Finite Group Databasebeta

← #10697501

Subgroups — #10697501

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
1C2×#5087C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3#74368C211yesmaximal, characteristic
4C3×#1990C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#10231—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C2×#895—21no—
12C2×#895—21no—
13C2×C4×D12C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C3×#384C2211yescharacteristic
17C2×#835—21no—
18#5087—22no—
19#5087—22no—
20#5083—22no—
21#5083—22no—
22#4991—22no—
23#4987—21no—
24#4987—21no—
25C2×#809—21no—
26C2×#807—21no—
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C3×#340—22no—
30#4459—22no—
31#4459—22no—
32C2×#474—31no—
33C2×#461—31no—
34#2537—31no—
35#1990D311yescharacteristic, complemented
36C2×D4⋊C8—31no—
37C2×D4⋊C8—31no—
38#1296—31no—
39C22×C3×Q8—21no—
40C22×D12—22no—
41C22×D12D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×C4×D3—21no—
45C22×C3×C8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D412yescomplemented
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×#215—24no—
57C2×#215—42no—
58C2×#215—42no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67#895—42no—
68#895—42no—
69#895—42no—
70#895—42no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12—21no—
78C4×D12—21no—
79C3×C4×C8—21no—
80C3×C4×C8—21no—
81#835—24no—
82C3⋊#95—22no—
83#809—24no—
84#807—24no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yescomplemented
88C4⋊(C3⋊C8)D412yescomplemented
89C22×SD16—61no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×#121—61no—
92C2×#121—61no—
93C2×C4×D4—31no—
94C2×C4×D4—31no—
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×C8C2×D311yescharacteristic
105#394—62no—
106#384C2×D311yescharacteristic
107#380—61no—
108#380—61no—
109#340—22no—
110D4⋊C8—62no—
111D4⋊C8—62no—
112C23×D3—21no—
113C23×D3—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C22×C3×C4—22no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—44no—
138C2×D12—44no—
139C2×D12—82no—
140C2×D12D812yes—
141C2×D12SD1612yescomplemented
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—41no—
148C2×C4×D3—41no—
149C2×C4×D3—41no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C3×C8—22no—
155C2×C3×C8—41no—
156C2×C3×C8—41no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—41no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42#4111yescharacteristic
170#215—48no—
171(C2×D3)⋊C4—44no—
172(C2×D3)⋊C4—44no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177C4⋊Dic3—21no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C22×Q8—21no—
188C22×D4—31no—
189C22×D4—31no—
190C22×D4—62no—
191C23×C4—61no—
192C23×C4—61no—
193C2×SD16—64no—
194C2×SD16—122no—
195C2×SD16—122no—
196C22×C8—21no—
197C22×C8—61no—
198C22×C8—61no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8D1212yescomplemented
203#121—122no—
204#121—122no—
205#121—122no—
206#121—122no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C4×D4—61no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4#6712yes—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
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—21no—
235C4×C8—21no—
236#95—62no—
237C23×C3#12011yescharacteristic
238C22×D3—22no—
239C22×D3—22no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C3×Q8—42no—
253C3×Q8—42no—
254C3×Q8—81no—
255C3×Q8—81no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12011yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12711yescharacteristic
270C2×C3×C4C2×D811yescharacteristic
271C2×C3×C4C2×SD1611yescharacteristic
272C2×C3×C4#13511yescharacteristic
273C2×C3×C4#13511yescharacteristic
274C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
277C2×C3×C4#13711yescharacteristic
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284D12—22no—
285D12—22no—
286D12—22no—
287D12—22no—
288D12—41no—
289D12—41no—
290D12—41no—
291D12—41no—
292D12—88no—
293C4×D3—81no—
294C4×D3—81no—
295C4×D3—81no—
296C4×D3—81no—
297C4×D3—81no—
298C4×D3—81no—
299C4×D3—81no—
300C4×D3—81no—
301C3×C8—42no—
302C3⋊C8—42no—
303C3⋊C8—42no—
304C24—61no—
305C24—61no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×D4—32no—
313C2×D4—32no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—61no—
