Finite Group Databasebeta

← #10695721

Subgroups — #10695721

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
1C2×C3×#450C211yesmaximal, characteristic, complemented
2C2×#5217C211yesmaximal, characteristic
3#82824C211yesmaximal, characteristic, complemented
4#72131C211yesmaximal, characteristic, complemented
5C2×#4538C211yesmaximal, characteristic, complemented
6C2×#4470C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9963—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#217—22no—
13C22×#213—21no—
14C2×#930C2211yescharacteristic
15C3×#450—22no—
16C3×#450—22no—
17C2×C3×#102—21no—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5231—21no—
20#5217—22no—
21#5217—22no—
22C2×#832—21no—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4992—21no—
27#4991—21no—
28#4986—21no—
29#4958—22no—
30#4538—22no—
31#4470—22no—
32C2×#450D311yescharacteristic, complemented
33C2×#447—31no—
34#2639—31no—
35C2×#328—31no—
36C2×#327—31no—
37#1438—31no—
38#1280—31no—
39C25×C3—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—44no—
47C22×C3×D4D412yes—
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C23×Dic3—21no—
51C23×Dic3D412yes—
52C22×D12D411yescharacteristic
53C2×C3×SD16—24no—
54C2×C3×SD16—44no—
55C22×C3×C8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×#217—44no—
64C2×#217—44no—
65C2×#217—44no—
66C2×#213—24no—
67C2×#213—44no—
68#930—21no—
69#930—21no—
70#930—21no—
71#930—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3D411yescharacteristic
76C22×C3⋊C8—21no—
77C3×#102—24no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8D412yes—
80#832—24no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830D412yes—
89C23×D4C2×D311yescharacteristic
90C22×SD16—21no—
91C22×D8—61no—
92C2×C22⋊Q8C2×D311yescharacteristic
93C2×#121—31no—
94C22×C22⋊C4—62no—
95#450—22no—
96#450—22no—
97#447—62no—
98#447—62no—
99C2×#102—21no—
100C2×#102—61no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#400—61no—
104#397—61no—
105#394—61no—
106#393—61no—
107#386—62no—
108C2×#95—31no—
109C2×#95—31no—
110#328—62no—
111#327—62no—
112C24×C3—24no—
113C24×C3—22no—
114C24×C3—42no—
115C24×C3—42no—
116C24×C3—42no—
117C24×C3—41no—
118C24×C3—41no—
119C24×C3—42no—
120C24×C3—42no—
121C24×C3C2×D411yescharacteristic
122C23×D3—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—42no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—44no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—42no—
150C2×#67—42no—
151C22×Dic3—22no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—42no—
157C22×Dic3—42no—
158C22×Dic3—42no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—42no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—42no—
165C3×SD16—48no—
166C2×C3×C8—22no—
167C2×C3×C8—42no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—41no—
174#217—416no—
175#213—48no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—41no—
178(C2×D3)⋊C4—82no—
179(C2×D3)⋊C4—82no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—42no—
184C25—21no—
185C22×Q8#6711yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—31no—
192C22×D4—44no—
193C22×D4—61no—
194C22×D4#6712yes—
195C23×C4—32no—
196C23×C4—61no—
197C23×C4C22×D311yescharacteristic
198C2×SD16—24no—
199C2×SD16—44no—
200C2×D8—64no—
201C2×D8—124no—
202C22×C8—21no—
203C22×C8—61no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208#121—61no—
209#121—61no—
210#121—61no—
211#121—61no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4#6711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—124no—
220C2×C22⋊C4—124no—
221C2×C22⋊C4—124no—
222#102—24no—
223#102—64no—
224C22⋊C8—21no—
225C22⋊C8—32no—
226C22⋊C8—61no—
227C22⋊C8#6712yes—
228#95—62no—
229#95—62no—
230#95—61no—
231#95—61no—
232#95—61no—
233#95—61no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—22no—
237C23×C3—24no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—42no—
241C23×C3—44no—
242C23×C3—44no—
243C23×C3—44no—
244C23×C3—44no—
245C23×C3—42no—
246C23×C3—44no—
247C23×C3—44no—
248C23×C3—82no—
249C23×C3—82no—
250C23×C3—82no—
251C23×C3—82no—
252C23×C3—82no—
253C23×C3—82no—
254C23×C3—81no—
255C23×C3—81no—
256C23×C3—81no—
257C23×C3—81no—
258C23×C3#12011yescharacteristic
259C23×C3#12111yescharacteristic
260C23×C3#12711yescharacteristic
261C23×C3#13512yes—
262C23×C3AGL(1,ℤ/8)12yes—
