Finite Group Databasebeta

← #10799090

Subgroups — #10799090

Order 768.

1518 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C3×#2310C211yesmaximal, characteristic, complemented
3C22×#835C211yesmaximal, characteristic, complemented
4#82270C211yesmaximal, characteristic
5C2×#5002C211yesmaximal, characteristic
6C2×#4995C211yesmaximal, characteristic, complemented
7C2×#4997C211yesmaximal, characteristic, complemented
8#69679C212yesmaximal, complemented
9#69678C212yesmaximal, complemented
10#69677C212yesmaximal, complemented
11#69676C212yesmaximal, complemented
12#21234—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#216C2212yes—
18C2×#942—24no—
19C2×#942—24no—
20C2×#942C418yescomplemented
21C22×#215C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#391—22no—
26C3×#391C2212yes—
27C3×#391C2212yes—
28C3×#381—22no—
29C2×#835—24no—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5002—24no—
33#4997—24no—
34#4995—24no—
35#4987—22no—
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2212yes—
41C2×#807C2212yes—
42#4958—22no—
43#4939C2212yes—
44#4929C2212yes—
45C2×#803C2212yes—
46C2×C4×C3⋊C8C2212yes—
47C22×#137—31no—
48C2×#428—31no—
49C2×#416—31no—
50C22×#103—31no—
51C2×#405—31no—
52#2311—31no—
53#2310D311yescharacteristic, complemented
54#1973—32no—
55#1971—32no—
56#1969—32no—
57#1968—32no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8C2×C414yes—
65C22×C3×Q8C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C22×#67—21no—
69C23×Dic3—21no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235C2×C414yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C3×#117—22no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4D412yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4D412yes—
88C2×C3×C42—21no—
89C2×C3×C42C2312yes—
90C2×#216—24no—
91C2×#216—24no—
92C2×#216C2×C418yescomplemented
93#942—232no—
94C2×#215—24no—
95C2×#215—24no—
96C2×#215C2×C418yescomplemented
97C2×(C2×D3)⋊C4—24no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2312yes—
102C2×#208C2×C414yes—
103C2×#208D414yes—
104C2×#208D414yes—
105C22×C3⋊C8—22no—
106C22×C3⋊C8—22no—
107C22×C3⋊C8D414yes—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111C3×#95—24no—
112C3×#95—24no—
113#835—24no—
114#835—24no—
115C3⋊#95—24no—
116#810—22no—
117#810—22no—
118#809—22no—
119#809—22no—
120#808—24no—
121#807—24no—
122#803—24no—
123C4×C3⋊C8—24no—
124C22×#41—31no—
125C23×Q8C2×D311yescharacteristic
126C2×#137—38no—
127C2×#137—64no—
128C2×#137—64no—
129C22×Q16—32no—
130C22×SD16—32no—
131C22×M16—31no—
132C23×C8—31no—
133C2×#117C2×D311yescharacteristic
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4—31no—
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—64no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—64no—
145C2×#102—31no—
146C2×#102—31no—
147#405—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—22no—
151#391—32no—
152#391C2×D312yes—
153#391C2×D312yes—
154#390—32no—
155#386—62no—
156#381—22no—
157#380—62no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—28no—
163C2×C3×Q8—28no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—44no—
169C2×C3×Q8—42no—
170C2×C3×Q8—44no—
171C2×C3×Q8—42no—
172C2×C3×Q8C22×C414yes—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—24no—
189C2×#67—42no—
190C2×#67—42no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—42no—
194C22×Dic3—82no—
195#235—28no—
196#235—28no—
197#235—44no—
198#235—44no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12C22×C412yes—
205C2×C4×D3—24no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6C22×C412yes—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—42no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42—42no—
224#216—216no—
225#215—216no—
226(C2×D3)⋊C4—84no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—42no—
230#208C22⋊C4116yescomplemented
231C2×C3⋊C8—28no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8C22×C414yes—
237C2×C3⋊C8C2×D412yes—
238C2×C3⋊C8C2×D412yes—
239C2×C3⋊C8C2×D414yes—
240C2×C3⋊C8C2×D414yes—
241C2×#41—34no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C22×Q8—22no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—31no—
254C22×Q8C4×D314yes—
255C22×Q8C22×D312yes—
256C22×D4—31no—
257C22×D4—61no—
258C23×C4—21no—
259C23×C4—61no—
260C23×C4—61no—
261C23×C4C22×D311yescharacteristic
262#137—632no—
263C2×Q16—38no—
264C2×Q16—64no—
265C2×Q16—64no—
266C2×SD16—38no—
267C2×SD16—64no—
268C2×SD16—64no—
269C2×M16—34no—
270C2×M16—34no—
271C2×M16—34no—
272C22×C8—32no—
273C22×C8—34no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—62no—
277C22×C8—62no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—22no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4#6712yes—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—64no—
291C2×C22⋊C4#6712yes—
292C2×C42—21no—
293C2×C42C22×D312yes—
294#107—64no—
295#106—64no—
296#103—62no—
297#103—62no—
298#103—64no—
