Finite Group Databasebeta

← #10823593

Subgroups — #10823593

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5518C211yesmaximal, characteristic
2#82235C211yesmaximal, characteristic
3C2×#4716C211yesmaximal, characteristic, complemented
4C2×C4×Dic12C211yesmaximal, characteristic, complemented
5C3×#1750C211yesmaximal, characteristic
6#69964C211yesmaximal, characteristic
7#69964C211yesmaximal, characteristic
8#69937C211yesmaximal, characteristic
9#69936C211yesmaximal, characteristic
10#69813C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#68690C211yesmaximal, characteristic, complemented
13#68688C211yesmaximal, characteristic, complemented
14#68634C211yesmaximal, characteristic, complemented
15#68631C211yesmaximal, characteristic, complemented
16#23586—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22C22×Dic12C2211yescharacteristic
23#5518—24no—
24#5518—24no—
25C2×#889—21no—
26C2×#889—21no—
27C2×#879—21no—
28C2×#879—21no—
29C2×#870C2211yescharacteristic
30C2×#869C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C2×C3×C4×C8C2211yescharacteristic
38#5212C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×#816C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#816C2211yescharacteristic
48C2×#816C2211yescharacteristic
49#4978—21no—
50#4976—21no—
51#4945—21no—
52#4944—21no—
53C4×C4⋊Dic3C2211yescharacteristic
54#4929C2211yescharacteristic
55#4928C2211yescharacteristic
56#4716—24no—
57C4×Dic12—24no—
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62#4551C2211yescharacteristic
63#4551C2211yescharacteristic
64C2×#541—31no—
65C2×#441—31no—
66C2×C4×Q16—31no—
67#2323—31no—
68#2059—31no—
69#2043—31no—
70#2036—31no—
71#2035—31no—
72#2026—31no—
73#2016—31no—
74#2001—31no—
75#2001—31no—
76#1763—31no—
77C4×#103—31no—
78#1750D311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—22no—
107#924—22no—
108#924—22no—
109#924—22no—
110#924—22no—
111#924—22no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×Dic12—22no—
126C2×Dic12—22no—
127C2×Dic12—22no—
128C2×Dic12—22no—
129C2×Dic12—22no—
130C2×Dic12—22no—
131#889—28no—
132#889—28no—
133#879—28no—
134#879—28no—
135#870—22no—
136#870—22no—
137#869—22no—
138#869—22no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C4×Dic6D414yes—
144C4×Dic6D414yes—
145C3×C22⋊C8—22no—
146C3×C22⋊C8—22no—
147C3×#97—22no—
148C3×C4×C8—22no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154#818—22no—
155#817—22no—
156#816—22no—
157#816—22no—
158#816—22no—
159#816—22no—
160#816—22no—
161#816—22no—
162#816—22no—
163#816—22no—
164C22×Q16—31no—
165C22×Q16—31no—
166#541—64no—
167#541—64no—
168C2×C4⋊Q8—31no—
169C2×#126—61no—
170C2×#126—61no—
171C2×#125—31no—
172C2×#125—61no—
173C2×#125—61no—
174C2×C22⋊Q8—31no—
175C2×C22⋊Q8—31no—
176C2×C22⋊Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×Q8—31no—
179C2×C4×Q8—31no—
180C2×#117C2×D311yescharacteristic
181#441—64no—
182C4×Q16—64no—
183C2×#107—31no—
184C2×#106—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103—31no—
192C2×#103—31no—
193C2×C22⋊C8C2×D311yescharacteristic
194C2×C22⋊C8C2×D311yescharacteristic
195C2×#97C2×D311yescharacteristic
196C2×C4×C8C2×D311yescharacteristic
197#391—31no—
198#389—31no—
199#389—61no—
200#388—61no—
201#388—61no—
202#387—61no—
203#380—31no—
204C4×C4⋊C4—31no—
205#340—31no—
206#340—31no—
207#340—31no—
208#340—31no—
209#336C2×D311yescharacteristic
210#336C2×D311yescharacteristic
211C24×C3C2×D411yescharacteristic
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6C2×D412yes—
248C2×Dic6C2×D412yes—
249C2×Dic6#4112yes—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
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#4112yes—
261C2×C3×C8#4112yes—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42C2×D412yes—
269C3×C42C2×D412yes—
270#217—24no—
271#217—24no—
272#217—24no—
273#217—81no—
274C4⋊Dic3—24no—
275C4⋊Dic3—24no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3C2×D412yes—
284C4⋊Dic3C2×D412yes—
285C4⋊Dic3#4112yes—
286Dic3⋊C4—24no—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—24no—
292Dic3⋊C4—24no—
293Dic3⋊C4—42no—
294Dic3⋊C4—81no—
295C4×Dic3—24no—
296C4×Dic3—24no—
297C4×Dic3—42no—
298C4×Dic3—81no—
299Dic12—28no—
300Dic12—28no—
301C22×Q8—31no—
302C22×Q8—31no—
303C22×Q8—31no—
304C23×C4C22×D311yescharacteristic
305C2×Q16—62no—
306C2×Q16—62no—
307C2×Q16—62no—
308C2×Q16—62no—
309C2×Q16—62no—
310C2×Q16—62no—
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C22×C8C22×D311yescharacteristic
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—62no—
316C4⋊Q8—62no—
317#126—68no—
318#126—68no—
319#125—62no—
320#125—62no—
321#125—68no—
322#125—68no—
323C22⋊Q8—62no—
324C22⋊Q8—62no—
325C22⋊Q8—62no—
326C22⋊Q8—62no—
327C22⋊Q8—62no—
328C22⋊Q8—62no—
329C4×Q8—34no—
330C4×Q8—34no—
331C4×Q8—62no—
332C4×Q8—62no—
333C4×Q8—62no—
334C4×Q8—62no—
335#117—22no—
336#117—22no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4D1211yescharacteristic
354C2×C4⋊C4D1211yescharacteristic
355C2×C22⋊C4—61no—
