Finite Group Databasebeta

← #10834015

Subgroups — #10834015

Order 768.

944 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82186C211yesmaximal, characteristic
2C2×#4733C211yesmaximal, characteristic
3C2×#4720C211yesmaximal, characteristic, complemented
4#73951C211yesmaximal, characteristic, complemented
5C3×#1852C211yesmaximal, characteristic, complemented
6#71559C211yesmaximal, characteristic, complemented
7#70643C211yesmaximal, characteristic, complemented
8#70639C211yesmaximal, characteristic, complemented
9#70629C211yesmaximal, characteristic, complemented
10#70627C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic
12#68864C211yesmaximal, characteristic, complemented
13#68863C211yesmaximal, characteristic, complemented
14#68855C211yesmaximal, characteristic, complemented
15#68853C211yesmaximal, characteristic, complemented
16#24515—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22C22×#204C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27#5213—21no—
28#5213—21no—
29C2×#830C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38#4987—21no—
39#4987—21no—
40#4984C2211yescharacteristic
41#4984C2211yescharacteristic
42#4973C2211yescharacteristic
43#4973C2211yescharacteristic
44#4973C2211yescharacteristic
45#4973C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52#4953—21no—
53#4953—21no—
54#4733—24no—
55#4720—24no—
56#4661—21no—
57#4659—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#452—31no—
63C2×#433—31no—
64#2625—31no—
65#2311—31no—
66#2068—31no—
67#2067—31no—
68#2056—31no—
69#2055—31no—
70#1973—31no—
71#1969—31no—
72#1935—31no—
73#1934—31no—
74#1872—31no—
75#1852D311yescharacteristic, complemented
76#1788—31no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C22×#67—21no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235—41no—
85C2×#235—41no—
86C22×D12C2311yescharacteristic
87C22×C4×D3—21no—
88C22×Dic6C2311yescharacteristic
89C22×Dic6C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—41no—
94C2×C3×C4⋊C4—41no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×#217—21no—
101C2×#217D411yescharacteristic
102C2×#217D411yescharacteristic
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107#926—22no—
108#926—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114#924—22no—
115#924—22no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C2×Dic12—22no—
122C2×Dic12—22no—
123C2×Dic12—41no—
124C2×#204—22no—
125C2×#204—22no—
126C2×#204—41no—
127C3×#107—41no—
128C3×#107—41no—
129C3×C22⋊C8D414yes—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#830—22no—
142#821—22no—
143#816—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148#810—22no—
149#810—41no—
150#810—41no—
151#810—41no—
152#809—22no—
153#809—41no—
154C22×#41—31no—
155C22×Q16—31no—
156C22×SD16—31no—
157C2×C22⋊Q8—31no—
158C2×#117—31no—
159C22×C4⋊C4C2×D311yescharacteristic
160#452—64no—
161#433—64no—
162C2×#107C2×D311yescharacteristic
163C2×#107C2×D311yescharacteristic
164C2×C4⋊C8—31no—
165C2×C4⋊C8—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175C2×C22⋊C8C2×D311yescharacteristic
176C2×#97—31no—
177C2×C4×C8—31no—
178#400—61no—
179#398—61no—
180#397—61no—
181#397—61no—
182#391—31no—
183#391—31no—
184#391—31no—
185#391—31no—
186#390—31no—
187#390—31no—
188#389—31no—
189#389—31no—
190#386—61no—
191#386—61no—
192#380—61no—
193#380—61no—
194#380C2×D311yescharacteristic
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—81no—
211C22×C3×C4—81no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4#4112yes—
215C22×C3×C4C2411yescharacteristic
216C2×#67—24no—
217C2×#67—42no—
218C2×#67—42no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226#235—44no—
227#235—44no—
228#235—44no—
229#235—44no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12#4112yes—
236C2×C4×D3—24no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6#4112yes—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8C2×D412yes—
256C2×C3×C8C2×D412yes—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—42no—
266C3×C22⋊C4—81no—
267#217—22no—
268#217—22no—
269#217—42no—
270(C2×D3)⋊C4—81no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273(C2×D3)⋊C4—81no—
274C4⋊Dic3—22no—
275C4⋊Dic3—22no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—81no—
279C4⋊Dic3—81no—
280Dic3⋊C4—42no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291Dic12—44no—
292#204—44no—
293C2×#41—62no—
294C2×#41—62no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—121no—
298C2×#41—121no—
299C22×Q8—31no—
300C22×Q8—31no—
301C22×D4—31no—
302C22×D4—61no—
303C23×C4—21no—
304C23×C4—61no—
305C23×C4C22×D311yescharacteristic
306C2×Q16—62no—
307C2×Q16—62no—
308C2×Q16—121no—
309C2×SD16—62no—
310C2×SD16—62no—
311C2×SD16—121no—
312C22×C8—31no—
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C22×C8C22×D311yescharacteristic
316C22⋊Q8—62no—
317C22⋊Q8—62no—
318#117—62no—
319#117—62no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—41no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—31no—
335C2×C22⋊C4—31no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—31no—
342C2×C42—31no—
343#107—41no—
344#107—41no—
