Finite Group Databasebeta

← #10849696

Subgroups — #10849696

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86223C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#78506C211yesmaximal, characteristic, complemented
4#73800C211yesmaximal, characteristic, complemented
5C3×#1980C211yesmaximal, characteristic, complemented
6#70363C214yesmaximal, complemented
7#70359C214yesmaximal, complemented
8C4×#823C211yesmaximal, characteristic, complemented
9#69493C211yesmaximal, characteristic, complemented
10#25694—31nomaximal, Sylow
11C3×#532C2211yescharacteristic
12#5830C2212yes—
13#5827C2212yes—
14#5732—21no—
15#5718—21no—
16#5681—22no—
17#5617—22no—
18#5606—22no—
19#5565—22no—
20#5563—21no—
21#5555—21no—
22C2×#897—22no—
23C2×C4×D12C2211yescharacteristic
24#5470C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#104C2214yes—
27C3×C4×M16C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5232C2214yes—
30C2×#823C2212yes—
31#5121C2214yes—
32#5079—21no—
33#5070—21no—
34#5055—22no—
35#4939C2212yes—
36#4935C2212yes—
37C4×#208C2211yescharacteristic
38#4702—22no—
39#4682—21no—
40#4680—21no—
41#4677—22no—
42#4554C2212yes—
43#4523C2214yes—
44#3506—31no—
45#3165—31no—
46#3141—31no—
47#2542—31no—
48#2037—34no—
49#1986—31no—
50#1980D311yescharacteristic, complemented
51#1859—34no—
52C4×#100—31no—
53C2×C3×#41C2312yes—
54C2×#237—22no—
55C22×D12C2311yescharacteristic
56C22×C4×D3D412yes—
57C22×Dic6—21no—
58C2×C3×M16C2312yes—
59C3×C4⋊Q8D411yescharacteristic
60C3×#127D411yescharacteristic
61C3×#124D412yes—
62C3×#121—21no—
63C3×#121—21no—
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C42C2312yes—
68C2×C3×C42C2311yescharacteristic
69#947—22no—
70Q8×Dic3—22no—
71#940—22no—
72#939—24no—
73#938—22no—
74#937—24no—
75D4×Dic3—22no—
76D4×Dic3—24no—
77#930—22no—
78#930—22no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4D412yes—
81#926—22no—
82#926C2312yes—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88#924—21no—
89#924—21no—
90C2×#208C2312yes—
91#898—24no—
92#897—22no—
93#897—48no—
94#892—22no—
95#886—24no—
96#885—24no—
97#883—24no—
98#67⋊C4—24no—
99C4×D12—21no—
100C4×D12—21no—
101C4×D12—21no—
102C4×D12—21no—
103#870D412yes—
104#869—21no—
105#869—21no—
106#869—21no—
107#869—21no—
108#869D411yescharacteristic
109#869D411yescharacteristic
110C4×Dic6—21no—
111C4×Dic6—21no—
112C3×#104—28no—
113C3×#97—21no—
114C3×C4×C8—21no—
115C3×#95—21no—
116#835—24no—
117#832—24no—
118#823—24no—
119#818—24no—
120#817—24no—
121#803—21no—
122C4×C3⋊C8—21no—
123#559—62no—
124#556—61no—
125#555—61no—
126#553—62no—
127#550—61no—
128#547—61no—
129#538—62no—
130#535—32no—
131#533—31no—
132#532C2×D311yescharacteristic
133C2×C4⋊Q8—31no—
134#525—62no—
135C2×C22⋊Q8—62no—
136#520—32no—
137C2×C4×D4—31no—
138#501—62no—
139#500—61no—
140#498—61no—
141#489—62no—
142#487—61no—
143#486—61no—
144#428—34no—
145C2×#104C2×D314yes—
146#417—34no—
147C2×#100—32no—
148C4×M16—31no—
149C4×M16C2×D311yescharacteristic
150#399—62no—
151#390—32no—
152#384—32no—
153C4×C4⋊C4C2×D311yescharacteristic
154#343—32no—
155#339—34no—
156C23×D3C2×D412yes—
157C3×#41—24no—
158C3×#41—24no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—24no—
170C2×#67—24no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174#237—84no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—22no—
178C2×D12—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—44no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—44no—
185C2×Dic6—41no—
186C2×Dic6—41no—
187C3×M16—22no—
188C3×M16#4114yescomplemented
189C2×C3×C8—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C42—24no—
196C3×C42C2×D411yescharacteristic
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200#217—24no—
201#217—22no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—22no—
205(C2×D3)⋊C4—22no—
206(C2×D3)⋊C4—48no—
207C4⋊Dic3—24no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—48no—
215C4⋊Dic3—41no—
216C4⋊Dic3—41no—
217C4⋊Dic3C2×D412yes—
218C4⋊Dic3C2×D412yes—
219Dic3⋊C4—22no—
220Dic3⋊C4—22no—
221C4×Dic3—24no—
222C4×Dic3—22no—
223#208—22no—
224#208—22no—
225C2×C3⋊C8—22no—
226C2×#41—62no—
227C2×#41C22×D312yes—
228C22×Q8—61no—
229C22×D4—31no—
230C23×C4—32no—
231C2×M16—32no—
232C2×M16C22×D312yes—
233C4⋊Q8—31no—
234C4⋊Q8—31no—
235C4⋊Q8—61no—
236C4⋊Q8—61no—
237C4⋊Q8—61no—
238C4⋊Q8—61no—
239C4⋊Q8D1211yescharacteristic
240#127—62no—
241#127D1211yescharacteristic
242#126—64no—
243#125—32no—
244#124—64no—
245#124—62no—
246#124D1212yes—
247#123—64no—
248#123—64no—
249C22⋊Q8—62no—
250C22⋊Q8—61no—
251C22⋊Q8—61no—
252C22⋊Q8—128no—
253#121—21no—
254#121—21no—
255#121—62no—
256#121—62no—
257#121—62no—
258#121—64no—