324C2×D4—124no—
325C2×D4—124no—
326C2×D4—242no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4D3×D411yescharacteristic
344C22×C4D3×D411yescharacteristic
345C22×C4C2×#6711yescharacteristic
346SD16—128no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—62no—
351C2×C8—62no—
352C2×C8—121no—
353C2×C8—121no—
354C2×C8—121no—
355C2×C8—121no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C22⋊C4—124no—
370C22⋊C4—124no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C42C2×D1211yescharacteristic
376C42#23511yescharacteristic
377C42D3×D411yescharacteristic
378C42D3×D411yescharacteristic
379C22×C3#39211yescharacteristic
380C22×C3#44711yescharacteristic
381C22×C3#44911yescharacteristic
382C22×C3#45011yescharacteristic
383C22×C3#45211yescharacteristic
384C22×C3#45511yescharacteristic
385C22×C3#45611yescharacteristic
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—42no—
392C2×D3—81no—
393C2×D3—81no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—81no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—82no—
403C3×C4—82no—
404C3×C4#44711yescharacteristic
405C3×C4#44811yescharacteristic
406C3×C4#44911yescharacteristic
407C3×C4#44911yescharacteristic
408C3×C4#46511yescharacteristic
409C3×C4#46611yescharacteristic
410C3×C4#46811yescharacteristic
411C3×C4#46911yescharacteristic
412Dic3—81no—
413Dic3—81no—
414C23—62no—
415C23—62no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23#93711yescharacteristic
429Q8—42no—
430Q8—42no—
431Q8—81no—
432Q8—81no—
433D4—62no—
434D4—62no—
435D4—62no—
436D4—62no—
437D4—121no—
438D4—121no—
439D4—121no—
440D4—121no—
441D4—248no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—241no—
457C2×C4—241no—
458C2×C4—241no—
459C2×C4—241no—
460C2×C4—241no—
461C2×C4—241no—
462C2×C4—241no—
463C2×C4—241no—
464C2×C4#88211yescharacteristic
465C2×C4#88211yescharacteristic
466C2×C4#88411yescharacteristic
467C2×C4#88411yescharacteristic
468C2×C4D3×D811yescharacteristic
469C2×C4#91111yescharacteristic
470C2×C4D3×SD1611yescharacteristic
471C2×C4#91411yescharacteristic
472C2×C4#91411yescharacteristic
473C2×C4#91611yescharacteristic
474C2×C4#91811yescharacteristic
475C2×C4#91911yescharacteristic
476C2×C4#93011yescharacteristic
477C2×C4#94011yescharacteristic
478C8—42no—
479C8—122no—
480C8—122no—
481C2×C3#163911yescharacteristic
482C2×C3#164611yescharacteristic
483C2×C3#164711yescharacteristic
484C2×C3#165411yescharacteristic
485C2×C3#199911yescharacteristic
486C2×C3#200011yescharacteristic
487C2×C3#200711yescharacteristic
488D3—82no—
489D3—82no—
490C22—122no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—241no—
497C22—241no—
498C22—241no—
499C22—241no—
500C22—241no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22#496011yescharacteristic
505C22#506111yescharacteristic
506C22#509511yescharacteristic
507C22#515611yescharacteristic
508C22#516911yescharacteristic
509C22#517211yescharacteristic
510C22#518711yescharacteristic
511C4—21no—
512C4—21no—
513C4—82no—
514C4—82no—
515C4—241no—
516C4—241no—
517C4#477311yescharacteristic
518C4#477511yescharacteristic
519C4#478111yescharacteristic
520C4#478211yescharacteristic
521C4#480611yescharacteristic
522C4#480711yescharacteristic
523C4#481011yescharacteristic
524C4#481211yescharacteristic
525C3#1023111yescharacteristic, complemented, Sylow
526C2—242no—
527C2—242no—
528C2#6969411yescharacteristic
529C2#7020411yescharacteristic
530C2#7027211yescharacteristic
531C2#7119811yescharacteristic
532C2#7123111yescharacteristic
533C2#7134311yescharacteristic
534C2#7136911yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1990C2
socleC23×C3#120
prime radical#74368C2
hypercenter#1990D3
hypocenterC3#10231
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×#5087C2semidirect
2. C2×#5083C2semidirect
4. C3×#1990C2semidirect
5. C2×#4459C2semidirect
6. C2×#4459C2semidirect
7. #67634C2semidirect
35. #1990D3semidirect
49. C3×C4⋊Q8D4semidirect
87. C4⋊(C3⋊C8)D4semidirect
88. C4⋊(C3⋊C8)D4semidirect
141. C2×D12SD16semidirect
202. C4⋊Q8D12semidirect
525. C3#10231semidirect

Maximal subgroups

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

subgroupindex
C2×#50872
C2×#50832
#743682
C3×#19902
C2×#44592
#676342
#102313