Finite Group Databasebeta

← #10855238

Subgroups — #10855238

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82358C211yesmaximal, characteristic
2C2×C4×#214C211yesmaximal, characteristic, complemented
3C2×#4762C211yesmaximal, characteristic
4#75981C211yesmaximal, characteristic, complemented
5C3×#1864C211yesmaximal, characteristic, complemented
6#70588C211yesmaximal, characteristic, complemented
7#70587C211yesmaximal, characteristic, complemented
8#70213C211yesmaximal, characteristic, complemented
9#70177C211yesmaximal, characteristic
10#70165C211yesmaximal, characteristic, complemented
11#69615C211yesmaximal, characteristic, complemented
12#68909C211yesmaximal, characteristic, complemented
13#68809C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic, complemented
15#68655C211yesmaximal, characteristic, complemented
16#26163—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C2×#929—21no—
21C22×#204C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5213—21no—
32C2×#821C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C4×#214—24no—
36#4988C2211yescharacteristic
37#4984C2211yescharacteristic
38#4976—21no—
39#4975—21no—
40#4975C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4959—21no—
47#4959—21no—
48#4939C2211yescharacteristic
49C4×Dic3⋊C4C2211yescharacteristic
50C2×C4×C3⋊C8C2211yescharacteristic
51#4762—24no—
52#4649—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460—31no—
63C2×C4×SD16—31no—
64#2445—31no—
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2088—31no—
68#2071—31no—
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1865—31no—
76#1864D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×#67—21no—
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×D4—22no—
86C3×C4×D4—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#940—22no—
96#940—41no—
97C2×#214—22no—
98C2×#214—22no—
99C2×#214D414yes—
100#929—44no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C2×#204—22no—
124C2×#204—22no—
125C2×#204—22no—
126#887—22no—
127#887—22no—
128#887—22no—
129#887—41no—
130C4×Dic6—22no—
131C4×Dic6—41no—
132C3×#106—41no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#95—21no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#821—41no—
142#816—22no—
143Dic3⋊C8—22no—
144#810—22no—
145#809—22no—
146#809—22no—
147#808—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#123—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×D4C2×D311yescharacteristic
157#460—64no—
158C4×SD16—64no—
159C2×#106—31no—
160C2×#106C2×D311yescharacteristic
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389—61no—
179#389—61no—
180#389C2×D311yescharacteristic
181#386—61no—
182#384—31no—
183#384—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C2×D412yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—81no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—42no—
215C2×#67—81no—
216C2×#67—81no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6C2×D412yes—
234C2×Dic6#4112yes—
235C2×Dic6#4112yes—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4#4112yes—
249C3×C22⋊C4—42no—
250C3×C42—21no—
251C3×C42—21no—
252#217—42no—
253#217—81no—
254#214—28no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—81no—
264Dic3⋊C4—24no—
265Dic3⋊C4—41no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—81no—
271Dic3⋊C4SD1614yescomplemented
272C4×Dic3—24no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8C2×D412yes—
285#204—44no—
286C22×Q8—31no—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4—61no—
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302#127—62no—
303#127—121no—
304#123—124no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—121no—
310C4×Q8—121no—
311C4×D4—22no—
312C4×D4—41no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#106—41no—
342#106—62no—
343C4⋊C8—62no—
344#103—62no—
345#103—62no—
346#102—22no—
347#102—22no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—121no—
352C4×C8—62no—
353#95—21no—
354#95—31no—
355#95—31no—
356#95—61no—
357#95—61no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3C22×D411yescharacteristic
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
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—84no—
392C2×C3×C4C2×SD1612yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399#67—84no—
400#67—84no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3C2×SD1612yes—
421C2×Dic3C2×SD1612yes—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426Dic6—22no—
427Dic6—22no—
428Dic6—22no—
429Dic6#12314yescomplemented
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—41no—
434C24—21no—
435C24—61no—
436C2×Q8—32no—
437C2×Q8—32no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C2×D4D3×D412yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—81no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477SD16—124no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—34no—
496C4⋊C4—64no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—81no—
502C4⋊C4—121no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4#23512yes—
510C22⋊C4—42no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C42—21no—
518C42—21no—
519C42—64no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—122no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3#52511yescharacteristic
534C22×C3D4211yescharacteristic
535C22×C3#54811yescharacteristic
536C22×C3C22×SD1611yescharacteristic
537C22×C3#57511yescharacteristic
538C22×C3#57611yescharacteristic
539C22×C3#57711yescharacteristic
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4C2×#12311yescharacteristic
553C3×C4#53911yescharacteristic
554C3×C4#54711yescharacteristic
555C3×C4#55311yescharacteristic
556Dic3—22no—
557Dic3—24no—
558Dic3—22no—
559Dic3—41no—
560Dic3—41no—
561Dic3—81no—
562Dic3—81no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23C2×D3×D411yescharacteristic
578Q8—34no—
579Q8—62no—
580Q8—62no—
581Q8—62no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586D4—121no—
587D4—121no—
588D4—121no—
589D4—121no—
590D4—244no—
591D4—244no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
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—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—62no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—84no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4D3×SD1612yes—
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4#100411yescharacteristic
632C2×C4#100711yescharacteristic
633C2×C4D3×#4111yescharacteristic
634C2×C4D3×#4111yescharacteristic
635C8—41no—
636C8—41no—
637C8—62no—
638C8—121no—
639C2×C3—82no—
640C2×C3#244311yescharacteristic
641C2×C3#310311yescharacteristic
642C2×C3#311911yescharacteristic
643C2×C3#326811yescharacteristic
644C2×C3D4×SD1611yescharacteristic
645C2×C3#336211yescharacteristic
646C2×C3#337111yescharacteristic
647D3—161no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22—241no—
661C22—241no—
662C22#554711yescharacteristic
663C22#559011yescharacteristic
664C22#563311yescharacteristic
665C22C2×D3×SD1611yescharacteristic
666C22#576811yescharacteristic
667C22#577411yescharacteristic
668C22#577511yescharacteristic
669C4—22no—
670C4—41no—
671C4—42no—
672C4—62no—
673C4—62no—
674C4—64no—
675C4—81no—
676C4—81no—
677C4—121no—
678C4—121no—
679C4—241no—
680C4—241no—
681C4#550711yescharacteristic
682C4#555811yescharacteristic
683C4#563811yescharacteristic
684C4D3×#12311yescharacteristic
685C3#2616311yescharacteristic, complemented, Sylow
686C2—82no—
687C2—481no—
688C2#7460811yescharacteristic
689C2#7965311yescharacteristic
690C2#7987311yescharacteristic
691C2#8012811yescharacteristic
692C2#8017111yescharacteristic
693C2#8109011yescharacteristic
694C2#8120311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1864D3
hypocenterC3#26163
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×C4×#214C2semidirect
4. #75981C2semidirect
5. C3×#1864C2semidirect
6. #70588C2semidirect
7. #70587C2semidirect
8. #70213C2semidirect
10. #70165C2semidirect
11. #69615C2semidirect
12. #68909C2semidirect
13. #68809C2semidirect
14. #68686C2semidirect
15. #68655C2semidirect
76. #1864D3semidirect
271. Dic3⋊C4SD16semidirect
429. Dic6#123semidirect
685. C3#26163semidirect

Maximal subgroups

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

subgroupindex
#823582
C2×C4×#2142
C2×#47622
#759812
C3×#18642
#705882
#705872
#702132
#701772
#701652
#696152
#689092
#688092
#686862
#686552
#261633