259#120—64no—
260C4×Q8—62no—
261C4×Q8—61no—
262C4×Q8—61no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—64no—
266C4×D4—64no—
267C4×D4—62no—
268C4×D4—61no—
269C4×D4—61no—
270C4×D4—61no—
271C4×D4—61no—
272#117—32no—
273#117—62no—
274#117—62no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—61no—
281C2×C22⋊C4—32no—
282C2×C22⋊C4—64no—
283C2×C42C22×D312yes—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#106—64no—
287#104—28no—
288#103—64no—
289#102—64no—
290#100—64no—
291#97—21no—
292#97—61no—
293C4×C8—21no—
294C4×C8—61no—
295#95—21no—
296C23×C3—22no—
297C23×C3C22×D411yescharacteristic
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—42no—
302C22×D3—42no—
303C3×Q8—24no—
304C3×D4—24no—
305C3×D4—42no—
306C3×D4—42no—
307C3×D4—42no—
308C3×D4—81no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12011yescharacteristic
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×C3×C42−1+412yes—
324#67—84no—
325#67—84no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—42no—
330C2×Dic3—44no—
331C2×Dic3—42no—
332C2×Dic3—84no—
333D12—42no—
334D12—42no—
335C4×D3—44no—
336C4×D3—82no—
337Dic6—42no—
338Dic6—81no—
339Dic6—82no—
340Dic6—82no—
341C3×C8—24no—
342C3⋊C8—42no—
343C24—32no—
344#41—24no—
345#41—24no—
346#41—244no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—124no—
350C2×Q8—121no—
351C2×Q8—121no—
352C2×Q8C2×D1212yes—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4C2×D1212yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—32no—
366C22×C4—64no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—124no—
371C22×C4D3×D411yescharacteristic
372C22×C4D3×D411yescharacteristic
373C22×C4C23×D311yescharacteristic
374M16—22no—
375M16—62no—
376M16—62no—
377M16#23714yescomplemented
378C2×C8—22no—
379C2×C8—62no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—32no—
384C4⋊C4—32no—
385C4⋊C4—64no—
386C4⋊C4—61no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—128no—
395C4⋊C4—121no—
396C4⋊C4—121no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—62no—
405C22⋊C4—128no—
406C42—24no—
407C42—64no—
408C42—62no—
409C42C2×D1211yescharacteristic
410C42C2×D1211yescharacteristic
411C42D3×D411yescharacteristic
412C42D3×D411yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3C2×#12011yescharacteristic
419C22×C3#53211yescharacteristic
420C22×C3#53611yescharacteristic
421C2×D3—44no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—82no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4#45312yes—
432C3×C4#54712yes—
433C3×C4#54812yes—
434Dic3—42no—
435Dic3—42no—
436Dic3—42no—
437Dic3—82no—
438C23—22no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23—122no—
443C23—122no—
444C23C2×D3×D411yescharacteristic
445Q8—24no—
446Q8—122no—
447Q8—241no—
448Q8—242no—
449Q8—242no—
450D4—24no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—81no—
455D4—122no—
456D4—122no—
457D4—244no—
458D4—244no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—124no—
470C2×C4—122no—
471C2×C4—124no—
472C2×C4—122no—
473C2×C4—242no—
474C2×C4—244no—
475C2×C4#88211yescharacteristic
476C2×C4#88211yescharacteristic
477C2×C4#88211yescharacteristic
478C2×C4#88211yescharacteristic
479C2×C4C22×D1211yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C2×C4C2×#23712yes—
482C2×C4#101012yes—
483C8—24no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—82no—
488C2×C3#243211yescharacteristic
489C2×C3C2×#45311yescharacteristic
490C2×C3#302211yescharacteristic
491D3—82no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—124no—
498C22—122no—
499C22—122no—
500C22—122no—
501C22—242no—
502C22C2×#88211yescharacteristic
503C22#567511yescharacteristic
504C22#571611yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—44no—
509C4—42no—
510C4—122no—
511C4—122no—
512C4—122no—
513C4—242no—
514C4#482212yes—
515C4#555512yes—
516C4#559812yes—
517C3#2569411yescharacteristic, complemented, Sylow
518C2—21no—
519C2—21no—
520C2—82no—
521C2—242no—
522C2#7644511yescharacteristic
523C2C2×#482211yescharacteristic
524C2#7899511yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1980C2
socleC22×C3C2×#120
prime radical#86223C2
hypercenter#1980D3
hypocenterC3#25694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #78506C2semidirect
4. #73800C2semidirect
5. C3×#1980C2semidirect
6. #70363C2semidirect
7. #70359C2semidirect
8. C4×#823C2semidirect
9. #69493C2semidirect
50. #1980D3semidirect
188. C3×M16#41semidirect
377. M16#237semidirect
517. C3#25694semidirect

Maximal subgroups

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

subgroupindex
#862232
#832832
#785062
#738002
C3×#19802
#703632
#703592
C4×#8232
#694932
#256943