Finite Group Databasebeta

← #10822719

Subgroups — #10822719

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5526C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic, complemented
3C2×#5120C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic
5#82186C211yesmaximal, characteristic
6C22×#810C211yesmaximal, characteristic, complemented
7#71580C211yesmaximal, characteristic, complemented
8#70635C214yesmaximal, complemented
9#70582C214yesmaximal, complemented
10#23484—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18#5526—24no—
19#5526—24no—
20C2×#897C2212yes—
21C2×#897C2212yes—
22C2×#886—22no—
23C2×#875—21no—
24C2×#869—21no—
25C3×#428—24no—
26C2×C3×#107C2212yes—
27C2×C3×#106C2212yes—
28#5213—22no—
29#5129—24no—
30#5120—24no—
31C2×(C2×D3)⋊C8C2214yes—
32C2×#816C2214yes—
33#4988—21no—
34#4987C2214yes—
35#4973—21no—
36C2×#810—24no—
37C2×#810C2214yes—
38#4529C2212yes—
39#4529C2212yes—
40C2×#536—31no—
41C2×#428D311yescharacteristic, complemented
42C2×#419—31no—
43C22×#103—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2035—34no—
48#1735—31no—
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67D412yes—
52C23×Dic3—21no—
53C2×#235—22no—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C22×D12D411yescharacteristic
60C22×C4×D3C2312yes—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—21no—
66C22×Dic6—22no—
67C22×Dic6D411yescharacteristic
68C22×Dic6D414yes—
69C22×Dic6C2312yes—
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16D414yes—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117D414yes—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×#217—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—24no—
88#926—22no—
89#926—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4×Dic3—21no—
94C22×C3⋊C8—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8C2312yes—
98#897—24no—
99#897—24no—
100#897—24no—
101#897—24no—
102#886—216no—
103#875—28no—
104#869—28no—
105C3×#107—24no—
106C3×#106—24no—
107C3⋊#95—24no—
108(C2×D3)⋊C8—28no—
109#816—28no—
110#810—28no—
111C22×#41—31no—
112C23×Q8—31no—
113C22×M16C2×D311yescharacteristic
114C23×C8—31no—
115#536—64no—
116#536—64no—
117C2×C4⋊Q8—61no—
118C2×#124—61no—
119C2×#123—62no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—32no—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4C2×D311yescharacteristic
125#428—24no—
126C2×#107C2×D312yes—
127C2×#106C2×D312yes—
128#419—64no—
129C2×#103—34no—
130C2×#103—34no—
131C2×#103—64no—
132#408—64no—
133C2×C22⋊C8—34no—
134#391—61no—
135#390—61no—
136#386—62no—
137#380—34no—
138#340—32no—
139#340—32no—
140C24×C3C2×D411yescharacteristic
141C23×D3C2×D412yes—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—42no—
146C22×C3×C4—82no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4#4112yes—
152C22×C3×C4#4112yes—
153C22×C3×C4#4112yes—
154C22×C3×C4C2411yescharacteristic
155C2×#67—24no—
156C2×#67—24no—
157C2×#67—24no—
158C22×Dic3—24no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3C2×D412yes—
164#235—44no—
165#235—44no—
166#235—44no—
167#235—44no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×Dic6—28no—
178C2×Dic6—28no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—24no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6—42no—
189C2×Dic6—44no—
190C2×Dic6—42no—
191C2×Dic6—44no—
192C2×Dic6C2×D414yes—
193C3×M16—28no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8C2×D414yes—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D412yes—
204C3×C22⋊C4—24no—
205C3×C42—24no—
206#217—42no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—84no—
210C4⋊Dic3—28no—
211C4⋊Dic3—28no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4×Dic3—42no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—44no—
219C2×#41—62no—
220C2×#41—62no—
221C2×#41—62no—
222C2×#41—62no—
223C2×#41—62no—
224C2×#41—62no—
225C22×Q8—31no—
226C22×Q8—34no—
227C22×Q8—32no—
228C22×Q8—61no—
229C22×Q8—62no—
230C22×Q8—62no—
231C22×Q8—62no—
232C22×Q8—62no—
233C22×Q8—62no—
234C22×D4—32no—
235C22×D4—31no—
236C23×C4—21no—
237C23×C4—32no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C2×M16—22no—
241C2×M16—22no—
242C2×M16D1214yes—
243C22×C8—32no—
244C22×C8—62no—
245C22×C8—62no—
246C22×C8—62no—
247C22×C8C22×D312yes—
248C4⋊Q8—68no—
249#124—68no—
250#123—616no—
251C22⋊Q8—64no—
252C22⋊Q8—64no—
253C22⋊Q8—64no—
254C22⋊Q8—64no—
255#117—22no—
256#117—62no—
257#117—62no—
258#117#6714yes—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4C22×D312yes—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—64no—
273C2×C42—21no—
274C2×C42—61no—
275#107—24no—
276#106—24no—
277#103—68no—
278#103—68no—
279C22⋊C8—68no—