345C4⋊C8—62no—
346C4⋊C8—62no—
347#103—62no—
348#103—62no—
349#103—121no—
350#103—121no—
351#103—121no—
352#102—62no—
353#102—62no—
354#102—121no—
355C22⋊C8—62no—
356C22⋊C8D1214yes—
357#97—62no—
358C4×C8—62no—
359#95—21no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3#12112yes—
377C23×C3C22×D411yescharacteristic
378C22×D3—22no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—84no—
404C2×C3×C4#12112yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411#67—48no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433D12—22no—
434D12—22no—
435D12—41no—
436D12—41no—
437C4×D3—48no—
438Dic6—22no—
439Dic6—22no—
440Dic6—41no—
441Dic6—41no—
442Dic6—41no—
443Dic6—41no—
444Dic6—41no—
445Dic6—41no—
446C3×C8—22no—
447C3×C8—22no—
448C3⋊C8—41no—
449C3⋊C8—41no—
450C24—61no—
451C24D3×D411yescharacteristic
452#41—124no—
453#41—124no—
454#41—124no—
455#41—124no—
456C2×Q8—32no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×D4—32no—
469C2×D4—64no—
470C2×D4—61no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—61no—
474C2×D4—61no—
475C2×D4—122no—
476C2×D4—122no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—31no—
486C22×C4—31no—
487C22×C4—31no—
488C22×C4—31no—
489C22×C4—42no—
490C22×C4—64no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—122no—
497C22×C4—122no—
498C22×C4C2×D1212yes—
499C22×C4D3×D411yescharacteristic
500C22×C4#23912yes—
501C22×C4C23×D311yescharacteristic
502Q16—124no—
503SD16—124no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8C2×D1212yes—
515C2×C8C2×D1212yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—62no—
526C4⋊C4—62no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—122no—
530C4⋊C4—241no—
531C4⋊C4—241no—
532C22⋊C4—62no—
533C22⋊C4—62no—
534C22⋊C4—81no—
535C22⋊C4—122no—
536C22⋊C4—241no—
537C22⋊C4—241no—
538C22⋊C4—241no—
539C22⋊C4—241no—
540C42—61no—
541C42—61no—
542C42—61no—
543C42—61no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—22no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3C2×#12111yescharacteristic
552C22×C3#52511yescharacteristic
553C22×C3#53511yescharacteristic
554C22×C3#57611yescharacteristic
555C22×C3#57711yescharacteristic
556C22×C3#57811yescharacteristic
557C22×C3#57811yescharacteristic
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C3×C4—22no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4—81no—
570C3×C4#54611yescharacteristic
571C3×C4#54611yescharacteristic
572C3×C4#55111yescharacteristic
573C3×C4#55111yescharacteristic
574Dic3—42no—
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3—41no—
579Dic3—81no—
580Dic3—81no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—62no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23#89512yes—
595C23C2×D3×D411yescharacteristic
596Q8—62no—
597Q8—62no—
598Q8—121no—
599Q8—121no—
600Q8—121no—
601Q8—121no—
602Q8—121no—
603Q8—121no—
604D4—62no—
605D4—62no—
606D4—128no—
607D4—121no—
608D4—121no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—62no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—84no—
641C2×C4—128no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4#89512yes—
651C2×C4C22×D1211yescharacteristic
652C2×C4#100411yescharacteristic
653C2×C4#100411yescharacteristic
654C2×C4C2×#23911yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4D3×#4111yescharacteristic
657C8—22no—
658C8—22no—
659C8—121no—
660C8—121no—
661C2×C3—42no—
662C2×C3#241111yescharacteristic
663C2×C3#306111yescharacteristic
664C2×C3#306311yescharacteristic
665C2×C3#312011yescharacteristic
666C2×C3#312411yescharacteristic
667C2×C3#322311yescharacteristic
668C2×C3#322411yescharacteristic
669D3—82no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22C2×#89511yescharacteristic
685C22#559211yescharacteristic
686C22#565611yescharacteristic
687C22#575711yescharacteristic
688C22#576611yescharacteristic
689C22#577511yescharacteristic
690C22#577911yescharacteristic
691C4—22no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—121no—
697C4—121no—
698C4—122no—
699C4—121no—
700C4—121no—
701C4—241no—
702C4—241no—
703C4#555411yescharacteristic
704C4#555411yescharacteristic
705C4#563711yescharacteristic
706C4#563711yescharacteristic
707C3#2451511yescharacteristic, complemented, Sylow
708C2—42no—
709C2—242no—
710C2#7460211yescharacteristic
711C2#7915011yescharacteristic
712C2#7921411yescharacteristic
713C2#7999911yescharacteristic
714C2#8018011yescharacteristic
715C2#8127311yescharacteristic
716C2#8145611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1852D3
hypocenterC3#24515
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
3. C2×#4720C2semidirect
4. #73951C2semidirect
5. C3×#1852C2semidirect
6. #71559C2semidirect
7. #70643C2semidirect
8. #70639C2semidirect
9. #70629C2semidirect
10. #70627C2semidirect
12. #68864C2semidirect
13. #68863C2semidirect
14. #68855C2semidirect
15. #68853C2semidirect
75. #1852D3semidirect
707. C3#24515semidirect

Maximal subgroups

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

subgroupindex
#821862
C2×#47332
C2×#47202
#739512
C3×#18522
#715592
#706432
#706392
#706292
#706272
#695222
#688642
#688632
#688552
#688532
#245153