Finite Group Databasebeta

← #10695445

Subgroups — #10695445

Order 768.

982 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#82788C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic
3C2×#4672C211yesmaximal, characteristic
4C2×C3×#328C211yesmaximal, characteristic, complemented
5#69959C211yesmaximal, characteristic
6C2×#4469C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9923—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#215—21no—
12C22×(C2×D3)⋊C4—21no—
13C22×(C2×D3)⋊C4—21no—
14C22×Dic3⋊C4C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5230—21no—
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#821—22no—
21#5046—22no—
22#5046—22no—
23#4985—21no—
24#4982—22no—
25C2×#809—21no—
26#4946—21no—
27#4943—21no—
28#4672—22no—
29#4672—22no—
30C3×#328—22no—
31#4469—22no—
32C2×#450—31no—
33#2627—31no—
34C2×#394—31no—
35#1786—31no—
36C2×#328D311yescharacteristic, complemented
37C2×#324—31no—
38#1294—31no—
39C24×D3—21no—
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—44no—
51C22×D12D412yes—
52C22×C3×C8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#215—24no—
62C2×#215—44no—
63C2×(C2×D3)⋊C4—24no—
64C2×(C2×D3)⋊C4—24no—
65C2×(C2×D3)⋊C4—42no—
66C2×(C2×D3)⋊C4—44no—
67C2×(C2×D3)⋊C4—42no—
68C2×C4⋊Dic3D412yes—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—21no—
75C2×Dic3⋊C4—22no—
76C22×C3⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×#95—21no—
80C3×#95D412yes—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—21no—
85#830—21no—
86#830D412yes—
87#821—44no—
88#809—24no—
89C23×D4—31no—
90C22×SD16—61no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C22×C4⋊C4—31no—
93C22×C22⋊C4—61no—
94C22×C22⋊C4—61no—
95#450—62no—
96#450—62no—
97C2×#102—61no—
98C2×#102—62no—
99C2×C22⋊C8—31no—
100C2×C22⋊C8C2×D311yescharacteristic
101#400—61no—
102#398—62no—
103#395—61no—
104#394—62no—
105#394—62no—
106#385—61no—
107#381—61no—
108#380—31no—
109C2×#95C2×D311yescharacteristic
110#328—22no—
111#324—62no—
112C24×C3C2×D411yescharacteristic
113C23×D3—24no—
114C23×D3—22no—
115C23×D3—42no—
116C23×D3—42no—
117C23×D3—42no—
118C23×D3—41no—
119C23×D3—41no—
120C23×D3—41no—
121C23×D3—41no—
122C23×D3—41no—
123C23×D3—41no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—42no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—42no—
152C22×Dic3SD1612yescomplemented
153C2×D12—24no—
154C2×D12—24no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—44no—
158C2×D12—44no—
159C2×D12—44no—
160C2×D12—42no—
161C2×D12—42no—
162C2×D12—44no—
163C2×D12—41no—
164C2×D12—41no—
165C2×C3×C8—42no—
166C2×C3×C8—41no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—41no—
173#217—41no—
174#217—41no—
175#215—48no—
176(C2×D3)⋊C4—48no—
177(C2×D3)⋊C4—48no—
178C4⋊Dic3—42no—
179Dic3⋊C4—41no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182Dic3⋊C4—41no—
183Dic3⋊C4—82no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—42no—
186C25—61no—
187C22×Q8D1211yescharacteristic
188C22×D4—32no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—61no—
192C22×D4—61no—
193C22×D4—61no—
194C22×D4—124no—
195C23×C4—21no—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4C22×D311yescharacteristic
199C2×SD16—64no—
200C2×SD16—124no—
201C22×C8—21no—
202C22×C8—61no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—61no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4D1211yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—122no—
222C2×C22⋊C4—122no—
223C2×C22⋊C4—124no—
224#102—64no—
225#102—124no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—32no—
229C22⋊C8—61no—
230#95—21no—
231#95—62no—
232#95—62no—
233#95—62no—
234#95—61no—
235#95#6712yes—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3#12011yescharacteristic
239C23×C3#12011yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3C2×SD1611yescharacteristic
242C23×C3#13511yescharacteristic
243C23×C3AGL(1,ℤ/8)11yescharacteristic
244C23×C3#13711yescharacteristic
245C22×D3—24no—
246C22×D3—42no—
247C22×D3—44no—
248C22×D3—44no—
249C22×D3—44no—
250C22×D3—44no—
251C22×D3—44no—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—44no—
258C22×D3—82no—
259C22×D3—82no—
260C22×D3—82no—
261C22×D3—82no—
262C22×D3—82no—
263C22×D3—81no—
264C22×D3—81no—
265C22×D3—82no—
266C22×D3—81no—
267C22×D3—81no—
268C3×Q8—42no—
269C3×Q8—81no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—82no—