356C2×C22⋊C4—61no—
357C2×C22⋊C4—61no—
358C2×C22⋊C4—61no—
359C2×C22⋊C4D1211yescharacteristic
360C2×C22⋊C4D1211yescharacteristic
361C2×C42—61no—
362C2×C42—61no—
363C2×C42—61no—
364C2×C42—61no—
365C2×C42C22×D311yescharacteristic
366C2×C42C22×D311yescharacteristic
367#107—62no—
368#106—62no—
369#103—62no—
370#103—62no—
371#103—62no—
372#103—62no—
373#103—62no—
374#103—62no—
375#103—62no—
376#103—62no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379#97—22no—
380C4×C8—22no—
381#95—61no—
382#95—61no—
383#95—61no—
384#95—61no—
385#95—61no—
386C23×C3—21no—
387C23×C3—21no—
388C23×C3—21no—
389C23×C3—21no—
390C23×C3—21no—
391C23×C3—21no—
392C23×C3—21no—
393C23×C3C22×D411yescharacteristic
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—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#12012yes—
412C2×C3×C4#12012yes—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
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—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443Dic6—22no—
444Dic6—22no—
445Dic6—22no—
446Dic6—22no—
447Dic6—22no—
448Dic6—22no—
449Dic6—41no—
450Dic6—41no—
451Dic6—41no—
452Dic6—41no—
453Dic6—41no—
454Dic6—41no—
455C3×C8—22no—
456C3×C8—22no—
457C3×C8—41no—
458C3×C8—41no—
459C24C2×D1211yescharacteristic
460C2×Q8—32no—
461C2×Q8—32no—
462C2×Q8—32no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8—61no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4C2×D1211yescharacteristic
493C22×C4C2×D1211yescharacteristic
494C22×C4C2×D1211yescharacteristic
495C22×C4C2×D1211yescharacteristic
496C22×C4C2×D1211yescharacteristic
497C22×C4C23×D311yescharacteristic
498Q16—68no—
499Q16—68no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8#23512yes—
511C2×C8#23512yes—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—32no—
515C4⋊C4—32no—
516C4⋊C4—32no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—64no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—122no—
534C4⋊C4—241no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—64no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—241no—
541C42—21no—
542C42—21no—
543C42—64no—
544C42—64no—
545C42—122no—
546C42—241no—
547C42D3×D412yes—
548C42D3×D412yes—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—41no—
556C22×C3C2×#12011yescharacteristic
557C22×C3#52511yescharacteristic
558C22×C3#53211yescharacteristic
559C22×C3#57711yescharacteristic
560C22×C3#57811yescharacteristic
561C22×C3#57811yescharacteristic
562C22×C3#57811yescharacteristic
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4#52511yescharacteristic
569C3×C4#52511yescharacteristic
570C3×C4#53211yescharacteristic
571C3×C4#53211yescharacteristic
572Dic3—42no—
573Dic3—42no—
574Dic3—42no—
575Dic3—42no—
576Dic3—42no—
577Dic3—42no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23C22×D1211yescharacteristic
588Q8—62no—
589Q8—62no—
590Q8—62no—
591Q8—62no—
592Q8—62no—
593Q8—62no—
594Q8—121no—
595Q8—121no—
596Q8—121no—
597Q8—121no—
598Q8—121no—
599Q8—121no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4#88212yes—
642C2×C4#88212yes—
643C2×C4C2×#23511yescharacteristic
644C2×C4C2×#23511yescharacteristic
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×D3×D411yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C8—22no—
650C8—22no—
651C8—41no—
652C8—41no—
653C2×C3—81no—
654C2×C3#238811yescharacteristic
655C2×C3#301111yescharacteristic
656C2×C3#301311yescharacteristic
657C2×C3#312011yescharacteristic
658C2×C3#312711yescharacteristic
659C2×C3#317711yescharacteristic
660C2×C3#317811yescharacteristic
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#547711yescharacteristic
669C22C2×#88211yescharacteristic
670C22#552411yescharacteristic
671C22#575111yescharacteristic
672C22#575311yescharacteristic
673C22#575311yescharacteristic
674C22#575311yescharacteristic
675C4—22no—
676C4—22no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—122no—
681C4—122no—
682C4—122no—
683C4—122no—
684C4—122no—
685C4—122no—
686C4—241no—
687C4—241no—
688C4#554711yescharacteristic
689C4#554711yescharacteristic
690C4#554811yescharacteristic
691C4#554811yescharacteristic
692C3#2358611yescharacteristic, complemented, Sylow
693C2—81no—
694C2#7358011yescharacteristic
695C2#7857011yescharacteristic
696C2#7857711yescharacteristic
697C2#7867911yescharacteristic
698C2#7868011yescharacteristic
699C2#7938511yescharacteristic
700C2#7939611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1750C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1750D3
hypocenterC3#23586
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
3. C2×#4716C2semidirect
4. C2×C4×Dic12C2semidirect
10. #69813C2semidirect
11. C4×#816C2semidirect
12. #68690C2semidirect
13. #68688C2semidirect
14. #68634C2semidirect
15. #68631C2semidirect
692. C3#23586semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
#822352
C2×#47162
C2×C4×Dic122
C3×#17502
#699642
#699372
#699362
#698132
C4×#8162
#686902
#686882
#686342
#686312
#235863