Finite Group Databasebeta

← #10691605

Subgroups — #10691605

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4731C211yesmaximal, characteristic
2C22⋊C4×D12C211yesmaximal, characteristic
3#69959C211yesmaximal, characteristic
4#69946C211yesmaximal, characteristic, complemented
5#69457C211yesmaximal, characteristic, complemented
6C3×#1282C211yesmaximal, characteristic, complemented
7#67676C211yesmaximal, characteristic, complemented
8#9255—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C22×#204C412yescomplemented
12C2×D3×C22⋊C4—21no—
13C2×C4×D12—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5212C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#821—22no—
20C2×#821C412yescomplemented
21#4955—21no—
22#4939—21no—
23#4731—21no—
24#4731—22no—
25#4731—21no—
26#4665—22no—
27#4527—21no—
28#4527—21no—
29#4467—22no—
30C2×#450—31no—
31D4×C22⋊C4—31no—
32#1996—31no—
33#1786—31no—
34#1520—31no—
35#1289—31no—
36#1282D311yescharacteristic, complemented
37C24×D3—21no—
38C23×C3×C4C2311yescharacteristic
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—44no—
43C22×D12C2×C412yes—
44C22×D12D411yescharacteristic
45C22×D12D412yes—
46C22×D12D411yescharacteristic
47C22×C4×D3—22no—
48C22×Dic6C2×C411yescharacteristic, complemented
49C22×C3×C8—21no—
50C22×C3×C8C2×C412yescomplemented
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C42—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic, complemented
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×#204—22no—
69C2×#204—22no—
70C2×#204—44no—
71D3×C22⋊C4—42no—
72D3×C22⋊C4—42no—
73C4×D12—44no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#821—22no—
82#821—22no—
83#821—44no—
84C23×D4—31no—
85C22×SD16—32no—
86C2×C22⋊Q8—31no—
87C2×C4×D4—61no—
88C22×C22⋊C4—61no—
89#450—61no—
90#450—62no—
91#450—61no—
92C2×#102—32no—
93C2×#102—62no—
94C2×C22⋊C8C2×D311yescharacteristic
95C2×C22⋊C8C2×D311yescharacteristic
96#390—61no—
97#386—62no—
98#380—31no—
99#380—31no—
100#379—61no—
101C4×C22⋊C4C2×D311yescharacteristic
102#342—61no—
103#342—61no—
104#328—62no—
105C24×C3C2×D411yescharacteristic
106C23×D3—21no—
107C23×D3—21no—
108C23×D3—22no—
109C23×D3—22no—
110C23×D3—44no—
111C23×D3—41no—
112C23×D3—41no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3—82no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—42no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—41no—
129C2×D12—21no—
130C2×D12—24no—
131C2×D12—24no—
132C2×D12—24no—
133C2×D12—22no—
134C2×D12—22no—
135C2×D12—21no—
136C2×D12—42no—
137C2×D12—42no—
138C2×D12—42no—
139C2×D12—44no—
140C2×D12—44no—
141C2×D12—44no—
142C2×D12C22⋊C412yes—
143C2×D12C22⋊C412yes—
144C2×C4×D3—44no—
145C2×C4×D3—44no—
146C2×C4×D3—82no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C2×C3×C8—22no—
151C2×C3×C8—42no—
152C2×C3×C8—42no—
153C2×C3×C8—41no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4SD1612yescomplemented
158C3×C42—42no—
159#217—41no—
160#217—41no—
161#217—41no—
162#217—41no—
163#217—41no—
164#217—41no—
165(C2×D3)⋊C4—44no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—82no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—42no—
171C4⋊Dic3—42no—
172Dic3⋊C4—41no—
173Dic3⋊C4—41no—
174#204—44no—
175#204—44no—
176C25—61no—
177C22×Q8—31no—
178C22×D4—32no—
179C22×D4—31no—
180C22×D4—32no—
181C22×D4—31no—
182C22×D4—61no—
183C22×D4—61no—
184C22×D4—61no—
185C22×D4—124no—
186C23×C4—62no—
187C23×C4C22×D311yescharacteristic
188C2×SD16—62no—
189C2×SD16—62no—
190C2×SD16—124no—
191C22×C8—21no—
192C22×C8C4×D312yescomplemented
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C4×D4—124no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—122no—
209C2×C22⋊C4—122no—
210C2×C22⋊C4D1211yescharacteristic
211C2×C22⋊C4D1211yescharacteristic
212C2×C42—21no—
213#102—62no—
214#102—62no—
215#102—124no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220#95—21no—
221#95—32no—
222#95—61no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3C4×D411yescharacteristic
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3C2×SD1611yescharacteristic
229C23×C3#13511yescharacteristic
230C23×C3AGL(1,ℤ/8)11yescharacteristic
231C23×C3#13711yescharacteristic
232C22×D3—24no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—44no—
236C22×D3—44no—
237C22×D3—42no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—82no—
249C22×D3—82no—
250C22×D3—84no—
251C22×D3—84no—
252C22×D3—84no—
253C22×D3—84no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4C2×C22⋊C411yescharacteristic
267C2×C3×C4C4×D411yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12311yescharacteristic
