Finite Group Databasebeta

← #10697502

Subgroups — #10697502

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2537C211yesmaximal, characteristic
2C2×#5019C211yesmaximal, characteristic, complemented
3C2×#5007C211yesmaximal, characteristic, complemented
4C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#69582C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#10231—31nomaximal, Sylow
9C2×C3×#121—21no—
10C2×C3×#121—21no—
11C2×C3×C4×D4C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C22×#214—21no—
14C2×#869C2211yescharacteristic
15C3×#394—22no—
16C3×#384C2211yescharacteristic
17C3×#380—21no—
18C3×#380—21no—
19C2×#832—21no—
20#5019—22no—
21#5019—22no—
22#5007—22no—
23#5007—22no—
24C2×#810—21no—
25C2×#807—21no—
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4×C3⋊C8C2211yescharacteristic
29D4⋊(C3⋊C8)—22no—
30D4⋊(C3⋊C8)—22no—
31#4529—22no—
32C2×#474—31no—
33C2×#461—31no—
34#2537D311yescharacteristic
35#1990—31no—
36C2×D4⋊C8—31no—
37C2×D4⋊C8—31no—
38#1296—31no—
39C22×C3×D4—22no—
40C22×C3×D4D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C22×Dic6—21no—
45C3×#121—42no—
46C3×#121—42no—
47C3×#121—42no—
48C3×#121—42no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C4⋊C4D412yes—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#214—24no—
68C2×#214—42no—
69C2×#214—42no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C22×C3⋊C8—21no—
75#869—21no—
76#869—21no—
77#869—21no—
78#869D412yescomplemented
79C3×#95—22no—
80#832—24no—
81#810—24no—
82#807—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yescomplemented
86C4⋊(C3⋊C8)D412yescomplemented
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89C22×SD16—61no—
90C2×C4⋊Q8—31no—
91C2×#121—21no—
92C2×#121—21no—
93C2×C4×D4C2×D311yescharacteristic
94C2×C4×D4C2×D311yescharacteristic
95#474—62no—
96#474—62no—
97#461—62no—
98#461—62no—
99C2×#107—61no—
100C2×C4⋊C8—31no—
101C2×C4⋊C8—31no—
102C2×#103—61no—
103C2×#102—61no—
104C2×C4×C8—31no—
105#394—22no—
106#384C2×D311yescharacteristic
107#380—21no—
108#380—21no—
109#340—62no—
110D4⋊C8—62no—
111D4⋊C8—62no—
112C24×C3—21no—
113C24×C3—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—44no—
125C2×C3×D4—44no—
126C2×C3×D4—82no—
127C2×C3×D4D812yes—
128C2×C3×D4SD1612yescomplemented
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×Dic3—41no—
148C2×Dic6—22no—
149C2×Dic6—41no—
150C2×Dic6—41no—
151C2×Dic6—41no—
152C2×Dic6—41no—
153C2×Dic6—41no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—42no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C22⋊C4—44no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D411yescharacteristic
172C3×C42#4111yescharacteristic
173#214—48no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—41no—
177C4⋊Dic3—41no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C22×Q8—61no—
188C22×D4—22no—
189C22×D4#6711yescharacteristic
190C22×D4#6711yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C2×SD16—64no—
194C2×SD16—122no—
195C2×SD16—122no—
196C22×C8—61no—
197C22×C8—61no—
198C22×C8—61no—
199C4⋊Q8—32no—
200C4⋊Q8—61no—
201C4⋊Q8—61no—
202C4⋊Q8—61no—
203#121—42no—
204#121—42no—
205#121—42no—
206#121—42no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4#6711yescharacteristic
220C2×C4⋊C4#6711yescharacteristic
221C2×C4⋊C4#6712yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42C22×D311yescharacteristic
227#107—64no—
228C4⋊C8—32no—
229C4⋊C8—32no—
230C4⋊C8—61no—
231C4⋊C8—61no—
232#103—64no—
233#102—64no—
234C4×C8—61no—
235C4×C8—61no—
236#95—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3#12011yescharacteristic
252C3×D4—22no—
253C3×D4—22no—
254C3×D4—22no—
255C3×D4—22no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—88no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4—81no—
279C2×C3×C4—81no—
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4#12711yescharacteristic
286C2×C3×C4C2×D811yescharacteristic
287C2×C3×C4C2×SD1611yescharacteristic
288C2×C3×C4#13511yescharacteristic
289C2×C3×C4#13511yescharacteristic
290C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
291C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
