497D3—81no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22—81no—
518C22—81no—
519C22—122no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—241no—
524C22#472611yescharacteristic
525C22#472711yescharacteristic
526C22#472911yescharacteristic
527C22#473011yescharacteristic
528C22#495611yescharacteristic
529C22#511111yescharacteristic
530C22#512811yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—81no—
536C4—122no—
537C4—121no—
538C4—121no—
539C4—121no—
540C4—241no—
541C3#910911yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—82no—
549C2—241no—
550C2#6995811yescharacteristic
551C2#7003411yescharacteristic
552C2#7008911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC22×C3#386
prime radicalC2×#953C4
cosocleC23×C3×C4C23
hypercenterC2×#323D3
hypocenterC3#9109
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #78602C2semidirect
3. C2×C3×#323C2semidirect
4. C2×#4525C2semidirect
5. #69919C2semidirect
6. #67678C2semidirect
7. #67677C2semidirect
36. C2×#323D3semidirect
77. C3×C22⋊C8D4semidirect
84. #830D4semidirect
119. C24×C3M16semidirect
166. #217M16semidirect
225. C22⋊C8#67semidirect
311. C24ΓL(1,𝔽25)semidirect
541. C3#9109semidirect

Maximal subgroups

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

subgroupindex
#820962
#786022
C2×C3×#3232
C2×#45252
#699192
#676782
#676772
#91093