Finite Group Databasebeta

← #10695722

Subgroups — #10695722

Order 768.

1046 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1046 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
1C3×#2639C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic, complemented
3C2×#5036C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic, complemented
6#69426C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9963—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C22×#215—21no—
13C22×#213—21no—
14C22×(C2×D3)⋊C4—22no—
15C3×#400—21no—
16C3×#397—21no—
17C3×#394—21no—
18C3×#393—21no—
19C3×#386—22no—
20C2×C3×#95C2211yescharacteristic
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5046—22no—
25#5046—22no—
26#5036—22no—
27#5036—22no—
28C2×#809—21no—
29C2×#809—21no—
30#4538—22no—
31#4537—22no—
32C2×#450—31no—
33C2×#447—31no—
34#2639D311yescharacteristic, complemented
35C2×#328—31no—
36C2×#327—31no—
37#1438—31no—
38#1280—31no—
39C24×D3—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4D412yes—
44C23×C3×C4C2311yescharacteristic
45C23×Dic3—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12—44no—
52C22×D12D412yes—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×#215—24no—
69C2×#215—44no—
70C2×#213—24no—
71C2×#213—44no—
72C2×(C2×D3)⋊C4—44no—
73C2×(C2×D3)⋊C4—44no—
74C2×(C2×D3)⋊C4—44no—
75C22×C3⋊C8—21no—
76C22×C3⋊C8—21no—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95—21no—
80C3×#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830—21no—
85#830D412yes—
86#830D412yes—
87#809—24no—
88#809—24no—
89C23×D4—31no—
90C22×SD16—61no—
91C22×D8—61no—
92C2×C22⋊Q8C2×D311yescharacteristic
93C2×#121C2×D311yescharacteristic
94C22×C22⋊C4—62no—
95#450—62no—
96#450—62no—
97#447—62no—
98#447—62no—
99C2×#102—61no—
100C2×#102—61no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#400—21no—
104#397—21no—
105#394—21no—
106#393—21no—
107#386—22no—
108C2×#95—31no—
109C2×#95C2×D311yescharacteristic
110#328—62no—
111#327—62no—
112C24×C3—21no—
113C24×C3C2×D411yescharacteristic
114C23×D3—24no—
115C23×D3—22no—
116C23×D3—42no—
117C23×D3—42no—
118C23×D3—42no—
119C23×D3—41no—
120C23×D3—41no—
121C23×D3—42no—
122C23×D3—42no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—42no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—42no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—42no—
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—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—82no—
176C3×C22⋊C4—82no—
177#215—48no—
178#213—48no—
179(C2×D3)⋊C4—416no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—42no—
183C2×C3⋊C8—42no—
184C25—61no—
185C22×Q8#6711yescharacteristic
186C22×D4—21no—
187C22×D4—32no—
188C22×D4—61no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—61no—
192C22×D4—61no—
193C22×D4—124no—
194C22×D4#6711yescharacteristic
195C23×C4—61no—
196C23×C4#6712yes—
197C23×C4C22×D311yescharacteristic
198C2×SD16—64no—
199C2×SD16—124no—
200C2×D8—64no—
201C2×D8—124no—
202C22×C8—61no—
203C22×C8—61no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208#121—21no—
209#121—21no—
210#121—21no—
211#121—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4#6711yescharacteristic
214C2×C4⋊C4#6711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—124no—
220C2×C22⋊C4—124no—
221C2×C22⋊C4—124no—
222#102—64no—
223#102—64no—
224C22⋊C8—32no—
225C22⋊C8—32no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—61no—
233#95—61no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—22no—
237C23×C3—41no—
238C23×C3—42no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—42no—
242C23×C3#12011yescharacteristic
243C23×C3#12111yescharacteristic
244C23×C3#12711yescharacteristic
245C23×C3#13512yes—
246C23×C3AGL(1,ℤ/8)12yes—
247C22×D3—24no—
248C22×D3—44no—
249C22×D3—44no—
250C22×D3—42no—
251C22×D3—44no—
252C22×D3—44no—
253C22×D3—44no—
254C22×D3—44no—
255C22×D3—42no—
256C22×D3—44no—
257C22×D3—44no—
258C22×D3—82no—
259C22×D3—82no—
260C22×D3—82no—
261C22×D3—82no—
262C22×D3—82no—
263C22×D3—82no—
264C22×D3—81no—
265C22×D3—81no—
266C22×D3—81no—
267C22×D3—81no—
268C3×Q8—42no—
269C3×Q8—81no—
270C3×D4—42no—
271C3×D4—81no—
272C3×D4—82no—
273C3×D4—81no—
274C3×D4—81no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—44no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—82no—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12011yescharacteristic