280#95—64no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3#12012yes—
284C23×C3#12112yes—
285C23×C3#12312yes—
286C23×C3C22×D411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—24no—
291C22×D3—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—44no—
298C2×C3×C4—84no—
299C2×C3×C4#12012yes—
300C2×C3×C4#12114yes—
301C2×C3×C4#12112yes—
302C2×C3×C4#12114yes—
303C2×C3×C4#12314yes—
304C2×C3×C4#12312yes—
305C2×C3×C4#12414yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312#67—48no—
313C2×Dic3—22no—
314C2×Dic3—24no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—48no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323D12—44no—
324C4×D3—42no—
325C4×D3—42no—
326C4×D3—42no—
327C4×D3—42no—
328Dic6—24no—
329Dic6—24no—
330Dic6—42no—
331Dic6—42no—
332Dic6—48no—
333Dic6—44no—
334C3×C8—24no—
335C3⋊C8—42no—
336C24—32no—
337C24D3×D411yescharacteristic
338#41—124no—
339#41—124no—
340#41—124no—
341#41—124no—
342C2×Q8—34no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—64no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×Q8—68no—
349C2×Q8—68no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—122no—
355C2×Q8—122no—
356C2×Q8—124no—
357C2×Q8—124no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—62no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—32no—
368C22×C4—42no—
369C22×C4—64no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—82no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4C2×D1212yes—
382C22×C4#23512yes—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D412yes—
385C22×C4#23912yes—
386C22×C4#23912yes—
387C22×C4C2×#6712yes—
388C22×C4C23×D311yescharacteristic
389M16—28no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—124no—
396C2×C8C2×D1214yes—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—68no—
403C4⋊C4—68no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4C2×#6712yes—
407C4⋊C4C2×#6712yes—
408C22⋊C4—24no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—122no—
412C22⋊C4—244no—
413C42—24no—
414C42—122no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3#39414yes—
420C22×C3C2×#12011yescharacteristic
421C22×C3C2×#12111yescharacteristic
422C22×C3C2×#12311yescharacteristic
423C22×C3#57712yes—
424C22×C3#57812yes—
425C2×D3—44no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—44no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—82no—
433C3×C4#39414yes—
434C3×C4C2×#12111yescharacteristic
435C3×C4C2×#12111yescharacteristic
436C3×C4C2×#12311yescharacteristic
437C3×C4C2×#12411yescharacteristic
438Dic3—44no—
439Dic3—42no—
440Dic3—82no—
441C23—22no—
442C23—22no—
443C23—62no—
444C23—62no—
445C23—64no—
446C23—64no—
447C23—62no—
448C23#89412yes—
449C23#89512yes—
450C23#93712yes—
451C23C2×D3×D411yescharacteristic
452Q8—64no—
453Q8—64no—
454Q8—128no—
455Q8—124no—
456Q8—122no—
457Q8—122no—
458D4—128no—
459D4—124no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—64no—
469C2×C4—62no—
470C2×C4—84no—
471C2×C4—128no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4#89412yes—
482C2×C4#89414yes—
483C2×C4#89512yes—
484C2×C4#89514yes—
485C2×C4#93014yes—
486C2×C4#93712yes—
487C2×C4#94714yes—
488C2×C4C22×D1211yescharacteristic
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×#23911yescharacteristic
492C2×C4C2×#23911yescharacteristic
493C2×C4C22×#6711yescharacteristic
494C8—24no—
495C8—122no—
496C2×C3—24no—
497C2×C3C2×#39411yescharacteristic
498C2×C3#301011yescharacteristic
499C2×C3#301111yescharacteristic
500C2×C3#306312yes—
501C2×C3#309712yes—
502D3—82no—
503C22—22no—
504C22—24no—
505C22—24no—
506C22—22no—
507C22—122no—
508C22—122no—
509C22—124no—
510C22—124no—
511C22—122no—
512C22#499114yes—
513C22C2×#89411yescharacteristic
514C22C2×#89511yescharacteristic
515C22#577512yes—
516C22#577912yes—
517C22C2×#93711yescharacteristic
518C4—42no—
519C4—42no—
520C4—82no—
521C4—122no—
522C4—124no—
523C4—242no—
524C4#499114yes—
525C4C2×#89411yescharacteristic
526C4C2×#89511yescharacteristic
527C4C2×#93011yescharacteristic
528C4C2×#94711yescharacteristic
529C3#2348411yescharacteristic, complemented, Sylow
530C2—24no—
531C2—242no—
532C2#7921412yes—
533C2#7922012yes—
534C2C2×#499111yescharacteristic
535C2#8394811yescharacteristic
536C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#428D3
hypocenterC3#23484
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C3×#428C2semidirect
3. C2×#5120C2semidirect
6. C22×#810C2semidirect
7. #71580C2semidirect
8. #70635C2semidirect
9. #70582C2semidirect
41. C2×#428D3semidirect
529. C3#23484semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
C2×C3×#4282
C2×#51202
C2×#51292
#821862
C22×#8102
#715802
#706352
#705822
#234843