Finite Group Databasebeta

← #10690339

Subgroups — #10690339

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4736C211yesmaximal, characteristic
2#73873C211yesmaximal, characteristic
3#69501C211yesmaximal, characteristic, complemented
4#69423C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9043—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#67⋊C4—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#821—21no—
20C2×#818—21no—
21#4990—21no—
22#4989—21no—
23#4958—21no—
24#4736—22no—
25#4736C414yescomplemented
26#4665—21no—
27#4654—21no—
28#4529—21no—
29#4529—21no—
30#4451—22no—
31C2×#480—31no—
32#2325—31no—
33#1823—31no—
34C2×#327—31no—
35#1445—31no—
36#1294D311yescharacteristic, complemented
37#1280—31no—
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C23×Dic3D411yescharacteristic
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50#930—21no—
51#930—21no—
52#930C2×C412yes—
53#930D412yes—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2×C412yescomplemented
64C2×C4⋊Dic3D411yescharacteristic
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—21no—
68#67⋊C4—42no—
69#67⋊C4—42no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8C2×C412yescomplemented
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81#830—21no—
82#830—21no—
83#821—24no—
84#818—24no—
85C2×#121—31no—
86C2×C4×D4—61no—
87C22×C4⋊C4—31no—
88#480—34no—
89#480—62no—
90C2×#107—61no—
91C2×#102—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8C2×D311yescharacteristic
94#388—61no—
95#387—61no—
96#386—61no—
97#386—61no—
98#385—61no—
99#380C2×D311yescharacteristic
100C2×#95—31no—
101C2×#95—61no—
102C2×#95—61no—
103C2×#95C2×D311yescharacteristic
104#340—61no—
105#340—61no—
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—22no—
127C2×#67—22no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3D812yes—
150C2×D12—21no—
151C2×D12—41no—
152C2×D12—41no—
153C2×D12C22⋊C412yes—
154C2×C4×D3—42no—
155C2×C4×D3—81no—
156C2×C4×D3—81no—
157C2×C3×C8—22no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164(C2×D3)⋊C4—22no—
165(C2×D3)⋊C4—22no—
166(C2×D3)⋊C4—42no—
167(C2×D3)⋊C4—81no—
168(C2×D3)⋊C4—81no—
169C4⋊Dic3—21no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—41no—
173C4⋊Dic3—41no—
174C4⋊Dic3C22⋊C412yes—
175Dic3⋊C4—42no—
176C4×Dic3—42no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C22×D4—31no—
181C22×D4—61no—
182C23×C4—21no—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—61no—
186C23×C4—61no—
187C23×C4C22×D311yescharacteristic
188C22×C8—21no—
189C22×C8—61no—
190#121—32no—
191#121—32no—
192#121—61no—
193#121—61no—
194C4×D4—122no—
195C4×D4—122no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4D1211yescharacteristic
205C2×C4⋊C4D1211yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C42—61no—
212#107—64no—
213#102—64no—
214C22⋊C8—21no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8C4×D312yescomplemented
218#95—21no—
219#95—21no—
220#95—21no—
221#95—61no—
222#95—61no—
223#95—61no—
224#95—61no—
225#95—122no—
226#95—122no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3C4×D411yescharacteristic
232C23×C3#12011yescharacteristic
233C23×C3#12111yescharacteristic
234C23×C3C2×D811yescharacteristic
235C23×C3#13511yescharacteristic
236C23×C3AGL(1,ℤ/8)11yescharacteristic
237C23×C3AGL(1,ℤ/8)11yescharacteristic
238C22×D3—22no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4C2×C22⋊C411yescharacteristic
267C2×C3×C4C4×D411yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12311yescharacteristic
270#67—42no—
271#67—42no—
272#67—42no—
273#67—42no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—81no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291C2×Dic3—81no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294D12—42no—
295D12—81no—
296C4×D3—84no—
297C3×C8—42no—
298C3⋊C8—81no—
299C24—61no—
300C24D3×D411yescharacteristic
301C2×D4—32no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—61no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—121no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—121no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—32no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—122no—
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—241no—
342C22×C4—241no—
343C22×C4C2×C4×D311yescharacteristic
344C22×C4C2×D1211yescharacteristic
345C22×C4#23511yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4#23911yescharacteristic
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—121no—
353C2×C8—121no—
354C2×C8—121no—
355C4⋊C4—32no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—61no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—122no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—62no—
367C22⋊C4—62no—
368C22⋊C4—122no—
369C22⋊C4—241no—
370C22⋊C4—241no—
371C42—122no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3#38611yescharacteristic
385C22×C3C4×D811yescharacteristic
386C22×C3#44211yescharacteristic
387C22×C3#44711yescharacteristic
388C22×C3#44911yescharacteristic
389C22×C3#45511yescharacteristic
390C22×C3#45611yescharacteristic
391C22×C3#45811yescharacteristic
392C22×C3#45811yescharacteristic
393C22×C3#45911yescharacteristic
394C22×C3#46311yescharacteristic
395C2×D3—42no—
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—82no—
407C3×C4—81no—
408C3×C4—81no—
409Dic3—42no—
410Dic3—42no—
411Dic3—81no—
412Dic3—82no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—62no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—121no—
424C23#88211yescharacteristic
425C23#88411yescharacteristic
426C23#89311yescharacteristic
427C23D3×D811yescharacteristic
428C23#91411yescharacteristic
429C23#91411yescharacteristic
430C23#91911yescharacteristic
431D4—122no—
432D4—122no—
433D4—122no—
434D4—122no—
435D4—122no—
436D4—241no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—244no—
473C2×C4—241no—
474C2×C4—241no—
475C2×C4—241no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4C4×D1211yescharacteristic
480C2×C4D3×C22⋊C411yescharacteristic
481C2×C4#89411yescharacteristic
482C2×C4#89511yescharacteristic
483C8—42no—
484C8—241no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3#160711yescharacteristic
490C2×C3#160911yescharacteristic
491C2×C3#161011yescharacteristic
492C2×C3#161211yescharacteristic
493C2×C3#187911yescharacteristic
494C2×C3#188211yescharacteristic
495C2×C3#189011yescharacteristic
496D3—82no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—241no—
513C22—241no—
514C22—241no—
515C22—241no—
516C22#466511yescharacteristic
517C22#474211yescharacteristic
518C22#474211yescharacteristic
519C22#477311yescharacteristic
520C22#478611yescharacteristic
521C22#481011yescharacteristic
522C22#481911yescharacteristic
523C22#482511yescharacteristic
524C22#482611yescharacteristic
525C22#486911yescharacteristic
526C22#487211yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—82no—
532C4—81no—
533C4—81no—
534C4—122no—
535C4—122no—
536C4—242no—
537C4—241no—
538C3#904311yescharacteristic, complemented, Sylow
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—242no—
544C2#6828811yescharacteristic
545C2#6829711yescharacteristic
546C2#6832111yescharacteristic
547C2#6832811yescharacteristic
548C2#6877511yescharacteristic
549C2#6889711yescharacteristic
550C2#6896311yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9043
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69501C2semidirect
4. #69423C2semidirect
6. C3×#1294C2semidirect
7. #67674C2semidirect
25. #4736C4semidirect
36. #1294D3semidirect
63. C2×C4⋊Dic3C2×C4semidirect
71. C3×C22⋊C8C2×C4semidirect
217. C22⋊C8C4×D3semidirect
538. C3#9043semidirect

Maximal subgroups

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

subgroupindex
C2×#47362
#738732
#695012
#694232
C2×#44512
C3×#12942
#676742
#90433