Finite Group Databasebeta

← #10695410

Subgroups — #10695410

Order 768.

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

normal subgroups only

This group has 812 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#82710C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic, complemented
3C2×#4674C211yesmaximal, characteristic, complemented
4C2×C3×#327C211yesmaximal, characteristic, complemented
5#69924C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9918—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C22×(C2×D3)⋊C4—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5223—21no—
17#5220—21no—
18C2×#832—21no—
19#5212C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×#830C2211yescharacteristic
22#5033—22no—
23#5033—22no—
24#4992—21no—
25C2×#807—21no—
26#4961—22no—
27#4674—22no—
28#4674—22no—
29C3×#327—22no—
30#4551—22no—
31#4468—22no—
32C2×#480—31no—
33#2617—31no—
34C2×#397—31no—
35#1823—31no—
36C2×#327D311yescharacteristic, complemented
37C2×#323—31no—
38#1294—31no—
39C24×D3D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4C2311yescharacteristic
44C22×#67—22no—
45C23×Dic3—21no—
46C23×Dic3D411yescharacteristic
47C22×C3×C8—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×#217—22no—
56#937—42no—
57#937—42no—
58C2×(C2×D3)⋊C4—24no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—42no—
67C2×(C2×D3)⋊C4—42no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3D412yes—
74C2×C4⋊Dic3D412yescomplemented
75C22×C3⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79C3×#95D412yescomplemented
80#832—24no—
81C3⋊#95—24no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yescomplemented
86#807—24no—
87C2×#121C2×D311yescharacteristic
88C2×#120—61no—
89C22×C4⋊C4—31no—
90C22×C22⋊C4—31no—
91C22×C22⋊C4—61no—
92#480—62no—
93#480—62no—
94C2×#107—61no—
95C2×#102—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98#397—62no—
99#397—61no—
100#397—62no—
101#394—62no—
102#392—61no—
103#386—61no—
104#380—31no—
105C2×#95—61no—
106C2×#95C2×D311yescharacteristic
107#340—62no—
108#327—22no—
109#323—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—21no—
116C23×D3—21no—
117C23×D3—21no—
118C23×D3—42no—
119C23×D3—42no—
120C23×D3D812yes—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—44no—
144C2×#67—82no—
145C2×#67—82no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—42no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157C22×Dic3—41no—
158C2×C3×C8—42no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—41no—
165#217—41no—
166#217—82no—
167(C2×D3)⋊C4—48no—
168(C2×D3)⋊C4—44no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—41no—
176C4⋊Dic3—41no—
177C4⋊Dic3—42no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—42no—
180C25—31no—
181C22×D4—21no—
182C22×D4—62no—
183C22×D4D1211yescharacteristic
184C23×C4—21no—
185C23×C4—31no—
186C23×C4—61no—
187C23×C4C22×D311yescharacteristic
188C22×C8—21no—
189C22×C8—61no—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#120—122no—
195#120—122no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4D1211yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—64no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—62no—
215C2×C22⋊C4—122no—
216C2×C22⋊C4—122no—
217#107—64no—
218#102—64no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—32no—
222C22⋊C8—61no—
223#95—21no—
224#95—64no—
225#95—62no—
226#95—61no—
227#95#6712yescomplemented
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3#12011yescharacteristic
236C23×C3#12011yescharacteristic
237C23×C3#12111yescharacteristic
238C23×C3C2×D811yescharacteristic
239C23×C3#13511yescharacteristic
240C23×C3AGL(1,ℤ/8)11yescharacteristic
241C23×C3AGL(1,ℤ/8)11yescharacteristic
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—22no—
245C22×D3—44no—
246C22×D3—44no—
247C22×D3—42no—
248C22×D3—42no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—41no—
255C22×D3—82no—
256C22×D3—82no—
257C22×D3—81no—
258C22×D3—81no—
259C3×D4—42no—
260C3×D4—81no—
261C3×D4—81no—
262C3×D4—81no—
263C3×D4—81no—
264C3×D4—81no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—82no—
278C2×C3×C4—82no—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12711yescharacteristic
