Finite Group Databasebeta

← #10822717

Subgroups — #10822717

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×#5825C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic
3C2×#5234C211yesmaximal, characteristic
4C2×#5004C211yesmaximal, characteristic, complemented
5C22×#810C211yesmaximal, characteristic, complemented
6C2×#4964C211yesmaximal, characteristic, complemented
7#71580C211yesmaximal, characteristic, complemented
8#69527C214yesmaximal, complemented
9#69526C214yesmaximal, complemented
10#23484—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15#5825—24no—
16#5825—24no—
17C2×#944—21no—
18C2×#936—21no—
19C2×#935—22no—
20C2×#926C2211yescharacteristic
21C2×#924C2212yes—
22C2×#924C2212yes—
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#391—21no—
26C3×#390—21no—
27C3×#386—22no—
28C3×#380C2214yes—
29#5234—24no—
30C2×#830C2214yes—
31#5004—24no—
32C2×#810—24no—
33C2×#810C2214yes—
34C2×#810C2214yes—
35#4964—24no—
36C2×#808C2212yes—
37C2×#807C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40C2×#536—31no—
41C2×#428—31no—
42C2×#419—31no—
43C22×#103—31no—
44C2×#408—31no—
45#2311D311yescharacteristic
46#2036—34no—
47#2035—34no—
48#1735—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4D412yes—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—21no—
67C22×Dic6—22no—
68C22×Dic6D414yes—
69C22×Dic6C2312yes—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C42—21no—
80C2×#217—22no—
81C2×#217—22no—
82C2×#217—21no—
83#944—28no—
84#936—28no—
85#935—216no—
86#926—22no—
87#926D414yes—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90#924—24no—
91#924—24no—
92#924—24no—
93#924—24no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208D414yes—
100C22×C3⋊C8—22no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105C3×#95—24no—
106#830—28no—
107#810—28no—
108#810—28no—
109#808—24no—
110#807—24no—
111C22×#41C2×D311yescharacteristic
112C23×Q8—31no—
113C22×M16—31no—
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—64no—
126C2×#107—32no—
127C2×#106—32no—
128#419—64no—
129C2×#103—34no—
130C2×#103—34no—
131C2×#103—64no—
132#408—64no—
133C2×C22⋊C8—34no—
134#391—21no—
135#390—21no—
136#386—22no—
137#380C2×D314yes—
138#340—32no—
139#340—32no—
140C24×C3C2×D412yes—
141C24×C3C2×D411yescharacteristic
142C3×#41—44no—
143C3×#41—44no—
144C3×#41—44no—
145C3×#41—44no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—42no—
164C22×C3×C4—82no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4#4112yes—
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—24no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—42no—
179C22×Dic3—42no—
180C2×Dic6—28no—
181C2×Dic6—28no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—24no—
187C2×Dic6—22no—
188C2×Dic6—44no—
189C2×Dic6—42no—
190C2×Dic6—42no—
191C2×Dic6—44no—
192C2×Dic6C2×D414yes—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—84no—
200C3×C42—42no—
201#217—28no—
202#217—28no—
203#217—24no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3C2×D412yes—
207C4⋊Dic3C2×D412yes—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210C4×Dic3—24no—
211#208—28no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—44no—
218C2×C3⋊C8C2×D414yes—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C22×Q8—34no—
226C22×Q8—32no—
227C22×Q8—62no—
228C22×Q8—61no—
229C22×Q8—62no—
230C22×Q8—62no—
231C22×Q8—62no—
232C22×Q8—62no—
233C22×Q8#6711yescharacteristic
234C22×D4#6712yes—
235C22×D4#6711yescharacteristic
236C23×C4—21no—
237C23×C4—61no—
238C23×C4C22×D312yes—
239C23×C4C22×D311yescharacteristic
240C2×M16—34no—
241C2×M16—62no—
242C2×M16—62no—
243C22×C8—32no—
244C22×C8—32no—
245C22×C8—62no—
246C22×C8—62no—
247C22×C8—62no—
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—22no—
257#117—34no—
258#117—62no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C42—21no—
274C2×C42—61no—
275#107—64no—
276#106—64no—
277#103—68no—
278#103—68no—
279C22⋊C8—68no—
280#95—24no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3#12012yes—
