Finite Group Databasebeta

← #10849406

Subgroups — #10849406

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#86205C211yesmaximal, characteristic
2C3×#2862C211yesmaximal, characteristic, complemented
3#83212C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5#71790C211yesmaximal, characteristic, complemented
6#69563C214yesmaximal, complemented
7#69562C214yesmaximal, complemented
8#69395C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic, complemented
10#25667—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5817—22no—
14#5734—21no—
15#5733—22no—
16#5732—21no—
17#5730—21no—
18#5728—21no—
19#5726—21no—
20#5725—21no—
21#5514—22no—
22#5510C2212yes—
23#5504C2212yes—
24#5477C2211yescharacteristic
25C4×#235C2211yescharacteristic
26C3×#391—22no—
27C3×#391—22no—
28C3×#382C2214yes—
29C3×C43C2211yescharacteristic
30#5094—21no—
31#5093—21no—
32#5091—21no—
33#5089—22no—
34#5088—21no—
35#5087—21no—
36#5084—21no—
37#4966C2212yes—
38#4966C2214yes—
39#4964C2212yes—
40C2×#805C2212yes—
41C2×#805C2214yes—
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4530C2214yes—
45#4523C2212yes—
46#3511—31no—
47#3141—31no—
48#3079—31no—
49#2862D311yescharacteristic, complemented
50#2040—34no—
51#1997—31no—
52#1962—31no—
53#1860—34no—
54C4×#104—31no—
55C22×C3×Q8—21no—
56C22×C3×Q8D412yes—
57C2×#239—22no—
58C2×#235C2311yescharacteristic
59C2×#235C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76#947—22no—
77#947—24no—
78#946—22no—
79#946—24no—
80Q8×Dic3—22no—
81#930—21no—
82#930—21no—
83#930—24no—
84C4×#67—21no—
85C4×#67—21no—
86C4×#67—24no—
87#926C2311yescharacteristic
88C2×#208C2312yes—
89C2×#208C2312yes—
90#898—24no—
91#897—22no—
92#897—22no—
93#895—22no—
94#895—22no—
95#895—22no—
96#894—24no—
97#893—22no—
98#893—22no—
99#892—22no—
100#892—22no—
101D3×C4⋊C4—22no—
102#890—22no—
103#887—22no—
104#875D412yes—
105#874D411yescharacteristic
106C4×D12—21no—
107C4×D12—21no—
108C4×D12D411yescharacteristic
109C4×D12D411yescharacteristic
110#872—21no—
111C42×D3—21no—
112#869D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C4×Dic6D411yescharacteristic
115C3×#95—28no—
116#810—24no—
117#810—22no—
118#809—22no—
119#809—24no—
120#808—22no—
121#807—22no—
122#805—24no—
123#805—28no—
124C4⋊(C3⋊C8)—21no—
125C4⋊(C3⋊C8)—21no—
126#803—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—31no—
138#532—61no—
139C2×C4⋊Q8C2×D311yescharacteristic
140C2×C4⋊Q8C2×D311yescharacteristic
141#520—32no—
142C4×#41—31no—
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—22no—
158#391—22no—
159#382C2×D314yes—
160C43C2×D311yescharacteristic
161#343—34no—
162#339—32no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—44no—
166C2×C3×Q8—44no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—21no—
176C2×#67—22no—
177#239—84no—
178#235—22no—
179#235—24no—
180#235—22no—
181#235—24no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—24no—
185C2×D12C2×D411yescharacteristic
186C2×D12C2×D412yes—
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×C4×D3—22no—
191C2×C4×D3—24no—
192C2×Dic6C2×D411yescharacteristic
193C2×Dic6C2×D412yes—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C42—22no—
203C3×C42—42no—
204C3×C42—42no—
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209C3×C42#4114yescomplemented
210#217—21no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—22no—
218C4⋊Dic3—24no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3C2×D411yescharacteristic
222C4⋊Dic3C2×D411yescharacteristic
223Dic3⋊C4—24no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226C4×Dic3—24no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230#208—24no—
231#208—22no—
232#208—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×#41—31no—
236C2×#41—32no—
237C2×#41—62no—
238C22×Q8—21no—
239C22×Q8#6712yes—
240C2×M16—32no—
241C2×M16—32no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—31no—
251#127—31no—
252#126—64no—
253#125—62no—
254#124—32no—
255#124—62no—
256#124—64no—
257#123—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—64no—
262#121—62no—
263#121—62no—
264#121—62no—
265#121—61no—
266#121—61no—
267#121—64no—
268C4×Q8—31no—
269C4×Q8—31no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×D4—31no—
273C4×D4—31no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—61no—
277C4×D4—61no—
278C4×D4—64no—
