Finite Group Databasebeta

← #10820041

Subgroups — #10820041

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5127C211yesmaximal, characteristic
2C2×#4830C211yesmaximal, characteristic, complemented
3#73481C211yesmaximal, characteristic
4C3×#1851C211yesmaximal, characteristic, complemented
5#71983C211yesmaximal, characteristic, complemented
6#70477C211yesmaximal, characteristic, complemented
7#70450C211yesmaximal, characteristic, complemented
8#70442C211yesmaximal, characteristic, complemented
9#69916C211yesmaximal, characteristic
10#69912C211yesmaximal, characteristic, complemented
11#69451C211yesmaximal, characteristic, complemented
12#68647C211yesmaximal, characteristic, complemented
13#68620C211yesmaximal, characteristic, complemented
14#68615C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#23210—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C3×C4×C22⋊C4C2211yescharacteristic
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5127—24no—
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#4961C411yescharacteristic
41#4959C411yescharacteristic
42#4952C411yescharacteristic
43#4947C411yescharacteristic
44C4×(C2×D3)⋊C4C2211yescharacteristic
45C4×(C2×D3)⋊C4C2211yescharacteristic
46#4932C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48C2×C4×C3⋊C8C2211yescharacteristic
49#4830—24no—
50#4675C411yescharacteristic
51#4673C411yescharacteristic
52#4660C411yescharacteristic
53#4658C411yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×C8×Q8—31no—
63C2×#435—31no—
64#2398—31no—
65#1985—31no—
66#1954—31no—
67#1924—31no—
68#1923—31no—
69#1923—31no—
70#1923—31no—
71#1851D311yescharacteristic, complemented
72#1850—31no—
73#1768—31no—
74#1768—31no—
75#1753—31no—
76C8×C22⋊C4—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67C2×C411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6C2×C411yescharacteristic
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C2×C3×M16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C22⋊C4C2×C411yescharacteristic
89C2×C3×C22⋊C4C2×C411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217C2×C411yescharacteristic
93C2×#217C2×C411yescharacteristic
94C2×#217C2×C411yescharacteristic
95C4×#67—22no—
96C4×#67—22no—
97C2×(C2×D3)⋊C4C2×C411yescharacteristic
98C2×(C2×D3)⋊C4C2×C411yescharacteristic
99C2×(C2×D3)⋊C4C2×C411yescharacteristic
100C2×(C2×D3)⋊C4C2×C411yescharacteristic
101C2×(C2×D3)⋊C4C2×C411yescharacteristic
102C2×C4⋊Dic3C2×C411yescharacteristic
103C2×Dic3⋊C4C2×C411yescharacteristic
104C2×Dic3⋊C4C2×C411yescharacteristic
105C2×Dic3⋊C4C2×C411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D3×C8—22no—
114C2×D3×C8—22no—
115C2×D3×C8—41no—
116C4×Dic6—22no—
117C4×Dic6—22no—
118C3×C4⋊C8—22no—
119C3×C4⋊C8—22no—
120C3×C22⋊C8—41no—
121C3×#95C2×C411yescharacteristic
122C3×#95C2×C411yescharacteristic
123C3⋊#95C2×C411yescharacteristic
124C3⋊#95C2×C411yescharacteristic
125C3⋊#95C2×C411yescharacteristic
126C3⋊#95C2×C411yescharacteristic
127C3⋊#95C2×C411yescharacteristic
128C3⋊#95C2×C411yescharacteristic
129C3⋊#95C2×C411yescharacteristic
130C3⋊#95C2×C411yescharacteristic
131C3⋊#95C2×C411yescharacteristic
132C3⋊#95C2×C411yescharacteristic
133#830—22no—
134#830—22no—
135(C2×D3)⋊C8—22no—
136(C2×D3)⋊C8—22no—
137(C2×D3)⋊C8—41no—
138#815—22no—
139#815—41no—
140Dic3⋊C8—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8D414yes—
143C8×Dic3—22no—
144C8×Dic3—22no—
145C8×Dic3—41no—
146C4⋊(C3⋊C8)—22no—
147C4×C3⋊C8—22no—
148C22×M16C2×D311yescharacteristic
149C23×C8—31no—
150C2×C4×Q8—31no—
151C2×C4×D4—31no—
152C8×Q8—64no—
153#435—64no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×C4⋊C8—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×C22⋊C8—31no—
161C2×C22⋊C8—31no—
162C2×C22⋊C8—31no—
163C2×C22⋊C8—31no—
164C2×C22⋊C8—31no—
165C2×C22⋊C8C2×D311yescharacteristic
166C2×#97—31no—
167C2×#97—31no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171C2×C4×C8—31no—
172#400—31no—
173#400—31no—
174#397—31no—
175#397—31no—
176#394—31no—
177#394—31no—
178#393—31no—
179#393—31no—
180C4×C22⋊C4—31no—
181C4×C22⋊C4—31no—
182C4×C22⋊C4—31no—
183C4×C22⋊C4C2×D311yescharacteristic
184#376—31no—
185#336—31no—
186#336—31no—
187#336—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C24×C3C22×C411yescharacteristic
194C23×D3C22×C411yescharacteristic
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C22×C411yescharacteristic
202C22×C3×C4C22×C411yescharacteristic
203C22×C3×C4C22×C411yescharacteristic
204C22×C3×C4C22×C411yescharacteristic
205C22×C3×C4C22×C411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—41no—
210C22×Dic3C22×C411yescharacteristic
211C22×Dic3C22×C411yescharacteristic
212C22×Dic3C22×C411yescharacteristic
213C22×Dic3C22×C411yescharacteristic
214C22×Dic3C22×C411yescharacteristic
215C22×Dic3C22×C411yescharacteristic
216C22×Dic3C22×C411yescharacteristic
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C3×M16—44no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C2×C3×C8#4112yes—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—41no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245#217—22no—
246#217—22no—
247#217—41no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—22no—
253C4⋊Dic3—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256Dic3⋊C4—22no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3C2×D412yes—
262C4×Dic3#4112yes—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8#4112yes—
275D3×C8—44no—
