Finite Group Databasebeta

← #10957252

Subgroups — #10957252

Order 768.

2860 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#67C211yesmaximal, characteristic
2C2×#5812C211yesmaximal, characteristic, complemented
3C2×#5572C211yesmaximal, characteristic, complemented
4C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5Q8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C4×Q8×Dic3C211yesmaximal, characteristic, complemented
8#76611C213yesmaximal, complemented
9#76492C213yesmaximal, complemented
10#76491C213yesmaximal, complemented
11#76481C213yesmaximal, complemented
12#73885C213yesmaximal
13#73871C213yesmaximal, complemented
14#73468C213yesmaximal, complemented
15#73439C213yesmaximal
16#34311—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×D3×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21Q8×#67C4116yescomplemented
22C2×#946C2213yes—
23#5812—28no—
24C2×Q8×Dic3C2211yescharacteristic
25C2×Q8×Dic3C2213yes—
26C2×Q8×Dic3C2213yes—
27C2×Q8×Dic3C2211yescharacteristic
28C2×Q8×Dic3C2211yescharacteristic
29C2×#944C2213yes—
30C2×C4×#67C2213yes—
31C2×#926C2213yes—
32C2×#926C2213yes—
33C2×#924C2213yes—
34#5572—28no—
35C2×#892C2213yes—
36C2×#887C2213yes—
37C2×#872C2213yes—
38C2×C4×Dic6C2213yes—
39C3×#391C2213yes—
40C3×#391C2213yes—
41C3×#380C2213yes—
42C3×C4×C22⋊C4C2213yes—
43#4987C2213yes—
44#4975C2213yes—
45Dic3×C4⋊C4C2213yes—
46Dic3×C4⋊C4C2213yes—
47#4973C2213yes—
48#4972C2213yes—
49#4957C2213yes—
50#4957C2211yescharacteristic
51#4945C2216yes—
52#4944C2213yes—
53#4944C2213yes—
54C4×(C2×D3)⋊C4C2213yes—
55C4×C4⋊Dic3C2213yes—
56C4×Dic3⋊C4C2213yes—
57C42×Dic3C2213yes—
58#4929C2213yes—
59#4928C2213yes—
60#4670C2213yes—
61#4670C2213yes—
62#4669C2216yes—
63#4655C2216yes—
64#4652C2216yes—
65#4649C2216yes—
66#4649C2216yes—
67#4647C2216yes—
68#4647C2213yes—
69#4647C2213yes—
70C2×D4×Q8—31no—
71C2×#519—31no—
72C2×#519—31no—
73#2365—33no—
74#2362—33no—
75#2362—33no—
76#2354—33no—
77Q8×C4⋊C4—31no—
78Q8×C22⋊C4—31no—
79Q8×C22⋊C4D311yescharacteristic, complemented
80#2323—33no—
81#2323—33no—
82C4×C4⋊Q8—33no—
83#2293—33no—
84C42×Q8—31no—
85C22×C3×Q8—24no—
86C22×C3×Q8—24no—
87C22×C3×Q8C2×C418yes—
88C22×C3×Q8C2313yes—
89C22×C3×Q8C2311yescharacteristic
90C23×C3×C4C2313yes—
91C22×#67C2311yescharacteristic
92C2×D3×Q8—24no—
93C2×D3×Q8—24no—
94C2×D3×Q8C2×C418yes—
95C22×C4×D3C2313yes—
96C22×Dic6C2313yes—
97C3×C4×Q8—24no—
98C3×C4×Q8—24no—
99C2×C3×C4⋊C4C2313yes—
100C2×C3×C4⋊C4C2313yes—
101C2×C3×C22⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4C2313yes—
103C2×C3×C42C2313yes—
104C2×C3×C42C2313yes—
105C2×#217C2311yescharacteristic
106C2×#217C2313yes—
107C2×#217C2313yes—
108#946C2×C4124yes—
