Finite Group Databasebeta

← #10698470

Subgroups — #10698470

Order 768.

632 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82199C211yesmaximal, characteristic
2C2×C3×#324C211yesmaximal, characteristic, complemented
3#71565C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic
5#69932C211yesmaximal, characteristic
6#69928C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#10396—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#929—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15#5213—21no—
16#5212C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#821—21no—
21C2×#817—21no—
22#4994—21no—
23#4994—21no—
24#4991—21no—
25#4981—21no—
26C3×#324—22no—
27#4551—21no—
28#4551—21no—
29#4470—22no—
30#2618—31no—
31#1874—31no—
32#1853—31no—
33C2×#327—31no—
34C2×#324D311yescharacteristic, complemented
35#1441—31no—
36#1294—31no—
37C23×C3×C4D411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C23×Dic3—21no—
43C22×D12D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×#217—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57#929—42no—
58#929—42no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—42no—
67C2×C4⋊Dic3D411yescharacteristic
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82C3⋊#95—42no—
83#821—42no—
84#817—42no—
85C2×#123—61no—
86C2×#121—31no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4C2×D311yescharacteristic
89C2×#106—61no—
90C2×#102—61no—
91C2×C22⋊C8C2×D311yescharacteristic
92C2×C22⋊C8C2×D311yescharacteristic
93#398—61no—
94#396—61no—
95#396—61no—
96#394—61no—
97#386—61no—
98#380—31no—
99C2×#95—31no—
100C2×#95—61no—
101C2×#95—61no—
102#340—61no—
103#340—61no—
104#327—62no—
105#324—22no—
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4SD1612yes—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—41no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—41no—
154C2×D12—41no—
155C2×C3×C8—22no—
156C2×C3×C8—41no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C2×C3×C8—41no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165#217—41no—
166#217—41no—
167(C2×D3)⋊C4—42no—
168(C2×D3)⋊C4—42no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—81no—
172(C2×D3)⋊C4—81no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—41no—
178C4⋊Dic3—41no—
179C4⋊Dic3—41no—
180C4⋊Dic3—41no—
181Dic3⋊C4—42no—
182Dic3⋊C4—42no—
183C22×D4—31no—
184C22×D4—61no—
185C23×C4—61no—
186C23×C4—61no—
187C23×C4D1211yescharacteristic
188C23×C4D1211yescharacteristic
189C23×C4C22×D311yescharacteristic
190C22×C8—21no—
191C22×C8—21no—
192#123—122no—
193#123—122no—
194#121—61no—
195#121—61no—
196#121—61no—
197#121—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—122no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218#106—122no—
219#102—122no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8#6712yes—
224#95—61no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—61no—
229#95—122no—
230#95—122no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3#12011yescharacteristic
236C23×C3#12111yescharacteristic
237C23×C3#12311yescharacteristic
238C23×C3C2×SD1611yescharacteristic
239C23×C3#13511yescharacteristic
240C23×C3AGL(1,ℤ/8)11yescharacteristic
241C23×C3AGL(1,ℤ/8)11yescharacteristic
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—81no—
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12311yescharacteristic
272C2×C3×C4#12411yescharacteristic
273#67—81no—
274#67—81no—
275#67—81no—
276#67—81no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299C2×Dic3—81no—
300D12—42no—
301D12—81no—
302C3×C8—42no—
303C3×C8—81no—
304C24—61no—
305C24C2×D1211yescharacteristic
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—121no—
311C2×D4—121no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4#20412yes—
347C22×C4#23511yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4#23711yescharacteristic
351C22×C4#23711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—122no—
372C4⋊C4—122no—
373C22⋊C4—122no—
374C22⋊C4—122no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—241no—
380C22⋊C4—241no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3#39411yescharacteristic
394C22×C3#44911yescharacteristic
395C22×C3#45011yescharacteristic
396C22×C3#45411yescharacteristic
397C22×C3#45611yescharacteristic
398C22×C3B(4,𝔽2)11yescharacteristic
399C22×C3#45811yescharacteristic
400C22×C3#46511yescharacteristic
401C22×C3#46911yescharacteristic
402C22×C3#48111yescharacteristic
403C22×C3#48211yescharacteristic
404C2×D3—81no—
405C2×D3—81no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—42no—
416C3×C4—81no—
417Dic3—82no—
418Dic3—82no—
419Dic3—81no—
420Dic3—81no—
421Dic3—81no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23—121no—
432C23—121no—
433C23#88211yescharacteristic
434C23#88611yescharacteristic
435C23C2×#20411yescharacteristic
436C23#90411yescharacteristic
437C23#90811yescharacteristic
438C23#90811yescharacteristic
439C23#93011yescharacteristic
440D4—122no—
441D4—241no—
442D4—241no—
443D4—241no—
444D4—241no—
445D4—241no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—81no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4#88311yescharacteristic
490C2×C4#88511yescharacteristic
491C2×C4#93811yescharacteristic
492C2×C4#93911yescharacteristic
493C8—42no—
494C8—81no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3#159711yescharacteristic
500C2×C3#160411yescharacteristic
501C2×C3#160911yescharacteristic
502C2×C3#161611yescharacteristic
503C2×C3#203411yescharacteristic
504C2×C3#203811yescharacteristic
505C2×C3#204411yescharacteristic
506D3—161no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—241no—
520C22—241no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22—241no—
525C22—241no—
526C22#473111yescharacteristic
527C22#473411yescharacteristic
528C22#473511yescharacteristic
529C22#473711yescharacteristic
530C22#474111yescharacteristic
531C22#474211yescharacteristic
532C22#475211yescharacteristic
533C22#475311yescharacteristic
534C22#496111yescharacteristic
535C22#511411yescharacteristic
536C22#513511yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—81no—
543C4—242no—
544C4—242no—
545C4—241no—
546C4—241no—
547C4—241no—
548C3#1039611yescharacteristic, complemented, Sylow
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—481no—
554C2#6813011yescharacteristic
555C2#6813311yescharacteristic
556C2#6813411yescharacteristic
557C2#6813711yescharacteristic
558C2#6993511yescharacteristic
559C2#7003211yescharacteristic
560C2#7010011yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenterC2×#324D3
hypocenterC3#10396
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×C3×#324C2semidirect
3. #71565C2semidirect
6. #69928C2semidirect
7. #67675C2semidirect
34. C2×#324D3semidirect
548. C3#10396semidirect

Maximal subgroups

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

subgroupindex
#821992
C2×C3×#3242
#715652
C2×#44702
#699322
#699282
#676752
#103963