Finite Group Databasebeta

← #10834019

Subgroups — #10834019

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
1C3×#2625C211yesmaximal, characteristic, complemented
2C2×#5047C211yesmaximal, characteristic
3#82270C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic, complemented
5#71692C211yesmaximal, characteristic, complemented
6#69681C211yesmaximal, characteristic, complemented
7#69680C211yesmaximal, characteristic, complemented
8#69679C211yesmaximal, characteristic, complemented
9#69677C211yesmaximal, characteristic, complemented
10#69635C211yesmaximal, characteristic, complemented
11#69630C211yesmaximal, characteristic, complemented
12#69616C211yesmaximal, characteristic, complemented
13#69615C211yesmaximal, characteristic, complemented
14#69522C211yesmaximal, characteristic
15#69504C211yesmaximal, characteristic, complemented
16#24515—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C3×#400—21no—
24C3×#398—21no—
25C3×#397—21no—
26C3×#397—21no—
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#5047—24no—
37#5000—24no—
38#4987—21no—
39#4987—21no—
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4958—21no—
49#4958—21no—
50#4939C2211yescharacteristic
51#4939C2211yescharacteristic
52#4929C2211yescharacteristic
53#4929C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#452—31no—
63C2×#433—31no—
64#2625D311yescharacteristic, complemented
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#1852—31no—
76#1788—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×#67—21no—
80C23×Dic3—21no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—22no—
84C2×#235—22no—
85C2×#235—41no—
86C2×#235—41no—
87C22×D12C2311yescharacteristic
88C22×C4×D3—21no—
89C22×Dic6C2311yescharacteristic
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×#117—22no—
93C3×#117—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×#216—22no—
109C2×#216—22no—
110C2×#216—41no—
111C2×#215—22no—
112C2×#215—22no—
113C2×#215—41no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—41no—
120C2×C4⋊Dic3—41no—
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—41no—
135#835—41no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141#830D414yes—
142#810—22no—
143#810—22no—
144#810—41no—
145#809—22no—
146#809—22no—
147#809—41no—
148#807—41no—
149#807—41no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)—22no—
152#803—22no—
153C4×C3⋊C8—22no—
154C22×#41—31no—
155C22×Q16—31no—
156C22×SD16—31no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×#117C2×D311yescharacteristic
159C22×C4⋊C4—31no—
160#452—64no—
161#433—64no—
162C2×#107—31no—
163C2×#107—31no—
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⋊C8—31no—
176C2×#97—31no—
177C2×C4×C8—31no—
178#400—21no—
179#398—21no—
180#397—21no—
181#397—21no—
182#391—31no—
183#391—31no—
184#391C2×D311yescharacteristic
185#391C2×D311yescharacteristic
186#390—31no—
187#390—31no—
188#389C2×D311yescharacteristic
189#389C2×D311yescharacteristic
190#386—61no—
191#386—61no—
192#380—61no—
193#380—61no—
194#380C2×D311yescharacteristic
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4112yes—
222C22×C3×C4C2411yescharacteristic
223C2×#67—24no—
224C2×#67—42no—
225C2×#67—42no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—42no—
230C22×Dic3—81no—
231C22×Dic3—81no—
232#235—44no—
233#235—44no—
234#235—44no—
235#235—44no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12#4112yes—
242C2×C4×D3—24no—
243C2×C4×D3—42no—
244C2×C4×D3—42no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6#4112yes—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—42no—
260C3×C4⋊C4—81no—
261C3×C4⋊C4—81no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—42no—
265C3×C22⋊C4—81no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270#216—44no—
271#215—44no—
272(C2×D3)⋊C4—81no—
273(C2×D3)⋊C4—81no—
274(C2×D3)⋊C4—81no—
275(C2×D3)⋊C4—81no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—42no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—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—
291C2×C3⋊C8C2×D412yes—
292C2×C3⋊C8C2×D412yes—
293C2×#41—62no—
294C2×#41—62no—
295C2×#41—62no—
296C2×#41—62no—
297C2×#41—121no—
298C2×#41—121no—
299C22×Q8—31no—
300C22×Q8C22×D311yescharacteristic
301C22×D4—31no—
302C22×D4—61no—
303C23×C4—61no—
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×C8—31no—
