Finite Group Databasebeta

← #10822744

Subgroups — #10822744

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3C22×#835C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic
5#82270C211yesmaximal, characteristic
6C2×#4998C211yesmaximal, characteristic, complemented
7#71985C211yesmaximal, characteristic, complemented
8#70275C214yesmaximal, complemented
9#70274C214yesmaximal, complemented
10#23487—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5816—24no—
16#5816—24no—
17C2×#947—21no—
18C2×#946C2214yes—
19C2×#944—21no—
20C2×#929—22no—
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C3×#419—24no—
25C2×C3×#103C2214yes—
26C2×#835—24no—
27C2×#835C2212yes—
28C2×#835C2212yes—
29#5129—24no—
30C2×(C2×D3)⋊C8C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32#4998—24no—
33#4987C2214yes—
34C2×#808C2212yes—
35C2×#807C2212yes—
36#4958—22no—
37#4939—21no—
38#4929—21no—
39#4551C2214yes—
40C2×#536—31no—
41C2×#427—31no—
42C2×#419D311yescharacteristic, complemented
43C22×#103—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2035—34no—
48#1736—31no—
49C22×C3×Q8—21no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8D414yes—
56C22×C3×Q8C2312yes—
57C23×C3×C4—21no—
58C23×C3×C4C2311yescharacteristic
59C22×#67D412yes—
60C23×Dic3—21no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C22×D12D411yescharacteristic
68C22×C4×D3C2312yes—
69C22×Dic6D411yescharacteristic
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×#217—21no—
80#947—28no—
81#946—28no—
82#946—28no—
83#944—28no—
84#929—216no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87#926—22no—
88#926D414yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94C2×Dic3⋊C4—24no—
95C2×C4×Dic3—21no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8D414yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C3×#103—28no—
102#835—24no—
103#835—24no—
104C3⋊#95—24no—
105(C2×D3)⋊C8—24no—
106(C2×D3)⋊C8—24no—
107#808—24no—
108#807—24no—
109C22×#41—31no—
110C23×Q8C2×D311yescharacteristic
111C22×M16C2×D311yescharacteristic
112C23×C8—31no—
113#536—64no—
114#536—64no—
115C2×C4⋊Q8—61no—
116C2×#124—61no—
117C2×#123—62no—
118C2×C22⋊Q8—34no—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4—31no—
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#419—24no—
126C2×#103—32no—
127C2×#103—32no—
128C2×#103—64no—
129C2×#103C2×D314yes—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—61no—
134#390—61no—
135#386—62no—
136#380—34no—
137#340—34no—
138C24×C3C2×D411yescharacteristic
139C23×D3C2×D412yes—
140C2×C3×Q8—24no—
141C2×C3×Q8—22no—
142C2×C3×Q8—28no—
143C2×C3×Q8—28no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—42no—
149C2×C3×Q8—44no—
150C2×C3×Q8—42no—
151C2×C3×Q8—44no—
152C2×C3×Q8C2×D414yes—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C2×#67—24no—
167C2×#67—24no—
168C2×#67—24no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—42no—
172C22×Dic3—82no—
173C22×Dic3C2×D412yes—
174#235—44no—
175#235—44no—
176#235—44no—
177#235—44no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C3×M16—44no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C22⋊C4—42no—
197C3×C42—42no—
198#217—24no—
199(C2×D3)⋊C4—216no—
200(C2×D3)⋊C4—84no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—42no—
205C4⋊Dic3C2×D414yes—
206Dic3⋊C4—216no—
207C4×Dic3—24no—
208C2×C3⋊C8—28no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8C2×D412yes—
214C2×C3⋊C8C2×D412yes—
215C2×#41—62no—
216C2×#41—62no—
217C2×#41—62no—
218C2×#41—62no—
219C2×#41—62no—
220C2×#41—62no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—31no—
228C22×Q8#6714yes—
229C22×Q8C22×D312yes—
230C22×D4—32no—
231C22×D4—31no—
232C23×C4—21no—
233C23×C4—32no—
234C23×C4—61no—
235C23×C4C22×D311yescharacteristic
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C22×C8—34no—
240C22×C8—31no—
241C22×C8—31no—
242C22×C8—62no—
243C22×C8—62no—
244C22×C8C22×D312yes—
245C4⋊Q8—68no—
246#124—68no—
247#123—616no—
248C22⋊Q8—68no—
249C22⋊Q8—68no—
250#117—22no—
251#117—22no—
252#117—34no—
253#117—62no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—64no—
265C2×C42—21no—
266C2×C42—61no—
267#107—64no—
268#106—64no—
269#103—28no—
270#103—64no—
271#103—64no—
272C22⋊C8—64no—
273C22⋊C8—64no—
274#95—64no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3#12012yes—
