Finite Group Databasebeta

← #10858333

Subgroups — #10858333

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#5021C211yesmaximal, characteristic
3C2×C4×#216C211yesmaximal, characteristic
4#73674C211yesmaximal, characteristic
5#71632C211yesmaximal, characteristic
6#71632C211yesmaximal, characteristic
7#71628C211yesmaximal, characteristic
8#71628C211yesmaximal, characteristic
9#69635C211yesmaximal, characteristic
10#69631C211yesmaximal, characteristic
11#69619C211yesmaximal, characteristic
12#69604C211yesmaximal, characteristic
13#69590C211yesmaximal, characteristic
14#69569C211yesmaximal, characteristic
15#69482C211yesmaximal, characteristic
16#26381—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#389—21no—
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#384—21no—
28C3×#384C2211yescharacteristic
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C4×#216—24no—
35#5021—24no—
36#4980—21no—
37#4979—21no—
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4929C2211yescharacteristic
47#4929C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4657—21no—
53#4646—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#477—31no—
63C2×C4×Q16—31no—
64#2451—31no—
65Q8×C4⋊C4D311yescharacteristic
66#2095—31no—
67#2083—31no—
68#2068—31no—
69#2057—31no—
70#1942—31no—
71#1916—31no—
72#1878—31no—
73#1878—31no—
74#1863—31no—
75#1863—31no—
76#1863—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—41no—
85C3×C4×Q8—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#216—22no—
105C2×#216—22no—
106C2×#216Q814yescomplemented
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#869—22no—
120#869—41no—
121C4×Dic6—22no—
122C4×Dic6—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127#835—22no—
128#835—22no—
129#835D414yes—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#810—22no—
137#810—22no—
138#810—41no—
139#807—22no—
140#807—22no—
141#807—22no—
142#807—22no—
143#807—41no—
144C4⋊(C3⋊C8)—22no—
145C4×C3⋊C8—22no—
146C22×Q16—31no—
147C2×C4⋊Q8—31no—
148C2×C4×Q8—31no—
149C2×C4×Q8C2×D311yescharacteristic
150C2×C4×Q8C2×D311yescharacteristic
151C2×C4×Q8C2×D311yescharacteristic
152#477—64no—
153C4×Q16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C4×C8—31no—
167#398—61no—
168#395—61no—
169#391—31no—
170#391—31no—
171#391—61no—
172#389—21no—
173#389—21no—
174#389—31no—
175#389—31no—
176#389C2×D311yescharacteristic
177#389C2×D311yescharacteristic
178#387—61no—
179#384—21no—
180#384C2×D311yescharacteristic
181C4×C4⋊C4—21no—
182C4×C4⋊C4C2×D311yescharacteristic
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#336—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8C2×D412yes—
195C2×C3×Q8C2×Q812yes—
196C2×C3×Q8#4112yes—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—42no—
218C2×Dic6—42no—
219C2×Dic6—81no—
220C2×Dic6C2×Q812yes—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4Q1614yescomplemented
235C3×C4⋊C4#4112yes—
236C3×C42—24no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—42no—
244C3×C42—42no—
245#216—28no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3C2×D412yes—
253Dic3⋊C4—42no—
254Dic3⋊C4—81no—
255Dic3⋊C4—81no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C2×D412yes—
268C2×C3⋊C8C2×Q812yes—
269C22×Q8—31no—
270C22×Q8—61no—
271C22×Q8C22×D311yescharacteristic
272C2×Q16—34no—
273C2×Q16—62no—
274C2×Q16—62no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8—31no—
279C4⋊Q8—62no—
280C4⋊Q8—121no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—41no—
286C4×Q8—41no—
287C4×Q8—62no—
288C4×Q8—121no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#107—62no—
317#107—62no—
318#107—62no—
319#107—121no—
320C4⋊C8—62no—
321#103—34no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—121no—
327C4×C8—62no—
328#95—21no—
329#95—21no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95#6711yescharacteristic
337#95#6711yescharacteristic
338C23×C3C22×D411yescharacteristic
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8C22⋊Q814yes—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4C2×Q1612yes—
371C2×C3×C4C2×Q1612yes—
372C2×C3×C4C2×Q1612yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×Q811yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C42−1+411yescharacteristic
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391Dic6—22no—
392Dic6—22no—
393Dic6—41no—
394Dic6—41no—
395Dic6—84no—
396C3⋊C8—22no—
397C3⋊C8—22no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—32no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—122no—
410C2×Q8—122no—
411C2×Q8—241no—
412C2×Q8#23512yes—
413C2×Q8D3×D412yes—
414C2×Q8D3×Q812yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C23×D311yescharacteristic
430Q16—68no—
431C2×C8—32no—
432C2×C8—32no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4#21614yescomplemented
467C4⋊C4#23512yes—
468C42—24no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—42no—
476C42—42no—
477C42—122no—
478C22×C3#52511yescharacteristic
479C22×C3D4211yescharacteristic
480C22×C3D4×Q811yescharacteristic
481C22×C3C22×Q1611yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57611yescharacteristic
484C22×C3#57811yescharacteristic
485C3×C4—22no—
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4C2×C22⋊Q811yescharacteristic
496C3×C4#53911yescharacteristic
497C3×C4#54711yescharacteristic
498C3×C4#55411yescharacteristic
499Dic3—42no—
500Dic3—42no—
501Dic3—81no—
502Dic3—81no—
503C23C22×#6711yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—62no—
508Q8—62no—
509Q8—121no—
510Q8—121no—
511Q8—244no—
512Q8#89614yes—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4C2×#21612yes—
553C2×C4C2×#21612yes—
554C2×C4C2×#21612yes—
555C2×C4C2×#23511yescharacteristic
556C2×C4C2×#23511yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4#100411yescharacteristic
559C2×C4C2×D3×Q811yescharacteristic
560C2×C4#100711yescharacteristic
561C8—62no—
562C8—62no—
563C8—121no—
564C8—121no—
565C2×C3#244911yescharacteristic
566C2×C3#310411yescharacteristic
567C2×C3#312411yescharacteristic
568C2×C3#326811yescharacteristic
569C2×C3D4×Q1611yescharacteristic
570C2×C3Q8×Q1611yescharacteristic
571C2×C3#339011yescharacteristic
572C22#579111yescharacteristic
573C22D4×#6711yescharacteristic
574C22C22×#21611yescharacteristic
575C22Q8×#6711yescharacteristic
576C22#582211yescharacteristic
577C22#583511yescharacteristic
578C22#583811yescharacteristic
579C4—22no—
580C4—22no—
581C4—22no—
582C4—24no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—122no—
590C4—122no—
591C4—241no—
592C4—241no—
593C4#550711yescharacteristic
594C4C2×#89611yescharacteristic
595C4#551111yescharacteristic
596C4#551511yescharacteristic
597C3#2638111yescharacteristic, complemented, Sylow
598C2#8217311yescharacteristic
599C2#8244711yescharacteristic
600C2#8256911yescharacteristic
601C2D4×#21611yescharacteristic
602C2#8357711yescharacteristic
603C2#8368411yescharacteristic
604C2Q8×#21611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×Q8×C4⋊C4C2
socle, prime radicalC23×C3C22×D4
hypercenterQ8×C4⋊C4D3
hypocenterC3#26381
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
106. C2×#216Q8semidirect
234. C3×C4⋊C4Q16semidirect
466. C4⋊C4#216semidirect
597. C3#26381semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C4⋊C42
C2×#50212
C2×C4×#2162
#736742
#716322
#716282
#696352
#696312
#696192
#696042
#695902
#695692
#694822
#263813