109Q8×Dic3—24no—
110Q8×Dic3—212no—
111Q8×Dic3—212no—
112Q8×Dic3C2×C418yes—
113Q8×Dic3D418yescomplemented
114#944C2×C4124yes—
115C4×#67C2×C4124yes—
116C2×(C2×D3)⋊C4C2313yes—
117C2×(C2×D3)⋊C4C2311yescharacteristic
118C2×(C2×D3)⋊C4C2311yescharacteristic
119C2×(C2×D3)⋊C4C2313yes—
120C2×(C2×D3)⋊C4C2313yes—
121#926—212no—
122#926—212no—
123C2×C4⋊Dic3C2313yes—
124C2×C4⋊Dic3C2313yes—
125C2×C4⋊Dic3C2313yes—
126C2×C4⋊Dic3C2313yes—
127C2×C4⋊Dic3C2313yes—
128#924C2×C4124yes—
129C2×Dic3⋊C4C2311yescharacteristic
130C2×Dic3⋊C4C2313yes—
131C2×Dic3⋊C4C2313yes—
132C2×Dic3⋊C4C2313yes—
133C2×Dic3⋊C4C2316yes—
134C2×Dic3⋊C4C2313yes—
135C2×Dic3⋊C4C2316yes—
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2313yes—
138C2×C4×Dic3C2316yes—
139C2×C4×Dic3C2316yes—
140C2×C4×Dic3C2313yes—
141C2×C4×Dic3C2313yes—
142C2×C4×Dic3C2313yes—
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2313yes—
145#892—212no—
146#887—212no—
147#872—212no—
148C4×Dic6—212no—
149C3×#95C2313yes—
150C3×#95C2316yes—
151C3×#95C2313yes—
152C3⋊#95C2313yes—
153C3⋊#95C2316yes—
154C3⋊#95C2313yes—
155C3⋊#95C2313yes—
156C3⋊#95C2316yes—
157C3⋊#95C2313yes—
158C23×Q8—31no—
159C23×Q8C2×D311yescharacteristic
160D4×Q8—316no—
161C2×C4⋊Q8—33no—
162C2×C22⋊Q8—33no—
163C2×C22⋊Q8—33no—
164#519—68no—
165#519—68no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—33no—
169C2×C4×Q8—31no—
170C2×C4×Q8—33no—
171C2×C4×Q8—33no—
172C2×C4×Q8—33no—
173C2×C4×Q8C2×D311yescharacteristic
174C2×C4×Q8C2×D311yescharacteristic
175C2×C4×D4—33no—
176C2×#117—33no—
177C2×#117—33no—
178C2×#117—33no—
179C2×#117—33no—
180#391—33no—
181#391—33no—
182#391C2×D313yes—
183#391C2×D313yes—
184#390—33no—
185#390—31no—
186#389—33no—
187#389—33no—
188#389—33no—
189#389—36no—
190#389—33no—
191#388—33no—
192#388—36no—
193#388—33no—
194#388—33no—
195#388—36no—
196#388—33no—
197#387—36no—
198#387—36no—
199#384—33no—
200#380—33no—
201#380C2×D313yes—
202C4×C4⋊C4—33no—
203C4×C4⋊C4—33no—
204C4×C4⋊C4—33no—
205C4×C4⋊C4—33no—
206C4×C4⋊C4—36no—
207C4×C4⋊C4—36no—
208C4×C22⋊C4—33no—
209C4×C22⋊C4C2×D313yes—
210C43—33no—
211C24×C3C2411yescharacteristic
212C23×D3C2411yescharacteristic
213C2×C3×Q8—232no—
214C2×C3×Q8—26no—
215C2×C3×Q8—26no—
216C2×C3×Q8C22×C4112yes—
217C2×C3×Q8C22×C414yes—
218C2×C3×Q8C2×D414yes—
219C2×C3×Q8#4114yes—
220C22×C3×C4—26no—
221C22×C3×C4—26no—
222C22×C3×C4C22×C4112yes—
223C22×C3×C4C2413yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2413yes—
226C22×C3×C4C2413yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2413yes—
229C2×#67—22no—
230C2×#67—22no—
231C2×#67C22×C414yes—
232C22×Dic3C2411yescharacteristic
233C22×Dic3C2413yes—
234C22×Dic3C2413yes—
235C22×Dic3C2413yes—
