Finite Group Databasebeta

← #10838947

Subgroups — #10838947

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4738C211yesmaximal, characteristic
2C2×C4×#204C211yesmaximal, characteristic, complemented
3#73584C211yesmaximal, characteristic
4C22⋊C4×Dic6C211yesmaximal, characteristic, complemented
5C3×#1923C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#70482C211yesmaximal, characteristic, complemented
8#69963C211yesmaximal, characteristic, complemented
9#69940C211yesmaximal, characteristic, complemented
10#69934C211yesmaximal, characteristic
11#69767C211yesmaximal, characteristic, complemented
12#68693C211yesmaximal, characteristic, complemented
13#68686C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68672C211yesmaximal, characteristic, complemented
16#24894—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×C4×C22⋊C4C2211yescharacteristic
29#5212C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4959—21no—
42#4956—21no—
43#4942—21no—
44Dic3×C22⋊C4—21no—
45#4939C2211yescharacteristic
46#4936C2211yescharacteristic
47#4929C2211yescharacteristic
48#4929C2211yescharacteristic
49#4928C2211yescharacteristic
50#4738—24no—
51C4×#204—24no—
52#4675—21no—
53#4664—21no—
54#4646—21no—
55#4646—21no—
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#448—31no—
63C2×C4×SD16—31no—
64#2420—31no—
65Q8×C22⋊C4—31no—
66#2097—31no—
67#2077—31no—
68#2055—31no—
69#2033—31no—
70#1933—31no—
71#1923D311yescharacteristic, complemented
72#1880—31no—
73#1877—31no—
74#1863—31no—
75#1805—31no—
76#1787—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C23×Dic3—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C22×Dic6—22no—
84C22×Dic6—22no—
85C22×Dic6—22no—
86C22×Dic6—22no—
87C22×Dic6—41no—
88C22×Dic6—41no—
89C22×Dic6C2311yescharacteristic
90C22×Dic6C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2311yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101#930—22no—
102#930—41no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×#204—22no—
118C2×#204—22no—
119C2×#204—22no—
120C2×#204—22no—
121C2×#204—41no—
122C2×#204D414yes—
123C4×D12—22no—
124C4×D12—41no—
125C4×Dic6—22no—
126C4×Dic6—22no—
127C4×Dic6—41no—
128C4×Dic6—41no—
129C3×C4⋊C8—41no—
130C3×C22⋊C8—22no—
131C3×C22⋊C8—41no—
132C3×C4×C8—22no—
133C3×#95D411yescharacteristic
134C3×#95D411yescharacteristic
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821—41no—
146#818—41no—
147#817—22no—
148#816—22no—
149#816—22no—
150#816—22no—
151#816—22no—
152#816—22no—
153#816—41no—
154C23×Q8—31no—
155C22×SD16—31no—
156C22×SD16—31no—
157C2×#121—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C2×C4×D4—31no—
161#448—64no—
162C4×SD16—64no—
163C2×#107—31no—
164C2×#106—31no—
165C2×C4⋊C8C2×D311yescharacteristic
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8C2×D311yescharacteristic
175C2×C22⋊C8C2×D311yescharacteristic
176C2×C4×C8C2×D311yescharacteristic
177#397—61no—
178#394—61no—
179#391—31no—
180#391—31no—
181#391—61no—
182#391—61no—
183#390—31no—
184#389—31no—
185#386—61no—
186#382—31no—
187#381—61no—
188#380—31no—
189#380—31no—
190#380—61no—
191C4×C22⋊C4—61no—
192C4×C22⋊C4C2×D311yescharacteristic
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#336C2×D311yescharacteristic
198#336C2×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×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—42no—
214C2×#67—42no—
215C2×#67—81no—
216C22×Dic3—21no—
217C22×Dic3—24no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12C2×D412yes—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—24no—
243C2×Dic6—21no—
244C2×Dic6—44no—
245C2×Dic6—44no—
246C2×Dic6—44no—
