Finite Group Databasebeta

← #10695424

Subgroups — #10695424

Order 768.

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

normal subgroups only

This group has 856 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×#5034C211yesmaximal, characteristic, complemented
3C2×#4677C211yesmaximal, characteristic, complemented
4C2×C3×#327C211yesmaximal, characteristic, complemented
5#69925C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9920—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C22×(C2×D3)⋊C4—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C22×C4⋊Dic3—21no—
14C22×C4⋊Dic3C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5223—21no—
18#5220—21no—
19C2×#832—21no—
20#5212C2211yescharacteristic
21C2×#830C2211yescharacteristic
22#5034—22no—
23#5034—22no—
24#4992—21no—
25C2×#808—21no—
26#4961—22no—
27#4677—22no—
28#4677—22no—
29C3×#327—22no—
30#4551—22no—
31#4468—22no—
32C2×#481—31no—
33#2617—31no—
34C2×#399—31no—
35#1824—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—44no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—24no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—44no—
74C2×C4⋊Dic3D412yes—
75C2×C4⋊Dic3D412yescomplemented
76C22×C3⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×#95—21no—
80C3×#95D412yescomplemented
81#832—24no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yescomplemented
86#808—24no—
87C2×#121C2×D311yescharacteristic
88C2×#120—61no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—61no—
91C22×C22⋊C4—31no—
92C22×C22⋊C4—61no—
93#481—62no—
94#481—62no—
95C2×#106—61no—
96C2×#102—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#399—62no—
100#399—62no—
101#397—61no—
102#394—62no—
103#392—61no—
104#386—61no—
105#380—31no—
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×D3SD1612yes—
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—24no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
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—22no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C2×C3×C8—42no—
157C2×C3×C8—41no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162#217—41no—
163#217—41no—
164#217—82no—
165(C2×D3)⋊C4—48no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173C4⋊Dic3—48no—
174C4⋊Dic3—41no—
175C4⋊Dic3—41no—
176C4⋊Dic3—42no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—42no—
179C25—31no—
180C22×D4—21no—
181C22×D4—62no—
182C22×D4D1211yescharacteristic
183C23×C4—21no—
184C23×C4—31no—
185C23×C4—61no—
186C23×C4C22×D311yescharacteristic
187C22×C8—21no—
188C22×C8—61no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121—21no—
193#120—122no—
194#120—122no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—124no—
204C2×C4⋊C4D1211yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—62no—
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—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—124no—
217#106—64no—
218#102—64no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—32no—
222C22⋊C8—61no—
223#95—21no—
224#95—62no—
225#95—61no—
226#95#6712yescomplemented
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3#12011yescharacteristic
235C23×C3#12011yescharacteristic
236C23×C3#12111yescharacteristic
237C23×C3C2×SD1611yescharacteristic
238C23×C3#13511yescharacteristic
239C23×C3AGL(1,ℤ/8)11yescharacteristic
240C23×C3#13711yescharacteristic
241C22×D3—22no—
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—44no—
245C22×D3—42no—
246C22×D3—44no—
247C22×D3—42no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—42no—
252C22×D3—42no—
253C22×D3—41no—
254C22×D3—82no—
255C22×D3—82no—
256C22×D3—81no—
257C22×D3—81no—
258C3×D4—42no—
259C3×D4—81no—
260C3×D4—81no—
261C3×D4—81no—
262C3×D4—81no—
263C3×D4—81no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12711yescharacteristic
282#67—88no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—82no—
293C2×Dic3—82no—
294C2×Dic3—81no—
295C3×C8—81no—
296C3⋊C8—42no—
297C24—21no—
298C24—32no—
299C24—61no—
300C24—61no—
301C24—61no—
302C24—61no—
303C24—61no—
304C24—61no—
305C24—122no—
306C24—122no—
307C24D3×D411yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—124no—
318C2×D4—242no—
319C2×D4—242no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4—122no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4C2×D1211yescharacteristic
338C22×C4#23511yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—62no—
346C2×C8—122no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—122no—
352C4⋊C4—128no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—128no—
358C22⋊C4—124no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C22⋊C4—242no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—82no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3#39211yescharacteristic
380C22×C3#44811yescharacteristic
381C22×C3#44911yescharacteristic
382C22×C3#45011yescharacteristic
383C22×C3#45211yescharacteristic
384C22×C3#45312yes—
385C22×C3#45812yes—
386C22×C3#46511yescharacteristic
387C22×C3#46811yescharacteristic
388C2×D3—44no—
389C2×D3—44no—
390C2×D3—42no—
391C2×D3—82no—
392C2×D3—82no—
393C2×D3—82no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—82no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—44no—
402C3×C4—81no—
403Dic3—82no—
404Dic3—81no—
405Dic3—82no—
406C23—22no—
407C23—22no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—62no—
414C23—62no—
415C23—62no—
416C23—124no—
417C23—122no—
418C23—122no—
419C23—124no—
420C23—121no—
421C23—121no—
422C23—122no—
423C23—121no—
424C23—121no—
425C23—122no—
426C23—242no—
427C23—241no—
428C23—242no—
429C23—241no—
430C23#88211yescharacteristic
431C23#88411yescharacteristic
432C23#91111yescharacteristic
433C23#91211yescharacteristic
434C23D3×SD1611yescharacteristic
435C23#91511yescharacteristic
436C23#93711yescharacteristic
437D4—42no—
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—248no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—242no—
468C2×C4—242no—
469C2×C4—241no—
470C2×C4#88211yescharacteristic
471C2×C4#88411yescharacteristic
472C2×C4#93011yescharacteristic
473C2×C4#94011yescharacteristic
474C8—81no—
475C8—122no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—161no—
481C2×C3#161312yes—
482C2×C3#161812yes—
483C2×C3#200011yescharacteristic
484C2×C3#200211yescharacteristic
485C2×C3#202211yescharacteristic
486D3—82no—
487C22—21no—
488C22—21no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—82no—
498C22—82no—
499C22—81no—
500C22—124no—
501C22—122no—
502C22—124no—
503C22—242no—
504C22—242no—
505C22—242no—
506C22—241no—
507C22—242no—
508C22—241no—
509C22#474212yes—
510C22#477611yescharacteristic
511C22#477811yescharacteristic
512C22#478011yescharacteristic
513C22#478311yescharacteristic
514C22#482212yes—
515C22#496011yescharacteristic
516C22#515811yescharacteristic
517C22#517011yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—44no—
523C4—81no—
524C4—242no—
525C4—241no—
526C4—242no—
527C3#992011yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—161no—
533C2—242no—
534C2#6829012yes—
535C2#6829112yes—
536C2#7003111yescharacteristic
537C2#7020511yescharacteristic
538C2#7038511yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radical#82710C2
hypercenterC2×#327D3
hypocenterC3#9920
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×#5034C2semidirect
3. C2×#4677C2semidirect
4. C2×C3×#327C2semidirect
5. #69925C2semidirect
6. C2×#4468C2semidirect
7. #67612C2semidirect
36. C2×#327D3semidirect
75. C2×C4⋊Dic3D4semidirect
80. C3×#95D4semidirect
85. #830D4semidirect
226. #95#67semidirect
527. C3#9920semidirect

Maximal subgroups

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

subgroupindex
#827102
C2×#50342
C2×#46772
C2×C3×#3272
#699252
C2×#44682
#676122
#99203