295C2×C3×C4#12111yescharacteristic
296C2×C3×C4#12111yescharacteristic
297C2×Dic3—44no—
298C2×Dic3—42no—
299C2×Dic3—82no—
300C2×Dic3—81no—
301C2×Dic3—81no—
302D12—24no—
303D12—24no—
304D12—42no—
305D12—42no—
306D12—44no—
307D12—44no—
308C3⋊C8—42no—
309C3⋊C8—42no—
310C24—21no—
311C24—64no—
312C24—62no—
313C24—122no—
314C24—122no—
315C24—122no—
316C24—121no—
317C24—122no—
318C24—122no—
319C24—121no—
320C24C2×#6711yescharacteristic
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—64no—
332C2×D4—64no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—122no—
336C2×D4—122no—
337C2×D4—124no—
338C2×D4—124no—
339C2×D4—124no—
340C2×D4—124no—
341C2×D4—121no—
342C2×D4—121no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—62no—
353C22×C4—122no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—122no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4D3×D411yescharacteristic
360C22×C4#23711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363SD16—128no—
364D8—128no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—122no—
368C2×C8—122no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—1216no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3#39211yescharacteristic
396C22×C3#44912yes—
397C22×C3#45311yescharacteristic
398C22×C3#45411yescharacteristic
399C22×C3B(4,𝔽2)12yes—
400C22×C3#46312yes—
401C22×C3#49511yescharacteristic
402C22×C3#49611yescharacteristic
403C2×D3—48no—
404C2×D3—44no—
405C2×D3—84no—
406C2×D3—84no—
407C2×D3—84no—
408C2×D3—84no—
409C2×D3—84no—
410C2×D3—82no—
411C2×D3—82no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—82no—
417C3×C4—81no—
418C3×C4—81no—
419C3×C4—81no—
420Dic3—82no—
421C23—21no—
422C23—22no—
423C23—21no—
424C23—41no—
425C23—42no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—64no—
430C23—124no—
431C23—124no—
432C23—122no—
433C23—124no—
434C23—124no—
435C23—124no—
436C23—124no—
437C23—122no—
438C23—124no—
439C23—124no—
440C23—242no—
441C23—242no—
442C23—242no—
443C23—242no—
444C23—242no—
445C23—242no—
446C23—241no—
447C23—241no—
448C23—241no—
449C23—241no—
450C23#93811yescharacteristic
451C23#94011yescharacteristic
452C23#94912yes—
453C23#95012yes—
454C23#95311yescharacteristic
455Q8—42no—
456Q8—81no—
457D4—42no—
458D4—64no—
459D4—64no—
460D4—81no—
461D4—82no—
462D4—81no—
463D4—81no—
464D4—122no—
465D4—122no—
466D4—124no—
467D4—124no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—124no—
487C2×C4—122no—
488C2×C4—241no—
489C2×C4—242no—
490C2×C4—241no—
491C2×C4#93711yescharacteristic
492C2×C4#93711yescharacteristic
493C2×C4#93911yescharacteristic
494C2×C4#93911yescharacteristic
495C8—122no—
496C8—122no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—161no—
501C2×C3#159712yes—
502C2×C3#160012yes—
503C2×C3#200611yescharacteristic
504C2×C3#201211yescharacteristic
505C2×C3#202111yescharacteristic
506D3—84no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22—128no—
521C22—124no—
522C22—244no—
523C22—244no—
524C22—244no—
525C22—242no—
526C22—242no—
527C22—244no—
528C22—244no—
529C22#505712yes—
530C22#505811yescharacteristic
531C22#505911yescharacteristic
532C22#522311yescharacteristic
533C22#523812yes—
534C22#524111yescharacteristic
535C22#524412yes—
536C22#524511yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4—82no—
545C4—242no—
546C3#996311yescharacteristic, complemented, Sylow
547C2—21no—
548C2—22no—
549C2—21no—
550C2—161no—
551C2—244no—
552C2#7172811yescharacteristic
553C2#7185111yescharacteristic
554C2#7187911yescharacteristic
555C2#7188412yes—
556C2#7188812yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2639C2
socleC23×C3#120
prime radicalC2×#5036C2
hypercenter#2639D3
hypocenterC3#9963
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2639

Complements

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

NH ≅ G/Ntype
1. C3×#2639C2semidirect
2. C2×#5046C2semidirect
4. C2×#4538C2semidirect
5. C2×#4537C2semidirect
6. #69426C2semidirect
7. #67613C2semidirect
34. #2639D3semidirect
546. C3#9963semidirect

Maximal subgroups

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

subgroupindex
C3×#26392
C2×#50462
C2×#50362
C2×#45382
C2×#45372
#694262
#676132
#99633