299#103—64no—
300#102—62no—
301#102—62no—
302#97—64no—
303C4×C8—64no—
304#95—24no—
305#95—24no—
306#95—64no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3C4×D412yes—
310C23×C3#12112yes—
311C23×C3#12712yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—42no—
317C22×D3—42no—
318C3×Q8—24no—
319C3×Q8—24no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—48no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—48no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4C2×C22⋊C414yes—
335C2×C3×C4C4×D414yes—
336C2×C3×C4C4×D412yes—
337C2×C3×C4#12114yes—
338C2×C3×C4#12112yes—
339C2×C3×C4#12414yes—
340C2×C3×C4#12712yes—
341C2×C3×C4C23×C411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347#67—48no—
348C2×Dic3—22no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—42no—
352C2×Dic3—44no—
353C2×Dic3—84no—
354D12—22no—
355D12—22no—
356D12—41no—
357D12—41no—
358C4×D3—48no—
359Dic6—22no—
360Dic6—22no—
361Dic6—41no—
362Dic6—41no—
363C3⋊C8—22no—
364C3⋊C8—22no—
365C3⋊C8C2×C22⋊C418yes—
366C24—61no—
367C24C2×#6711yescharacteristic
368#41—68no—
369#41—68no—
370#41—124no—
371#41—124no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—32no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8C2×C4×D314yes—
391C2×D4—32no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—122no—
399C2×D4—122no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—61no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—242no—
413C22×C4C2×C4×D312yes—
414C22×C4#23512yes—
415C22×C4D3×D412yes—
416C22×C4D3×D412yes—
417C22×C4D3×D412yes—
418C22×C4#23712yes—
419C22×C4C2×#6714yes—
420C22×C4C2×#6711yescharacteristic
421C22×C4C23×D311yescharacteristic
422Q16—616no—
423SD16—616no—
424M16—316no—
425C2×C8—34no—
426C2×C8—34no—
427C2×C8—34no—
428C2×C8—32no—
429C2×C8—32no—
430C2×C8—68no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—122no—
443C22⋊C4—24no—
444C22⋊C4—42no—
445C22⋊C4—244no—
446C42—22no—
447C42—22no—
448C42—42no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3#39014yes—
454C22×C3C2×C4×D411yescharacteristic
455C22×C3C2×#12111yescharacteristic
456C22×C3C2×#12711yescharacteristic
457C22×C3#57712yes—
458C22×C3#57812yes—
459C2×D3—44no—
460C2×D3—42no—
461C2×D3—81no—
462C2×D3—81no—
463C2×D3—82no—
464C3×C4—44no—
465C3×C4—44no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4#39014yes—
469C3×C4C22×C22⋊C411yescharacteristic
470C3×C4C2×C4×D411yescharacteristic
471C3×C4C2×#12111yescharacteristic
472C3×C4C2×#12411yescharacteristic
473Dic3—42no—
474Dic3—82no—
475C23—22no—
476C23—22no—
477C23—62no—
478C23—121no—
479C23—121no—
480C23—122no—
481C23—122no—
482C23C4×#6712yes—
483C23#93912yes—
484C23#94012yes—
485C23C22×#6711yescharacteristic
486Q8—24no—
487Q8—24no—
488Q8—48no—
489Q8—42no—
490Q8—42no—
491Q8—62no—
492Q8—62no—
493Q8—121no—
494Q8—121no—
495D4—62no—
496D4—62no—
497D4—128no—
498D4—121no—
499D4—121no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—62no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—128no—
515C2×C4—122no—
516C2×C4—124no—
517C2×C4—244no—
518C2×C4D3×C22⋊C414yes—
519C2×C4#67⋊C414yes—
520C2×C4#88414yes—
521C2×C4#88514yes—
522C2×C4C4×#6712yes—
523C2×C4#93912yes—
524C2×C4#94012yes—
525C2×C4C22×C4×D311yescharacteristic
526C2×C4C2×#23511yescharacteristic
527C2×C4C2×D3×D411yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C8—38no—
532C8—62no—
533C8—62no—
534C2×C3—24no—
535C2×C3C2×#39011yescharacteristic
536C2×C3#294712yes—
537C2×C3#306312yes—
538C2×C3#315411yescharacteristic
539C2×C3#315611yescharacteristic
540D3—82no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—124no—
546C22—122no—
547C22—242no—
548C22—241no—
549C22—241no—
550C22#495714yes—
551C22C2×C4×#6711yescharacteristic
552C22C2×#93911yescharacteristic
553C22C2×#94011yescharacteristic
554C22#583712yes—
555C22#583812yes—
556C4—44no—
557C4—44no—
558C4—81no—
559C4—81no—
560C4—122no—
561C4—242no—
562C4#495714yes—
563C4C2×D3×C22⋊C411yescharacteristic
564C4C2×#67⋊C411yescharacteristic
565C4C2×#88411yescharacteristic
566C4C2×#88511yescharacteristic
567C3#2123411yescharacteristic, complemented, Sylow
568C2—24no—
569C2—242no—
570C2C2×#495711yescharacteristic
571C2#8254712yes—
572C2#8340012yes—
573C2#8340211yescharacteristic
574C2#8340311yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC23×C3C22×D4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#2310D3
hypocenterC3#21234
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C3×#2310C2semidirect
3. C22×#835C2semidirect
6. C2×#4995C2semidirect
7. C2×#4997C2semidirect
8. #69679C2semidirect
9. #69678C2semidirect
10. #69677C2semidirect
11. #69676C2semidirect
20. C2×#942C4semidirect
53. #2310D3semidirect
92. C2×#216C2×C4semidirect
96. C2×#215C2×C4semidirect
230. #208C22⋊C4semidirect
567. C3#21234semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C3×#23102
C22×#8352
#822702
C2×#50022
C2×#49952
C2×#49972
#696792
#696782
#696772
#696762
#212343