Finite Group Databasebeta

← #10692506

Subgroups — #10692506

Order 768.

640 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78601C211yesmaximal, characteristic, complemented
2#73378C211yesmaximal, characteristic
3#69944C211yesmaximal, characteristic, complemented
4#69530C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9384—31nomaximal, Sylow
9C2×#930—21no—
10C2×#930C2211yescharacteristic
11C2×C4×#67C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5213—21no—
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C8×#67—22no—
19C2×#806—21no—
20#67⋊C8—22no—
21D3×C22⋊C8—21no—
22#4718—21no—
23#4674C411yescharacteristic, complemented
24#4672C411yescharacteristic, complemented
25#4671C411yescharacteristic, complemented
26#4670—21no—
27#4665—21no—
28#4661C411yescharacteristic, complemented
29#4527—21no—
30#4470—22no—
31#4451—22no—
32#2972—31no—
33#2543—31no—
34#1799—31no—
35#1519—31no—
36#1519—31no—
37#1294—31no—
38#1294D311yescharacteristic, complemented
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×#67D411yescharacteristic
43C22×D12—21no—
44C22×D12C2×C411yescharacteristic
45C22×C4×D3D411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217D411yescharacteristic
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—42no—
56#930—42no—
57C4×#67—21no—
58C4×#67—21no—
59C4×#67—21no—
60C4×#67—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4C2×C411yescharacteristic
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3C2×C411yescharacteristic
70C2×Dic3⋊C4C2×C411yescharacteristic
71C2×C4×Dic3D411yescharacteristic
72C22×C3⋊C8—21no—
73C2×D3×C8—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yescomplemented
76C3×#95—21no—
77C3×#95—21no—
78C3×#95C2×C411yescharacteristic
79C3×#95C2×C411yescharacteristic
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830D412yescomplemented
85(C2×D3)⋊C8—22no—
86Dic3⋊C8—22no—
87C8×Dic3—22no—
88#806—42no—
89C2×#121—31no—
90C2×#121—61no—
91C2×C4×D4—31no—
92C8×D4—62no—
93C8×D4—62no—
94#431—61no—
95C2×#99—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—61no—
98C2×C22⋊C8C2×D311yescharacteristic
99#398—31no—
100#397—31no—
101#394—31no—
102#392—31no—
103#388—61no—
104#386—61no—
105#386—61no—
106#380—31no—
107#380C2×D311yescharacteristic
108C2×#95C2×D311yescharacteristic
109#342—61no—
110#327—62no—
111#327—62no—
112C24×C3C2×D411yescharacteristic
113C23×D3—21no—
114C23×D3C22⋊C411yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—81no—
122C22×C3×C4—81no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—21no—
130C2×#67—21no—
131C2×#67—21no—
132C2×#67—21no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C2×#67—41no—
139C2×#67D812yes—
140C2×#67SD1612yes—
141C22×Dic3—21no—
142C22×Dic3—41no—
143C22×Dic3C22⋊C411yescharacteristic
144C22×Dic3C22⋊C411yescharacteristic
145C22×Dic3C22⋊C411yescharacteristic
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—42no—
149C2×D12—42no—
150C2×D12—81no—
151C2×D12—81no—
152C2×C4×D3—21no—
153C2×C4×D3—21no—
154C2×C4×D3—81no—
155C2×C3×C8—22no—
156C2×C3×C8—81no—
157C3×C4⋊C4—81no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160#217—21no—
161#217—21no—
162#217—41no—
163#217—41no—
164(C2×D3)⋊C4—21no—
165(C2×D3)⋊C4—21no—
166(C2×D3)⋊C4—42no—
167(C2×D3)⋊C4—42no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—42no—
171(C2×D3)⋊C4—81no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—42no—
175C4⋊Dic3—81no—
176Dic3⋊C4—21no—
177Dic3⋊C4—21no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—81no—
182D3×C8—82no—
183C22×D4—31no—
184C22×D4—31no—
185C22×D4—61no—
186C22×D4—61no—
187C23×C4—21no—
188C23×C4—31no—
189C23×C4C22×D311yescharacteristic
190C22×C8—21no—
191C22×C8—62no—
192C22×C8—61no—
193#121—61no—
194#121—61no—
195#121—61no—
196#121—61no—
197#121—122no—
198#121—122no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—31no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C42—31no—
218C4⋊C8—62no—
219#99—122no—
220C22⋊C8—21no—
221C22⋊C8—32no—
222C22⋊C8—62no—
223C22⋊C8—61no—
224C22⋊C8D1212yescomplemented
225C4×C8—62no—
226#95—21no—
227#95—21no—
228#95—61no—
229#95—61no—
230#95—61no—
231#95C4×D311yescharacteristic
232#95C4×D311yescharacteristic
233C23×C3—41no—
234C23×C3—41no—
235C23×C3C2×C22⋊C411yescharacteristic
236C23×C3#12111yescharacteristic
237C23×C3#12111yescharacteristic
238C23×C3C2×D811yescharacteristic
239C23×C3C2×SD1611yescharacteristic
240C23×C3#13511yescharacteristic
241C23×C3#13511yescharacteristic
242C22×D3—21no—
243C22×D3—21no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3#10212yes—
254C2×C3×C4—22no—
255C2×C3×C4—41no—
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—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—82no—