315C22×C8—31no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318#117—22no—
319#117—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—121no—
327C2×C4⋊C4—121no—
328C2×C4⋊C4#6711yescharacteristic
329C2×C4⋊C4#6711yescharacteristic
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4#6711yescharacteristic
340C2×C22⋊C4#6711yescharacteristic
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#107—121no—
344#107—121no—
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—34no—
356C22⋊C8—62no—
357#97—62no—
358C4×C8—62no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
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—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—41no—
388C3×Q8—41no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4#12112yes—
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4H(5,𝔽2)11yescharacteristic
425C2×C3×C4H(5,𝔽2)11yescharacteristic
426#67—48no—
427C2×Dic3—22no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—84no—
437D12—22no—
438D12—22no—
439D12—41no—
440D12—41no—
441C4×D3—48no—
442Dic6—22no—
443Dic6—22no—
444Dic6—41no—
445Dic6—41no—
446C3⋊C8—22no—
447C3⋊C8—22no—
448C3⋊C8—41no—
449C3⋊C8—41no—
450C24—61no—
451C24C2×#6711yescharacteristic
452#41—124no—
453#41—124no—
454#41—124no—
455#41—124no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—32no—
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—64no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4—241no—
493C22×C4—241no—
494C22×C4D3×D412yes—
495C22×C4#23712yes—
496C22×C4C2×#6711yescharacteristic
497C22×C4C2×#6711yescharacteristic
498C22×C4C2×#6711yescharacteristic
499C22×C4C2×#6711yescharacteristic
500C22×C4C2×#6711yescharacteristic
501C22×C4C23×D311yescharacteristic
502Q16—124no—
503SD16—124no—
504C2×C8—32no—
505C2×C8—32no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—81no—
530C4⋊C4—81no—
531C4⋊C4—122no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—42no—
535C22⋊C4—81no—
536C22⋊C4—241no—
537C22⋊C4—241no—
538C22⋊C4—241no—
539C22⋊C4—241no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
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—41no—
567C3×C4—41no—
568C3×C4—41no—
569C3×C4—41no—
570C3×C4—81no—
571C3×C4—81no—
572C3×C4—81no—
573C3×C4—81no—
574C3×C4#54611yescharacteristic
575C3×C4#54611yescharacteristic
576C3×C4#55111yescharacteristic
577C3×C4#55111yescharacteristic
578Dic3—42no—
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#93912yes—
595C23C22×#6711yescharacteristic
596Q8—41no—
597Q8—41no—
598Q8—41no—
599Q8—41no—
600Q8—62no—
601Q8—62no—
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—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—62no—
640C2×C4—128no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—244no—
650C2×C4#93912yes—
651C2×C4C2×D3×D411yescharacteristic
652C2×C4C2×#23711yescharacteristic
653C2×C4D3×#4111yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C2×C4#100911yescharacteristic
656C2×C4#100911yescharacteristic
657C8—62no—
658C8—62no—
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×#93911yescharacteristic
685C22#582811yescharacteristic
686C22#583011yescharacteristic
687C22#583511yescharacteristic
688C22#583711yescharacteristic
689C22#583811yescharacteristic
690C22#583811yescharacteristic
691C4—22no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4#565411yescharacteristic
704C4#565411yescharacteristic
705C4#565811yescharacteristic
706C4#565811yescharacteristic
707C3#2451511yescharacteristic, complemented, Sylow
708C2—42no—
709C2—242no—
710C2#8289411yescharacteristic
711C2#8340011yescharacteristic
712C2#8342111yescharacteristic
713C2#8344611yescharacteristic
714C2#8345211yescharacteristic
715C2#8346111yescharacteristic
716C2#8346311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2625D3
hypocenterC3#24515
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
1. C3×#2625C2semidirect
4. C2×#5000C2semidirect
5. #71692C2semidirect
6. #69681C2semidirect
7. #69680C2semidirect
8. #69679C2semidirect
9. #69677C2semidirect
10. #69635C2semidirect
11. #69630C2semidirect
12. #69616C2semidirect
13. #69615C2semidirect
15. #69504C2semidirect
64. #2625D3semidirect
707. C3#24515semidirect

Maximal subgroups

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

subgroupindex
C3×#26252
C2×#50472
#822702
C2×#50002
#716922
#696812
#696802
#696792
#696772
#696352
#696302
#696162
#696152
#695222
#695042
#245153