Finite Group Databasebeta

← #10695549

Subgroups — #10695549

Order 768.

1008 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2539C211yesmaximal, characteristic, complemented
2#82624C211yesmaximal, characteristic, complemented
3#82125C211yesmaximal, characteristic, complemented
4#71674C211yesmaximal, characteristic, complemented
5#71539C211yesmaximal, characteristic, complemented
6#69454C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9938—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#123—21no—
11C2×C3×#121C2211yescharacteristic
12C2×C3×#120—21no—
13C22×C3×C22⋊C4C2211yescharacteristic
14C2×#953—21no—
15C2×#932—21no—
16C3×#399—21no—
17C3×#392—21no—
18C3×#390—22no—
19C3×C4×C22⋊C4C2211yescharacteristic
20#5213—21no—
21#5211C2211yescharacteristic
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5040—21no—
25#5037—22no—
26#5036—21no—
27#4966—21no—
28#4960—21no—
29#4956—22no—
30#4947—21no—
31C2×#805—22no—
32#4537—22no—
33#4525—22no—
34#3006—31no—
35#2539D311yescharacteristic, complemented
36#2404—31no—
37#1789—31no—
38#1510—31no—
39#1462—31no—
40#1284—31no—
41C25×C3D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4—22no—
44C22×C3×D4D411yescharacteristic, complemented
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C22×#67—21no—
48C22×#67—22no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—44no—
54C22×D12D411yescharacteristic, complemented
55C22×C4×D3—21no—
56C22×Dic6D411yescharacteristic, complemented
57C3×#123—44no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#120—44no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D411yescharacteristic, complemented
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42D412yescomplemented
76#953—44no—
77C2×#217—22no—
78C2×#217—21no—
79C2×#214—22no—
80#932—44no—
81C2×#213—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×#208—21no—
84C2×#208—21no—
85C3×#95—21no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88C3⋊#95D412yescomplemented
89#830—21no—
90#830—21no—
91#830—21no—
92#830D412yescomplemented
93#810—22no—
94#809—22no—
95#805—44no—
96C22×#41—31no—
97C2×AGL(1,ℤ/8)—61no—
98C2×#123—21no—
99C2×#121C2×D311yescharacteristic
100C2×#120—21no—
101C2×#120—61no—
102C22×C22⋊C4C2×D311yescharacteristic
103#449—62no—
104#448—61no—
105#447—61no—
106C2×#104—62no—
107#417—61no—
108#407—31no—
109#407—31no—
110#399—21no—
111#393—61no—
112#392—21no—
113#392—61no—
114#390—22no—
115#386—62no—
116#386—61no—
117#381—31no—
118C4×C22⋊C4C2×D311yescharacteristic
119#327—62no—
120#323—62no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—42no—
126C24×C3—42no—
127C24×C3—42no—
128C24×C3C2×D411yescharacteristic
129C23×D3—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—42no—
133C2×C3×D4—44no—
134C2×C3×D4—42no—
135C2×C3×D4—42no—
136C2×C3×D4—44no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—82no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C2×#67—24no—
154C2×#67—42no—
155C2×#67—41no—
156C2×#67—41no—
157C2×#67—44no—
158C2×#67—44no—
159C2×#67—82no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162#235—24no—
163#235—24no—
164#235—42no—
165#235—42no—
166#235—42no—
167#235—42no—
168#235—42no—
169#235—42no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—42no—
173C2×C4×D3—42no—
174C2×C4×D3—42no—
175C2×C4×D3—41no—
176C2×C4×D3—41no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—42no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—44no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—41no—
190C3×C22⋊C4—41no—
191C3×C22⋊C4—41no—
192C3×C42—42no—
193#217—44no—
194#217—42no—
195#214—44no—
196#213—44no—
197(C2×D3)⋊C4—82no—
198#208—42no—
199#208—42no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—41no—
202C25#6711yescharacteristic
203C2×#41—61no—
204C2×#41—61no—
205C2×#41—61no—
206C2×#41—61no—
207C2×#41—124no—
208C22×Q8—31no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—31no—
212C22×D4—61no—
213C22×D4—62no—
214C22×D4#6711yescharacteristic, complemented
215C23×C4—61no—
216C23×C4#6711yescharacteristic
217C23×C4C22×D311yescharacteristic
218AGL(1,ℤ/8)—124no—
219C2×SD16—62no—
220C2×D8—62no—
221C2×M16—61no—
222C2×M16—61no—
223#123—44no—
224#121—21no—
225#121—21no—
226#121—21no—
227#121—21no—
228#120—44no—
229#120—124no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4#6711yescharacteristic, complemented
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—62no—
245C2×C42#6712yescomplemented
246#104—124no—