276C22×Q8—31no—
277C22×D4—31no—
278C23×C4—31no—
279C23×C4C22×D311yescharacteristic
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—41no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—62no—
288C22×C8—62no—
289C22×C8—121no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4×Q8—62no—
295C4×Q8—62no—
296C4×D4—62no—
297C4×D4—62no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4C4×D311yescharacteristic
311C2×C22⋊C4C4×D311yescharacteristic
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—34no—
320C4⋊C8—62no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323C22⋊C8—41no—
324C22⋊C8—62no—
325C22⋊C8—62no—
326C22⋊C8—62no—
327C22⋊C8—62no—
328C22⋊C8—121no—
329#97—62no—
330#97—121no—
331C4×C8—62no—
332C4×C8—62no—
333C4×C8—62no—
334C4×C8—121no—
335#95—31no—
336#95—31no—
337#95—31no—
338#95—31no—
339#95—31no—
340#95—31no—
341#95—31no—
342#95—31no—
343#95—31no—
344#95—31no—
345#95C4×D311yescharacteristic
346#95C4×D311yescharacteristic
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3C23×C411yescharacteristic
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3#13112yes—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#13112yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387#67—44no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3C4×D412yes—
407C2×Dic3C4×D412yes—
408C2×Dic3#13112yes—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413Dic6—28no—
414C3×C8—22no—
415C3×C8—22no—
416C3×C8—41no—
417C3×C8—41no—
418C3⋊C8—22no—
419C3⋊C8—22no—
420C3⋊C8—41no—
421C3⋊C8—41no—
422C24—31no—
423C24C2×C4×D311yescharacteristic
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—121no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4C2×C4×D311yescharacteristic
450C22×C4C2×C4×D311yescharacteristic
451C22×C4C2×C4×D311yescharacteristic
452C22×C4C2×C4×D311yescharacteristic
453C22×C4C2×C4×D311yescharacteristic
454C22×C4C23×D311yescharacteristic
455M16—44no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—124no—
479C2×C8D3×D412yes—
480C2×C8#23712yes—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C22⋊C4—22no—
486C22⋊C4—41no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—62no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C22⋊C4—121no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—32no—
500C42—32no—
501C42—61no—
502C42—61no—
503C42—61no—
504C42—61no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3C2×C4×D411yescharacteristic
513C22×C3C4×#4111yescharacteristic
514C22×C3#52011yescharacteristic
515C22×C3C2×#13111yescharacteristic
516C22×C3C2×#13111yescharacteristic
517C22×C3C2×#13111yescharacteristic
518C22×C3#56811yescharacteristic
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C3×C4—22no—
527C3×C4—41no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4#54611yescharacteristic
532C3×C4#54811yescharacteristic
533C3×C4#55611yescharacteristic
534C3×C4#55911yescharacteristic
535Dic3—22no—
536Dic3—22no—
537Dic3—22no—
538Dic3—41no—
539Dic3—41no—
540Dic3—41no—
541Dic3—41no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—32no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23C22×C4×D311yescharacteristic
557Q8—68no—
558D4—124no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4#90712yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×#23711yescharacteristic
605C2×C4#100411yescharacteristic
606C2×C4#100711yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4D3×#4111yescharacteristic
609C8—22no—
610C8—22no—
611C8—41no—
612C8—41no—
613C8—62no—
614C8—62no—
615C8—121no—
616C8—121no—
617C2×C3—81no—
618C2×C3#236711yescharacteristic
619C2×C3#292911yescharacteristic
620C2×C3#293311yescharacteristic
621C2×C3#296611yescharacteristic
622C2×C3#296911yescharacteristic
623C2×C3#299311yescharacteristic
624C2×C3#299411yescharacteristic
625D3—42no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—62no—
634C22—62no—
635C22—62no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22#550411yescharacteristic
641C22C4×#23711yescharacteristic
642C22C4×D3×D411yescharacteristic
643C22C2×#90711yescharacteristic
644C22#574511yescharacteristic
645C22D3×#13111yescharacteristic
646C22D3×#13111yescharacteristic
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—62no—
653C4—62no—
654C4—62no—
655C4—121no—
656C4—121no—
657C4—121no—
658C4—121no—
659C4#559111yescharacteristic
660C4#563311yescharacteristic
661C4#564111yescharacteristic
662C4#566411yescharacteristic
663C3#2321011yescharacteristic, complemented, Sylow
664C2—81no—
665C2—122no—
666C2#7439811yescharacteristic
667C2#7905311yescharacteristic
668C2#7908411yescharacteristic
669C2#7945911yescharacteristic
670C2#7947811yescharacteristic
671C2#7951811yescharacteristic
672C2#7953911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1851C2
prime radicalC22×#67C2×C4
cosocleC22×C3×C4C24
hypercenter#1851D3
hypocenterC3#23210
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
4. C3×#1851C2semidirect
5. #71983C2semidirect
6. #70477C2semidirect
7. #70450C2semidirect
8. #70442C2semidirect
10. #69912C2semidirect
11. #69451C2semidirect
12. #68647C2semidirect
13. #68620C2semidirect
14. #68615C2semidirect
15. #68604C2semidirect
71. #1851D3semidirect
663. C3#23210semidirect

Maximal subgroups

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

subgroupindex
C2×#51272
C2×#48302
#734812
C3×#18512
#719832
#704772
#704502
#704422
#699162
#699122
#694512
#686472
#686202
#686152
#686042
#232103