289C23×C3#12112yes—
290C23×C3#12312yes—
291C23×C3C22×D411yescharacteristic
292C3×Q8—44no—
293C3×D4—48no—
294C3×D4—44no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—44no—
306C2×C3×C4—42no—
307C2×C3×C4—84no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12114yes—
310C2×C3×C4#12114yes—
311C2×C3×C4#12112yes—
312C2×C3×C4#12314yes—
313C2×C3×C4#12312yes—
314C2×C3×C4#12414yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—48no—
329Dic6—24no—
330Dic6—24no—
331Dic6—42no—
332Dic6—42no—
333Dic6—48no—
334C3⋊C8—24no—
335C3⋊C8—42no—
336C24C2×#6712yes—
337C24C2×#6711yescharacteristic
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—34no—
346C2×Q8—64no—
347C2×Q8—62no—
348C2×Q8—68no—
349C2×Q8—68no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—124no—
355C2×Q8—124no—
356C2×Q8—122no—
357C2×Q8—122no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—64no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—82no—
378C22×C4—122no—
379C22×C4—122no—
380C22×C4#23512yes—
381C22×C4#23512yes—
382C22×C4D3×D412yes—
383C22×C4D3×D412yes—
384C22×C4D3×D412yes—
385C22×C4#23712yes—
386C22×C4C2×#6712yes—
387C22×C4C2×#6711yescharacteristic
388C22×C4C23×D311yescharacteristic
389M16—68no—
390C2×C8—34no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—124no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—32no—
402C4⋊C4—32no—
403C4⋊C4—42no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C22⋊C4—42no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—84no—
413C42—42no—
414C42—64no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—44no—
424C22×C3#39414yes—
425C22×C3C2×#12011yescharacteristic
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12311yescharacteristic
428C22×C3#57712yes—
429C22×C3#57812yes—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4—82no—
433C3×C4#39414yes—
434C3×C4C2×#12111yescharacteristic
435C3×C4C2×#12111yescharacteristic
436C3×C4C2×#12311yescharacteristic
437C3×C4C2×#12411yescharacteristic
438Dic3—44no—
439Dic3—42no—
440Dic3—42no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23#92912yes—
449C23#93712yes—
450C23#93912yes—
451C23C22×#6711yescharacteristic
452Q8—44no—
453Q8—64no—
454Q8—64no—
455Q8—128no—
456Q8—122no—
457Q8—122no—
458D4—48no—
459D4—44no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—84no—
476C2×C4—128no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4#88314yes—
482C2×C4#88414yes—
483C2×C4#88414yes—
484C2×C4#88514yes—
485C2×C4#92912yes—
486C2×C4#93712yes—
487C2×C4#93912yes—
488C2×C4C2×#23511yescharacteristic
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×D3×D411yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×#23711yescharacteristic
494C8—64no—
495C8—122no—
496C2×C3—24no—
497C2×C3—82no—
498C2×C3C2×#39411yescharacteristic
499C2×C3#301011yescharacteristic
500C2×C3#301111yescharacteristic
501C2×C3#306312yes—
502C2×C3#309712yes—
503C22—24no—
504C22—22no—
505C22—22no—
506C22—24no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22#495914yes—
513C22C2×#92911yescharacteristic
514C22C2×#93711yescharacteristic
515C22C2×#93911yescharacteristic
516C22#583612yes—
517C22#583812yes—
518C4—42no—
519C4—82no—
520C4—82no—
521C4—122no—
522C4—124no—
523C4—122no—
524C4#495914yes—
525C4C2×#88311yescharacteristic
526C4C2×#88411yescharacteristic
527C4C2×#88411yescharacteristic
528C4C2×#88511yescharacteristic
529C3#2348411yescharacteristic, complemented, Sylow
530C2—24no—
531C2—82no—
532C2C2×#495911yescharacteristic
533C2#8257312yes—
534C2#8339011yescharacteristic
535C2#8341111yescharacteristic
536C2#8342312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#2311D3
hypocenterC3#23484
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. C2×#5004C2semidirect
5. C22×#810C2semidirect
6. C2×#4964C2semidirect
7. #71580C2semidirect
8. #69527C2semidirect
9. #69526C2semidirect
529. C3#23484semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
C3×#23112
C2×#52342
C2×#50042
C22×#8102
C2×#49642
#715802
#695272
#695262
#234843