247C2×Dic6—42no—
248C2×Dic6—42no—
249C2×Dic6—44no—
250C2×Dic6C2×D412yes—
251C2×Dic6#4112yes—
252C2×Dic6#4112yes—
253C2×Dic6#4112yes—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8C2×D412yes—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272#217—42no—
273#217—81no—
274(C2×D3)⋊C4—42no—
275(C2×D3)⋊C4—42no—
276(C2×D3)⋊C4—81no—
277(C2×D3)⋊C4—81no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287C4⋊Dic3—21no—
288Dic3⋊C4—42no—
289Dic3⋊C4—42no—
290Dic3⋊C4—81no—
291C4×Dic3—42no—
292C4×Dic3—42no—
293#204—28no—
294#204—44no—
295C22×Q8—31no—
296C22×Q8—31no—
297C22×Q8—62no—
298C22×Q8—62no—
299C22×Q8—62no—
300C22×Q8—62no—
301C22×Q8—61no—
302C22×Q8—121no—
303C22×Q8—121no—
304C22×D4—31no—
305C22×D4—61no—
306C23×C4—61no—
307C23×C4—61no—
308C23×C4C22×D311yescharacteristic
309C2×SD16—34no—
310C2×SD16—62no—
311C2×SD16—62no—
312C2×SD16—62no—
313C2×SD16—62no—
314C2×SD16—121no—
315C22×C8C22×D311yescharacteristic
316C22×C8C22×D311yescharacteristic
317C22×C8C22×D311yescharacteristic
318C22×C8C22×D311yescharacteristic
319#121—62no—
320#121—121no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—121no—
324C4×Q8—121no—
325C4×D4—62no—
326C4×D4—121no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4D1211yescharacteristic
342C2×C22⋊C4D1211yescharacteristic
343C2×C42—61no—
344C2×C42—61no—
345C2×C42C22×D311yescharacteristic
346C2×C42C22×D311yescharacteristic
347#107—121no—
348#106—62no—
349C4⋊C8—41no—
350#103—62no—
351#103—62no—
352#103—62no—
353#103—62no—
354#103—62no—
355#103—121no—
356#102—62no—
357#102—121no—
358C22⋊C8—22no—
359C22⋊C8—41no—
360C4×C8—22no—
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—
370#95D1211yescharacteristic
371#95D1211yescharacteristic
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3#13512yes—
379C23×C3C22×D411yescharacteristic
380C22×D3—22no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—41no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
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—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4#13712yes—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412#67—84no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—48no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435D12—22no—
436D12—22no—
437D12—41no—
438D12—41no—
439C4×D3—84no—
440Dic6—22no—
441Dic6—22no—
442Dic6—22no—
443Dic6—22no—
444Dic6—22no—
445Dic6—48no—
446Dic6—41no—
447Dic6—41no—
448Dic6#12314yes—
449C3×C8—22no—
450C3×C8—41no—
451C3×C8—41no—
452C3×C8—41no—
453C24—61no—
454C24C2×D1211yescharacteristic
455C2×Q8—32no—
456C2×Q8—32no—
457C2×Q8—32no—
458C2×Q8—32no—
459C2×Q8—64no—
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×Q8—122no—
469C2×Q8—122no—
470C2×Q8—124no—
471C2×Q8—124no—
472C2×Q8—124no—
473C2×Q8—124no—
474C2×D4—32no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—122no—
481C2×D4—122no—
482C2×D4—241no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—61no—
490C22×C4—64no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4—122no—
500C22×C4—122no—
501C22×C4—241no—
502C22×C4C2×D1211yescharacteristic
503C22×C4C2×D1211yescharacteristic
504C22×C4C2×D1211yescharacteristic
505C22×C4C2×D1211yescharacteristic
506C22×C4C2×D1211yescharacteristic
507C22×C4C23×D311yescharacteristic
508SD16—68no—
509SD16—124no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8D3×D412yes—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—122no—
533C4⋊C4—122no—
534C4⋊C4—241no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—122no—
538C22⋊C4—122no—
539C22⋊C4—122no—
540C22⋊C4—241no—
541C22⋊C4—241no—
542C22⋊C4—241no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—122no—
