Finite Group Databasebeta

← #10849405

Subgroups — #10849405

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#83252C211yesmaximal, characteristic, complemented
4#82508C211yesmaximal, characteristic, complemented
5#71877C214yesmaximal, complemented
6#71815C211yesmaximal, characteristic, complemented
7#71790C211yesmaximal, characteristic, complemented
8#71665C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25667—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×#869C2211yescharacteristic
26C2×#837C2214yes—
27C2×#837C2212yes—
28#5232C2212yes—
29#5232C2214yes—
30#5229—22no—
31#5079—21no—
32#5070—21no—
33#5055—22no—
34#5021—21no—
35#5020—21no—
36#5008—21no—
37#5007—21no—
38#4995C2212yes—
39#4977C2214yes—
40#4973—22no—
41C42×Dic3C2211yescharacteristic
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4532C2212yes—
45#4530C2214yes—
46#3511D311yescharacteristic
47#3141—31no—
48#3079—31no—
49#2862—31no—
50#2040—34no—
51#1997—31no—
52#1962—31no—
53#1860—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
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2312yes—
109C2×#208C2312yes—
110C2×#208C2312yes—
111#869—21no—
112#869—21no—
113#869—21no—
114#869—21no—
115#837—28no—
116#837—24no—
117#835—22no—
118#835—24no—
119#832—22no—
120#832—24no—
121C3⋊#95—28no—
122#808—22no—
123#807—22no—
124C4⋊(C3⋊C8)—21no—
125C4⋊(C3⋊C8)—21no—
126#803—21no—
127C4×C3⋊C8—21no—
128#556—21no—
129#555—21no—
130#554—21no—
131#553—22no—
132#548—22no—
133#547—21no—
134#542—22no—
135#537C2×D312yes—
136#533—21no—
137#532—21no—
138#532C2×D311yescharacteristic
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8C2×D311yescharacteristic
141#520C2×D312yes—
142C4×#41C2×D311yescharacteristic
143#489—62no—
144#487—61no—
145#486—61no—
146#477—61no—
147#476—61no—
148#475—61no—
149#474—61no—
150#428—32no—
151#423—31no—
152C2×#104—34no—
153C2×#104—32no—
154#417—32no—
155#417—34no—
156C4×M16—31no—
157#391—62no—
158#391—62no—
159#382—34no—
160C43—31no—
161#343—34no—
162#339—32no—
163C3×#41—24no—
164C3×#41—22no—
165C3×#41—24no—
166C3×#41—22no—
167C3×#41—84no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—44no—
171C2×C3×Q8C2×D412yes—
172C2×C3×Q8C2×D411yescharacteristic
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—24no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4C2×D412yes—
179C2×C3×D4C2×D411yescharacteristic
180C22×C3×C4—24no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D414yes—
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—22no—
190C22×Dic3—22no—
191C2×Dic6—44no—
192C2×Dic6—41no—
193C2×Dic6—41no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4C2×D411yescharacteristic
205C3×C4⋊C4C2×D411yescharacteristic
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—22no—
214C3×C42—24no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3—41no—
225C4⋊Dic3—41no—
226C4×Dic3—22no—
227C4×Dic3—42no—
228C4×Dic3—42no—
229C4×Dic3#4114yescomplemented
230#208—24no—
231#208—22no—
232#208—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×#41—22no—
236C2×#41C22×D311yescharacteristic
237C2×#41C22×D312yes—
238C22×Q8—61no—
239C22×Q8#6712yes—
240C2×M16—32no—
241C2×M16—32no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248C4⋊Q8—61no—
249C4⋊Q8—61no—
250C4⋊Q8#6711yescharacteristic
251#127#6711yescharacteristic
252#126—24no—
253#125—22no—
254#124—22no—
255#124—24no—
256#124#6712yes—
257#123—24no—
258C22⋊Q8—22no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—22no—
263#121—22no—
264#121—22no—
265#121—21no—
266#121—21no—
267#121—24no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8#6711yescharacteristic
271C4×Q8#6711yescharacteristic
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—24no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4#6711yescharacteristic
280C4×D4#6711yescharacteristic
281#117—22no—
282#117—22no—
283#117—21no—
284#117C22×D311yescharacteristic
285C2×C4⋊C4—22no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4#6712yes—
