Finite Group Databasebeta

← #10698515

Subgroups — #10698515

Order 768.

716 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82680C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic, complemented
3#71677C211yesmaximal, characteristic, complemented
4#69932C211yesmaximal, characteristic, complemented
5#69412C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#10403—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#939—21no—
11C2×#939—21no—
12C2×#930C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5223—21no—
17C2×#832—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#821—21no—
23#4993—21no—
24#4960—21no—
25#4949—21no—
26C3×#327—22no—
27#4551—21no—
28#4529—21no—
29#4451—22no—
30#2656—31no—
31#1874—31no—
32#1874—31no—
33C2×#327—31no—
34C2×#327D311yescharacteristic, complemented
35#1442—31no—
36#1280—31no—
37C22×C3×D4—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×#67—21no—
43C22×#67—21no—
44C23×Dic3—21no—
45C23×Dic3D411yescharacteristic
46C23×Dic3D411yescharacteristic
47C22×D12D411yescharacteristic
48C22×C3×C8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—21no—
57#939—42no—
58#939—42no—
59#939—42no—
60#939—42no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C22×C3⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×#95—21no—
78C3×#95D412yescomplemented
79#832—42no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84#830—21no—
85#830—21no—
86#821—42no—
87C2×#121—31no—
88C2×#121—61no—
89C2×#121—61no—
90C2×#121C2×D311yescharacteristic
91C22×C4⋊C4—31no—
92C2×#102—61no—
93C2×#102—61no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8C2×D311yescharacteristic
96#399—61no—
97#399—61no—
98#392—61no—
99#392—61no—
100C2×#95—31no—
101C2×#95—61no—
102C2×#95—61no—
103C2×#95C2×D311yescharacteristic
104#340—61no—
105#340—61no—
106#327—22no—
107#327—62no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C23×D3—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C2×#67—42no—
138C2×#67—42no—
139C2×#67—41no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—81no—
146C2×#67—81no—
147C2×#67—81no—
148C2×#67—81no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3—41no—
165C22×Dic3D812yes—
166C22×Dic3SD1612yes—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—41no—
170C2×D12—41no—
171C2×C3×C8—41no—
172C2×C3×C8—41no—
173C2×C3×C8—41no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178#217—42no—
179#217—42no—
180(C2×D3)⋊C4—42no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—41no—
183(C2×D3)⋊C4—41no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—41no—
189C4⋊Dic3—41no—
190Dic3⋊C4—42no—
191Dic3⋊C4—42no—
192C2×C3⋊C8—41no—
193C2×C3⋊C8—41no—
194C2×C3⋊C8—41no—
195C22×D4—21no—
196C22×D4—31no—
197C22×D4—61no—
198C22×D4—61no—
199C22×D4—61no—
200C22×D4D1211yescharacteristic
201C23×C4—21no—
202C23×C4—31no—
203C23×C4—31no—
204C23×C4—61no—
205C23×C4C22×D311yescharacteristic
206C22×C8—21no—
207C22×C8—61no—
208#121—21no—
209#121—21no—
210#121—21no—
211#121—21no—
212#121—61no—
213#121—61no—
214#121—61no—
215#121—61no—
216#121—122no—
217#121—122no—
218#121—122no—
219#121—122no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—122no—
226C2×C4⋊C4D1211yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—61no—
233#102—122no—
234#102—122no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—61no—
238C22⋊C8—61no—
239#95—21no—
240#95—61no—
241#95—61no—
242#95—122no—
243#95—122no—
244#95#6712yescomplemented
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3#12011yescharacteristic
257C23×C3#12111yescharacteristic
258C23×C3#12411yescharacteristic
259C23×C3C2×D811yescharacteristic
260C23×C3C2×SD1611yescharacteristic
261C23×C3AGL(1,ℤ/8)11yescharacteristic
262C23×C3AGL(1,ℤ/8)11yescharacteristic
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C3×D4—42no—
271C3×D4—81no—
272C3×D4—81no—
273C3×D4—81no—
274C3×D4—81no—
275C3×D4—81no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—81no—
291C2×C3×C4—81no—
292C2×C3×C4—81no—
293C2×C3×C4—81no—
294C2×C3×C4#12111yescharacteristic
295C2×C3×C4#12111yescharacteristic
296C2×C3×C4#12311yescharacteristic
297C2×C3×C4#12311yescharacteristic
298#67—84no—
299#67—84no—
300#67—81no—
301#67—81no—
302#67—81no—
303#67—81no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—81no—
319C2×Dic3—81no—
320C2×Dic3—81no—
321C2×Dic3—81no—
322C2×Dic3—81no—
323C2×Dic3—81no—
324D12—42no—
325D12—81no—