247#103—62no—
248#102—62no—
249C22⋊C8—32no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253#95—21no—
254#95—32no—
255#95—62no—
256#95—61no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—81no—
277C23×C3—81no—
278C23×C3—88no—
279C23×C3#12011yescharacteristic
280C23×C3#12011yescharacteristic
281C23×C3#12111yescharacteristic
282C22×D3—22no—
283C22×D3—41no—
284C22×D3—42no—
285C22×D3—41no—
286C22×D3—42no—
287C3×D4—42no—
288C3×D4—81no—
289C3×D4—88no—
290C3×D4—82no—
291C3×D4—82no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—81no—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12711yescharacteristic
312#67—44no—
313#67—44no—
314#67—88no—
315C2×Dic3—22no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321D12—22no—
322D12—22no—
323D12—41no—
324D12—41no—
325C4×D3—44no—
326C4×D3—44no—
327Dic6—22no—
328Dic6—22no—
329Dic6—41no—
330Dic6—41no—
331C3⋊C8—42no—
332C3⋊C8—81no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—61no—
341C24C2×#6711yescharacteristic
342#41—64no—
343#41—64no—
344#41—122no—
345#41—122no—
346#41—122no—
347#41—122no—
348#41—122no—
349#41—122no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—122no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—64no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—82no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—121no—
369C2×D4—121no—
370C2×D4—124no—
371C2×D4—124no—
372C2×D4—242no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—122no—
383C22×C4—122no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4#23711yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392SD16—124no—
393D8—124no—
394M16—122no—
395M16—122no—
396C2×C8—62no—
397C2×C8—121no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—44no—
401C22⋊C4—24no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—124no—
411C22⋊C4—122no—
412C22⋊C4—242no—
413C42—42no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—42no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3#39211yescharacteristic
437C22×C3#45312yes—
438C22×C3B(4,𝔽2)12yes—
439C2×D3—44no—
440C2×D3—42no—
441C2×D3—82no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—44no—
449C3×C4—81no—
450C3×C4—81no—
451Dic3—42no—
452Dic3—82no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—62no—
473C23—81no—
474C23—81no—
475C23—88no—
476C23—121no—
477C23—122no—
478C23—121no—
479C23—122no—
480C23#93711yescharacteristic
481C23#93911yescharacteristic
482C23#95311yescharacteristic
483Q8—62no—
484Q8—62no—
485Q8—121no—
486Q8—121no—
487D4—42no—
488D4—62no—
489D4—62no—
490D4—81no—
491D4—88no—
492D4—82no—
493D4—82no—
494D4—121no—
495D4—121no—
496D4—124no—
497D4—124no—
498D4—248no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—62no—
515C2×C4—81no—
516C2×C4—124no—
517C2×C4—124no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4#93711yescharacteristic
524C2×C4#93811yescharacteristic
525C2×C4#93911yescharacteristic
526C2×C4#94011yescharacteristic
527C8—122no—
528C8—241no—
529C2×C3—22no—
530C2×C3—22no—
531C2×C3—41no—
532C2×C3—82no—
533C2×C3—161no—
534C2×C3#200211yescharacteristic
535C2×C3#201011yescharacteristic
536C2×C3#202111yescharacteristic
537D3—82no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—124no—
561C22—122no—
562C22—242no—
563C22—241no—
564C22—241no—
565C22#507712yes—
566C22#522311yescharacteristic
567C22#524412yes—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—44no—
573C4—81no—
574C4—81no—
575C4—122no—
576C4—242no—
577C3#993811yescharacteristic, complemented, Sylow
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—161no—
583C2—242no—
584C2#7171511yescharacteristic
585C2#7176911yescharacteristic
586C2#7185111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2539C2
socleC22×C3#392
hypercenter#2539D3
hypocenterC3#9938
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. C3×#2539C2semidirect
2. #82624C2semidirect
3. #82125C2semidirect
4. #71674C2semidirect
5. #71539C2semidirect
6. #69454C2semidirect
7. #67616C2semidirect
35. #2539D3semidirect
44. C22×C3×D4D4semidirect
54. C22×D12D4semidirect
56. C22×Dic6D4semidirect
64. C2×C3×C4⋊C4D4semidirect
75. C2×C3×C42D4semidirect
88. C3⋊#95D4semidirect
92. #830D4semidirect
214. C22×D4#67semidirect
231. C2×C4⋊C4#67semidirect
245. C2×C42#67semidirect
577. C3#9938semidirect

Maximal subgroups

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

subgroupindex
C3×#25392
#826242
#821252
#716742
#715392
#694542
#676162
#99383