284C2×C3×C4—82no—
285C2×C3×C4#12011yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×C3×C4#12711yescharacteristic
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—81no—
303D12—24no—
304D12—24no—
305D12—42no—
306D12—44no—
307D12—42no—
308D12—44no—
309C3×C8—81no—
310C3⋊C8—42no—
311C24—64no—
312C24—62no—
313C24—122no—
314C24—122no—
315C24—122no—
316C24—121no—
317C24—121no—
318C24—121no—
319C24—121no—
320C24—121no—
321C24—121no—
322C24D3×D411yescharacteristic
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—42no—
326C2×D4—64no—
327C2×D4—64no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—122no—
331C2×D4—122no—
332C2×D4—124no—
333C2×D4—124no—
334C2×D4—124no—
335C2×D4—124no—
336C2×D4—121no—
337C2×D4—121no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—32no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—122no—
358C22×C4C2×D1211yescharacteristic
359C22×C4#23511yescharacteristic
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4D3×D411yescharacteristic
363C22×C4C2×#6711yescharacteristic
364SD16—128no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—62no—
368C2×C8—122no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—122no—
374C4⋊C4—121no—
375C4⋊C4—121no—
376C4⋊C4—121no—
377C4⋊C4—121no—
378C4⋊C4—242no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—128no—
382C22⋊C4—128no—
383C22⋊C4—121no—
384C22⋊C4—121no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3#39211yescharacteristic
394C22×C3#44811yescharacteristic
395C22×C3#44911yescharacteristic
396C22×C3#45011yescharacteristic
397C22×C3#45211yescharacteristic
398C22×C3#45412yes—
399C22×C3B(4,𝔽2)12yes—
400C22×C3#46511yescharacteristic
401C22×C3#46811yescharacteristic
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—84no—
408C2×D3—84no—
409C2×D3—84no—
410C2×D3—84no—
411C2×D3—82no—
412C2×D3—82no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—82no—
418C3×C4—81no—
419C3×C4—81no—
420Dic3—42no—
421Dic3—82no—
422Dic3—81no—
423C23—22no—
424C23—22no—
425C23—64no—
426C23—122no—
427C23—124no—
428C23—124no—
429C23—124no—
430C23—124no—
431C23—124no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23—122no—
436C23—122no—
437C23—124no—
438C23—242no—
439C23—242no—
440C23—242no—
441C23—242no—
442C23—242no—
443C23—241no—
444C23—241no—
445C23—241no—
446C23—241no—
447C23—242no—
448C23#88211yescharacteristic
449C23#88411yescharacteristic
450C23D3×SD1611yescharacteristic
451C23#91411yescharacteristic
452C23#91811yescharacteristic
453C23#91911yescharacteristic
454C23#93711yescharacteristic
455Q8—42no—
456Q8—81no—
457D4—64no—
458D4—64no—
459D4—124no—
460D4—122no—
461D4—122no—
462D4—124no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—241no—
492C2×C4#88211yescharacteristic
493C2×C4#88411yescharacteristic
494C2×C4#93011yescharacteristic
495C2×C4#94011yescharacteristic
496C8—81no—
497C8—122no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3#159712yes—
503C2×C3#159912yes—
504C2×C3#200011yescharacteristic
505C2×C3#200311yescharacteristic
506C2×C3#202111yescharacteristic
507D3—84no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—124no—
517C22—124no—
518C22—124no—
519C22—242no—
520C22—242no—
521C22—244no—
522C22—244no—
523C22—242no—
524C22—244no—
525C22—244no—
526C22—242no—
527C22#474112yes—
528C22#480611yescharacteristic
529C22#480711yescharacteristic
530C22#481011yescharacteristic
531C22#481211yescharacteristic
532C22#482712yes—
533C22#496011yescharacteristic
534C22#516911yescharacteristic
535C22#518711yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—82no—
541C4—81no—
542C4—81no—
543C4—122no—
544C4—241no—
545C4—242no—
546C3#992311yescharacteristic, complemented, Sylow
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—244no—
552C2#6832412yes—
553C2#6832512yes—
554C2#7003711yescharacteristic
555C2#7027211yescharacteristic
556C2#7038811yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC23×D12C22
hypercenterC2×#328D3
hypocenterC3#9923
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
1. #82788C2semidirect
4. C2×C3×#328C2semidirect
6. C2×#4469C2semidirect
7. #67612C2semidirect
36. C2×#328D3semidirect
152. C22×Dic3SD16semidirect
546. C3#9923semidirect

Maximal subgroups

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

subgroupindex
#827882
C2×#50462
C2×#46722
C2×C3×#3282
#699592
C2×#44692
#676122
#99233