Finite Group Databasebeta

← #10695551

Subgroups — #10695551

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×#2404C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic, complemented
3#82115C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71539C211yesmaximal, characteristic, complemented
6#69531C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9938—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#120—21no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×#953—21no—
13C2×#949—21no—
14C2×#930C2211yescharacteristic
15C2×#929—21no—
16C3×#393—21no—
17C3×#392—21no—
18C3×#386—21no—
19C3×#386—22no—
20C3×#381C2211yescharacteristic
21#5238—22no—
22C2×#837—22no—
23#5232—21no—
24#5227—21no—
25#5217—21no—
26C4×#217C2211yescharacteristic
27#5207C2211yescharacteristic
28#5207C2211yescharacteristic
29#4960—21no—
30#4957—22no—
31#4949—21no—
32#4525—22no—
33#4451—22no—
34#3006—31no—
35#2539—31no—
36#2404D311yescharacteristic, complemented
37#1789—31no—
38#1510—31no—
39#1462—31no—
40#1284—31no—
41C25×C3D411yescharacteristic
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—21no—
46C2×C3×#41—44no—
47C22×C3×Q8D411yescharacteristic, complemented
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4D411yescharacteristic, complemented
51C23×C3×C4—21no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×#67—21no—
55C22×#67—22no—
56C22×D12D411yescharacteristic, complemented
57C3×#120—44no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67#953—44no—
68C2×#217—21no—
69C2×#217—22no—
70#949—44no—
71C2×#215—22no—
72C2×#213—22no—
73#930—21no—
74#930—21no—
75#930—21no—
76#930—21no—
77#929—44no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3D411yescharacteristic, complemented
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3D412yescomplemented
83C2×#208—21no—
84C2×#208—21no—
85C3×#95—22no—
86C3×#95—21no—
87C3×#95D412yescomplemented
88#837—44no—
89#835—22no—
90#832—22no—
91C3⋊#95—21no—
92#830—21no—
93#830—21no—
94#830—21no—
95#830D412yescomplemented
96C22×#41C2×D311yescharacteristic
97C2×AGL(1,ℤ/8)—61no—
98C2×#123—61no—
99C2×#121—31no—
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—61no—
111#393—21no—
112#392—21no—
113#392—61no—
114#390—62no—
115#386—21no—
116#386—22no—
117#381C2×D311yescharacteristic
118C4×C22⋊C4—31no—
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—
130C3×#41—24no—
131C3×#41—24no—
132C3×#41—42no—
133C3×#41—42no—
134C3×#41—42no—
135C3×#41—42no—
136C3×#41—42no—
137C3×#41—42no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—42no—
141C2×C3×D4—24no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—82no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C2×#67—44no—
169C2×#67—42no—
170C2×#67—42no—
171C2×#67—44no—
172C2×#67—41no—
173C2×#67—41no—
174C2×#67—82no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—42no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—41no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—82no—
187#217—24no—
188#217—44no—
189#215—44no—
190#213—44no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—41no—
193(C2×D3)⋊C4—41no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196Dic3⋊C4—44no—
197C4×Dic3—42no—
198#208—42no—
199#208—42no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—41no—
202C25#6711yescharacteristic
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—21no—
206C2×#41—21no—
207C2×#41—44no—
208C22×Q8#6711yescharacteristic, complemented
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—31no—
212C22×D4—61no—
213C22×D4—62no—
214C22×D4#6711yescharacteristic, complemented
215C23×C4—21no—
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—124no—
224#121—61no—
225#121—61no—
226#121—61no—
227#121—61no—
228#120—44no—
229#120—124no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—62no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—61no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C42—32no—
246#104—124no—
247#103—62no—
248#102—62no—
249C22⋊C8—32no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253#95—22no—
254#95—21no—
255#95—61no—
256#95#6712yescomplemented
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—81no—
277C23×C3—81no—
278C23×C3—88no—
279C23×C3#12011yescharacteristic
280C23×C3#12011yescharacteristic
281C23×C3#12111yescharacteristic
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—42no—
285C22×D3—41no—
286C22×D3—42no—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—44no—
296C3×D4—44no—
297C3×D4—88no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—81no—
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12711yescharacteristic
320#67—88no—
321#67—82no—
322#67—82no—
323C2×Dic3—24no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329D12—42no—
330D12—81no—
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—24no—
343#41—24no—
344#41—42no—
345#41—42no—
346#41—42no—
347#41—42no—
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—42no—
353C2×D4—24no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—82no—
365C2×D4—122no—
366C2×D4—124no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4—124no—
370C2×D4—121no—
371C2×D4—121no—
372C2×D4—242no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—62no—
385C22×C4—61no—
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—61no—
399C4⋊C4—61no—
400C4⋊C4—124no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—64no—
408C22⋊C4—82no—
409C22⋊C4—124no—
410C22⋊C4—124no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C42—122no—
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—44no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
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—82no—
440C2×D3—82no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4—81no—
451Dic3—44no—
452Dic3—81no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—81no—
473C23—81no—
474C23—88no—
475C23—121no—
476C23—121no—
477C23—122no—
478C23—121no—
479C23—122no—
480C23#93711yescharacteristic
481C23#93911yescharacteristic
482C23#95311yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—41no—
486Q8—41no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—44no—
492D4—44no—
493D4—88no—
494D4—122no—
495D4—241no—
496D4—248no—
497D4—242no—
498D4—242no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—64no—
517C2×C4—81no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
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—82no—
534C2×C3#200211yescharacteristic
535C2×C3#201011yescharacteristic
536C2×C3#202111yescharacteristic
537D3—161no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—242no—
561C22—242no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22#503212yes—
566C22#522311yescharacteristic
567C22#524112yes—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—82no—
574C4—81no—
575C4—124no—
576C4—241no—
577C3#993811yescharacteristic, complemented, Sylow
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—82no—
583C2—481no—
584C2#7167111yescharacteristic
585C2#7171511yescharacteristic
586C2#7185611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2404C2
socleC22×C3#392
hypercenter#2404D3
hypocenterC3#9938
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C3×#2404C2semidirect
2. #84082C2semidirect
3. #82115C2semidirect
4. #72138C2semidirect
5. #71539C2semidirect
6. #69531C2semidirect
7. #67628C2semidirect
36. #2404D3semidirect
47. C22×C3×Q8D4semidirect
50. C22×C3×D4D4semidirect
56. C22×D12D4semidirect
80. C2×C4⋊Dic3D4semidirect
82. C2×C4×Dic3D4semidirect
87. C3×#95D4semidirect
95. #830D4semidirect
208. C22×Q8#67semidirect
214. C22×D4#67semidirect
256. #95#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×#24042
#840822
#821152
#721382
#715392
#695312
#676282
#99383