283#67—88no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—41no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C3×C8—81no—
299C3⋊C8—42no—
300C24—21no—
301C24—32no—
302C24—61no—
303C24—61no—
304C24—61no—
305C24—61no—
306C24—61no—
307C24—61no—
308C24—122no—
309C24—122no—
310C24D3×D411yescharacteristic
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—124no—
321C2×D4—242no—
322C2×D4—242no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—61no—
331C22×C4—62no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—61no—
335C22×C4—122no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4C2×D1211yescharacteristic
343C22×C4#23511yescharacteristic
344C22×C4D3×D411yescharacteristic
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—62no—
351C2×C8—122no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C4⋊C4—121no—
358C4⋊C4—121no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—128no—
362C22⋊C4—124no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C22⋊C4—121no—
366C22⋊C4—121no—
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—242no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3#39211yescharacteristic
384C22×C3#44711yescharacteristic
385C22×C3#44711yescharacteristic
386C22×C3#44911yescharacteristic
387C22×C3#44911yescharacteristic
388C22×C3#45512yes—
389C22×C3#45812yes—
390C22×C3#46611yescharacteristic
391C22×C3#46911yescharacteristic
392C2×D3—44no—
393C2×D3—44no—
394C2×D3—42no—
395C2×D3—82no—
396C2×D3—82no—
397C2×D3—82no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—82no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—82no—
406C3×C4—81no—
407Dic3—82no—
408Dic3—82no—
409Dic3—81no—
410C23—22no—
411C23—22no—
412C23—41no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—62no—
418C23—62no—
419C23—62no—
420C23—124no—
421C23—122no—
422C23—124no—
423C23—122no—
424C23—121no—
425C23—121no—
426C23—122no—
427C23—121no—
428C23—121no—
429C23—122no—
430C23—242no—
431C23—241no—
432C23—242no—
433C23—241no—
434C23#88211yescharacteristic
435C23#88411yescharacteristic
436C23D3×D811yescharacteristic
437C23#91111yescharacteristic
438C23#91411yescharacteristic
439C23#91611yescharacteristic
440C23#93711yescharacteristic
441D4—42no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446D4—81no—
447D4—248no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—241no—
472C2×C4—241no—
473C2×C4—241no—
474C2×C4—241no—
475C2×C4—241no—
476C2×C4#88211yescharacteristic
477C2×C4#88411yescharacteristic
478C2×C4#93011yescharacteristic
479C2×C4#94011yescharacteristic
480C8—81no—
481C8—122no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—161no—
487C2×C3#160712yes—
488C2×C3#161212yes—
489C2×C3#199911yescharacteristic
490C2×C3#200411yescharacteristic
491C2×C3#202211yescharacteristic
492D3—82no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22—81no—
506C22—124no—
507C22—122no—
508C22—124no—
509C22—242no—
510C22—242no—
511C22—242no—
512C22—241no—
513C22—242no—
514C22—241no—
515C22#474212yes—
516C22#477311yescharacteristic
517C22#477511yescharacteristic
518C22#478111yescharacteristic
519C22#478211yescharacteristic
520C22#482512yes—
521C22#496011yescharacteristic
522C22#515611yescharacteristic
523C22#517211yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—82no—
529C4—81no—
530C4—242no—
531C4—242no—
532C4—241no—
533C3#991811yescharacteristic, complemented, Sylow
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—161no—
539C2—242no—
540C2#6828812yes—
541C2#6828912yes—
542C2#7003111yescharacteristic
543C2#7020411yescharacteristic
544C2#7038611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radical#82710C2
hypercenterC2×#327D3
hypocenterC3#9918
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#5033C2semidirect
3. C2×#4674C2semidirect
4. C2×C3×#327C2semidirect
5. #69924C2semidirect
6. C2×#4468C2semidirect
7. #67612C2semidirect
36. C2×#327D3semidirect
74. C2×C4⋊Dic3D4semidirect
79. C3×#95D4semidirect
85. #830D4semidirect
227. #95#67semidirect
533. C3#9918semidirect

Maximal subgroups

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

subgroupindex
#827102
C2×#50332
C2×#46742
C2×C3×#3272
#699242
C2×#44682
#676122
#99183