Finite Group Databasebeta

← #10844714

Subgroups — #10844714

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5163C211yesmaximal, characteristic
2C2×#4830C211yesmaximal, characteristic, complemented
3#76548C211yesmaximal, characteristic, complemented
4#76353C211yesmaximal, characteristic
5C3×#2055C211yesmaximal, characteristic, complemented
6#72049C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic, complemented
8#70280C211yesmaximal, characteristic, complemented
9Dic3×#103C211yesmaximal, characteristic, complemented
10#70193C211yesmaximal, characteristic, complemented
11#70157C211yesmaximal, characteristic
12#69676C211yesmaximal, characteristic, complemented
13#69644C211yesmaximal, characteristic, complemented
14#68853C211yesmaximal, characteristic, complemented
15#68796C211yesmaximal, characteristic, complemented
16#25295—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C2×#885—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5163—24no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#4988C2211yescharacteristic
42#4984C2211yescharacteristic
43#4975C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4961—21no—
48#4959—21no—
49#4957—21no—
50#4939C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4830—24no—
54#4661—21no—
55#4657—21no—
56#4647—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460—31no—
63C2×C8×Q8—31no—
64#2582—31no—
65#2548—31no—
66#2065—31no—
67#2065—31no—
68#2055—31no—
69#2055D311yescharacteristic, complemented
70#1968—31no—
71#1968—31no—
72#1957—31no—
73#1948—31no—
74#1934—31no—
75#1934—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×Dic6D411yescharacteristic
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—41no—
100#940—22no—
101#940—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#885—44no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×C4⋊C8—22no—
127C3×#103—22no—
128C3×#102—41no—
129C3×#95—21no—
130C3×#95—21no—
131#835—22no—
132#832—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#821—41no—
141#817—41no—
142Dic3⋊C8—22no—
143Dic3⋊C8D414yes—
144C8×Dic3—22no—
145C8×Dic3—22no—
146#809—22no—
147#808—41no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16C2×D311yescharacteristic
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155#460—64no—
156C8×Q8—64no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171C2×C4×C8—31no—
172C2×C4×C8—31no—
173#398—61no—
174#398—61no—
175#394—61no—
176#394—61no—
177#391—31no—
178#391—61no—
179#391C2×D311yescharacteristic
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#390C2×D311yescharacteristic
185#389—31no—
186#389—31no—
187#389—61no—
188#384—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—42no—
218C2×#67—81no—
219C2×#67—81no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6SD1614yescomplemented
236C3×SD16—44no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D412yes—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—81no—
249C3×C42—21no—
250C3×C42—21no—
251#217—42no—
252#217—81no—
253#215—44no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258C4⋊Dic3—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266C4⋊Dic3—81no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269Dic3⋊C4—42no—
270Dic3⋊C4—81no—
271Dic3⋊C4—81no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8#4112yes—
284C22×Q8—31no—
285C22×Q8C22×D311yescharacteristic
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4C22×D311yescharacteristic
290C2×SD16—22no—
291C2×SD16—22no—
292C2×SD16—41no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300#127—62no—
301#127—62no—
302#124—124no—
303C4×Q8—62no—
304C4×Q8—62no—
305C4×Q8—62no—
306C4×Q8—62no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#106—121no—
334#106—121no—
335C4⋊C8—22no—
336C4⋊C8—34no—
337C4⋊C8—62no—
338#103—22no—
339#103—62no—
340#102—41no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—121no—
346C4×C8—62no—
347C4×C8—62no—
348C4×C8—62no—
349#95—21no—
350#95—21no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3C22×D411yescharacteristic
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4C2×SD1612yes—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4H(5,𝔽2)11yescharacteristic