326C3×C8—81no—
327C3⋊C8—81no—
328C24—21no—
329C24—61no—
330C24D3×D411yescharacteristic
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—61no—
342C2×D4—61no—
343C2×D4—121no—
344C2×D4—121no—
345C2×D4—122no—
346C2×D4—122no—
347C2×D4—121no—
348C2×D4—121no—
349C2×D4—121no—
350C2×D4—121no—
351C2×D4—121no—
352C2×D4—121no—
353C2×D4—241no—
354C2×D4—241no—
355C2×D4—241no—
356C2×D4—241no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—121no—
376C22×C4—121no—
377C22×C4—121no—
378C22×C4—121no—
379C22×C4—121no—
380C22×C4—121no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4C2×D1211yescharacteristic
386C22×C4#23511yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4#23711yescharacteristic
389C22×C4#23711yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—121no—
395C2×C8—121no—
396C2×C8—121no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—121no—
404C4⋊C4—121no—
405C4⋊C4—122no—
406C4⋊C4—122no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—122no—
410C22⋊C4—122no—
411C22⋊C4—122no—
412C22⋊C4—122no—
413C22⋊C4—121no—
414C22⋊C4—121no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3#39411yescharacteristic
435C22×C3#44711yescharacteristic
436C22×C3#45011yescharacteristic
437C22×C3#45411yescharacteristic
438C22×C3#45511yescharacteristic
439C22×C3B(4,𝔽2)11yescharacteristic
440C22×C3#45811yescharacteristic
441C22×C3#45911yescharacteristic
442C22×C3#46011yescharacteristic
443C22×C3#48611yescharacteristic
444C22×C3#49011yescharacteristic
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C2×D3—81no—
449C2×D3—81no—
450C2×D3—81no—
451C2×D3—81no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—82no—
457C3×C4—81no—
458Dic3—42no—
459Dic3—42no—
460Dic3—82no—
461Dic3—81no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—121no—
474C23—121no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—121no—
479C23—121no—
480C23#88211yescharacteristic
481C23#88511yescharacteristic
482C23D3×D811yescharacteristic
483C23#91111yescharacteristic
484C23D3×SD1611yescharacteristic
485C23#91411yescharacteristic
486C23#93911yescharacteristic
487D4—42no—
488D4—81no—
489D4—81no—
490D4—81no—
491D4—81no—
492D4—81no—
493D4—122no—
494D4—241no—
495D4—244no—
496D4—244no—
497D4—241no—
498D4—241no—
499D4—241no—
500D4—241no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—241no—
534C2×C4—241no—
535C2×C4—241no—
536C2×C4—241no—
537C2×C4—241no—
538C2×C4—241no—
539C2×C4#88311yescharacteristic
540C2×C4#88611yescharacteristic
541C2×C4#93011yescharacteristic
542C2×C4#93811yescharacteristic
543C8—81no—
544C8—241no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—161no—
550C2×C3#159811yescharacteristic
551C2×C3#160111yescharacteristic
552C2×C3#160711yescharacteristic
553C2×C3#161611yescharacteristic
554C2×C3#204011yescharacteristic
555C2×C3#204111yescharacteristic
556C2×C3#204411yescharacteristic
557D3—161no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—241no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22—241no—
582C22—241no—
583C22—241no—
584C22#474111yescharacteristic
585C22#474211yescharacteristic
586C22#476011yescharacteristic
587C22#476411yescharacteristic
588C22#477311yescharacteristic
589C22#477611yescharacteristic
590C22#482511yescharacteristic
591C22#482711yescharacteristic
592C22#496111yescharacteristic
593C22#515011yescharacteristic
594C22#516311yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—82no—
600C4—81no—
601C4—122no—
602C4—122no—
603C4—241no—
604C4—242no—
605C3#1040311yescharacteristic, complemented, Sylow
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—161no—
611C2—481no—
612C2#6828611yescharacteristic
613C2#6828811yescharacteristic
614C2#6829311yescharacteristic
615C2#6829911yescharacteristic
616C2#7003211yescharacteristic
617C2#7017711yescharacteristic
618C2#7036211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radical#82680C2
hypercenterC2×#327D3
hypocenterC3#10403
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×C3×#327C2semidirect
3. #71677C2semidirect
4. #69932C2semidirect
5. #69412C2semidirect
6. C2×#4451C2semidirect
7. #67674C2semidirect
34. C2×#327D3semidirect
78. C3×#95D4semidirect
244. #95#67semidirect
605. C3#10403semidirect

Maximal subgroups

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

subgroupindex
#826802
C2×C3×#3272
#716772
#699322
#694122
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