263C22×D3—41no—
264C22×D3—42no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—42no—
268C3×Q8—42no—
269C3×Q8—81no—
270C3×D4—24no—
271C3×D4—24no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—44no—
275C3×D4—44no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4#12011yescharacteristic
288C2×C3×C4#12011yescharacteristic
289C2×C3×C4#12111yescharacteristic
290C2×C3×C4#12111yescharacteristic
291#67—82no—
292#67—81no—
293#67—81no—
294C2×Dic3—44no—
295C2×Dic3—44no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—82no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305C2×Dic3—82no—
306D12—42no—
307D12—81no—
308C3×C8—42no—
309C3⋊C8—42no—
310C24—24no—
311C24—22no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—41no—
318C24—41no—
319C24—61no—
320C24D3×D411yescharacteristic
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—122no—
341C2×D4—121no—
342C2×D4—121no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4—121no—
355C22×C4—122no—
356C22×C4—121no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4#23711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363SD16—48no—
364D8—128no—
365C2×C8—22no—
366C2×C8—42no—
367C2×C8—62no—
368C2×C8—122no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—1216no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—242no—
381C22⋊C4—242no—
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—48no—
391C22×C3—44no—
392C22×C3—84no—
393C22×C3—84no—
394C22×C3—84no—
395C22×C3—84no—
396C22×C3—84no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3#39211yescharacteristic
400C22×C3#44912yes—
401C22×C3#45311yescharacteristic
402C22×C3#45411yescharacteristic
403C22×C3B(4,𝔽2)12yes—
404C22×C3#46312yes—
405C22×C3#49511yescharacteristic
406C22×C3#49611yescharacteristic
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—81no—
417C3×C4—81no—
418Dic3—82no—
419Dic3—82no—
420Dic3—81no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—24no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—44no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—81no—
441C23—82no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—122no—
446C23—121no—
447C23—121no—
448C23—121no—
449C23—122no—
450C23#91412yes—
451C23#91612yes—
452C23#93711yescharacteristic
453C23#93911yescharacteristic
454C23#94011yescharacteristic
455Q8—42no—
456Q8—81no—
457D4—24no—
458D4—24no—
459D4—42no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—122no—
464D4—241no—
465D4—242no—
466D4—241no—
467D4—241no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—124no—
480C2×C4—124no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—242no—
488C2×C4—241no—
489C2×C4—241no—
490C2×C4—242no—
491C2×C4#93711yescharacteristic
492C2×C4#93811yescharacteristic
493C2×C4#93911yescharacteristic
494C2×C4#95311yescharacteristic
495C8—42no—
496C8—122no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—84no—
501C2×C3#159712yes—
502C2×C3#160012yes—
503C2×C3#200611yescharacteristic
504C2×C3#201211yescharacteristic
505C2×C3#202111yescharacteristic
506D3—161no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—48no—
516C22—44no—
517C22—84no—
518C22—84no—
519C22—84no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—242no—
525C22—242no—
526C22—241no—
527C22—241no—
528C22—241no—
529C22#503212yes—
530C22#516512yes—
531C22#516811yescharacteristic
532C22#517212yes—
533C22#517611yescharacteristic
534C22#519811yescharacteristic
535C22#520111yescharacteristic
536C22#522311yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—81no—
542C4—81no—
543C4—242no—
544C4—242no—
545C4—241no—
546C3#996311yescharacteristic, complemented, Sylow
547C2—21no—
548C2—22no—
549C2—21no—
550C2—84no—
551C2—481no—
552C2#7097712yes—
553C2#7097812yes—
554C2#7167111yescharacteristic
555C2#7205511yescharacteristic
556C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#450C2
socleC23×C3#120
prime radicalC2×#5217C2
hypercenterC2×#450D3
hypocenterC3#9963
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C2×C3×#450C2semidirect
3. #82824C2semidirect
4. #72131C2semidirect
5. C2×#4538C2semidirect
6. C2×#4470C2semidirect
7. #67674C2semidirect
32. C2×#450D3semidirect
546. C3#9963semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4502
C2×#52172
#828242
#721312
C2×#45382
C2×#44702
#676742
#99633