292C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
293C2×C3×C4#13711yescharacteristic
294C2×Dic3—42no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297Dic6—42no—
298Dic6—42no—
299Dic6—81no—
300Dic6—81no—
301C3⋊C8—42no—
302C3⋊C8—42no—
303C3⋊C8—42no—
304C24—21no—
305C24—21no—
306C2×Q8—62no—
307C2×Q8—121no—
308C2×Q8—121no—
309C2×Q8—121no—
310C2×Q8—121no—
311C2×Q8—121no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—82no—
325C2×D4#21312yes—
326C2×D4#21412yescomplemented
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—121no—
343C22×C4C2×#6711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346SD16—128no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8—121no—
351C2×C8—121no—
352C2×C8—121no—
353C2×C8—121no—
354C2×C8—121no—
355C2×C8—121no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C42D3×D411yescharacteristic
376C42D3×D411yescharacteristic
377C42D3×D411yescharacteristic
378C42#23711yescharacteristic
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3#39211yescharacteristic
394C22×C3#44711yescharacteristic
395C22×C3#44911yescharacteristic
396C22×C3#45011yescharacteristic
397C22×C3#45211yescharacteristic
398C22×C3#45511yescharacteristic
399C22×C3#45611yescharacteristic
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—82no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4#44711yescharacteristic
406C3×C4#44811yescharacteristic
407C3×C4#44911yescharacteristic
408C3×C4#44911yescharacteristic
409C3×C4#46511yescharacteristic
410C3×C4#46611yescharacteristic
411C3×C4#46811yescharacteristic
412C3×C4#46911yescharacteristic
413Dic3—82no—
414C23—22no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23#95311yescharacteristic
429Q8—122no—
430Q8—122no—
431Q8—241no—
432Q8—241no—
433D4—22no—
434D4—22no—
435D4—22no—
436D4—22no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—88no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—122no—
462C2×C4—241no—
463C2×C4—241no—
464C2×C4C2×#21311yescharacteristic
465C2×C4#93211yescharacteristic
466C2×C4#93211yescharacteristic
467C2×C4C2×#21411yescharacteristic
468C2×C4#93711yescharacteristic
469C2×C4#93711yescharacteristic
470C2×C4#93811yescharacteristic
471C2×C4#93911yescharacteristic
472C2×C4#93911yescharacteristic
473C2×C4#94011yescharacteristic
474C2×C4#94911yescharacteristic
475C2×C4#95011yescharacteristic
476C2×C4#95011yescharacteristic
477C2×C4#95111yescharacteristic
478C8—122no—
479C8—122no—
480C8—122no—
481C2×C3—82no—
482C2×C3—82no—
483C2×C3#163911yescharacteristic
484C2×C3#164611yescharacteristic
485C2×C3#164711yescharacteristic
486C2×C3#165411yescharacteristic
487C2×C3#199911yescharacteristic
488C2×C3#200011yescharacteristic
489C2×C3#200711yescharacteristic
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#521711yescharacteristic
505C22#521811yescharacteristic
506C22#522311yescharacteristic
507C22#523811yescharacteristic
508C22#523911yescharacteristic
509C22#524211yescharacteristic
510C22#524611yescharacteristic
511C4—21no—
512C4—21no—
513C4—81no—
514C4—81no—
515C4—82no—
516C4—242no—
517C4#503611yescharacteristic
518C4#503711yescharacteristic
519C4#503711yescharacteristic
520C4#503811yescharacteristic
521C4#503911yescharacteristic
522C4#504011yescharacteristic
523C4#504111yescharacteristic
524C4#504211yescharacteristic
525C3#1023111yescharacteristic, complemented, Sylow
526C2—82no—
527C2—82no—
528C2#7168411yescharacteristic
529C2#7168511yescharacteristic
530C2#7188111yescharacteristic
531C2#7190811yescharacteristic
532C2#7191211yescharacteristic
533C2#7191311yescharacteristic
534C2#7192911yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3#120
hypercenter#2537D3
hypocenterC3#10231
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. C2×#5019C2semidirect
3. C2×#5007C2semidirect
4. C2×D4⋊(C3⋊C8)C2semidirect
5. C2×D4⋊(C3⋊C8)C2semidirect
6. #69582C2semidirect
7. #67634C2semidirect
78. #869D4semidirect
85. C4⋊(C3⋊C8)D4semidirect
86. C4⋊(C3⋊C8)D4semidirect
128. C2×C3×D4SD16semidirect
326. C2×D4#214semidirect
525. C3#10231semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
C2×#50192
C2×#50072
C2×D4⋊(C3⋊C8)2
#695822
#676342
#102313