Finite Group Databasebeta

← #10849396

Subgroups — #10849396

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#83253C211yesmaximal, characteristic, complemented
4#82508C211yesmaximal, characteristic, complemented
5#71876C214yesmaximal, complemented
6#71815C211yesmaximal, characteristic, complemented
7#71790C211yesmaximal, characteristic, complemented
8#71664C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25666—31nomaximal, Sylow
11C3×#556—21no—
12C3×#555—21no—
13C3×#554—21no—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537C2212yes—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#520C2212yes—
24C3×C4×#41C2211yescharacteristic
25C2×#944—22no—
26C2×#869C2211yescharacteristic
27C2×#837C2214yes—
28C2×#837C2212yes—
29#5232C2212yes—
30#5232C2214yes—
31#5079—21no—
32#5070—21no—
33#5055—22no—
34#5021—21no—
35#5020—21no—
36#5008—21no—
37#5007—21no—
38#4995C2212yes—
39#4975—22no—
40#4975C2214yes—
41C42×Dic3C2211yescharacteristic
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4532C2212yes—
45#4530C2214yes—
46#3511D311yescharacteristic
47#3141—31no—
48#3079—31no—
49#2868—31no—
50#2039—34no—
51#1997—31no—
52#1962—31no—
53#1859—34no—
54C4×#104—31no—
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8D412yes—
59C22×Dic6—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×#127D411yescharacteristic
66C3×#126—24no—
67C3×#125—22no—
68C3×#124—22no—
69C3×#124—24no—
70C3×#124D412yes—
71C3×#123—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—21no—
80C3×#121—21no—
81C3×#121—24no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D411yescharacteristic
85C3×C4×Q8D411yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—24no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4D411yescharacteristic
94C3×C4×D4D411yescharacteristic
95C3×#117—22no—
96C3×#117—22no—
97C3×#117—21no—
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4D412yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104#944—48no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×Dic3⋊C4—28no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2312yes—
111C2×#208C2312yes—
112C2×#208C2312yes—
113#869—21no—
114#869—21no—
115#869—21no—
116#869—21no—
117#837—28no—
118#837—24no—
119#835—22no—
120#835—24no—
121#832—22no—
122#832—24no—
123#808—22no—
124#807—22no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127#803—21no—
128C4×C3⋊C8—21no—
129#556—21no—
130#555—21no—
131#554—21no—
132#553—22no—
133#548—22no—
134#547—21no—
135#542—22no—
136#537C2×D312yes—
137#533—21no—
138#532—21no—
139#532C2×D311yescharacteristic
140C2×C4⋊Q8—31no—
141C2×C4⋊Q8—62no—
142C2×C4⋊Q8C2×D311yescharacteristic
143#520C2×D312yes—
144C4×#41C2×D311yescharacteristic
145#489—62no—
146#487—61no—
147#486—61no—
148#477—61no—
149#476—61no—
150#475—61no—
151#474—61no—
152#428—32no—
153#423—31no—
154C2×#104—34no—
155C2×#104—32no—
156#417—32no—
157#417—34no—
158C4×M16—31no—
159#384—34no—
160#384—62no—
161C43—31no—
162#343—34no—
163#339—32no—
164C3×#41—24no—
165C3×#41—22no—
166C3×#41—24no—
167C3×#41—22no—
168C3×#41—84no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8C2×D412yes—
173C2×C3×Q8C2×D411yescharacteristic
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4C2×D412yes—
180C2×C3×D4C2×D411yescharacteristic
181C22×C3×C4—24no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—22no—
191C22×Dic3—22no—
192C2×Dic6—44no—
193C2×Dic6—41no—
194C2×Dic6—41no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D411yescharacteristic
206C3×C4⋊C4C2×D411yescharacteristic
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—21no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C42—24no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C3×C42C2×D411yescharacteristic
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—41no—
226C4⋊Dic3—41no—
227Dic3⋊C4—416no—
228C4×Dic3—24no—
229C4×Dic3—24no—
230C4×Dic3—22no—
231C4×Dic3#4114yescomplemented
232#208—24no—
233#208—22no—
234#208—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×#41—22no—
238C2×#41C22×D311yescharacteristic
239C2×#41C22×D312yes—
240C22×Q8—61no—
241C22×Q8#6712yes—
242C2×M16—32no—
243C2×M16—32no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—61no—
249C4⋊Q8—61no—
250C4⋊Q8—61no—
251C4⋊Q8—61no—
252C4⋊Q8—128no—
253C4⋊Q8#6711yescharacteristic
254#127#6711yescharacteristic
255#126—24no—
256#125—22no—
257#124—22no—
258#124—24no—
259#124#6712yes—
260#123—24no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265#121—22no—
266#121—22no—
267#121—22no—