236C22×Dic3C2413yes—
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2411yescharacteristic
239D3×Q8—232no—
240C2×C4×D3—26no—
241C2×C4×D3—26no—
242C2×C4×D3C22×C4112yes—
243C2×Dic6—26no—
244C2×Dic6—26no—
245C2×Dic6C22×C4112yes—
246C3×C4⋊C4—26no—
247C3×C4⋊C4—26no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—43no—
250C3×C42—26no—
251C3×C42—26no—
252#217—26no—
253#217—22no—
254#217C22×C4112yes—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—26no—
258(C2×D3)⋊C4—43no—
259(C2×D3)⋊C4C22×C4112yes—
260C4⋊Dic3—26no—
261C4⋊Dic3—26no—
262C4⋊Dic3—26no—
263C4⋊Dic3C22×C4112yes—
264C4⋊Dic3C2×D4112yes—
265Dic3⋊C4—26no—
266Dic3⋊C4—26no—
267Dic3⋊C4—22no—
268Dic3⋊C4—26no—
269Dic3⋊C4—46no—
270Dic3⋊C4C22×C4124yes—
271Dic3⋊C4C22×C4112yes—
272C4×Dic3—26no—
273C4×Dic3—26no—
274C4×Dic3—212no—
275C4×Dic3—212no—
276C4×Dic3—26no—
277C4×Dic3—22no—
278C4×Dic3—43no—
279C4×Dic3C22×C4112yes—
280C4×Dic3C2×D4112yes—
281C22×Q8—24no—
282C22×Q8—24no—
283C22×Q8—38no—
284C22×Q8—33no—
285C22×Q8—64no—
286C22×Q8—64no—
287C22×Q8C4×D318yes—
288C22×Q8C22×D313yes—
289C22×Q8C22×D311yescharacteristic
290C22×D4—31no—
291C23×C4—33no—
292C23×C4C22×D313yes—
293C4⋊Q8—324no—
294C22⋊Q8—324no—
295C22⋊Q8—324no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—38no—
299C4×Q8—38no—
300C4×Q8—612no—
301C4×Q8—612no—
302C4×Q8—612no—
303C4×Q8—612no—
304C4×Q8—64no—
305C4×D4—324no—
306#117—612no—
307#117—612no—
308#117—612no—
309#117—612no—
310C2×C4⋊C4—33no—
311C2×C4⋊C4—33no—
312C2×C4⋊C4—33no—
313C2×C4⋊C4—36no—
314C2×C4⋊C4—33no—
315C2×C4⋊C4—33no—
316C2×C4⋊C4—33no—
317C2×C4⋊C4—33no—
318C2×C4⋊C4—33no—
319C2×C4⋊C4—33no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—36no—
322C2×C4⋊C4C22×D313yes—
323C2×C4⋊C4C22×D313yes—
324C2×C22⋊C4—33no—
325C2×C22⋊C4—33no—
326C2×C22⋊C4—33no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—33no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—33no—
332C2×C22⋊C4C22×D311yescharacteristic
333C2×C22⋊C4C22×D313yes—
334C2×C42—33no—
335C2×C42—33no—
336C2×C42—33no—
337C2×C42—36no—
338C2×C42—36no—
339C2×C42—31no—
340C2×C42—33no—
341C2×C42—33no—
342C2×C42—33no—
343C2×C42C22×D313yes—
344C2×C42C22×D313yes—
345#95—33no—
346#95—33no—
347#95—36no—
348#95—36no—
349#95—33no—
350#95—33no—
351#95C22×D313yes—
352#95C22×D313yes—
353#95C22×D316yes—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3C23×C412yes—
361C23×C3C2511yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3C23×C412yes—
369C3×Q8—224no—
370C3×Q8C4×D4116yes—
371C2×C3×C4—23no—
372C2×C3×C4—21no—
373C2×C3×C4—23no—
374C2×C3×C4—21no—
375C2×C3×C4—224no—
376C2×C3×C4—23no—
377C2×C3×C4—21no—
378C2×C3×C4—23no—
379C2×C3×C4—23no—