278C23×C3#12112yes—
279C23×C3#12312yes—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—24no—
285C22×D3—24no—
286C3×Q8—24no—
287C3×Q8—24no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×Q8—48no—
291C2×C3×C4—24no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—48no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4#12012yes—
299C2×C3×C4#12114yes—
300C2×C3×C4#12112yes—
301C2×C3×C4#12114yes—
302C2×C3×C4#12312yes—
303C2×C3×C4#12314yes—
304C2×C3×C4#12414yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311#67—48no—
312C2×Dic3—24no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—44no—
320C2×Dic3—84no—
321D12—44no—
322C4×D3—42no—
323C4×D3—42no—
324C4×D3—42no—
325C4×D3—42no—
326Dic6—44no—
327C3×C8—42no—
328C3⋊C8—22no—
329C3⋊C8—22no—
330C24—32no—
331C24D3×D411yescharacteristic
332#41—124no—
333#41—124no—
334#41—124no—
335#41—124no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8C2×#6714yes—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—32no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—122no—
373C22×C4—242no—
374C22×C4#23512yes—
375C22×C4#23512yes—
376C22×C4D3×D411yescharacteristic
377C22×C4D3×D412yes—
378C22×C4D3×D412yes—
379C22×C4#23712yes—
380C22×C4C2×#6712yes—
381C22×C4C23×D311yescharacteristic
382M16—44no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8—68no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—61no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—34no—
396C4⋊C4—616no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—122no—
401C22⋊C4—42no—
402C22⋊C4—616no—
403C22⋊C4—64no—
404C22⋊C4—244no—
405C42—42no—
406C42—64no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3#39414yes—
412C22×C3C2×#12011yescharacteristic
413C22×C3C2×#12111yescharacteristic
414C22×C3C2×#12311yescharacteristic
415C22×C3#57712yes—
416C22×C3#57812yes—
417C2×D3—44no—
418C2×D3—42no—
419C2×D3—44no—
420C2×D3—42no—
421C2×D3—42no—
422C3×C4—44no—
423C3×C4—82no—
424C3×C4#39414yes—
425C3×C4C2×#12111yescharacteristic
426C3×C4C2×#12111yescharacteristic
427C3×C4C2×#12311yescharacteristic
428C3×C4C2×#12411yescharacteristic
429Dic3—42no—
430Dic3—44no—
431Dic3—82no—
432C23—22no—
433C23—22no—
434C23—62no—
435C23—62no—
436C23—64no—
437C23—64no—
438C23—62no—
439C23#88312yes—
440C23#88412yes—
441C23#93712yes—
442C23C2×D3×D411yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—48no—
446Q8—42no—
447Q8—42no—
448Q8—124no—
449D4—128no—
450D4—124no—
451C2×C4—24no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—64no—
461C2×C4—62no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—124no—
470C2×C4—244no—
471C2×C4#88312yes—
472C2×C4#88412yes—
473C2×C4#88414yes—
474C2×C4#88514yes—
475C2×C4#92914yes—
476C2×C4#93712yes—
477C2×C4#93914yes—
478C2×C4C2×#23511yescharacteristic
479C2×C4C2×#23511yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×#23711yescharacteristic
483C2×C4C22×#6711yescharacteristic
484C8—42no—
485C8—62no—
486C8—62no—
487C2×C3—24no—
488C2×C3C2×#39411yescharacteristic
489C2×C3#301312yes—
490C2×C3#305711yescharacteristic
491C2×C3#305911yescharacteristic
492C2×C3#309712yes—
493D3—82no—
494C22—24no—
495C22—24no—
496C22—22no—
497C22—22no—
498C22—124no—
499C22—124no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22#495914yes—
504C22C2×#88311yescharacteristic
505C22C2×#88411yescharacteristic
506C22#577812yes—
507C22#577912yes—
508C22C2×#93711yescharacteristic
509C4—44no—
510C4—82no—
511C4—124no—
512C4—122no—
513C4—242no—
514C4#495914yes—
515C4C2×#88411yescharacteristic
516C4C2×#88511yescharacteristic
517C4C2×#92911yescharacteristic
518C4C2×#93911yescharacteristic
519C3#2348711yescharacteristic, complemented, Sylow
520C2—24no—
521C2—242no—
522C2#7895112yes—
523C2#7896311yescharacteristic
524C2#7896411yescharacteristic
525C2C2×#495911yescharacteristic
526C2#8398412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#419D3
hypocenterC3#23487
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
3. C22×#835C2semidirect
6. C2×#4998C2semidirect
7. #71985C2semidirect
8. #70275C2semidirect
9. #70274C2semidirect
42. C2×#419D3semidirect
519. C3#23487semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C2×C3×#4192
C22×#8352
C2×#51292
#822702
C2×#49982
#719852
#702752
#702742
#234873