267C2×C3×C4—82no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4—81no—
271C2×C3×C4C4×D411yescharacteristic
272C2×C3×C4C4×D411yescharacteristic
273C2×C3×C4#12011yescharacteristic
274C2×C3×C4#12311yescharacteristic
275#67—22no—
276#67—22no—
277#67—22no—
278#67—22no—
279#67—81no—
280#67—81no—
281#67—81no—
282#67—81no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3#10212yes—
296D12—81no—
297D12—81no—
298D12—84no—
299C4×D3—42no—
300C4×D3—81no—
301C3×C8—42no—
302C3⋊C8—42no—
303C24—31no—
304C24—61no—
305C24D3×D411yescharacteristic
306C2×D4—32no—
307C2×D4—32no—
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—61no—
312C2×D4—61no—
313C2×D4—61no—
314C2×D4—122no—
315C2×D4—122no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—121no—
319C2×D4—121no—
320C2×D4—121no—
321C2×D4—121no—
322C2×D4—241no—
323C2×D4—241no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—121no—
339C22×C4—241no—
340C22×C4C2×C4×D311yescharacteristic
341C22×C4C2×D1211yescharacteristic
342C22×C4#23511yescharacteristic
343C22×C4D3×D411yescharacteristic
344C22×C4D3×D411yescharacteristic
345C22×C4#23911yescharacteristic
346C2×C8—22no—
347C2×C8—62no—
348C2×C8—81no—
349C2×C8—242no—
350C2×C8—241no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—81no—
356C4⋊C4—122no—
357C4⋊C4—241no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—61no—
361C22⋊C4—61no—
362C22⋊C4—61no—
363C22⋊C4—61no—
364C22⋊C4—122no—
365C22⋊C4—122no—
366C22⋊C4—122no—
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—241no—
372C42—61no—
373C42—61no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3#38611yescharacteristic
385C22×C3C2×#10211yescharacteristic
386C22×C3#41611yescharacteristic
387C22×C3#46511yescharacteristic
388C22×C3#46511yescharacteristic
389C22×C3#46611yescharacteristic
390C22×C3#46611yescharacteristic
391C2×D3—22no—
392C2×D3—41no—
393C2×D3—41no—
394C2×D3—42no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—81no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—42no—
408C3×C4—81no—
409C3×C4—161no—
410Dic3—22no—
411Dic3—41no—
412Dic3—41no—
413Dic3—81no—
414Dic3—161no—
415C23—32no—
416C23—41no—
417C23—41no—
418C23—61no—
419C23—61no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23D3×C22⋊C411yescharacteristic
430C23#88411yescharacteristic
431C23#89511yescharacteristic
432C23D3×D811yescharacteristic
433C23D3×SD1611yescharacteristic
434C23#91611yescharacteristic
435C23#91911yescharacteristic
436D4—62no—
437D4—62no—
438D4—62no—
439D4—62no—
440D4—241no—
441D4—241no—
442D4—244no—
443D4—241no—
444D4—241no—
445D4—241no—
446D4—241no—
447C2×C4—22no—
448C2×C4—32no—
449C2×C4—41no—
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—41no—
459C2×C4—41no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—122no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—241no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4C4×D1211yescharacteristic
480C2×C4#88211yescharacteristic
481C2×C4#89311yescharacteristic
482C2×C4#89411yescharacteristic
483C8—42no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3#188411yescharacteristic
492C2×C3#189311yescharacteristic
493C2×C3#191311yescharacteristic
494D3—42no—
495D3—161no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—62no—
507C22—122no—
508C22—121no—
509C22—121no—
510C22—241no—
511C22—241no—
512C22—241no—
513C22—241no—
514C22—241no—
515C22—241no—
516C22—241no—
517C22—241no—
518C22#466511yescharacteristic
519C22D3×#10211yescharacteristic
520C22#477511yescharacteristic
521C22#480411yescharacteristic
522C22#480711yescharacteristic
523C22#486411yescharacteristic
524C22#488311yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—42no—
530C4—62no—
531C4—81no—
532C4—121no—
533C4—121no—
534C4—161no—
535C4—241no—
536C4—481no—
537C3#938411yescharacteristic, complemented, Sylow
538C2—21no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—122no—
545C2—481no—
546C2#6877711yescharacteristic
547C2#6888311yescharacteristic
548C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC22×C3#386
prime radical#73378C2
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9384
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
1. #78601C2semidirect
3. #69944C2semidirect
4. #69530C2semidirect
5. #69509C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
23. #4674C4semidirect
24. #4672C4semidirect
25. #4671C4semidirect
28. #4661C4semidirect
38. #1294D3semidirect
75. C3×C22⋊C8D4semidirect
84. #830D4semidirect
224. C22⋊C8D12semidirect
537. C3#9384semidirect

Maximal subgroups

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

subgroupindex
#786012
#733782
#699442
#695302
#695092
C3×#12942
#676122
#93843