380C2×C3×C4—23no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—23no—
384C2×C3×C4—23no—
385C2×C3×C4C23×C416yes—
386C2×C3×C4C23×C416yes—
387C2×C3×C4C22×D416yes—
388C2×C3×C4C2×#4116yes—
389C2×C3×C42−1+412yes—
390#67—28no—
391C2×Dic3—23no—
392C2×Dic3—23no—
393C2×Dic3—23no—
394C2×Dic3—23no—
395C2×Dic3—21no—
396C2×Dic3—23no—
397C2×Dic3—21no—
398C2×Dic3—23no—
399C2×Dic3—23no—
400C2×Dic3—21no—
401C2×Dic3—23no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—23no—
405C2×Dic3C23×C416yes—
406C2×Dic3C23×C416yes—
407C2×Dic3C23×C412yes—
408C2×Dic3C22×D416yes—
409C2×Dic3C22×D412yes—
410C2×Dic32−1+412yes—
411C2×Dic32−1+412yes—
412C4×D3—224no—
413Dic6—224no—
414C24—31no—
415C24C23×D311yescharacteristic
416C2×Q8—232no—
417C2×Q8—26no—
418C2×Q8—26no—
419C2×Q8—312no—
420C2×Q8—632no—
421C2×Q8—66no—
422C2×Q8—66no—
423C2×Q8C2×C4×D3112yes—
424C2×Q8C2×C4×D314yes—
425C2×Q8D3×D414yes—
426C2×Q8#23714yes—
427C2×D4—34no—
428C2×D4—62no—
429C2×D4—62no—
430C22×C4—26no—
431C22×C4—26no—
432C22×C4—31no—
433C22×C4—33no—
434C22×C4—33no—
435C22×C4—33no—
436C22×C4—31no—
437C22×C4—312no—
438C22×C4—31no—
439C22×C4—33no—
440C22×C4—66no—
441C22×C4—66no—
442C22×C4C2×C4×D3112yes—
443C22×C4C23×D313yes—
444C22×C4C23×D311yescharacteristic
445C22×C4C23×D313yes—
446C22×C4C23×D313yes—
447C22×C4C23×D311yescharacteristic
448C22×C4C23×D313yes—
449C4⋊C4—26no—
450C4⋊C4—26no—
451C4⋊C4—312no—
452C4⋊C4—312no—
453C4⋊C4—324no—
454C4⋊C4—312no—
455C4⋊C4—66no—
456C4⋊C4—66no—
457C4⋊C4—66no—
458C4⋊C4—66no—
459C4⋊C4—62no—
460C4⋊C4—66no—
461C4⋊C4—66no—
462C4⋊C4—126no—
463C22⋊C4—22no—
464C22⋊C4—312no—
465C22⋊C4—312no—
466C22⋊C4—43no—
467C22⋊C4—62no—
468C22⋊C4—66no—
469C22⋊C4—62no—
470C22⋊C4—66no—
471C22⋊C4—62no—
472C22⋊C4—123no—
473C42—26no—
474C42—26no—
475C42—312no—
476C42—312no—
477C42—612no—
478C42—612no—
479C42—62no—
480C42—66no—
481C42—66no—
482C42—66no—
483C42—123no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3C24×C411yescharacteristic
492C22×C3C23×D411yescharacteristic
493C22×C3C22×#4111yescharacteristic
494C22×C3C2×2−1+411yescharacteristic
495C22×C3C2×2−1+411yescharacteristic
496C22×C3C2×2−1+411yescharacteristic
497C22×C3#58511yescharacteristic
498C2×D3—22no—
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—41no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C3×C4—26no—
506C3×C4—22no—
507C3×C4—43no—
508C3×C4—41no—
509C3×C4—43no—
510C3×C4C2×C4×D4112yes—
511Dic3—26no—
512Dic3—26no—
513Dic3—22no—
514Dic3—26no—
515Dic3—43no—
516Dic3—41no—
517Dic3#51914yescomplemented
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—32no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23C22×C4×D312yes—
532C23C24×D311yescharacteristic