268#121—21no—
269#121—21no—
270#121—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8#6711yescharacteristic
274C4×Q8#6711yescharacteristic
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—24no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4#6711yescharacteristic
283C4×D4#6711yescharacteristic
284#117—22no—
285#117—22no—
286#117—21no—
287#117C22×D311yescharacteristic
288C2×C4⋊C4—22no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—68no—
292C2×C4⋊C4#6712yes—
293C2×C4⋊C4C22×D312yes—
294C2×C42—21no—
295C2×C42—32no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#106—62no—
301C4⋊C8—61no—
302C4⋊C8—61no—
303#104—68no—
304#104—64no—
305#103—62no—
306#103—64no—
307#102—62no—
308#102—64no—
309#97—61no—
310C4×C8—61no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3C22×D411yescharacteristic
314C3×Q8—21no—
315C3×Q8—21no—
316C3×Q8—24no—
317C3×Q8—44no—
318C3×Q8—82no—
319C3×D4—21no—
320C3×D4—21no—
321C3×D4—24no—
322C3×D4—44no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—84no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—24no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4—84no—
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C42−1+412yes—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—44no—
353C2×Dic3—42no—
354Dic6—82no—
355Dic6—82no—
356C3⋊C8—42no—
357C3⋊C8—42no—
358#41—22no—
359#41—24no—
360#41—22no—
361#41—24no—
362#41—84no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—44no—
366C2×Q8—124no—
367C2×Q8—121no—
368C2×Q8—121no—
369C2×Q8C2×#6712yes—
370C2×Q8C2×#6711yescharacteristic
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C2×#6712yes—
377C2×D4C2×#6711yescharacteristic
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6714yes—
388C22×C4C23×D311yescharacteristic
389M16—62no—
390M16—64no—
391M16—62no—
392C2×C8—62no—
393C2×C8—62no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—1216no—
407C4⋊C4—121no—
408C4⋊C4—121no—
409C4⋊C4C2×#6711yescharacteristic
410C4⋊C4C2×#6711yescharacteristic
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C42—24no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—34no—
424C42—64no—
425C42—64no—
426C42—62no—
427C42C2×#6711yescharacteristic
428C42C2×#6711yescharacteristic
429C42C2×#6711yescharacteristic
430C42C2×#6711yescharacteristic
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—42no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3C2×#12011yescharacteristic
440C22×C3#54812yes—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—42no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—44no—
447C3×C4—41no—
448C3×C4#45312yes—
449C3×C4#45612yes—
450C3×C4#53211yescharacteristic
451C3×C4#53611yescharacteristic
452C3×C4#54712yes—
453Dic3—28no—
454Dic3—82no—
455C23—21no—
456C23—22no—
457C23C22×#6711yescharacteristic
458Q8—24no—
459Q8—21no—
460Q8—21no—
461Q8—44no—
462Q8—82no—
463Q8—242no—
464Q8—242no—
465D4—24no—
466D4—21no—
467D4—21no—
468D4—42no—
469D4—44no—
470D4—42no—
471D4—42no—
472D4—84no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—84no—
490C2×C4—124no—
491C2×C4—122no—
492C2×C4#95311yescharacteristic
493C2×C4#95311yescharacteristic
494C2×C4#95311yescharacteristic
495C2×C4#95311yescharacteristic
496C2×C4D3×#4112yes—
497C2×C4#101012yes—
498C2×C4C22×#6711yescharacteristic
499C2×C4C22×#6711yescharacteristic
500C8—122no—
501C8—122no—
502C2×C3—22no—
503C2×C3—81no—
504C2×C3—82no—
505C2×C3#243211yescharacteristic
506C2×C3C2×#45311yescharacteristic
507C2×C3C2×#45611yescharacteristic
508C22—21no—
509C22—21no—
510C22—41no—
511C22—42no—
512C22—41no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22#583112yes—
517C22C2×#95311yescharacteristic
518C4—21no—
519C4—21no—
520C4—41no—
521C4—42no—
522C4—41no—
523C4—44no—
524C4—41no—
525C4—68no—
526C4—242no—
527C4#524112yes—
528C4#524612yes—
529C4#582412yes—
530C4#582511yescharacteristic
531C4#583211yescharacteristic
532C3#2566611yescharacteristic, complemented, Sylow
533C2—22no—
534C2—81no—
535C2—82no—
536C2#8409311yescharacteristic
537C2C2×#524111yescharacteristic
538C2C2×#524611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#25666
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #83253C2semidirect
4. #82508C2semidirect
5. #71876C2semidirect
6. #71815C2semidirect
7. #71790C2semidirect
8. #71664C2semidirect
9. C4×#837C2semidirect
231. C4×Dic3#41semidirect
532. C3#25666semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#832832
#832532
#825082
#718762
#718152
#717902
#716642
C4×#8372
#256663