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280D12—24no—
281D12—22no—
282D12—22no—
283D12—22no—
284D12—44no—
285D12—44no—
286D12—42no—
287D12#9914yes—
288C4×D3—88no—
289Dic6—42no—
290Dic6—81no—
291C3×C8—42no—
292C3×C8—81no—
293C24—61no—
294C24—61no—
295C24—62no—
296C24—62no—
297C24—124no—
298C24—121no—
299C24—121no—
300C24—121no—
301C24—121no—
302C24—242no—
303C24C2×D1211yescharacteristic
304C2×Q8—61no—
305C2×Q8—61no—
306C2×Q8—122no—
307C2×D4—32no—
308C2×D4—32no—
309C2×D4—64no—
310C2×D4—64no—
311C2×D4—64no—
312C2×D4—62no—
313C2×D4—62no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—122no—
317C2×D4—122no—
318C2×D4—122no—
319C2×D4—124no—
320C2×D4—124no—
321C2×D4—124no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—62no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—124no—
329C22×C4—124no—
330C22×C4—121no—
331C22×C4—242no—
332C22×C4C2×C4×D311yescharacteristic
333C22×C4#23511yescharacteristic
334C22×C4D3×D411yescharacteristic
335C22×C4D3×D411yescharacteristic
336C22×C4D3×D411yescharacteristic
337C22×C4#23911yescharacteristic
338SD16—124no—
339SD16—124no—
340C2×C8—22no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4—121no—
349C4⋊C4—121no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—124no—
354C22⋊C4—124no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C22⋊C4—121no—
358C22⋊C4—121no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C22⋊C4—242no—
362C22⋊C4#20412yescomplemented
363C42—42no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3#38611yescharacteristic
375C22×C3C4×SD1611yescharacteristic
376C22×C3#44011yescharacteristic
377C22×C3#45011yescharacteristic
378C22×C3#45211yescharacteristic
379C22×C3#46111yescharacteristic
380C22×C3#46311yescharacteristic
381C2×D3—24no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—84no—
390C2×D3—84no—
391C2×D3—84no—
392C2×D3—84no—
393C2×D3—82no—
394C2×D3—82no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—42no—
399C3×C4—81no—
400C3×C4C2×#9912yes—
401Dic3—82no—
402Dic3—81no—
403Dic3—81no—
404Dic3—161no—
405C23—41no—
406C23—41no—
407C23—64no—
408C23—62no—
409C23—62no—
410C23—124no—
411C23—124no—
412C23—122no—
413C23—122no—
414C23—122no—
415C23—121no—
416C23—121no—
417C23—122no—
418C23—122no—
419C23—121no—
420C23—121no—
421C23—122no—
422C23—122no—
423C23—242no—
424C23—242no—
425C23—244no—
426C23—244no—
427C23—244no—
428C23—244no—
429C23C4×D1211yescharacteristic
430C23#88211yescharacteristic
431C23#89511yescharacteristic
432C23C2×#20411yescharacteristic
433C23#90411yescharacteristic
434C23#90811yescharacteristic
435C23#90911yescharacteristic
436Q8—122no—
437Q8—241no—
438D4—34no—
439D4—64no—
440D4—62no—
441D4—62no—
442D4—62no—
443D4—124no—
444D4—124no—
445D4—122no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—248no—
464C2×C4—241no—
465C2×C4—241no—
466C2×C4—241no—
467C2×C4—241no—
468C2×C4—241no—
469C2×C4D3×C22⋊C411yescharacteristic
470C2×C4#88411yescharacteristic
471C2×C4#89311yescharacteristic
472C2×C4#89411yescharacteristic
473C8—42no—
474C8—81no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3#187711yescharacteristic
482C2×C3#189811yescharacteristic
483C2×C3#190411yescharacteristic
484D3—44no—
485D3—82no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—64no—
497C22—124no—
498C22—124no—
499C22—122no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—244no—
505C22—244no—
506C22—244no—
507C22—244no—
508C22—242no—
509C22—242no—
510C22#466511yescharacteristic
511C22C4×#20411yescharacteristic
512C22#471011yescharacteristic
513C22#473111yescharacteristic
514C22#473311yescharacteristic
515C22#484111yescharacteristic
516C22#484411yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—242no—
523C4—241no—
524C4—241no—
525C4—481no—
526C4D3×#9912yes—
527C3#925511yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—124no—
535C2—242no—
536C2#6867211yescharacteristic
537C2#6897011yescharacteristic
538C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9255
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
4. #69946C2semidirect
5. #69457C2semidirect
6. C3×#1282C2semidirect
7. #67676C2semidirect
11. C22×#204C4semidirect
20. C2×#821C4semidirect
36. #1282D3semidirect
48. C22×Dic6C2×C4semidirect
50. C22×C3×C8C2×C4semidirect
62. C2×C4⋊Dic3C2×C4semidirect
157. C3×C22⋊C4SD16semidirect
192. C22×C8C4×D3semidirect
362. C22⋊C4#204semidirect
527. C3#9255semidirect

Maximal subgroups

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

subgroupindex
C2×#47312
C22⋊C4×D122
#699592
#699462
#694572
C3×#12822
#676762
#92553