Finite Group Databasebeta

← #10849412

Subgroups — #10849412

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
1#86225C211yesmaximal, characteristic
2C3×#3141C211yesmaximal, characteristic, complemented
3#83252C211yesmaximal, characteristic, complemented
4#83212C211yesmaximal, characteristic, complemented
5#71815C211yesmaximal, characteristic, complemented
6#70312C214yesmaximal, complemented
7#70298C214yesmaximal, complemented
8#69879C211yesmaximal, characteristic, complemented
9C4×#825C211yesmaximal, characteristic, complemented
10#25667—31nomaximal, Sylow
11C3×#532C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5833C2212yes—
14#5827C2212yes—
15#5728—21no—
16#5716—21no—
17#5682—22no—
18#5618—22no—
19#5598—22no—
20#5563—21no—
21#5555—21no—
22#5543—21no—
23#5542—21no—
24C4×#235C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#489—22no—
27C3×#487—21no—
28C3×#486—21no—
29C3×#417C2214yes—
30C3×C4×M16C2211yescharacteristic
31C2×#837C2214yes—
32#5229—22no—
33C2×#825C2212yes—
34#5094—21no—
35#5091—21no—
36#5087—21no—
37#5084—21no—
38#4977C2214yes—
39#4973—22no—
40#4966C2212yes—
41#4964C2212yes—
42C42×Dic3C2211yescharacteristic
43#4924C2211yescharacteristic
44#4555C2212yes—
45#4554C2214yes—
46#3511—31no—
47#3141D311yescharacteristic, complemented
48#3079—31no—
49#2862—31no—
50#2040—34no—
51#1997—31no—
52#1962—31no—
53#1860—34no—
54C4×#104—31no—
55C2×C3×#41C2312yes—
56C22×C3×Q8—21no—
57C2×#237—22no—
58C2×#235C2311yescharacteristic
59C22×Dic6D412yes—
60C2×C3×M16C2312yes—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8D411yescharacteristic
66C3×#127D411yescharacteristic
67C3×#124D412yes—
68C3×#121—21no—
69C3×#121—21no—
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42C2311yescharacteristic
75#946—22no—
76Q8×Dic3—22no—
77#939—24no—
78#938—22no—
79#938—22no—
80#936—22no—
81D4×Dic3—22no—
82D4×Dic3—22no—
83D4×Dic3—24no—
84#930—22no—
85C4×#67—21no—
86C4×#67—21no—
87#926—22no—
88#926C2311yescharacteristic
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3D412yes—
91C2×C4⋊Dic3C2312yes—
92#924—22no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3C2312yes—
96C2×#208C2312yes—
97#897—22no—
98#892—22no—
99#890—22no—
100#887—22no—
101#886—24no—
102#885—24no—
103#879—24no—
104#878—24no—
105C4×D12D411yescharacteristic
106C4×D12D411yescharacteristic
107#872—21no—
108C42×D3—21no—
109#869—21no—
110#869—21no—
111#869—21no—
112#869—21no—
113C4×Dic6D411yescharacteristic
114C4×Dic6D411yescharacteristic
115C3×#103—24no—
116C3×#102—24no—
117C3×#97—21no—
118C3×C4×C8—21no—
119#837—28no—
120C3⋊#95—28no—
121#825—24no—
122#810—22no—
123#809—22no—
124#808—22no—
125#807—22no—
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)—21no—
128#556—61no—
129#555—61no—
130#554—61no—
131#553—62no—
132#548—62no—
133#547—61no—
134#542—62no—
135#537—32no—
136#533—61no—
137#532—61no—
138#532C2×D311yescharacteristic
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8C2×D311yescharacteristic
141#520—32no—
142C4×#41—31no—
143#489—22no—
144#487—21no—
145#486—21no—
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#417C2×D314yes—
156C4×M16C2×D311yescharacteristic
157#391—62no—
158#391—62no—
159#382—34no—
160C43—31no—
161#343—34no—
162#339—32no—
163C3×#41—24no—
164C3×#41—24no—
165C2×C3×Q8—44no—
166C2×C3×Q8—41no—
167C2×C3×Q8—41no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—21no—
179C2×#67—24no—
180C22×Dic3—22no—
181C22×Dic3—24no—
182C22×Dic3C2×D414yes—
183#237—84no—
184#235—22no—
185#235—22no—
186C2×D12C2×D411yescharacteristic
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—44no—
193C2×Dic6C2×D411yescharacteristic
194C3×M16—22no—
195C3×M16—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—41no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209#217—24no—
210#217—24no—
211#217—21no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—21no—
214(C2×D3)⋊C4—21no—
215C4⋊Dic3—24no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3C2×D411yescharacteristic
221C4⋊Dic3C2×D411yescharacteristic
222Dic3⋊C4—24no—
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225C4×Dic3—24no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C4×Dic3—22no—
230C4×Dic3—42no—
231C4×Dic3—42no—
232C4×Dic3#4114yescomplemented
233#208—24no—
234C2×C3⋊C8—22no—
235C2×#41—31no—
236C2×#41—62no—
237C2×#41C22×D312yes—
238C22×Q8—21no—
239C22×Q8—32no—
240C2×M16—32no—
241C2×M16C22×D312yes—
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—64no—
253#125—62no—
254#124—64no—
255#124—62no—
256#124#6712yes—
257#123—64no—