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—62no—
287C2×C4⋊C4#6712yes—
288C2×C4⋊C4C22×D312yes—
289C2×C4⋊C4C22×D312yes—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—61no—
293C2×C42C22×D312yes—
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—28no—
308C23×C3C22×D411yescharacteristic
309C22×D3—21no—
310C22×D3—22no—
311C3×Q8—44no—
312C3×Q8—82no—
313C3×Q8—82no—
314C3×Q8—82no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4#12011yescharacteristic
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4112yes—
331C2×C3×C42−1+412yes—
332#67—42no—
333#67—44no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—22no—
337C2×Dic3—21no—
338D12—21no—
339D12—21no—
340D12—24no—
341D12—42no—
342D12—42no—
343D12—84no—
344C4×D3—42no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—84no—
350Dic6—21no—
351Dic6—21no—
352Dic6—24no—
353C3⋊C8—42no—
354C3⋊C8—42no—
355#41—64no—
356#41—62no—
357#41—64no—
358#41—62no—
359#41—244no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—31no—
363C2×Q8—32no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×D4—31no—
369C2×D4—32no—
370C2×D4—64no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—61no—
374C2×D4—61no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—64no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6714yes—
385C22×C4C23×D311yescharacteristic
386M16—64no—
387M16—62no—
388M16—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—31no—
398C4⋊C4—31no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C22⋊C4—64no—
408C22⋊C4—64no—
409C22⋊C4—64no—
410C22⋊C4—61no—
411C22⋊C4—61no—
412C22⋊C4—61no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C42—22no—
416C42—42no—
417C42—42no—
418C42—64no—
419C42—61no—
420C42—61no—
421C42—61no—
422C42D3×D411yescharacteristic
423C42D3×D411yescharacteristic
424C42D3×D411yescharacteristic
425C42D3×D411yescharacteristic
426C42#23914yescomplemented
427C22×C3—21no—
428C22×C3—21no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#54812yes—
431C2×D3—42no—
432C2×D3—41no—
433C2×D3—41no—
434C2×D3—42no—
435C2×D3—41no—
436C2×D3—42no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—44no—
440C3×C4—44no—
441C3×C4—82no—
442C3×C4#45412yes—
443C3×C4#45512yes—
444C3×C4#53211yescharacteristic
445C3×C4#53611yescharacteristic
446C3×C4#54712yes—
447Dic3—41no—
448Dic3—42no—
449Dic3—41no—
450Dic3—41no—
451C23—61no—
452C23—62no—
453C23C22×#6711yescharacteristic
454Q8—44no—
455Q8—61no—
456Q8—61no—
457Q8—64no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461D4—61no—
462D4—61no—
463D4—64no—
464D4—124no—
465D4—122no—
466D4—122no—
467D4—122no—
468D4—244no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—61no—
479C2×C4—62no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—244no—
488C2×C4#93711yescharacteristic
489C2×C4#93711yescharacteristic
490C2×C4#93711yescharacteristic
491C2×C4#93711yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23912yes—
495C2×C4#100712yes—
496C8—122no—
497C8—122no—
498C2×C3—22no—
499C2×C3#243211yescharacteristic
500C2×C3C2×#45511yescharacteristic
501C2×C3C2×#45411yescharacteristic
502D3—81no—
503D3—82no—
504C22—21no—
505C22—21no—
506C22—121no—
507C22—122no—
508C22—121no—
509C22—122no—
510C22—122no—
511C22—121no—
512C22C2×#93711yescharacteristic
513C22#581712yes—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—44no—
518C4—82no—
519C4—121no—
520C4—121no—
521C4—122no—
522C4—121no—
523C4#506112yes—
524C4#506212yes—
525C4#572411yescharacteristic
526C4#572511yescharacteristic
527C4#572612yes—
528C3#2566711yescharacteristic, complemented, Sylow
529C2—22no—
530C2—241no—
531C2—242no—
532C2C2×#506111yescharacteristic
533C2C2×#506211yescharacteristic
534C2#8325511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2862C2
socleC22×C3C2×#120
prime radical#86205C2
hypercenter#2862D3
hypocenterC3#25667
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
2. C3×#2862C2semidirect
3. #83212C2semidirect
4. #83199C2semidirect
5. #71790C2semidirect
6. #69563C2semidirect
7. #69562C2semidirect
8. #69395C2semidirect
9. C4×#805C2semidirect
49. #2862D3semidirect
209. C3×C42#41semidirect
426. C42#239semidirect
528. C3#25667semidirect

Maximal subgroups

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

subgroupindex
#862052
C3×#28622
#832122
#831992
#717902
#695632
#695622
#693952
C4×#8052
#256673