548C42—122no—
549C22×C3—22no—
550C22×C3—22no—
551C22×C3—22no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—41no—
556C22×C3#52511yescharacteristic
557C22×C3D4211yescharacteristic
558C22×C3#54611yescharacteristic
559C22×C3C2×#13511yescharacteristic
560C22×C3C2×#13711yescharacteristic
561C22×C3#57711yescharacteristic
562C22×C3#57711yescharacteristic
563C2×D3—42no—
564C2×D3—42no—
565C2×D3—42no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C3×C4—22no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—41no—
574C3×C4—41no—
575C3×C4C2×#12311yescharacteristic
576C3×C4#53811yescharacteristic
577C3×C4#54611yescharacteristic
578C3×C4#55111yescharacteristic
579Dic3—24no—
580Dic3—42no—
581Dic3—81no—
582Dic3—81no—
583Dic3—81no—
584Dic3—81no—
585Dic3—81no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—62no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23#90412yes—
600C23C22×D1211yescharacteristic
601Q8—34no—
602Q8—62no—
603Q8—62no—
604Q8—62no—
605Q8—62no—
606Q8—62no—
607Q8—128no—
608Q8—121no—
609Q8—121no—
610D4—62no—
611D4—62no—
612D4—121no—
613D4—121no—
614D4—244no—
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—21no—
628C2×C4—21no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—128no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
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—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—244no—
656C2×C4#90912yes—
657C2×C4C2×D3×D411yescharacteristic
658C2×C4#100411yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4D3×#4111yescharacteristic
661C2×C4D3×#4111yescharacteristic
662C2×C4#100911yescharacteristic
663C8—22no—
664C8—41no—
665C8—41no—
666C8—41no—
667C2×C3—42no—
668C2×C3#241811yescharacteristic
669C2×C3#310811yescharacteristic
670C2×C3#312611yescharacteristic
671C2×C3#326611yescharacteristic
672C2×C3#328211yescharacteristic
673C2×C3#330111yescharacteristic
674C2×C3#332911yescharacteristic
675D3—82no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—122no—
684C22—122no—
685C22—122no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22#552411yescharacteristic
691C22D4×D1211yescharacteristic
692C22#555411yescharacteristic
693C22C2×#90411yescharacteristic
694C22C2×#90911yescharacteristic
695C22#575111yescharacteristic
696C22#575111yescharacteristic
697C4—22no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—64no—
703C4—122no—
704C4—241no—
705C4—241no—
706C4—241no—
707C4—241no—
708C4—241no—
709C4#561211yescharacteristic
710C4D3×#12311yescharacteristic
711C4#567811yescharacteristic
712C4#570511yescharacteristic
713C3#2489411yescharacteristic, complemented, Sylow
714C2—42no—
715C2—242no—
716C2#7584411yescharacteristic
717C2#7931111yescharacteristic
718C2#7939511yescharacteristic
719C2#7943511yescharacteristic
720C2#7945011yescharacteristic
721C2#7949511yescharacteristic
722C2#7956411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1923C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1923D3
hypocenterC3#24894
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
2. C2×C4×#204C2semidirect
4. C22⋊C4×Dic6C2semidirect
5. C3×#1923C2semidirect
6. #70486C2semidirect
7. #70482C2semidirect
8. #69963C2semidirect
9. #69940C2semidirect
11. #69767C2semidirect
12. #68693C2semidirect
13. #68686C2semidirect
14. #68683C2semidirect
15. #68672C2semidirect
71. #1923D3semidirect
713. C3#24894semidirect

Maximal subgroups

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

subgroupindex
C2×#47382
C2×C4×#2042
#735842
C22⋊C4×Dic62
C3×#19232
#704862
#704822
#699632
#699402
#699342
#697672
#686932
#686862
#686832
#686722
#248943