258C22⋊Q8—64no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262#121—21no—
263#121—21no—
264#121—62no—
265#121—64no—
266#121—62no—
267#121—62no—
268C4×Q8—31no—
269C4×Q8—31no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—31no—
275C4×D4—31no—
276C4×D4—64no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—61no—
280C4×D4—61no—
281#117—31no—
282#117—62no—
283#117—62no—
284#117—61no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4C22×D312yes—
290C2×C42—32no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297C4⋊C8—61no—
298C4⋊C8—61no—
299#104—64no—
300#104—68no—
301#103—24no—
302#103—62no—
303#102—24no—
304#102—62no—
305#97—21no—
306C4×C8—21no—
307#95—68no—
308C23×C3—22no—
309C23×C3C22×D411yescharacteristic
310C22×D3—21no—
311C3×Q8—24no—
312C3×Q8—82no—
313C3×Q8—82no—
314C3×D4—24no—
315C3×D4—44no—
316C3×D4—42no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—42no—
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4#12011yescharacteristic
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C42−1+412yes—
333#67—42no—
334#67—84no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—44no—
343C2×Dic3—84no—
344D12—21no—
345D12—21no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349Dic6—21no—
350Dic6—21no—
351Dic6—44no—
352Dic6—82no—
353C3×C8—42no—
354C3⋊C8—42no—
355#41—24no—
356#41—24no—
357#41—62no—
358#41—62no—
359#41—244no—
360C2×Q8—31no—
361C2×Q8—44no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×Q8—124no—
367C2×Q8C2×#6712yes—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—31no—
372C2×D4—61no—
373C2×D4—64no—
374C2×D4C2×#6712yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—34no—
378C22×C4—62no—
379C22×C4—64no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386M16—22no—
387M16—22no—
388M16—64no—
389C2×C8—22no—
390C2×C8—62no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—31no—
396C4⋊C4—31no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—64no—
400C4⋊C4—64no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—61no—
413C22⋊C4—61no—
414C22⋊C4—61no—
415C42—34no—
416C42—64no—
417C42—61no—
418C42—61no—
419C42—62no—
420C42—61no—
421C42—122no—
422C42—122no—
423C42D3×D411yescharacteristic
424C42D3×D411yescharacteristic
425C42C2×#6711yescharacteristic
426C42C2×#6711yescharacteristic
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54812yes—
434C2×D3—41no—
435C2×D3—41no—
436C2×D3—41no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—42no—
440C3×C4—82no—
441C3×C4#45412yes—
442C3×C4#45512yes—
443C3×C4#53211yescharacteristic
444C3×C4#53611yescharacteristic
445C3×C4#54712yes—
446Dic3—44no—
447Dic3—41no—
448Dic3—44no—
449Dic3—41no—
450Dic3—41no—
451C23—22no—
452C23—61no—
453C23C2×D3×D411yescharacteristic
454Q8—24no—
455Q8—61no—
456Q8—61no—
457Q8—82no—
458Q8—82no—
459Q8—124no—
460Q8—242no—
461D4—24no—
462D4—44no—
463D4—42no—
464D4—42no—
465D4—61no—
466D4—61no—
467D4—122no—
468D4—244no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—64no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—64no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—124no—
487C2×C4—244no—
488C2×C4#93711yescharacteristic
489C2×C4#93711yescharacteristic
490C2×C4#93711yescharacteristic
491C2×C4#93711yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4D3×#4112yes—
494C2×C4#101012yes—
495C2×C4C22×#6711yescharacteristic
496C8—42no—
497C8—122no—
498C2×C3—22no—
499C2×C3—82no—
500C2×C3#243211yescharacteristic
501C2×C3C2×#45511yescharacteristic
502C2×C3C2×#45411yescharacteristic
503D3—81no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—121no—
510C22—121no—
511C22—121no—
512C22#567712yes—
513C22C2×#93711yescharacteristic
514C4—21no—
515C4—21no—
516C4—42no—
517C4—82no—
518C4—124no—
519C4—121no—
520C4—121no—
521C4—124no—
522C4—121no—
523C4#520112yes—
524C4#520212yes—
525C4#566811yescharacteristic
526C4#567511yescharacteristic
527C4#582412yes—
528C3#2566711yescharacteristic, complemented, Sylow
529C2—22no—
530C2—82no—
531C2—241no—
532C2#8295911yescharacteristic
533C2C2×#520111yescharacteristic
534C2C2×#520211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3141C2
socleC22×C3C2×#120
prime radical#86225C2
hypercenter#3141D3
hypocenterC3#25667
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. C3×#3141C2semidirect
3. #83252C2semidirect
4. #83212C2semidirect
5. #71815C2semidirect
6. #70312C2semidirect
7. #70298C2semidirect
8. #69879C2semidirect
9. C4×#825C2semidirect
47. #3141D3semidirect
232. 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
#862252
C3×#31412
#832522
#832122
#718152
#703122
#702982
#698792
C4×#8252
#256673