533Q8—224no—
534Q8—624no—
535Q8#67⋊C4116yes—
536D4—68no—
537C2×C4—23no—
538C2×C4—21no—
539C2×C4—23no—
540C2×C4—21no—
541C2×C4—224no—
542C2×C4—21no—
543C2×C4—23no—
544C2×C4—23no—
545C2×C4—21no—
546C2×C4—23no—
547C2×C4—23no—
548C2×C4—23no—
549C2×C4—23no—
550C2×C4—21no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—36no—
555C2×C4—36no—
556C2×C4—32no—
557C2×C4—36no—
558C2×C4—63no—
559C2×C4—63no—
560C2×C4—63no—
561C2×C4—63no—
562C2×C4—61no—
563C2×C4—63no—
564C2×C4—61no—
565C2×C4—63no—
566C2×C4—624no—
567C2×C4—63no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—63no—
571C2×C4—63no—
572C2×C4—61no—
573C2×C4C22×C4×D316yes—
574C2×C4C22×C4×D316yes—
575C2×C4C2×D3×D416yes—
576C2×C4C2×#23716yes—
577C2×C4#100712yes—
578C2×C3—42no—
579C2×C3C22×C4×D411yescharacteristic
580C2×C3C2×#51911yescharacteristic
581C2×C3C4×2−1+411yescharacteristic
582C2×C3#349711yescharacteristic
583C2×C3#349911yescharacteristic
584C2×C3#351811yescharacteristic
585C2×C3#352311yescharacteristic
586D3—42no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—62no—
595C22—62no—
596C22—62no—
597C22—121no—
598C22—121no—
599C22—121no—
600C22—121no—
601C22C23×C4×D311yescharacteristic
602C22C22×D3×D411yescharacteristic
603C22C22×#23711yescharacteristic
604C22C2×#100711yescharacteristic
605C22#596411yescharacteristic
606C22D3×2−1+411yescharacteristic
607C22D3×2−1+411yescharacteristic
608C4—26no—
609C4—22no—
610C4—34no—
611C4—43no—
612C4—43no—
613C4—41no—
614C4—66no—
615C4—66no—
616C4—62no—
617C4—66no—
618C4—123no—
619C4—121no—
620C4C2×#67⋊C4112yes—
621C3#3431111yescharacteristic, complemented, Sylow
622C2—42no—
623C2—122no—
624C2C22×#67⋊C411yescharacteristic
625C2C4×#100711yescharacteristic
626C2D3×#51911yescharacteristic
627C2#8672011yescharacteristic
628C2#8673711yescharacteristic
629C2#8674611yescharacteristic
630C2#8675211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×Q8×C22⋊C4C2
socle, cosocleC23×C3C25
prime radicalC22×#67C23
hypercenterQ8×C22⋊C4D3
hypocenterC3#34311
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5812C2semidirect
3. C2×#5572C2semidirect
4. C3×Q8×C22⋊C4C2semidirect
5. Q8×(C2×D3)⋊C4C2semidirect
6. Q8×Dic3⋊C4C2semidirect
7. C4×Q8×Dic3C2semidirect
8. #76611C2semidirect
9. #76492C2semidirect
10. #76491C2semidirect
11. #76481C2semidirect
13. #73871C2semidirect
14. #73468C2semidirect
21. Q8×#67C4semidirect
79. Q8×C22⋊C4D3semidirect
113. Q8×Dic3D4semidirect
517. Dic3#519semidirect
621. C3#34311semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#672
C2×#58122
C2×#55722
C3×Q8×C22⋊C42
Q8×(C2×D3)⋊C42
Q8×Dic3⋊C42
C4×Q8×Dic32
#766112
#764922
#764912
#764812
#738852
#738712
#734682
#734392
#343113