289C2×C4⋊C4C22×D312yes—
290C2×C42—21no—
291C2×C42—32no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297C4⋊C8—61no—
298C4⋊C8—61no—
299#104—68no—
300#104—64no—
301#103—62no—
302#103—64no—
303#102—62no—
304#102—64no—
305#97—61no—
306C4×C8—61no—
307#95—68no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3C22×D411yescharacteristic
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8—24no—
314C3×Q8—44no—
315C3×Q8—82no—
316C3×D4—21no—
317C3×D4—21no—
318C3×D4—24no—
319C3×D4—44no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—84no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—24no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—84no—
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4112yes—
345C2×C3×C42−1+412yes—
346C2×Dic3—24no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—44no—
350C2×Dic3—42no—
351Dic6—82no—
352Dic6—82no—
353C3⋊C8—42no—
354C3⋊C8—42no—
355#41—22no—
356#41—24no—
357#41—22no—
358#41—24no—
359#41—84no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—44no—
363C2×Q8—124no—
364C2×Q8—121no—
365C2×Q8—121no—
366C2×Q8C2×#6712yes—
367C2×Q8C2×#6711yescharacteristic
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4C2×#6712yes—
374C2×D4C2×#6711yescharacteristic
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6714yes—
385C22×C4C23×D311yescharacteristic
386M16—62no—
387M16—64no—
388M16—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—121no—
404C4⋊C4—121no—
405C4⋊C4C2×#6711yescharacteristic
406C4⋊C4C2×#6711yescharacteristic
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—34no—
420C42—62no—
421C42—122no—
422C42—122no—
423C42C2×#6711yescharacteristic
424C42C2×#6711yescharacteristic
425C42C2×#6711yescharacteristic
426C42C2×#6711yescharacteristic
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3C2×#12011yescharacteristic
436C22×C3#54812yes—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—42no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—44no—
443C3×C4—41no—
444C3×C4#45412yes—
445C3×C4#45512yes—
446C3×C4#53211yescharacteristic
447C3×C4#53611yescharacteristic
448C3×C4#54712yes—
449Dic3—44no—
450Dic3—82no—
451C23—21no—
452C23—22no—
453C23C22×#6711yescharacteristic
454Q8—24no—
455Q8—21no—
456Q8—21no—
457Q8—44no—
458Q8—82no—
459Q8—242no—
460Q8—242no—
461D4—24no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—84no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—64no—
483C2×C4—84no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—124no—
487C2×C4—122no—
488C2×C4#95311yescharacteristic
489C2×C4#95311yescharacteristic
490C2×C4#95311yescharacteristic
491C2×C4#95311yescharacteristic
492C2×C4D3×#4112yes—
493C2×C4#101012yes—
494C2×C4C22×#6711yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—122no—
497C8—122no—
498C2×C3—22no—
499C2×C3—81no—
500C2×C3—82no—
501C2×C3#243211yescharacteristic
502C2×C3C2×#45511yescharacteristic
503C2×C3C2×#45411yescharacteristic
504C22—21no—
505C22—21no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—42no—
510C22—41no—
511C22—42no—
512C22#583112yes—
513C22C2×#95311yescharacteristic
514C4—21no—
515C4—21no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—44no—
520C4—41no—
521C4—124no—
522C4—242no—
523C4#524212yes—
524C4#524512yes—
525C4#582412yes—
526C4#582511yescharacteristic
527C4#583211yescharacteristic
528C3#2566711yescharacteristic, complemented, Sylow
529C2—22no—
530C2—81no—
531C2—82no—
532C2#8409311yescharacteristic
533C2C2×#524211yescharacteristic
534C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#25667
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. #83252C2semidirect
4. #82508C2semidirect
5. #71877C2semidirect
6. #71815C2semidirect
7. #71790C2semidirect
8. #71665C2semidirect
9. C4×#837C2semidirect
229. C4×Dic3#41semidirect
528. C3#25667semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#832832
#832522
#825082
#718772
#718152
#717902
#716652
C4×#8372
#256673