399C2×C3×C42−1+411yescharacteristic
400#67—84no—
401#67—84no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3#12112yes—
419C2×Dic3#12112yes—
420C2×Dic3C2×SD1612yes—
421C2×Dic3C2×SD1612yes—
422C2×Dic3#13512yes—
423D12—41no—
424D12—41no—
425D12—41no—
426D12—41no—
427Dic6—28no—
428C3×C8—22no—
429C3×C8—41no—
430C3⋊C8—22no—
431C3⋊C8—22no—
432C24—21no—
433C24—61no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—34no—
440C2×Q8—62no—
441C2×Q8—62no—
442C2×Q8#23712yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—241no—
460C2×D4—241no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—44no—
475SD16—124no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—32no—
482C2×C8—32no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8D3×D412yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4—241no—
504C4⋊C4—241no—
505C4⋊C4—241no—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—241no—
512C22⋊C4—241no—
513C22⋊C4—241no—
514C42—21no—
515C42—21no—
516C42—32no—
517C42—32no—
518C42—61no—
519C42—61no—
520C42—122no—
521C42—122no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3C2×#12111yescharacteristic
530C22×C3#54811yescharacteristic
531C22×C3#55011yescharacteristic
532C22×C3C22×SD1611yescharacteristic
533C22×C3C2×#13511yescharacteristic
534C22×C3#57611yescharacteristic
535C22×C3#57711yescharacteristic
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C3×C4—22no—
544C3×C4—42no—
545C3×C4—41no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4#52911yescharacteristic
549C3×C4#54211yescharacteristic
550C3×C4#54611yescharacteristic
551C3×C4#54811yescharacteristic
552Dic3—22no—
553Dic3—22no—
554Dic3—41no—
555Dic3—81no—
556Dic3—81no—
557Dic3—81no—
558Dic3#46514yes—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23C2×D3×D411yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—68no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—121no—
584D4—121no—
585D4—121no—
586D4—121no—
587D4—244no—
588D4—244no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4D3×SD1612yes—
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×#23711yescharacteristic
627C2×C4#100411yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4#101011yescharacteristic
631C8—22no—
632C8—41no—
633C8—62no—
634C8—62no—
635C2×C3—161no—
636C2×C3#242611yescharacteristic
637C2×C3C2×#46511yescharacteristic
638C2×C3#306211yescharacteristic
639C2×C3#336211yescharacteristic
640C2×C3#336511yescharacteristic
641C2×C3#339211yescharacteristic
642C2×C3#339611yescharacteristic
643D3—161no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22#559811yescharacteristic
659C22#559911yescharacteristic
660C22D3×#12111yescharacteristic
661C22C2×D3×SD1611yescharacteristic
662C22D3×#13511yescharacteristic
663C22#577411yescharacteristic
664C22#577811yescharacteristic
665C4—22no—
666C4—34no—
667C4—42no—
668C4—41no—
669C4—62no—
670C4—62no—
671C4—81no—
672C4—81no—
673C4—121no—
674C4—241no—
675C4—241no—
676C4—241no—
677C4#559111yescharacteristic
678C4#564911yescharacteristic
679C4#566211yescharacteristic
680C4#567711yescharacteristic
681C3#2529511yescharacteristic, complemented, Sylow
682C2—161no—
683C2—481no—
684C2#7690111yescharacteristic
685C2#8036911yescharacteristic
686C2#8042011yescharacteristic
687C2#8052511yescharacteristic
688C2#8058111yescharacteristic
689C2D3×#46511yescharacteristic
690C2#8080911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2055C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2055D3
hypocenterC3#25295
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
3. #76548C2semidirect
5. C3×#2055C2semidirect
6. #72049C2semidirect
7. #70483C2semidirect
8. #70280C2semidirect
9. Dic3×#103C2semidirect
10. #70193C2semidirect
12. #69676C2semidirect
13. #69644C2semidirect
14. #68853C2semidirect
15. #68796C2semidirect
69. #2055D3semidirect
235. C2×Dic6SD16semidirect
681. C3#25295semidirect

Maximal subgroups

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

subgroupindex
C2×#51632
C2×#48302
#765482
#763532
C3×#20552
#720492
#704832
#702802
Dic3×#1032
#701932
#701572
#696762
#696442
#688532
#687962
#252953